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+{"text":"\\section{Vertical Defect Fusion} \\label{sec:vertical_defect_fusion}\n\nIn this section, we present several of the more complicated vertical defect fusion computations.\n\n\\begin{exmp}[$\\defect{F_q}{X_l}{}{}{} \\circ \\defect{X_l}{F_s}{}{}{}$]\n\nConsider the defect fusion\n\\begin{align}\n\t\\defect{F_q}{X_l}{}{}{} \\circ \\defect{X_l}{F_s}{}{}{}\\to \n\t\\begin{cases} \n\t\t\\defect{F_s}{F_s}{\\alpha}{\\beta}{} & q = s\\\\\n\t\t\\defect{F_q}{F_s}{}{}{} & q\\neq s\n\t\\end{cases}.\n\\end{align}\n\n\n\\subsubsection*{Case I: $q=s$}\n\nWhen $q=s$, the vertical fusion looks like\n\\begin{align}\n \\defect{F_q}{X_l}{x}{}{}  \\circ \\defect{X_l}{F_s}{z}{}{}  \\to \\defect{F_s}{F_s}{\\zeta}{\\eta}{}\n  \\end{align}\nThe general pants absorbing $\\defect{X_l}{F_s}{z}{}{}$ and $\\defect{F_q}{X_l}{x}{}{}$ on the legs and $\\defect{F_s}{F_s}{\\zeta}{\\eta}{}$ on the waist looks like\n\\begin{align}\n  \\frac{1}{p^4}\\sum_{g,h,\\gamma,\\delta} \\omega^{\\gamma\\zeta+\\delta\\eta+s\\gamma(h-g)} \\Theta_{z,ls}(h)\\Theta_{x,-ql}(g)\n\t\\begin{array}{c}\n\t\t\\includeTikz{XlFs_FqXl_vert_1}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t  \\vpantsparams{$*$}{$0$}{$*$};\n        \\vpantsstp{$lg{+}\\gamma$}{$-g{-}\\delta$}{$lh{+}\\gamma$}{$-h{-}\\delta$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\\end{align}\nIf we set\n\\begin{align}\n  a &= -g -\\delta \\\\\n  b &= -h - \\delta \\\\\n  c &= lh + \\gamma\n  \\end{align}\nthen this pair of pants becomes\n\\begin{align}\n  \\frac{1}{p^4} \\sum_{a,b,c,\\delta}\\omega^{\\rm exponent}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XlFs_FqXl_vert_2}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t  \\vpantsparams{$*$}{$0$}{$*$};\n        \\vpantsstp{$c{+}l(b{-}a)$}{$a$}{$c$}{$b$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\\end{align}\nwhere\n\\begin{align}\n{\\rm exponent}=-2^{-1} a^2 l s+a (b l s+c s-x)-2^{-1} b^2 l s-b (c s-\\zeta  l+z)+c \\zeta +\\delta  (\\eta +\\zeta  l-x-z).\n\\end{align}\nThis implies $\\eta = x + z - l\\zeta$, so\n\\begin{align}\n \\defect{F_q}{X_l}{x}{}{} \\circ   \\defect{X_l}{F_s}{z}{}{} \\to \\oplus_{\\zeta} \\defect{F_s}{F_s}{\\zeta}{x+z-l\\zeta}{}\n\\end{align}\n\\subsubsection*{Case II: $q\\neq s$}\nWhen $q=s$, the vertical fusion looks like\n\\begin{align}\n  \\defect{F_q}{X_l}{x}{}{} \\circ    \\defect{X_l}{F_s}{z}{}{} \\to \\defect{F_q}{F_s}{}{}{}\n  \\end{align}\nAll the variables in general pants transform into global phases, so there is no multiplicity.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{X_l}{}{}{} \\circ  \\defect{X_l}{X_m}{}{}{}$]\n\t\n\t\n\tConsider the defect fusion\n\t\\begin{align}\n\t\t\\defect{X_k}{X_l}{}{}{} \\circ \\defect{X_l}{X_m}{}{}{}\\to \n\t\t\\begin{cases} \n\t\t\t\\defect{X_k}{X_k}{\\alpha}{\\beta}{} & k = m\\\\\n\t\t\t\\defect{X_k}{X_m}{}{}{} & k\\neq m\n\t\t\\end{cases}.\n\t\\end{align}\n\t\n\\subsubsection*{Case I: $k=m$}\nThe general pants absorbing $\\defect{X_l}{X_m}{}{}{}$ and $\\defect{X_k}{X_l}{}{}{}$ on the legs and $\\defect{X_k}{X_k}{\\alpha}{\\zeta}{}$ on the waist is\n\\begin{align}\n  \\frac{1}{p} \\sum_{\\gamma}\\omega^{\\gamma\\zeta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XlXm_XkXk_vert_1}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t  \\vpantsparams{$0$}{$0$}{$\\alpha$};\n        \\vpantsstp{$k\\gamma{+}k_2$}{$k_3{-}\\gamma$}{$k\\gamma{+}k_0$}{$k_1{-}\\gamma$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\\end{align}\nwhere\n\\begin{align}\n  k_0 + kk_1 &= \\alpha \\\\\n  k_0 + lk_1 - k_2 - lk_3 &= 0 \\\\\n  k_2 + k k_3 &= 0.\n\\end{align}\nThe solution to these equations is\n\\begin{align*}\n  k_1 &= k^{-1}(\\alpha-k_0) \\\\\n  k_2 &= l\\alpha(k-l)^{-1}+k_0 \\\\\n  k_3 &= - l\\alpha(k-l)^{-1}k^{-1}-k_0k^{-1}\n\\end{align*}\nso the substitution $\\gamma \\to \\gamma - k^{-1}t$ transforms $k_0$ into a global phase. Therefore\n\\begin{align}\n \\defect{X_k}{X_l}{}{}{}  \\circ  \\defect{X_l}{X_k}{}{}{}  = \\oplus_{\\alpha,\\zeta}\\defect{X_k}{X_k}{\\alpha}{\\zeta}{}\n\\end{align}\n\\subsubsection*{Case II: $k\\not=m$}\nThe general pant absorbing $\\defect{X_l}{X_m}{}{}{}$ and $\\defect{X_k}{X_l}{}{}{}$ on the legs and $\\defect{X_k}{X_m}{}{}{}$ on the waist is\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XlXm_XkXk_vert_2}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t  \\vpantsparams{$0$}{$0$}{$0$};\n        \\vpantsstp{$k_2$}{$k_3$}{$k_0$}{$k_1$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\\end{align}\nwhere\n\\begin{align}\n  k_0 + mk_1 &= 0 \\\\\n  k_0 + lk_1 - k_2 - lk_3 &= 0 \\\\\n  k_2 + k k_3 &= 0.\n\\end{align}\nThis system has rank 3, which implies\n\\begin{align}\n \\defect{X_k}{X_l}{}{}{}  \\circ  \\defect{X_l}{X_m}{}{}{}  = p\\cdot\\defect{X_k}{X_m}{}{}{}\n\\end{align}\n\\end{exmp}\n\\section{Vertical fusion outcomes} \\label{sec:vertical_fusion_table}\n\\newlength{\\tabwidth}\n\\setlength{\\tabwidth}{.49\\textheight}\n\\newlength{\\tabheight}\n\\setlength{\\tabheight}{.3\\textwidth}\n\\vspace*{10mm}\n\\begin{table}[h] \n\t\\begin{minipage}[t][.85\\textheight][c]{\\textwidth}\n\t\\renewcommand{\\arraystretch}{2.9}\n\\rotatebox[]{90}{\n\t\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$L$&$\\defect{L}{T}{c}{}{}$&$\\defect{L}{L}{c}{z}{}$&$\\defect{L}{R}{}{}{}$&$\\defect{L}{F_{0}}{z}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$\\defect{L}{F_{s}}{z}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{T}{L}{a}{}{}$&$\\oplus_{\\alpha}\\defect{T}{T}{\\alpha}{a+c}{}$&$\\defect{T}{L}{a+c}{}{}$&$\\oplus_{\\alpha}\\defect{T}{R}{\\alpha}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\oplus_{\\alpha}\\defect{T}{X_{m}}{\\alpha}{}{}$&$\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{L}{L}{a}{x}{}$&$\\defect{L}{T}{a+c}{}{}$&$\\defect{L}{L}{a+c}{x+z}{}$&$\\defect{L}{R}{}{}{}$&$\\defect{L}{F_{0}}{x+z}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$\\defect{L}{F_{s}}{x+z+sa}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{R}{L}{}{}{}$&$\\oplus_{\\alpha}\\defect{R}{T}{\\alpha}{}{}$&$\\defect{R}{L}{}{}{}$&$\\oplus_{\\alpha,\\zeta}\\defect{R}{R}{\\alpha}{\\zeta}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{0}}{\\zeta}{}{}$&$p \\cdot \\defect{R}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{F_{0}}{L}{x}{}{}$&$\\defect{F_{0}}{T}{}{}{}$&$\\defect{F_{0}}{L}{x+z}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{R}{\\zeta}{}{}$&$\\oplus_{\\eta}\\defect{F_{0}}{F_{0}}{x+z}{\\eta}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{X_{m}}{\\zeta}{}{}$&$\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{X_{k}}{L}{}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{T}{\\alpha}{}{}$&$\\defect{X_{k}}{L}{}{}{}$&$p \\cdot \\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{0}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\alpha,\\beta}\\defect{X_{k}}{X_{k}}{\\alpha}{\\beta}{}&k=m\\\\p\\cdot\\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\oplus_{\\alpha}\\defect{X_{k}}{F_{s}}{\\alpha}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{F_{q}}{L}{x}{}{}$&$\\defect{F_{q}}{T}{}{}{}$&$\\defect{F_{q}}{L}{x+z+qc}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{R}{\\zeta}{}{}$&$\\defect{F_{q}}{F_{0}}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{X_{m}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\alpha}\\defect{F_{q}}{F_{q}}{x+z}{\\alpha}{}&q=s\\\\\\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n}\n\\rotatebox[]{90}{\n\t\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$F_0$&$\\defect{F_{0}}{T}{}{}{}$&$\\defect{F_{0}}{L}{z}{}{}$&$\\defect{F_{0}}{R}{z}{}{}$&$\\defect{F_{0}}{F_{0}}{z}{w}{}$&$\\defect{F_{0}}{X_{m}}{z}{}{}$&$\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{T}{F_{0}}{}{}{}$&$\\oplus_{\\alpha,\\beta}\\defect{T}{T}{\\alpha}{\\beta}{}$&$\\oplus_{\\alpha}\\defect{T}{L}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{T}{R}{\\alpha}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\oplus_{\\alpha}\\defect{T}{X_{m}}{\\alpha}{}{}$&$p \\cdot \\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{L}{F_{0}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{L}{T}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{L}{L}{\\alpha}{x+z}{}$&$\\defect{L}{R}{}{}{}$&$\\defect{L}{F_{0}}{x+z}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{R}{F_{0}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{R}{T}{\\alpha}{}{}$&$\\defect{R}{L}{}{}{}$&$\\oplus_{\\alpha}\\defect{R}{R}{\\alpha}{x+z}{}$&$\\defect{R}{F_{0}}{x+w}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{F_{0}}{F_{0}}{x}{y}{}$&$\\defect{F_{0}}{T}{}{}{}$&$\\defect{F_{0}}{L}{x+z}{}{}$&$\\defect{F_{0}}{R}{y+z}{}{}$&$\\defect{F_{0}}{F_{0}}{x+z}{y+w}{}$&$\\defect{F_{0}}{X_{m}}{x+z+m^{-1}y}{}{}$&$\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{X_{k}}{F_{0}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{T}{\\alpha}{}{}$&$\\defect{X_{k}}{L}{}{}{}$&$\\defect{X_{k}}{R}{}{}{}$&$\\defect{X_{k}}{F_{0}}{x+z+k^{-1}w}{}{}$&$\\begin{cases}\\oplus_{\\alpha}\\defect{X_{k}}{X_{k}}{\\alpha}{k(x+z)}{}&k=m\\\\\\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$\\defect{F_{q}}{F_{0}}{}{}{}$&$p\\cdot\\defect{F_{q}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{L}{\\zeta}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{R}{\\zeta}{}{}$&$\\defect{F_{q}}{F_{0}}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{X_{m}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\zeta,\\eta}\\defect{F_{q}}{F_{q}}{\\zeta}{\\eta}{}&q=s\\\\ p \\cdot \\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n}\n\\rotatebox[]{90}{\n\t\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$F_r$&$\\defect{F_{r}}{T}{}{}{}$&$\\defect{F_{r}}{L}{z}{}{}$&$\\defect{F_{r}}{R}{z}{}{}$&$\\defect{F_{r}}{F_{0}}{}{}{}$&$\\defect{F_{r}}{X_{m}}{z}{}{}$&$\\defect{F_{r}}{F_{r}}{z}{w}{}$&$\\defect{F_{r}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{T}{F_{r}}{}{}{}$&$\\oplus_{\\alpha,\\beta}\\defect{T}{T}{\\alpha}{\\beta}{}$&$\\oplus_{\\alpha}\\defect{T}{L}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{T}{R}{\\alpha}{}{}$&$p\\cdot\\defect{T}{F_{0}}{}{}{}$&$\\oplus_{\\alpha}\\defect{T}{X_{m}}{\\alpha}{}{}$&$\\defect{T}{F_{r}}{}{}{}$&$p\\cdot\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\t\\greycline{2-8}\n\t\t\t$\\defect{L}{F_{r}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{L}{T}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{L}{L}{\\alpha}{x+z-r\\alpha}{}$&$\\defect{L}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{0}}{\\zeta}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$\\defect{L}{F_{r}}{x+z}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-8}\n\t\t\t$\\defect{R}{F_{r}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{R}{T}{}{}{}$&$\\defect{R}{L}{}{}{}$&$\\oplus_{\\alpha}\\defect{R}{R}{\\alpha}{x+z-r\\alpha}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{0}}{\\zeta}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$\\defect{R}{F_{r}}{x+w}{}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-8}\n\t\t\t$\\defect{F_{0}}{F_{r}}{}{}{}$&$p\\cdot\\defect{F_{0}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{L}{\\zeta}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{R}{\\zeta}{}{}$&$\\oplus_{\\zeta,\\eta}\\defect{F_{0}}{F_{0}}{\\zeta}{\\eta}{}$&$\\sum_\\alpha \\defect{F_{0}}{X_{m}}{\\alpha}{}{}$&$\\defect{F_{0}}{F_{r}}{}{}{}$&$p\\cdot\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$\\defect{X_{k}}{F_{r}}{x}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{T}{\\alpha}{}{}$&$\\defect{X_{k}}{L}{}{}{}$&$\\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{0}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\alpha}\\defect{X_{k}}{X_{k}}{\\alpha}{x+z-r\\alpha}{}&k=m\\\\\\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\defect{X_{k}}{F_{r}}{w+x+kz}{}{}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\t\\greycline{2-8}\n\t\t\t$\\defect{F_{r}}{F_{r}}{x}{y}{}$&$\\defect{F_{r}}{T}{}{}{}$&$\\defect{F_{q}}{L}{x+z}{}{}$&$\\defect{F_{q}}{R}{y+z}{}{}$&$\\defect{F_{q}}{F_{0}}{}{}{}$&$\\defect{F_{q}}{X_{m}}{y+z+mx}{}{}$&$\\defect{F_{r}}{F_{r}}{x+z}{y+w}{}$&$\\defect{F_{r}}{F_{s}}{}{}{}$\\\\\n\t\t\t\\greycline{2-8}\n\t\t\t$\\defect{F_{q}}{F_{r}}{}{}{}$&$p\\cdot\\defect{F_{q}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{L}{\\zeta}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{R}{\\zeta}{}{}$&$p\\cdot\\defect{F_{q}}{F_{0}}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{X_{m}}{\\zeta}{}{}$&$\\defect{F_{q}}{F_{r}}{}{}{}$&$\\begin{cases}\\oplus_{\\zeta,\\eta}\\defect{F_{q}}{F_{q}}{\\zeta}{\\eta}{}&q=s\\\\p\\cdot\\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n}\n\\rotatebox[]{90}{\n\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$T$&$\\defect{T}{T}{c}{d}{}$&$\\defect{T}{L}{c}{}{}$&$\\defect{T}{R}{c}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\defect{T}{X_{m}}{c}{}{}$&$\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{T}{T}{a}{b}{}$&$\\defect{T}{T}{a+c}{b+d}{}$&$\\defect{T}{L}{b+c}{}{}$&$\\defect{T}{R}{a+c}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\defect{T}{X_{m}}{a+c+mb}{}{}$&$\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\t\t\\greycline{2-7}\n\t\t$\\defect{L}{T}{a}{}{}$&$\\defect{L}{T}{a+d}{}{}$&$\\oplus_{\\beta}\\defect{L}{L}{a+c}{\\beta}{}$&$\\defect{L}{R}{}{}{}$&$\\oplus_{\\alpha}\\defect{L}{F_{0}}{\\alpha}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$\\oplus_{\\alpha}\\defect{L}{F_{s}}{\\alpha}{}{}$\\\\\n\t\t\t\t\\greycline{2-7}\n\t\t$\\defect{R}{T}{a}{}{}$&$\\defect{R}{T}{a+c}{}{}$&$\\defect{R}{L}{}{}{}$&$\\oplus_{\\beta}\\defect{R}{R}{a+c}{\\beta}{}$&$\\oplus_\\alpha \\defect{R}{F_{0}}{\\alpha}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$\\oplus_{\\alpha}\\defect{R}{F_{s}}{\\alpha}{}{}$\\\\\n\t\t\t\t\\greycline{2-7}\n\t\t$\\defect{F_{0}}{T}{}{}{}$&$\\defect{F_{0}}{T}{}{}{}$&$\\oplus_{\\alpha}\\defect{F_{0}}{L}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{F_{0}}{R}{\\alpha}{}{}$&$\\oplus_{\\alpha,\\beta}\\defect{F_{0}}{F_{0}}{\\alpha}{\\beta}{}$&$\\oplus_{\\alpha}\\defect{F_{0}}{X_{m}}{\\alpha}{}{}$&$p\\cdot\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{X_{k}}{T}{a}{}{}$&$\\defect{X_{k}}{T}{a+c+kd}{}{}$&$\\defect{X_{k}}{L}{}{}{}$&$\\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{F_{0}}{\\alpha}{}{}$&$\\begin{cases}\\oplus_{\\beta}\\defect{X_{k}}{X_{k}}{a+c}{\\beta}{}&k=m\\\\\\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\oplus_{\\alpha}\\defect{X_{k}}{F_{s}}{\\alpha}{}{}$\\\\\n\t\t\t\t\\greycline{2-7}\n\t\t$\\defect{F_{q}}{T}{}{}{}$&$\\defect{F_{q}}{T}{}{}{}$&$\\oplus_{\\alpha}\\defect{F_{q}}{L}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{F_{q}}{R}{\\alpha}{}{}$&$p\\cdot\\defect{F_{q}}{F_{0}}{}{}{}$&$\\oplus_{\\alpha}\\defect{F_{q}}{X_{m}}{\\alpha}{}{}$&$\\begin{cases}\\oplus_{\\alpha,\\beta}\\defect{F_{q}}{F_{q}}{\\alpha}{\\beta}{}&q=s\\\\p\\cdot\\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n}\n}\n\\rotatebox[]{90}{\n\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$R$&$\\defect{R}{T}{c}{}{}$&$\\defect{R}{L}{}{}{}$&$\\defect{R}{R}{c}{z}{}$&$\\defect{R}{F_{0}}{z}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$\\defect{R}{F_{s}}{z}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{T}{R}{a}{}{}$&$\\oplus_{\\beta}\\defect{T}{T}{a+c}{\\beta}{}$&$\\oplus_{\\alpha}\\defect{T}{L}{\\alpha}{}{}$&$\\defect{T}{R}{a+c}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\oplus_{\\alpha}\\defect{T}{X_{m}}{\\alpha}{}{}$&$\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\\greycline{2-7}\n\t\t$\\defect{L}{R}{}{}{}$&$\\oplus_{\\alpha}\\defect{L}{T}{\\alpha}{}{}$&$\\oplus_{\\alpha,\\zeta}\\defect{L}{L}{\\alpha}{\\zeta}{}$&$\\defect{L}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{0}}{\\zeta}{}{}$&$p\\cdot\\defect{L}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\\greycline{2-7}\n\t\t$\\defect{R}{R}{a}{x}{}$&$\\defect{R}{T}{a+c}{}{}$&$\\defect{R}{L}{}{}{}$&$\\defect{R}{R}{a+c}{x+z}{}$&$\\defect{R}{F_{0}}{x+z}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$\\defect{R}{F_{s}}{x+z+sa}{}{}$\\\\\n\t\t\\greycline{2-7}\n\t\t$\\defect{F_{0}}{R}{x}{}{}$&$\\defect{F_{0}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{L}{\\zeta}{}{}$&$\\defect{F_{0}}{R}{x+z}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{F_{0}}{\\zeta}{x+z}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{X_{m}}{\\zeta}{}{}$&$\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{T}{\\alpha}{}{}$&$p\\cdot\\defect{X_{k}}{L}{}{}{}$&$\\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{0}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\alpha,\\zeta}\\defect{X_{k}}{X_{k}}{\\alpha}{\\zeta}{}&k=m\\\\ p\\cdot \\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\\greycline{2-7}\n\t\t$\\defect{F_{q}}{R}{x}{}{}$&$\\defect{F_{q}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{L}{\\zeta}{}{}$&$\\defect{F_{q}}{R}{x+z+qc}{}{}$&$\\defect{F_{q}}{F_{0}}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{X_{m}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\zeta}\\defect{F_{q}}{F_{q}}{\\zeta}{x+z}{}&q=s\\\\\\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n}\n}\n\\rotatebox[]{90}{\n\\resizebox*{\\tabwidth}{\\tabheight}{\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\vrule width 1pt} c!{\\color[gray]{.8}\\vrule}c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\\cellcolor{red!20}$X_{l}$&$\\defect{X_{l}}{T}{c}{}{}$&$\\defect{X_{l}}{L}{}{}{}$&$\\defect{X_{l}}{R}{}{}{}$&$\\defect{X_{l}}{F_{0}}{z}{}{}$&$\\defect{X_{l}}{X_{l}}{c}{z}{}$&$\\defect{X_{l}}{X_{m}}{}{}{}$&$\\defect{X_{l}}{F_{s}}{z}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{T}{X_{l}}{a}{}{}$&$\\oplus_{\\beta}\\defect{T}{T}{a+c-l\\beta}{\\beta}{}$&$\\oplus_{\\alpha}\\defect{T}{L}{\\alpha}{}{}$&$\\oplus_{\\alpha}\\defect{T}{R}{\\alpha}{}{}$&$\\defect{T}{F_{0}}{}{}{}$&$\\defect{T}{X_{m}}{a+c}{}{}$&$\\oplus_{\\alpha}\\defect{T}{X_{m}}{\\alpha}{}{}$&$\\defect{T}{F_{s}}{}{}{}$\\\\\n\t\t\\greycline{2-8}\n\t\t$\\defect{L}{X_{l}}{}{}{}$&$\\oplus_{\\alpha}\\defect{L}{T}{\\alpha}{}{}$&$\\oplus_{\\alpha,\\zeta}\\defect{L}{L}{\\alpha}{\\zeta}{}$&$p\\cdot\\defect{L}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{0}}{\\zeta}{}{}$&$\\defect{L}{X_{m}}{}{}{}$&$p\\cdot\\defect{L}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{L}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\\greycline{2-8}\n\t\t$\\defect{R}{X_{l}}{}{}{}$&$\\oplus_{\\alpha}\\defect{R}{T}{\\alpha}{}{}$&$p\\cdot\\defect{R}{L}{}{}{}$&$\\oplus_{\\alpha,\\zeta}\\defect{R}{R}{\\alpha}{\\zeta}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{0}}{\\zeta}{}{}$&$\\defect{R}{X_{m}}{}{}{}$&$p\\cdot\\defect{R}{X_{m}}{}{}{}$&$\\oplus_{\\zeta}\\defect{R}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\\greycline{2-8}\n\t\t$\\defect{F_{0}}{X_{l}}{x}{}{}$&$\\defect{F_{0}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{L}{\\zeta}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{R}{\\zeta}{}{}$&$\\oplus_{\\eta}\\defect{F_{0}}{F_{0}}{x+z-l^{-1}\\eta}{\\eta}{}$&$\\defect{F_{0}}{X_{l}}{x+l^{-1}z}{}{}$&$\\oplus_{\\zeta}\\defect{F_{0}}{X_{m}}{\\zeta}{}{}$&$\\defect{F_{0}}{F_{s}}{}{}{}$\\\\\n\t\t\\toprule[1pt]\n\t\t$\\defect{X_{l}}{X_{l}}{a}{x}{}$&$\\defect{X_{k}}{T}{a+c}{}{}$&$\\defect{X_{k}}{L}{}{}{}$&$\\defect{X_{k}}{R}{}{}{}$&$\\defect{X_{k}}{F_{0}}{l^{-1}x+z}{}{}$&$\\defect{X_{l}}{X_{l}}{a+c}{x+z}{}$&$\\defect{X_{l}}{X_{m}}{}{}{}$&$\\defect{X_{k}}{F_{s}}{x+z+as}{}{}$\\\\\n\t\t\\greycline{2-8}\n\t\t$\\defect{X_{k}}{X_{l}}{}{}{}$&$\\oplus_{\\alpha}\\defect{X_{k}}{T}{\\alpha}{}{}$&$p\\cdot\\defect{X_{k}}{L}{}{}{}$&$p\\cdot\\defect{X_{k}}{R}{}{}{}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{0}}{\\zeta}{}{}$&$\\defect{X_{k}}{X_{l}}{}{}{}$&$\\begin{cases}\\oplus_{\\alpha,\\zeta}\\defect{X_{k}}{X_{k}}{\\alpha}{\\zeta}{}&k=m\\\\p\\cdot\\defect{X_{k}}{X_{m}}{}{}{}&k\\neq m\\end{cases}$&$\\oplus_{\\zeta}\\defect{X_{k}}{F_{s}}{\\zeta}{}{}$\\\\\n\t\t\\greycline{2-8}\n\t\t$\\defect{F_{q}}{X_{l}}{x}{}{}$&$\\defect{F_{q}}{T}{}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{L}{\\zeta}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{R}{\\zeta}{}{}$&$\\defect{F_{q}}{F_{0}}{}{}{}$&$\\defect{F_{q}}{X_{m}}{x+z+qc}{}{}$&$\\oplus_{\\zeta}\\defect{F_{q}}{X_{m}}{\\zeta}{}{}$&$\\begin{cases}\\oplus_{\\zeta}\\defect{F_{q}}{F_{q}}{\\zeta}{x+z-l\\zeta}{}&q=s\\\\\\defect{F_{q}}{F_{s}}{}{}{}&q\\neq s\\end{cases}$\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n}\n}\n\\end{minipage}\n\\vspace*{14mm}\n\t\\caption{Vertical fusion tables}\\label{tab:vertical_fusion_tables}\n\t\\renewcommand{\\arraystretch}{1}\n\\vspace*{-20mm}\n\\end{table}\n\n\\section{The Horizontal Fusion Algorithm} \\label{sec:hotizontal_defect_fusion_algorithm}\n\nIn this section, we explain the algorithm used to compute horizontal defect fusion. Due to the diagrammatic nature of the algorithm and the variety of phenomena that can occur during the computation, we are not going to give a formal specification, suitable for computer implementation. Instead, we shall explain how to proceed by hand in a specific example. The procedure can vary between examples and we demonstrate this in Section~\\ref{sec:horizontal_defect_fusion}. This variability only occurs after all the necessary information has been extracted from the correct tables, so for the purpose of explaining how to navigate the tables of data which appear in this paper, a specific example is easier to understand than a general algorithm.\n\nThe algorithm proceeds in four key steps:\n\\begin{enumerate} \n\t\\item Determine idempotents corresponding to source defects using Table~\\ref{tab:idempotents}.\n\t\\item Determine idempotent for target defect using Tables~\\ref{tab:zptable} and \\ref{tab:idempotents}.\n\t\\item Inflate the target idempotent to a 4-string annulus using Tables~\\ref{tab:inflation_1}-\\ref{tab:inflation_2}.\n\t\\item Find a nonzero pants diagram that absorbs the source idempotents on the legs and (inflated) target on the waist.\n\\end{enumerate}\n\n\\begin{exmp}[$\\defect{X_k}{T}{}{}{} \\otimes \\defect{F_s}{R}{}{}{}$]\nWe shall give the step by step procedure for the fusion\n\\begin{align}\n\\defect{X_k}{T}{a}{}{} \\otimes \\defect{F_s}{R}{z}{}{}.\n\\end{align}\n\\subsubsection{Writing down the source defect idempotents}\nThe first step in the procedure is to look up the idempotents representing the defects. These are found in Table~\\ref{tab:idempotents}. For our current example, we have\n\\begin{align}\n\\defect{X_k}{T}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$(a,0)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n        \\end{array}, \\quad\n\\defect{F_s}{R}{z}{}{}=\\frac{1}{p}\\sum_g\\omega^{gz}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqR_idempotent_diff_variable_names}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}.\n        \\end{align}\n\\subsubsection{Writing down the target defect idempotent}\nTo decide the defect species resulting from the fusion, we need to look up the relevant domain wall fusions. These are contained in Table~\\ref{tab:zptable}. For our current example, we have\n\\begin{align}\n\tT \\otimes_{\\vvec{\\ZZ{p}}} R &= p \\cdot R  \\label{eq:top_domain_wall_fusion}\\\\\n\tX_k \\otimes_{\\vvec{\\ZZ{p}}} F_s &= F_{k^{-1}s}. \\label{eq:bottom_domain_wall_fusion}\n\\end{align}\n\nFrom Eqns.~\\eqref{eq:top_domain_wall_fusion} and \\eqref{eq:bottom_domain_wall_fusion}, we can read of the target defect up to the labels. In this case it is $\\defect{F_{k^{-1}s}}{R}{\\zeta}{}{}$, where $\\zeta$ is yet to be determined. From Table~\\ref{tab:idempotents}, we find that the defect idempotent is\n\\begin{align}\n\t\\defect{F_{k^{-1}s}}{R}{\\zeta}{}{}=\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma \\zeta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{FqR_idempotent_diff_variable_names_2}\n\t\t{\n\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\\annst{}{$-\\gamma$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array},\n\\end{align}\nAt this stage of the computation, the multiplicities in the domain wall fusion don't play a role. They show up in the inflation stage.\n\\subsubsection{Inflating the target idempotent}\nIn order for the target defect idempotent and the initial defect idempotents to interact with each other on a pair-of-pants, we need to \\emph{inflate} the target idempotent so that it has four vertical strings. The information required to do this is contained in Table~\\ref{tab:inflation_1}-\\ref{tab:inflation_2}. These tables contain the information required to explicitly decompose the tensor products $\\mathcal{M}\\otimes_{\\vvec{\\ZZ{p}}}\\mathcal{N}$ into simple bimodule categories. In our example, we need the entries corresponding to $R \\to T \\otimes_{\\vvec{\\ZZ{p}}} R$ and $F_{k^{-1}s} \\to X_k \\otimes_{\\vvec{\\ZZ{p}}} F_s$. These entries tell how to replace the trivalent vertices in our target idempotent, giving us\n\\begin{align}\\frac{1}{p} \\sum_{\\gamma} \\omega^{\\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{XkT_FsR_inflation}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamstpq{$0$}{$*$}{$(0,v)$}{$0$}{}{}{}{};\n\t\\annstpq{}{}{$-\\gamma$}{};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nThe $\\nu$ appearing corresponds to the multiplicity in the tensor product $T \\otimes_{\\vvec{\\ZZ{p}}} R = p \\cdot R$.\n\n\\subsubsection{Decorating the pants}\nAt this point, we have extracted all the data we need from the tables. Now we need to find all the pants diagrams which absorb our source defect idempotents on the legs and our target inflated defect idempotent at the waist. The most general pair-of-pants which we need to consider looks like:\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{generalpants_basis}{\n\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\\pantsparams{}{}{}{}{}{}{}{};\n\t\\pantsstpq{$k_0$}{$k_1$}{$k_3$}{$k_4$}{$k_2$};\n\t\\end{tikzpicture}}\n\\end{array}.\n\\end{align}\nWe use the following equation to bring every horizontal pair-of-pants into the standard form\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{generalpants}{\n\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\\pantsparams{$m$}{$n$}{$p$}{$q$}{}{}{}{};\n\t\t\\generalpantsstpq{$g_0$}{$g_1$}{$g_2$}{$g_3$}{$h_0$}{$h_1$}{$h_2$}{$h_3$};\n\t\\end{tikzpicture}}\n\\end{array}\n&=\\frac{\\Omega_M(h_0^{-1},g_1^{-1})\\Omega_Q(h_1,h_2)\\Omega_N(h_3,(g_3h_2)^{-1})\\Omega_Q(h_3,g_3h_2)}{\\Omega_P(h_0,g_1)}\n\\begin{array}{c}\n\\includeTikz{generalpants_canonical}{\n\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\\pantsparams{$m$}{$n$}{$p$}{$q$}{}{}{}{};\n\t\\pantsstpq{$h_0g_0$}{$g_1h_3$}{$h_3^{-1}g_2$}{$g_3h_2$}{$h_1h_3$};\n\n\t\\end{tikzpicture}}\n\\end{array},\n\\end{align}\nwhere $\\Omega_{X}(\\bullet,\\bullet)$ is the associator for the bimodule $X$\\cite{1806.01279}.\nNow, when we insert our source defect idempotents on the legs and our inflated target idempotent on the belt, we get\n\\begin{align}\\frac{1}{p^2} \\sum_{g,\\gamma} \\omega^{gz+\\gamma\\zeta+s\\gamma k_3}\n\\begin{array}{c}\n\\includeTikz{generalpants_basis_with_idempotents}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\t\\pantsparams{$0$}{$*$}{$(0,\\nu)$}{$0$}{}{}{}{};\n\t\t\\generalpantsstpq{$k_0$}{$k_1$}{$k_3$}{$-\\gamma-g+k_4$}{}{$k_2$}{}{};\n\t\\end{tikzpicture}}\n\\end{array}\n\\end{align}\nup to a global phase. The term $\\omega^{s\\gamma k_3}$ appears because we need to use the middle associator on $F_s$ to bring the diagram into the standard form. We have omitted the labels on the leg holes to make the diagram less cluttered. They match the labels on the source defect idempotents. To make all the labels match up, we must have\n\\begin{align*}\n  k_0 &= -a \\\\\n  k_1 &= k^{-1}a \\\\\n  k_2 &= k^{-1}a - \\nu \\\\\n  k_3 &= \\nu - k^{-1}a\n  \\end{align*}\nThe transformation $g \\to g + k_4$ only changes the expression by a phase. The whole expression is zero unless $z = \\zeta + sk_3$. Rearranging this equation gives\n\\begin{align}\n  \\zeta = z + s (k^{-1}a - v).\n\\end{align}\nSo we have\n\\begin{align}\n  \\defect{X_k}{T}{a}{}{} \\otimes \\defect{F_s}{R}{z}{}{} = \\defect{F_{k^{-1}s}}{R}{z+s(k^{-1}a-\\nu)}{}{\\nu}\n\\end{align}\nAs explained above, the superscript $\\nu$ indexes the multiplicity in the top domain wall fusion.\n\\end{exmp}\n\\section{Defect idempotents} \\label{sec:idempotents_table}\n\\vspace*{20mm}\n\t\\begin{table}[h]\n\\begin{minipage}[t][.8\\textheight][c]{\\textwidth}\n\t\\rotatebox[]{90}{\n\\resizebox{.98\\textheight}{!}{\n\t\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\t\t\\toprule[1pt]\n\t\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$T$&$L$&$R$&$F_0$&$X_l$&$F_r$\\\\\n\t\t\t\t\\toprule[1pt]\n\t\t\t\t$T$&$\\defect{T}{T}{a}{b}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamss{$(0,0)$}{$(a,b)$}{}{};\n\t\t\t\t\t\\annss{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{T}{L}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TL_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$(0,0)$}{$a$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{T}{R}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TR_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$(0,0)$}{$a$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{T}{F_0}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TF0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$(0,0)$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{T}{X_l}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TXl_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$(0,0)$}{$a$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{T}{F_r}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{TFr_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$(0,0)$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\\\\\n\\greycline{2-7}\n\t\t\t\t$L$&\n\t\t\t\t$\\defect{L}{T}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$(0,a)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\t&$\\defect{L}{L}{a}{x}{}=\\frac{1}{p}\\sum_{g}\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LL_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamss{$0$}{$a$}{}{};\n\t\t\t\t\t\\annss{$g$}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{L}{R}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LR_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{L}{F_0}{x}{}{}=\\frac{1}{p}\\sum_g \\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LF0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{$g$}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{L}{X_l}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LXl_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{L}{F_r}{x}{}{}=\\frac{1}{p}\\sum_g \\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{LFr_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{$g$}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\\\\\n\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t\t$R$&$\\defect{R}{T}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$(a,0)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{R}{L}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RL_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{R}{R}{a}{x}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RR_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamss{$0$}{$a$}{}{};\n\t\t\t\t\t\\annss{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{R}{F_0}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RF0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{R}{X_l}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RXl_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{R}{F_r}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RFr_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\\\\\n\\greycline{2-7}\n\t\t\t\t$F_0$&$\\defect{F_0}{T}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0T_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$(0,0)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_0}{L}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0L_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{$g$}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_0}{R}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0R_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_0}{F_0}{x}{y}{}=\\frac{1}{p^2}\\sum_{g,h}\\omega^{gx+hy}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0F0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamss{$*$}{$*$}{}{};\n\t\t\t\t\t\\annss{$g$}{$-h$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_0}{X_l}{x}{}{}=\\frac{1}{p}\\sum_{g}\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0Xl_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.5}]\n\t\t\t\t\t\\annparamss{$*$}{$0$}{}{};\n\t\t\t\t\t\\annss{$g$}{$-l^{-1}g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_0}{F_r}{}{}{}=\\frac{1}{p}\\sum_{g}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{F0Fr_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\\\\\n\\greycline{2-7}\n\t\t\t\t$X_k$&$\\defect{X_k}{T}{a}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$(a,0)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{X_k}{L}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkL_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{X_k}{R}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkR_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{X_k}{F_0}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkF0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.5}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{$g$}{$-k^{-1}g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&\n\t\t\t\t\\begin{tabular}{c}\n\t\t\t\t\t$\\defect{X_k}{X_k}{a}{x}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\includeTikz{XkXk_idempotent}\n\t\t\t\t\t{\n\t\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.65}]\n\t\t\t\t\t\t\\annparamss{$0$}{$a$}{}{};\n\t\t\t\t\t\t\\annss{$kg$}{$-g$};\n\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{array}$\n\t\t\t\t\t\\\\\n\t\t\t\t\t\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t\t\t$\\defect{X_k}{X_l}{}{}{}=\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\includeTikz{XkXl_idempotent}\n\t\t\t\t\t{\n\t\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\t\\annparamst{$0$}{$0$}{}{};\n\t\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{array}$\n\t\t\t\t\\end{tabular}\n\t\t\t\t&\t\n\t\t\t\t$\\defect{X_k}{F_r}{x}{}{}=\\frac{1}{p}\\sum_g\\Theta_{x,kr}(g)\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkFr_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{$kg$}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\t\\\\\n\\greycline{2-7}\n\t\t\t\t$F_q$&$\\defect{F_q}{T}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqT_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$(0,0)$}{}{};\n\t\t\t\t\t\\annst{}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_q}{L}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqL_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{$g$}{};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_q}{R}{x}{}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqR_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&$\\defect{F_q}{F_0}{}{}{}=\\frac{1}{p}\\sum_{g}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqF0_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$&\n\t\t\t\t$\\defect{F_q}{X_l}{x}{}{}=\\frac{1}{p}\\sum_g\\Theta_{x,-ql}(g)\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqXl_idempotent}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$*$}{$0$}{}{};\n\t\t\t\t\t\\annst{$lg$}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\t&\n\t\t\t\t\\begin{tabular}{c}\n\t\t\t\t\t$\\defect{F_q}{F_q}{x}{y}{}=\\frac{1}{p^2}\\sum_{g,h}\\omega^{gx+hy}\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\includeTikz{FqFq_idempotent}\n\t\t\t\t\t{\n\t\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\t\\annparamss{$*$}{$*$}{}{};\n\t\t\t\t\t\t\\annss{$g$}{$-h$};\n\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{array}$\\\\\n\t\t\t\t\t\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t\t\t$\\defect{F_q}{F_r}{}{}{}=\\frac{1}{p}\\sum_{g}\n\t\t\t\t\t\\begin{array}{c}\n\t\t\t\t\t\\includeTikz{FqFr_idempotent}\n\t\t\t\t\t{\n\t\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\t\\annparamst{$*$}{$*$}{}{};\n\t\t\t\t\t\t\\annst{}{$-g$};\n\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{array}$\n\t\t\t\t\\end{tabular}\n\t\t\t\t\\\\\n\t\t\t\t\\toprule[1pt]\n\t\t\t\\end{tabular}\n\t\t}\n\t}\n\t\\caption{Indecomposable idempotents for 2-string annuli of all domain walls, corresponding to defects. For $p=2$, $\\Theta_{x,a}(g)=(-1)^{gx} i^{ag}$, whilst for odd $p$ $\\Theta_{x,a}(g)=\\omega^{gx+ag^2 2^{-1}}$, where $2^{-1}$ is the modular inverse of 2.}\\label{tab:idempotents}\n\\end{minipage}\n\t\n\t\\end{table}\n\n\\section{Natural Transformations} \\label{sec:natural_transformations}\n\nAs explained in \\onlinecite{MR2942952}, fusion category bimodules correspond to domain walls, and bimodule functors correspond to defects. On the mathematics side, we also have natural transformations. Using the diagrammatic framework from this paper, these natural transformations are easy to compute: they are just morphisms in the Karoubi envelope of the annular category $\\ann{M}{N}{}$. Since the annular category is semi-simple, there are no morphisms between distinct objects in the Karoubi envelope and the endomorphism algebra of any simple object is just $\\mathbb{C}$. Interesting natural transformations show up when we consider fusion. For example, in the vertical fusion\n\\begin{align}\n  \\defect{R}{L}{}{}{} \\circ \\defect{L}{X_m}{}{}{} = p \\cdot \\defect{R}{X_m}{}{}{}\n\\end{align}\nThere are $p$ distinct pants diagrams\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{distinct_natural_transformations}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t  \\vpantsparams{$0$}{}{$0$};\n        \\vpantsstp{}{$k$}{$-mk$}{$k$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\\end{align}\nparameterized by $k$, which absorb $\\defect{R}{L}{}{}{}, \\defect{L}{X_m}{}{}{}$ on the legs and $\\defect{R}{X_m}{}{}{}$ on the waist. These pants diagrams represent different natural transformations. For every defect fusion in this paper, the pants diagram which is computed represents a natural transformation.\n\nA reasonable physical interpretation is that natural transformations capture certain aspects of the renormalization process. When computing defect fusion, we are witnessing an isomorphism between the horizontal or vertical concatenation of the defects and another defect. Physically, this corresponds to bringing the defects close together and then locally renormalizing, or \\emph{zooming out}.\n\n\\section{Horizontal Defect Fusion} \\label{sec:horizontal_defect_fusion}\n\nIn this section, we present several horizontal defect fusion computations to elucidate some of the complications that arise.\n\n\\begin{exmp}[$\\defect{T}{T}{}{}{} \\otimes \\defect{T}{T}{}{}{}$]\n\nConsider the fusion of two $T$-$T$ defects $\\defect{T}{T}{a}{b}{}$ and $\\defect{T}{T}{c}{d}{}$. This case is interesting because there is multiplicity in the domain wall fusion $T \\otimes_{\\ZZ{p}} T = p \\cdot T$. The domain wall fusion and the defect fusion are correlated, therefore we represent this defect fusion as a $p \\times p$-matrix indexed by the components in the decomposition $T \\otimes_{\\ZZ{p}} T = p \\cdot T$: \n\\begin{align}\n\\left[ \\defect{T}{T}{a}{b}{}\\otimes \\defect{T}{T}{c}{d}{}\\right]_{\\mu,\\nu} \\to \\defect{T}{T}{\\alpha}{\\beta}{}.\n\\end{align}\nWe want to establish the possible values of $\\alpha$ and $\\beta$ and the associated multiplicities. We do this by decomposing the tensor product into simple bimodule categories using $T\\to T\\otimes_{\\vvec{\\ZZ{p}}}$. By fixing $\\mu$ and $\\nu$, we can inflate $\\defect{T}{T}{\\alpha}{\\beta}{}$ to a 4-string annulus\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{DTTab_A}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamss{$(0,0)$}{$(\\alpha,\\beta)$}{}{};\n\t\\annss{}{};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\n\\begin{array}{c}\n\\includeTikz{DTTab_B}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamstpq{$(0,\\mu)$}{$(0,0)$}{$(\\alpha,\\nu)$}{$(0,\\beta)$}{}{}{}{};\n\t\\annssss{}{}{}{};\n\t\\end{tikzpicture}\n}\n\\end{array},\n\\end{align} \nNext, we can find a pant mapping the pair of 2-string defects to the 4-string defect. The most general form of the pants is\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{DTTab_DTTcd_pants_A}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\pantsparams{$(0,\\mu)$}{$(0,0)$}{$(\\alpha,\\nu)$}{$(0,\\beta)$}{}{}{}{};\n\t\\generalpantsstpq{}{$g$}{$h$}{}{}{$k$}{}{$l$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n&=\n\\begin{array}{c}\n\\includeTikz{DTTab_DTTcd_pants_B}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\pantsparams{$(0,\\mu)$}{$(0,0)$}{$(\\alpha,\\nu)$}{$(0,\\beta)$}{}{}{}{};\n\t\\pantsstpq{}{$g{+}l$}{$h{-}l$}{}{$k+l$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n=\n\\begin{array}{c}\n\\includeTikz{DTTab_DTTcd_pants_C}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\pantsparams{$(0,\\mu)$}{$(0,0)$}{$(\\alpha,\\nu=b+c+\\mu)$}{$(0,\\beta)$}{}{}{}{};\n\t\\pantsstpq{}{$\\mu$}{}{}{$-c$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nFrom this, we conclude that $\\alpha = a$, $\\beta = d$ and $\\nu-\\mu \\equiv b + c \\mod p$, which implies that\n\\begin{align}\n\\left[ \\defect{T}{T}{a}{b}{}\\otimes \\defect{T}{T}{c}{d}{} \\right]_{\\mu,\\nu} = \\delta_{\\nu-\\mu}^{b+c} \\cdot \\defect{T}{T}{a}{d}{}.\n\\end{align}\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{X_k}{}{}{} \\otimes \\defect{X_l}{X_l}{}{}{}$, which includes fusing `anyons']\n\t\nThe bimodule $X_1$ corresponds to $\\vvec{\\ZZ{p}}$ as a self-bimodule. As such, $\\defect{X_1}{X_1}{}{}{}$ defects correspond to the excitations of the underlying topological phase. In the physics literature, these excitations are referred to as \\emph{anyons}. Consider the fusion\n\\begin{align}\n\\defect{X_k}{X_k}{a}{x}{}\\otimes \\defect{X_l}{X_l}{b}{y}{}\\to \\defect{X_{kl}}{X_{kl}}{\\alpha}{\\beta}{}.\n\\end{align}\n\nWe can inflate $ \\defect{X_{kl}}{X_{kl}}{\\alpha}{\\beta}{}$ to a 4-string annulus\n\\begin{align}\n\\frac{1}{p}\\sum_g\\omega^{g\\beta}\n\\begin{array}{c}\n\\includeTikz{DX1X1cz_A}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamss{$0$}{$c$}{}{};\n\t\\annss{$klg$}{$-g$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p}\\sum_g\\omega^{g\\beta}\n\\begin{array}{c}\n\\includeTikz{DX1X1cz_B}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\t\\annparamstpq{$0$}{$0$}{$\\alpha$}{$0$}{}{}{}{};\n\t\t\\annssss{$klg$}{$lg$}{$-g$}{$-lg$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\n\nNext, we can find a pant mapping the 2-string defects to the 4-string defect. We immediately observe that (for nonzero maps) this forces $\\alpha=a+kb$ \n\\begin{align}\n\\frac{1}{p^3}\\sum_{g,h,m}\\omega^{gx+hy+m\\beta}\n\\begin{array}{c}\n\\includeTikz{DX1X1ax_DX1X1by_pants_A}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.5}]\n\t\\pantsparams{$0$}{$0$}{$\\beta$}{$0$}{}{}{}{};\n\t\\generalpantsstpq{$kg$}{-$g$}{$lh$}{-$h$}{$klm$}{$lm{-}b$}{-$m$}{$-lm$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n&=\n\\frac{1}{p^3}\\sum_{g,h,m}\\omega^{gx+hy+m\\beta}\n\\begin{array}{c}\n\t\\includeTikz{DX1X1ax_DX1X1by_pants_B}{\n\t\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.5}]\n\t\t\\pantsparams{$0$}{$0$}{$\\beta$}{$0$}{}{}{}{};\n\t\t\\pantsstpq{$k(g{+}lm)$}{-$(g{+}lm)$}{$l(h{+}m)$}{-$(h{+}m)$}{$-b$};\n\t\t\\end{tikzpicture}\n\t}\n\\end{array}.\n\\end{align}\nMaking the replacements\n\\begin{align}\ng^\\prime&=g+lm,h^\\prime=h+m,\n\\end{align}\nwe obtain\n\\begin{align}\n\\frac{1}{p^2}\\sum_{g,h}\\omega^{gx+hy+m\\beta}\n\\begin{array}{c}\n\\includeTikz{DX1X1ax_DX1X1by_pants_C}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.5}]\n\t\\pantsparams{$0$}{$0$}{$\\beta$}{$0$}{}{}{}{};\n\t\\pantsstpq{$kg^\\prime$}{-$g^\\prime$}{$lh^\\prime$}{-$h^\\prime$}{$-b$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\times\n\\frac{1}{p}\\sum_m \\omega^{m(\\beta-lx-z)}.\n\\end{align}\nThe final sum is zero unless the exponent is 0. Therefore, $\\beta=lx+z$. The result of the fusion is\n\\begin{align}\n\\defect{X_k}{X_k}{a}{x}{}\\otimes \\defect{X_l}{X_l}{b}{y}{}=\\defect{X_{km}}{X_{kl}}{a+kb}{z+lx}{}.\n\\end{align}\nFor the special case $k=l=1$, we can identify this with the known anyon fusion rule\n\\begin{align}\nm^ae^x\\times m^be^y=m^{a+b}e^{x+y}.\n\\end{align}\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{L}{L}{}{}{}$]\n\nConsider the defect fusion\n\\begin{align}\n\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{L}{L}{c}{z}{} \\to \\defect{L}{L}{\\alpha}{\\zeta}{}\n\\end{align}\nFirst, we inflate $\\defect{L}{L}{\\alpha}{\\zeta}{}$ to a 4-string idempotent:\n\\begin{align}\n\\frac{1}{p}\\sum_h\\omega^{h\\zeta}\n\\begin{array}{c}\n\\includeTikz{XkXlLL_target_LL}{\n\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=.6}]\n\t\\annparamss{$0$}{$\\alpha$}{}{};\n\t\\annss{$h$}{};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p}\\sum_h\\omega^{h\\zeta}\n\\begin{array}{c}\n\\includeTikz{XkXlLL_target_inflated}{\n\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=0.6}]\n\t\t\\annparamstpq{$0$}{$0$}{$0$}{$\\alpha$}{}{}{}{};\n\t\t\\annstpq{$h$}{$l^{-1}h$}{}{$-k^{-1}h$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nIn order to find a pair-of-pants absorbing the inflated idempotent at the waist, and $\\defect{X_k}{X_l}{}{}{}$ and $\\defect{L}{L}{c}{z}{}$ on the legs respectively, we must have $\\alpha = c$. The general pair-of-pants looks like\n\\begin{align}\n  \\frac{1}{p^2} \\sum_{g,h} \\omega^{gz+h\\zeta}\n\\begin{array}{c}\n\\includeTikz{XkXlLL_pants}{\n\t\\begin{tikzpicture}[scale=1.2,,every node\/.style={scale=.6}]\n\t\\pantsparams{$0$}{$0$}{$0$}{$c$}{}{}{}{};\n\t\\pantsstpq{$-k(k_1-k^{-1}h)$}{$k_1 - k^{-1}h$}{$k_3+g+k^{-1}h$}{}{$(1-l^{-1}k)(k_1-k^{-1}h)$};\n\t\\end{tikzpicture}\n}\n\\end{array} = \\frac{\\omega^{k k_1 \\zeta - k_1 z - k_3 z}}{p^2} \\sum_{g,h} \\omega^{gz + h k \\zeta}\n\\begin{array}{c}\n\\includeTikz{XkXlLL_pants_substitution}{\n\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=.6}]\n\t\\pantsparams{$0$}{$0$}{$0$}{$c$}{}{}{}{};\n\t\\pantsstpq{$kh$}{$-h$}{$g + h$}{}{$(l^{-1}k-1)h$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nwhich is non-zero for all choices of $\\zeta$. Therefore\n\\begin{align}\n\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{L}{L}{c}{z}{} = \\oplus_{\\zeta} \\defect{L}{L}{c}{\\zeta}{}\n\\end{align}\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{F_0}{X_l}{}{}{} \\otimes \\defect{L}{L}{}{}{}$]\nConsider the defect fusion\n\\begin{align}\n  \\defect{F_0}{X_l}{x}{}{} \\otimes \\defect{L}{L}{c}{z}{} \\to \\defect{L}{L}{\\alpha}{\\zeta}{}\n\\end{align}\nInflating $\\defect{L}{L}{\\alpha}{\\zeta}{}$ to a 4-string idempotent gives\n\\begin{align}\n\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_target_LL}{\n\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=.6}]\n\t\\annparamss{$0$}{$\\alpha$}{}{};\n\t\\annss{$\\gamma$}{};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p^2}\\sum_{\\gamma,\\delta}\\omega^{\\delta \\mu + \\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_target_inflated}{\n\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=0.6}]\n\t\t\\annparamstpq{$*$}{$0$}{$0$}{$\\alpha$}{}{}{}{};\n\t\t\\annssss{$\\gamma$}{$l^{-1}\\gamma$}{}{$\\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nThe general pair-of-pants absorbing $\\defect{F_0}{X_l}{x}{}{}$ and $\\defect{L}{L}{c}{z}{}$ on the legs and the inflated idempotent on the belt is\n\\begin{align}\n  \\frac{1}{p^4} \\sum_{g,h,\\gamma,\\delta} \\omega^{gx + hz + \\delta \\mu + \\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_pants}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t  \\pantsparams{$*$}{$0$}{$0$}{$\\alpha$}{}{}{}{};\n  \\generalpantsstpq{$\\gamma + g + k_0$}{$k_1-l^{-1}g$}{$h$}{}{}{$l^{-1}\\gamma + k_2$}{}{$\\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array} = \\frac{1}{p^4} \\sum_{g,h} \\omega^{gx + hz + \\delta \\mu + \\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_simplified_1}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$c$}{}{}{}{};\n\t\\pantsstpq{$\\gamma + g + k_0$}{$k_1 - \\l^{-1}g + \\delta$}{$h - \\delta$}{}{$l^{-1}\\gamma + k_2 + \\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nIn order for the objects to match on the green string, we must have $k_0 = l(k_2 - k_1)$. This equation together with the transformations $\\gamma \\to \\gamma - l k_2$ and $g \\to g + l k_1$ let us phase away $k_0,k_1$ and $k_2$ giving the following pants:\n\\begin{align}\n\\frac{1}{p^4} \\sum_{g,h,\\gamma,\\delta} \\omega^{gx + hz + \\delta \\mu + \\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_simplified_2}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$c$}{}{}{}{};\n\t\\pantsstpq{$\\gamma + g$}{$- \\l^{-1}g + \\delta$}{$h - \\delta$}{}{$l^{-1}\\gamma + \\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nIf we define\n\\begin{align}\n  b &= l^{-1} \\gamma + \\delta \\\\\n  c &= -l^{-1} g + \\delta \\\\\n  d &= h - \\delta\n\\end{align}\nthen $(g,h,\\gamma,\\delta) \\to (b,c,d,\\gamma)$ is invertible over $\\ZZ{p}$ and we get the following pants:\n\\begin{align}\n\\frac{1}{p^3} \\sum_{b,c,d} \\omega^{(lx + z + \\mu)b - lcx + dz}\n\\begin{array}{c}\n\\includeTikz{F0Xl_LL_simplified_3}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$c$}{}{}{}{};\n\t\\pantsstpq{$l(b-c)$}{$c$}{$d$}{}{$b$};\n\t\\end{tikzpicture}\n}\n\\end{array}\\times\\frac{1}{p}\\sum_{\\gamma}\\omega^{(\\zeta - x - l^{-1}z - l^{-1}\\mu)\\gamma}\n\\end{align}\nwhich is nonzero only if $\\zeta = x + l^{-1}(\\mu + z)$. Therefore we have\n\\begin{align}\n\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{L}{L}{c}{z}{} \\to \\defect{L}{L}{c}{x + l^{-1}(\\mu + z)}{} \n\\end{align}\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{X_m}{X_n}{}{}{}$]\n\nConsider the defect fusion\n\\begin{align}\n\t\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{X_m}{X_n}{}{}{}\\to \n\t\\begin{cases} \n\t\t\\defect{X_{km}}{X_{ln}}{}{}{} & km \\neq ln\\\\\n\t\t\\defect{X_{km}}{X_{km}}{\\alpha}{\\beta}{} & km = ln\n\t\\end{cases}.\n\\end{align}\nwhere we assume $k\\neq l$ and $m\\neq n$. We will present the calculation for these two cases separately.\n\\subsubsection*{Case I: $km=ln$}\nInflating $\\defect{X_{km}}{X_{km}}{\\alpha}{\\beta}{}$ to a 4-string idempotent gives\n\\begin{align}\n\t\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma\\beta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_target_XkmXkm}{\n\t\t\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=.6}]\n\t\t\t\t\\annparamss{$0$}{$\\alpha$}{}{};\n\t\t\t\t\\annss{$km\\gamma$}{$-\\gamma$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}\n\t\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma\\beta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_target_1_inflated}{\n\t\t\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=0.6}]\n\t\t\t\\annparamstpq{$0$}{$0$}{$\\alpha$}{$0$}{}{}{}{};\n\t\t\t\\annstpq{$km\\gamma$}{$n\\gamma$}{$-\\gamma$}{$-m\\gamma$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\nThe general pair-of-pants absorbing $\\defect{X_k}{X_l}{}{}{}$ and $\\defect{X_m}{X_n}{}{}{}$ on the legs and the inflated idempotent on the belt is\n\\begin{align}\n\t\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma\\beta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_eq_pants}{\n\t\t\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\t\t\\pantsparams{$0$}{$0$}{$\\alpha$}{$0$}{}{}{}{};\n\t\t\t\\generalpantsstpq{$kx_1$}{$-x_1$}{$mx_2$}{$-x_2$}{$km\\gamma$}{$n\\gamma+q$}{$-\\gamma$}{$-m\\gamma$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array} = \\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma\\beta}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_eq_pants_simplified_1}{\n\t\t\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\t\t\\pantsparams{$0$}{$0$}{$\\alpha$}{$0$}{}{}{}{};\n\t\t\t\\pantsstpq{$k(m\\gamma+x_1)$}{$-(m\\gamma+x_1)$}{$m(\\gamma+x_2)$}{$-(\\gamma+x_2)$}{$(n-m)\\gamma+q$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array},\n\\end{align}\nwhere $q=(l^{-1}k-1)x_1-l^{-1}\\alpha$, $x_2=-q(m-n)^{-1}$.\nMaking the change of variables $\\gamma^\\prime=m\\gamma+x_1$ eliminate $x_1$ up to a global phase. Therefore, there is a single, nonzero map\n\\begin{align}\n\t\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{X_m}{X_n}{}{}{} \\to \\defect{X_{km}}{X_{km}}{\\alpha}{\\beta}{} \n\\end{align}\nfor all $\\alpha,\\,\\beta$.\n\n\\subsubsection*{Case II: $km\\neq ln$}\n\nInflating $\\defect{X_{km}}{X_{ln}}{}{}{}$ to a 4-string idempotent gives\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_target_XkmXln}{\n\t\t\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=.6}]\n\t\t\t\\annparamss{$0$}{$0$}{}{};\n\t\t\t\\annst{}{};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}\n\t\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_target_2_inflated}{\n\t\t\t\\begin{tikzpicture}[scale=0.8,,every node\/.style={scale=0.6}]\n\t\t\t\\annparamstpq{$0$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\t\t\\annstpq{}{}{}{};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\n\nThe general pair-of-pants absorbing $\\defect{X_k}{X_l}{}{}{}$ and $\\defect{X_m}{X_n}{}{}{}$ on the legs and the inflated idempotent on the belt is\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXl_XmXn_neq_pants}{\n\t\t\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\t\t\t\\pantsparams{$0$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\t\t\t\\pantsstpq{$kx_1$}{$-x_1$}{$mx_2$}{$-x_2$}{$(1-kl^{-1})x_1$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array},\n\\end{align}\nwhere $x_2=x_1(kl^{-1}-1)(m-n)^{-1}$.\nwhere $q=(l^{-1}k-1)x_1-l^{-1}\\alpha$, $x_2=-q(m-n)^{-1}$. There are no further constraints on the maps. Therefore, there are $p$ distinct maps between these defects\n\\begin{align}\n\t\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{X_m}{X_n}{}{}{} = p\\cdot \\defect{X_{km}}{X_{ln}}{}{}{}. \n\\end{align}\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{F_q}{X_l}{}{}{} \\otimes \\defect{F_s}{X_n}{}{}{}$, which includes fusing `twists']\n\nA particular set of defects of the $\\ZZ{2}$ Kitaev model identified in \\onlinecite{Bombin2010}, where they were referred to as \\emph{twists}. In our notation, twists occur at the interface of an $X_1$ and $F_1$ domain wall. In this example, we compute the fusion of generalizations of twists in the $\\ZZ{p}$ Kitaev model.\n\nConsider the defect fusion\n\\begin{align}\n\t\\defect{F_q}{X_l}{x}{}{} \\otimes \\defect{F_s}{X_n}{z}{}{}\\to \n\t\\begin{cases} \n\t\t\\defect{X_{q^{-1}s}}{X_{ln}}{}{}{} & q^{-1}s \\neq ln\\\\\n\t\t\\defect{X_{ln}}{X_{ln}}{\\alpha}{\\beta}{} & q^{-1}s = ln\n\t\\end{cases}.\n\\end{align}\n\n\\subsubsection*{Case I: $q^{-1}s\\neq ln$}\n\nWhen $q^{-1}s \\not= ln$, the fusion looks like\n\\begin{align}\n\\defect{F_q}{X_l}{x}{}{}\\otimes \\defect{F_s}{X_n}{z}{}{} \\to \\defect{X_{q^{-1}s}}{X_{ln}}{}{}{},\n\\end{align}\nbut we need to check for multiplicity. We can inflate $\\defect{X_{q^{-1}s}}{X_{ln}}{}{}{}$ to a 4-string annulus\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{DXqsXlna_A_1}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamst{$0$}{$0$}{}{};\n\t\\annst{}{};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p}\\sum_{\\gamma}\n\\begin{array}{c}\n\\includeTikz{DXqsXlna_B_1}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamstpq{$*$}{$*$}{$0$}{$0$}{}{}{}{};\n\t\\annstpq{}{}{}{$\\gamma$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nThis implies that the general pair-of-pants absorbing the 2-string defects on the legs and the 4-string defect on the waist is\n\\begin{align}\n\\frac{1}{p^3}\\sum_{g,h,\\gamma} \\Theta_{x,-ql}(g) \\Theta_{z,-ns}(h) \\omega^{-qk_0g - sk_3h}\n&\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_A_1}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$0$}{$0$}{}{}{}{};\n\t\\generalpantsstpq{$lg+k_0$}{$-g+k_1$}{$nh+k_3$}{$-h+k_4$}{}{$k_2$}{}{$\\gamma$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\\\=\n\\frac{1}{p^3}\\sum_{g,h,\\gamma} \\Theta_{x,-ql}(g) \\Theta_{z,-ns}(h) \\omega^{-qk_0g - sk_3h+s \\gamma (h-k_4)}\n&\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_B_1}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$0$}{$0$}{}{}{}{};\n\t\\pantsstpq{$lg+k_0$}{\\tiny$\\gamma-g+k_1$}{\\tiny$nh+k_3{-}\\gamma$}{$-h+k_4$}{$\\gamma+k_2$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nFor the labels to match, we must have\n\\begin{align}\n  k_0 + l k_1 - l k_2 &= 0 \\\\\n  k_3 + n k_4 + k_2 &= 0.\n  \\end{align}\nIf we make the transformation\n\\begin{align}\n  g &= a - l^{-1}k_0 \\\\\n  \\gamma &= b - k_2 \\\\\n  h &= c + k_4\n\\end{align}\nthen we get\n\\begin{align}\n  \\frac{1}{p^3}\\sum_{a,b,c} \\Theta_{x,-ql}(a)\\Theta_{z,-ns}(c) \\omega^{sbc}\n\\begin{array}{c}\n\\includeTikz{FqXl_XnFs_transformed_1}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$0$}{$0$}{}{}{}{};\n\t\\pantsstpq{$la$}{$b-a$}{$c-b$}{$-c$}{$b$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nSo there is 1 map \n\\begin{align}\n\\defect{F_q}{X_l}{x}{}{}\\otimes \\defect{F_s}{X_n}{z}{}{}\\to \\defect{X_{q^{-1}s}}{X_{ln}}{}{}{}.\n\\end{align}\n\n\\subsubsection*{Case II: $q^{-1}s=ln$}\nWhen $q^{-1}s=ln$, the fusion looks like\n\\begin{align}\n  \\defect{F_q}{X_l}{x}{}{}\\otimes \\defect{F_s}{X_n}{z}{}{}\\to \\defect{X_{ln}}{X_{ln}}{\\alpha}{\\zeta}{}.\n\\end{align}\nWe can inflate $\\defect{X_{ln}}{X_{ln}}{\\alpha}{\\zeta}{}$ to a 4-string annulus\n\\begin{align}\n\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{DXqsXlna_A_2}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamst{$0$}{$\\alpha$}{}{};\n\t\\annst{$ln\\gamma$}{$-\\gamma$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\\frac{1}{p^2}\\sum_{\\gamma,\\delta}\\omega^{\\gamma \\zeta}\n\\begin{array}{c}\n\\includeTikz{DXqsXlna_B_2}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamstpq{$*$}{$*$}{$\\alpha$}{$0$}{}{}{}{};\n\t\\annstpq{$ln\\gamma$}{$n\\gamma$}{$-\\gamma$}{$\\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\n\nNext, we can find the pant that absorbs the 2-string idempotents on the legs and 4-string idempotent on the waist. The most general pant that will absorb the appropriate idempotents is \n\\begin{align}\n\\frac{1}{p^4}\\sum_{g,h,\\gamma,\\delta}\\Theta_{x,-ql}(g) \\Theta_{z,-ns}(h) \\omega^{\\gamma \\zeta-hk_3s - gk_0q}\n\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_A_2}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$\\alpha$}{$0$}{}{}{}{};\n\t\\generalpantsstpq{\\tiny$lg+k_0$}{\\tiny$-g+k_1$}{\\tiny$nh+k_3$}{\\tiny$-h+k_4$}{$ln\\gamma$}{$n\\gamma + k_2$}{$-\\gamma$}{$\\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}\\\\\n=\n\\frac{1}{p^4}\\sum_{g,h,\\gamma,\\delta}\n\\Theta_{x,-ql}(g) \\Theta_{z,-ns}(h) \\omega^{\\gamma \\zeta-hk_3s - gk_0q + ln \\gamma(-g+k_1)q +\\delta(\\gamma + h - k_4)s}\n\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_B_2}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$\\alpha$}{$0$}{}{}{}{};\n\t\\pantsstpq{\\tiny$lg + ln\\gamma + k_0$}{\\tiny$\\delta-g+k_1$}{\\tiny$-\\delta+nh+k_3$}{\\tiny$-h-\\gamma+k_4$}{$n\\gamma+k_2+\\delta$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nFor the labels to match up\n\\begin{align}\n  k_0 + l k_1 - l k_2-\\alpha &= 0 \\\\\n  k_3 + n k_4 + k_2 &= 0.\n  \\end{align}\nIf we make the change of coordinates\n\\begin{align}\n  a &= \\delta - g \\\\\n  b &= n \\gamma + \\delta \\\\\n  c &= -h - \\gamma\n\\end{align}\nthen we get\n\\begin{align}\n  \\frac{1}{p^4}\\sum_{a,b,c,\\gamma}\\omega^{{\\rm exponent}}\n\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_B_3}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$\\alpha$}{$0$}{}{}{}{};\n\t\\pantsstpq{$l(b-a)+k_0$}{$a+k_1$}{$-nc-b+k_3$}{$c+k_4$}{$b+k_2$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nwhere\n\\begin{align*}\n  {\\rm exponent} = &-2^{-1} a^2 l q+k_0 q (a-b+\\gamma  n)+a b l q-a x-2^{-1} b^2 l q+ \\\\\n   &lnq \\left(k_4 (\\gamma  n-b)+k_3 (c+\\gamma )+\\gamma  k_1\\right)-b c l n q+bx-2^{-1} c^2 l n^2 q+\\gamma  \\zeta -c z-\\gamma  n x-\\gamma  z\n  \\end{align*}\nMaking the transformation\n\\begin{align}\n  a &\\to a - k_1 \\\\\n  b &\\to b - k_2 \\\\\n  c &\\to c - k_4\n\\end{align}\ngives\n\\begin{align}\n  \\frac{1}{p^4}\\sum_{a,b,c,\\gamma}\\omega^{{\\rm exponent}}\n\\begin{array}{c}\n\\includeTikz{DFqXlx_DFsXnz_pants_B_4}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.55}]\n\t\\pantsparams{$*$}{$*$}{$\\alpha$}{$0$}{}{}{}{};\n\t\\pantsstpq{$l(b-a)+\\alpha$}{$a$}{$-nc-b$}{$c$}{$b$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\end{align}\nwhere\n\\begin{align*}\n  {\\rm exponent} =& -2^{-1} a^2 l q+a b l q+\\alpha  a q-a x-2^{-1} b^2 l q-b c l n q-\\\\ \n  &\\alpha b q+b x-2^{-1} c^2 l n^2 q-c z+ \\gamma  (\\zeta +\\alpha  n q-n x-z)\n\\end{align*}\nThis implies that $\\zeta = nx + z - \\alpha n q$, so\n\\begin{align}\n\\defect{F_q}{X_l}{x}{}{} \\otimes \\defect{F_s}{X_n}{z}{}{} \\to \\oplus_{\\alpha} \\defect{X_{ln}}{X_{ln}}{\\alpha}{z+nx-nq\\alpha}{}\n  \\end{align}\nfor odd primes. The same calculation for $p=2$ yields\n\\begin{align}\n\t\\defect{F_1}{X_1}{x}{}{} \\otimes \\defect{F_1}{X_1}{z}{}{} \\to \\oplus_{\\alpha} \\defect{X_{1}}{X_{1}}{\\alpha}{z+x+\\alpha}{}.\n\\end{align}\nThis result was presented in \\onlinecite{Bombin2010} as $\\sigma^x\\sigma^z=\\sum_j e^{j+x+z} m^{j}$.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{F_q}{X_l}{}{}{} \\otimes \\defect{X_m}{X_m}{}{}{}$]\n\\begin{align}\n\\defect{F_q}{X_l}{x}{}{}\\otimes\\defect{X_m}{X_m}{c}{z}{}\\to \\defect{F_{qm}}{X_{lm}}{\\alpha}{}{}\n\\end{align}\n\nWe can inflate $\\defect{F_{qm}}{X_{lm}}{\\alpha}{}{}$ to a 4-string annulus\n\\begin{align}\n\\frac{1}{p}\\sum_g\\Theta_{\\alpha,-qlm^2}(g)\n\\begin{array}{c}\n\\includeTikz{DF1X1z_A}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamst{$*$}{$0$}{}{};\n\t\\annst{$lmg$}{$-g$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\,\\,\\rotatebox[origin=c]{-90}{$\\curlyveedownarrow$}\\,\\,\n\\frac{1}{p}\\sum_g\\Theta_{\\alpha,-qlm^2}(g)\n\\begin{array}{c}\n\\includeTikz{DF1X1z_B}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.6}]\n\t\\annparamstpq{$*$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\\annstpq{$lmg$}{$mg$}{$-g$}{$-mg$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\n\nNext, we can find a pant mapping the 2-string defects to the 4-string defect \n\\begin{align}\n\\frac{1}{p^3}&\\sum_{g,h,k}\\Theta_{x,-ql}(g)\\Theta_{\\alpha,-qlm^2}(k)\\omega^{hz-ql\\mu g}\n\\begin{array}{c}\n\\includeTikz{DX1X1ax_DF1X1y_pants_A}{\n\t\\begin{tikzpicture}[scale=.8,every node\/.style={scale=.5}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\\generalpantsstpq{$l(g{+}\\mu)$}{-$(g{+}\\mu{+}c)$}{$mh$}{$-h$}{$lmk$}{$mk-c$}{$-k$}{$-mk$};\n\t\\end{tikzpicture}\n}\n\\end{array}\\\\\n&=\n\\frac{1}{p^3}\\sum_{g,h,k}\\Theta_{x,-ql}(g)\\Theta_{\\alpha,-qlm^2}(k)\\omega^{hz-ql\\mu g-qlmk(g+\\mu+c)}\n\\begin{array}{c}\n\\includeTikz{DX1X1ax_DF1X1y_pants_B}{\n\t\\begin{tikzpicture}[scale=.75,,every node\/.style={scale=.5}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\\pantsstpq{$l(g{+}\\mu{+}mk)$}{-$(g$+$\\mu{+}mk$+$c)$}{$m(h{+}k)$}{-$(h{+}k)$}{$-c$};\n\t\\end{tikzpicture}\n}\n\\end{array}\\\\\n&\\propto  \n\\frac{1}{p^2}\\sum_{g,h}\\Theta_{x,-ql}(g)\\omega^{hz}\n\\begin{array}{c}\n\\includeTikz{DX1X1ax_DF1X1y_pants_C}{\n\t\\begin{tikzpicture}[scale=.6,,every node\/.style={scale=.5}]\n\t\\pantsparams{$*$}{$0$}{$0$}{$0$}{}{}{}{};\n\t\\pantsstpq{$lg$}{-$(g+c)$}{$mh$}{-$h$}{$-c$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n\\times \\frac{1}{p}\\sum_{k}\\omega^{k(\\alpha-z-m(x+qlc))}.\n\\end{align}\nTherefore\n\\begin{align}\n\\defect{F_q}{X_l}{x}{}{} \\otimes \\defect{X_m}{X_m}{c}{z}{}=\\defect{F_{qm}}{X_{lm}}{z+m(x+qlc)}{}{}.\n\\end{align}\nFor the special case $q=l=m=1$, this recovers the known fusion rule of `twists' with anyons\\cite{Bombin2010,Barkeshli2014,Bridgeman2017}\n\\begin{align}\n\\sigma^{x}\\times m^c e^z=\\sigma^{x+z+c}.\n\\end{align}\n\n\\end{exmp}\n\\section{Inflations} \\label{sec:inflations}\n\nWhen performing the horizontal fusion algorithm, the target idempotent needs to be inflated from a 2-string annulus onto a 4-string annulus. The tables in this section contain the data needed for this process. The procedure used to compute this data is explained in \\onlinecite{1806.01279}. For completeness, Table~\\ref{tab:zptable} contains the domain wall fusion data from \\onlinecite{1806.01279}.\n\n\\begin{table}[h]\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c c c c!{\\vrule width 1pt}c c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]$\\otimes_{\\vvec{\\ZZ{p}}}$&$T$&$L$&$R$&$F_0$&$X_l$&$F_r$\\\\\n\t\t\\toprule[1pt]\n\t\t$T$&$p\\cdot T$&$T$&$p\\cdot R$&$R$&$T$&$R$\\\\\n\t\t$L$&$p\\cdot L$&$L$&$p\\cdot F_0$&$F_0$&$L$&$F_0$\\\\\n\t\t$R$&$T$&$p\\cdot T$&$R$&$p\\cdot R$&$R$&$T$\\\\\n\t\t$F_0$&$L$&$p\\cdot L$&$F_0$&$p\\cdot F_0$&$F_0$&$L$\\\\\n\t\t\\toprule[1pt]\n\t\t$X_k$&$T$&$L$&$R$&$F_0$&$X_{kl}$&$F_{k^{-1}r}$\\\\\n\t\t$F_q$&$L$&$T$&$F_0$&$R$&$F_{ql}$&$X_{q^{-1}r}$\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n\t\\caption{Multiplication table for $\\protect\\bpr{\\protect\\vvec{\\ZZ{p}}}$, reproduced from \\onlinecite{1806.01279}.}\\label{tab:zptable}\n\\end{table}\n\n\n\\addtocounter{table}{1}\n\n\\newcounter{curtable}\n\\setcounter{curtable}{0}\n\\addtocounter{curtable}{\\value{table}}\n\n\\setcounter{table}{0}\n\\renewcommand{\\thetable}{\\Roman{curtable}(\\alph{table})}\n\n\\begin{table}[h!]\n\t\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\t\t\\toprule[1pt]\n&$T\\otimes_{\\vvec{\\ZZ{p}}}T$&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\t\\includeTikz{Tinflhsa}{\\begin{tikzpicture}\n\t\t\t\\inflationalhs{$g$}{$(a,b)$}{$h$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{TinfrhsaTT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhss{$g$}{$(a,\\mu)$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$\n&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\\includeTikz{Tinflhsb}{\\begin{tikzpicture}\n\t\t\\inflationblhs{$g$}{$(a,b)$}{$h$};\n\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{TinfrhsbTT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhss{$g$}{$(a,\\nu)$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$T\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\begin{array}{c}\\includeTikz{TinfrhsaTL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$(a,0)$}{}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{TinfrhsbTL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$(a,0)$}{}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$T\\otimes_{\\vvec{\\ZZ{p}}}X_l$&&$\\begin{array}{c}\\includeTikz{TinfrhsaTXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$(a,0)$}{}{$lb$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{TinfrhsbTXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$(a,0)$}{}{$lb$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$R\\otimes_{\\vvec{\\ZZ{p}}}T$&&$\\begin{array}{c}\\includeTikz{TinfrhsaRT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{TinfrhsbRT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$R\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\frac{1}{p}\\sum_k\\omega^{\\mu k}\\begin{array}{c}\\includeTikz{TinfrhsaRL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{$k$}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{\\nu k}\\begin{array}{c}\\includeTikz{TinfrhsbRL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{$k$}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$R\\otimes_{\\vvec{\\ZZ{p}}}F_r$&&$\\frac{1}{p}\\sum_k\\omega^{kbr}\\begin{array}{c}\\includeTikz{TinfrhsaRFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{kr(b-h)}\\begin{array}{c}\\includeTikz{TinfrhsbRFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$X_k\\otimes_{\\vvec{\\ZZ{p}}}T$&&$\\begin{array}{c}\\includeTikz{TinfrhsaXkT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{TinfrhsbXkT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{}{$(0,b)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t\\multirow{-20}{*}{$T$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\frac{1}{p}\\sum_k\\omega^{qak}\\begin{array}{c}\\includeTikz{TinfrhsaFqL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$k$}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{kqa}\\begin{array}{c}\\includeTikz{TinfrhsbFqL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$k$}{$b$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n&$L\\otimes_{\\vvec{\\ZZ{p}}}T$&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\\includeTikz{Linflhsa}{\\begin{tikzpicture}\n\t\t\\inflationalhs{$g$}{$a$}{$h$};\n\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{LinfrhsaLT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$\\mu$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\\includeTikz{Linflhsb}{\\begin{tikzpicture}\n\t\t\\inflationblhs{$g$}{$a$}{$h$};\n\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{LinfrhsbLT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$\\nu$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$L\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\begin{array}{c}\\includeTikz{LinfrhsaLL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhss{$g$}{$0$}{}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{LinfrhsbLL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhss{$g$}{$0$}{}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$L\\otimes_{\\vvec{\\ZZ{p}}}X_l$&&$\\begin{array}{c}\\includeTikz{LinfrhsaLXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{}{$la$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{LinfrhsbLXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{}{$la$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}T$&&$\\begin{array}{c}\\includeTikz{LinfrhsaF0T}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{LinfrhsbF0T}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\frac{1}{p}\\sum_k\\omega^{k \\mu}\\begin{array}{c}\\includeTikz{LinfrhsaF0L}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$k$}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{k \\nu}\\begin{array}{c}\\includeTikz{LinfrhsbF0L}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$k$}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}F_r$&&$\\frac{1}{p}\\sum_k\\omega^{kar}\\begin{array}{c}\\includeTikz{LinfrhsaF0Fr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{k(a-h)r}\\begin{array}{c}\\includeTikz{LinfrhsbF0Fr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t&$X_k\\otimes_{\\vvec{\\ZZ{p}}}L$&&$\\begin{array}{c}\\includeTikz{LinfrhsaXkL}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{$-k^{-1}g$}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{LinfrhsbXkL}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{$k^{-1}g$}{$a$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\t\t\\multirow{-20}{*}{$L$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}T$&&$\\begin{array}{c}\\includeTikz{LinfrhsaFqT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{LinfrhsbFqT}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{}{$(0,a)$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n\t\t\t\\end{tabular}\n\t\\caption{Inflations (part a). All $\\mu,\\,\\nu$ occurring label components of the tensor decomposition} \\label{tab:inflation_1}\n\t\\vspace*{-10mm}\n\\end{table}\n\n\\begin{table}\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n&$T\\otimes_{\\vvec{\\ZZ{p}}}R$&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\\includeTikz{Rinflhsa}{\\begin{tikzpicture}\n\t\t\\inflationalhs{$g$}{$a$}{$h$};\n\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{RinfrhsaTR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$(a,\\mu)$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&\\multirow{20}{*}{\n\t$\\begin{array}{c}\n\t\\includeTikz{Rinflhsb}{\\begin{tikzpicture}\n\t\t\\inflationblhs{$g$}{$a$}{$h$};\n\t\t\\end{tikzpicture}}\n\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{RinfrhsbTR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$(a,\\nu)$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$T\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\begin{array}{c}\\includeTikz{RinfrhsaTF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$(a,0)$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{RinfrhsbTF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$(a,0)$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$T\\otimes_{\\vvec{\\ZZ{p}}}F_r$&&$\\begin{array}{c}\\includeTikz{RinfrhsaTFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$(a,0)$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{RinfrhsbTFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$(a,0)$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$R\\otimes_{\\vvec{\\ZZ{p}}}R$&&$\\begin{array}{c}\\includeTikz{RinfrhsaRR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhss{$g$}{$a$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{RinfrhsbRR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhss{$g$}{$a$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$R\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\frac{1}{p}\\sum_k\\omega^{k \\mu}\\begin{array}{c}\\includeTikz{RinfrhsaRF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{k \\nu}\\begin{array}{c}\\includeTikz{RinfrhsbRF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$R\\otimes_{\\vvec{\\ZZ{p}}}X_l$&&$\\begin{array}{c}\\includeTikz{RinfrhsaRXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{$lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{RinfrhsbRXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{$-lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n&$X_k\\otimes_{\\vvec{\\ZZ{p}}}R$&&$\\begin{array}{c}\\includeTikz{RinfrhsaXkR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{RinfrhsbXkR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\\multirow{-20}{*}{$R$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\frac{1}{p}\\sum_k\\omega^{qak}\\begin{array}{c}\\includeTikz{RinfrhsaFqF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhss{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{qak}\\begin{array}{c}\\includeTikz{RinfrhsbFqF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n\t\t&$L\\otimes_{\\vvec{\\ZZ{p}}}R$&\\multirow{20}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{F0inflhsa}{\\begin{tikzpicture}\n\t\t\t\t\\inflationalhs{$g$}{$*$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{F0infrhsaLR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$\\mu$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&\\multirow{20}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{F0inflhsb}{\\begin{tikzpicture}\n\t\t\t\t\\inflationblhs{$g$}{$*$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{F0infrhsbLR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$\\nu$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$L\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\begin{array}{c}\\includeTikz{F0infrhsaLF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbLF0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$L\\otimes_{\\vvec{\\ZZ{p}}}F_r$&&$\\begin{array}{c}\\includeTikz{F0infrhsaLFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbLFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}R$&&$\\begin{array}{c}\\includeTikz{F0infrhsaF0R}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbF0R}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\frac{1}{p}\\sum_k\\omega^{k \\mu}\\begin{array}{c}\\includeTikz{F0infrhsaF0F0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhss{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{k \\nu}\\begin{array}{c}\\includeTikz{F0infrhsbF0F0}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhss{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$F_0\\otimes_{\\vvec{\\ZZ{p}}}X_l$&&$\\begin{array}{c}\\includeTikz{F0infrhsaF0Xl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$hl$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbF0Xl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$-hl$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t&$X_k\\otimes_{\\vvec{\\ZZ{p}}}F_0$&&$\\begin{array}{c}\\includeTikz{F0infrhsaXkF0}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{$-k^{-1}g$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbXkF0}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{$k^{-1}g$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\\multirow{-20}{*}{$F_0$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}R$&&$\\begin{array}{c}\\includeTikz{F0infrhsaFqR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{F0infrhsbFqR}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n\t\t&$X_k\\otimes_{\\vvec{\\ZZ{p}}}X_l,\\,m=kl$&\\multirow{2}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{Xminflhsa}{\\begin{tikzpicture}[scale=.5,every node\/.style={scale=.5}]\n\t\t\t\t\\inflationalhs{$g$}{$a$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{XminfrhsaXkXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$a$}{$lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&\\multirow{2}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{Xminflhsb}{\\begin{tikzpicture}[scale=.5,every node\/.style={scale=.5}]\n\t\t\t\t\\inflationblhs{$g$}{$a$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{XminfrhsbXkXl}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$a$}{$-lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\\multirow{-4}{*}{$X_m$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}F_r,\\,m=q^{-1}r$&&$\\frac{1}{p}\\sum_k\\omega^{qka}\\begin{array}{c}\\includeTikz{XminfrhsaFqFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\frac{1}{p}\\sum_k\\omega^{(qa-rh)k}\\begin{array}{c}\\includeTikz{XminfrhsbFqFr}{\\begin{tikzpicture}[xscale=.5,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$k$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n\t\t&$X_k\\otimes_{\\vvec{\\ZZ{p}}}F_r,\\,n=k^{-1}r$&\\multirow{2}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{Fninflhsa}{\\begin{tikzpicture}[scale=.5,every node\/.style={scale=.5}]\n\t\t\t\t\\inflationalhs{$g$}{$*$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\omega^{-ngh}\\begin{array}{c}\\includeTikz{FninfrhsaXkFr}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$0$}{$-k^{-1}g$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$&\\multirow{2}{*}{\n\t\t\t$\\begin{array}{c}\n\t\t\t\\includeTikz{Fninflhsb}{\\begin{tikzpicture}[scale=.5,every node\/.style={scale=.5}]\n\t\t\t\t\\inflationblhs{$g$}{$*$}{$h$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$}&$\\begin{array}{c}\\includeTikz{FninfrhsbXkFr}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$0$}{$k^{-1}g$}{$*$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\t\t\\greycline{2-2}\\greycline{4-4}\\greycline{6-6}\n\t\t\\multirow{-4}{*}{$F_n$}&$F_q\\otimes_{\\vvec{\\ZZ{p}}}X_l,\\,n=ql$&&$\\begin{array}{c}\\includeTikz{FninfrhsaFqXl}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationarhs{$g$}{$*$}{$lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$&&$\\begin{array}{c}\\includeTikz{FninfrhsbFqXl}{\\begin{tikzpicture}[xscale=.6,yscale=.25,every node\/.style={scale=.5}]\\inflationbrhs{$g$}{$*$}{$-lh$}{$0$}{$h$};\\end{tikzpicture}}\\end{array}$\\\\\n\\toprule[1pt]\n\t\\end{tabular}\n\t\\caption{Inflations (part b). All $\\mu,\\,\\nu$ occurring label components of the tensor decomposition} \\label{tab:inflation_2}\n\\end{table}\n\n\\setcounter{table}{\\value{curtable}}\n\\renewcommand{\\thetable}{\\Roman{table}}\n\n\\section{Conclusions}\n\nIn this work, we have studied binary interface defects and their fusion. Using string diagrams and the annular category, we have classified the full set of defects occurring interfacing a pair of (not necessarily invertible) domain walls for the tensor category $\\vvec{\\ZZ{p}}$. Further, we have provided algorithms for computing both horizontal (tensor product) and vertical (composition) fusion of arbitrary pairs of defects. For the theory $\\vvec{\\ZZ{p}}$, we have provided complete fusion tables. \n\nWe have specialized to $\\vvec{\\ZZ{p}}$ for simplicity. The framework outlined here is not restricted to this class of fusion categories. Of particular interest is the Color code\\cite{Bombin2007a,Yoshida2015a} ($\\vvec{\\ZZ{2}\\times\\ZZ{2}}$) due to its importance in quantum computation. Defects between invertible domain walls and the trivial wall ($X_1$ in this paper) were studied in \\onlinecite{Brown2018}, but the full theory is currently open. Additionally, the tools presented here are expected to be useful for studying nonabelian theories\\cite{PhysRevB.96.195129}. Additionally, one could study domain walls and defects between distinct phases, such as the Color code and $\\vvec{\\ZZ{4}}$, which may prove useful for quantum computing tasks.\nAlthough we restrict to binary interface defects, generalizations of the techniques developed here can be applied to higher defects such as those occurring at the interface of three or more domain walls. Such defects allow the meeting of many distinct topological phases.\n\nIn the physics literature, defect fusion is often synonymous with symmetry gauging\\cite{MR2677836,MR2609644,Barkeshli2014,SETPaper,1804.01657}. In this work, we have computed the fusions without consideration of gauging. It would be extremely useful if the techniques developed in this work can say something about (obstructions to) gauging invertible defects.\n\n\\section{Classifying defects} \\label{sec:constructing_idempotents}\n\nFor an underlying fusion category $\\mathcal{C}$, and given a pair of bimodule categories $\\mathcal{M},\\mathcal{N}$, the annular category $\\ann{M}{N}{C}$ is defined as follows.\nThe objects are pairs of simples $(m,n)$ and the morphisms are annular diagrams as shown in Eqn.~\\ref{eqn:generic_morphism_in_tube_category}.\nRepresentations of $\\ann{M}{N}{C}$ classify binary interface defects, which we denote $\\defect{\\mathcal{M}}{\\mathcal{N}}{}{}{}$. If $\\mathcal{C}$ is semi-simple, the category of representations is equivalent to the Karoubi envelope $\\kar{\\ann{M}{N}{C}}$. As in \\onlinecite{1806.01279}, we utilize this equivalence to classify the defects.\n\nObjects of $\\kar{\\ann{M}{N}{C}}$ are pairs $(A,e)$, where $A$ is an object from $\\ann{M}{N}{C}$, and $e:A\\to A$ is an idempotent annular diagram. The classification of defects is equivalent to construction of inequivalent idempotents. Two idempotent annular morphisms are equivalent if there exists a morphism absorbing the first idempotent on the inside and the second idempotent on the outside.\n\nIn the remainder of this section, we show how representative idempotents are constructed for $\\mathcal{C}=\\vvec{\\ZZ{p}}$. The domain wall labels used are those defined in \\onlinecite{1806.01279}.\n\n\\begin{exmp}[$\\defect{T}{T}{}{}{}$]\n\nIn many cases, the construction of representative idempotents is a simple task. For example, the basic annulus diagrams for $\\defect{T}{T}{}{}{}$ defects are\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{TTtube_example_1}{\n\t\t\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\t\t\t\\annparamss{$(a+g,b+h)$}{$(c+g,d+h)$}{$(a,b)$}{$(c,d)$};\n\t\t\t\t\\annss{$g$}{$h$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\\label{eqn:TTtube_example_1}\n\\end{align}\nFor such a diagram to contribute to an idempotent, the inner and outer bimodule labels must be the same. In this case, the only way for this to happen is $g=h=0$ (the group identity)\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{TTtube_example_2}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamss{$(a,b)$}{$(c,d)$}{$(a,b)$}{$(c,d)$};\n\t\\annss{}{};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nIt remains to find representatives of the isomorphism classes, where any annulus in Eqn.~\\ref{eqn:TTtube_example_1} defines an isomorphism between idempotents. Using Eqn.~\\ref{eqn:TTtube_example_1}, there is no way to change the value of $\\alpha=c-a$ or $\\beta=d-b$. These conserved quantities label the isomorphism classes, and we can pick representatives for each of the $p^2$ classes\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{TTtube_example_3}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamss{$(0,0)$}{$(\\alpha,\\beta)$}{$(0,0)$}{$(\\alpha,\\beta)$};\n\t\\annss{}{};\n\t\\end{tikzpicture}\n}\n\\end{array},\n\\end{align}\nwith $\\alpha,\\beta\\in \\ZZ{p}$.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{F_r}{}{}{}$: An unusual representation of $\\ZZ{p}$]\n\nA more complicated example involves $\\defect{X_k}{F_r}{}{}{}$ defects. In this case, the general annulus diagrams are\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{XkFrtube_example_1}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamst{$a+g+kh$}{$*$}{$a$}{$*$};\n\t\\annst{$g$}{$h$};\n\t\\end{tikzpicture}\n}\n\\end{array},\\label{eqn:XkFrtube_example_1}\n\\end{align}\nso the diagrams contributing to idempotents are\n\\begin{align}\nM_g&:=\n\\begin{array}{c}\n\\includeTikz{XkFrtube_example_2}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamst{$0$}{$*$}{$0$}{$*$};\n\t\\annst{$-kg$}{$g$};\n\t\\end{tikzpicture}\n}\n\\end{array},\\label{eqn:XkFrtube_example_2}\n\\end{align}\nwhere $a=0$ has been chosen using the isomorphism Eqn.~\\ref{eqn:XkFrtube_example_1}.\nDue to the nontrivial associator on $F_r$, the multiplication rule for these annuli is\n\\begin{align}\nM_gM_h&=\n\\begin{array}{c}\n\\includeTikz{XkFrtube_example_3}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\t\\draw[blue] (0,-.75) to[out=45,in=-90] (.5,0) to[out=90,in=-45](0,.75);\\node[blue,right,inline text] at (.5,0) {$g$};\n\t\t\\draw[blue] (0,-.5) to[out=180-45,in=-90] (-.3,0) to[out=90,in=180+45](0,.5);\\node[blue,left,inline text] at (-.3,0) {$-kg$};\n\t\t\\draw[blue] (0,-1.75) to[out=45,in=-90] (1.5,0) to[out=90,in=-45](0,1.75);\\node[blue,right,inline text] at (1.5,0) {$h$};\n\t\t\\draw[blue] (0,-1.5) to[out=180-45,in=-90] (-1.3,0) to[out=90,in=180+45](0,1.5);\\node[blue,left,inline text] at (-1.3,0) {$-kh$};\n\t\t\\draw[ultra thick,orange] (0,.25)--(0,2) node[above,inline text] {$*$};\n\t\t\\draw[ultra thick,red] (0,-.25)--(0,-2) node[below,inline text] {$0$};\n\t\t\\draw (0,0) circle (.25);\n\t\t\\draw (0,0) circle (1);\n\t\t\\draw (0,0) circle (2);\n\t\\end{tikzpicture}\n}\n\\end{array}=\n\\omega^{kghr}\n\\begin{array}{c}\n\\includeTikz{XkFrtube_example_4}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.6}]\n\t\\annparamst{$0$}{$*$}{}{};\n\t\\annst{-$k(g{+}h)$}{$g+h$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n=\\omega^{kr gh}M_{g+h},\n\\end{align}\nwhere $\\omega=\\exp(\\frac{2\\pi i}{p})$.\nThese annuli are therefore forming a twisted representation of $\\ZZ{p}$ with 2-cocycle $\\phi(g,h)=\\omega^{krgh}$. Since $H^2(G,U(1))\\cong \\{1\\}$, this is equivalent to a linear representation $U_g$ by some 1-cochain\n\\begin{align}\n\t\\beta_{kr}(g)M_g&=U_g.\n\\end{align}\nOne can obtain an explicit formula for $\\beta_{kr}$\n\\begin{align}\n\\beta_{kr}(g)&=\n\\begin{cases}\ni^{g}&p=2\\\\\n\\omega^{kr g^2 2^{-1}}&p>2\n\\end{cases},\n\\end{align}\nwhere $2^{-1}$ is the multiplicative inverse modulo $p$. With this explicit cochain, representatives for the inequivalent idempotents can be found\n\\begin{align}\n\t\\defect{X_k}{F_r}{x}{}{}&=\\frac{1}{p}\\sum_g \\omega^{gx} \\omega U_g\\\\\n\t&=\\frac{1}{p}\\sum_g \\omega^{gx}\\beta(g) M_g=:\\frac{1}{p}\\sum_g \\Theta_{x,kr}(g) M_g.\n\\end{align}\nThe equation $\\Theta_{x,a}(g+k) = \\Theta_{x,a}(g) \\; \\Theta_{x,a}(k) \\; \\omega^{agk}$ implies that the idempotent $\\defect{X_k}{F_r}{x}{}{}$ absorbs the diagram\n\\begin{align}\n\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkFr_XmFt_companion_1}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{$k k_0$}{$-k_0$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}\n\\end{align}\nup to a global phase.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{F_q}{F_r}{}{}{}$: A genuine projective representation]\n\nA particularly interesting example is given by $\\defect{F_q}{F_r}{}{}{}$ defects. Since there is a single object (denoted $*$) in the $F_x$ bimodule, all annulus diagrams potentially contribute to the idempotents\n\\begin{align}\nM_{g,h}&:=\n\\begin{array}{c}\n\\includeTikz{FqFrtube_example_1}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamst{$*$}{$*$}{}{};\n\t\\annst{$g$}{$h$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\nBoth modules (potentially) have nontrivial associators, leading to the multiplication rule\n\\begin{align}\nM_{g_0,h_0}M_{g_1,h_1}&=\n\\begin{array}{c}\n\\includeTikz{FqFrtube_example_2}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\draw[blue] (0,-.75) to[out=45,in=-90] (.5,0) to[out=90,in=-45](0,.75);\\node[blue,right,inline text] at (.5,0) {$h_0$};\n\t\\draw[blue] (0,-.5) to[out=180-45,in=-90] (-.5,0) to[out=90,in=180+45](0,.5);\\node[blue,left,inline text] at (-.5,0) {$g_0$};\n\t\\draw[blue] (0,-1.75) to[out=45,in=-90] (1.5,0) to[out=90,in=-45](0,1.75);\\node[blue,right,inline text] at (1.5,0) {$h_1$};\n\t\\draw[blue] (0,-1.5) to[out=180-45,in=-90] (-1.3,0) to[out=90,in=180+45](0,1.5);\\node[blue,left,inline text] at (-1.3,0) {$g_1$};\n\t\\draw[ultra thick,orange] (0,.25)--(0,2) node[above,inline text] {$*$};\n\t\\draw[ultra thick,red] (0,-.25)--(0,-2) node[below,inline text] {$*$};\n\t\\draw (0,0) circle (.25);\n\t\\draw (0,0) circle (1);\n\t\\draw (0,0) circle (2);\n\t\\end{tikzpicture}\n}\n\\end{array}=\n\\omega^{(q-r)h_0g_1}\n\\begin{array}{c}\n\\includeTikz{FqFrtube_example_3}{\n\t\\begin{tikzpicture}[scale=1,,every node\/.style={scale=.7}]\n\t\\annparamst{$*$}{$*$}{}{};\n\t\\annst{$g_0{+}g_1$}{$h_0{+}h_1$};\n\t\\end{tikzpicture}\n}\n\\end{array}\n=\\omega^{(q-r)h_0g_1}M_{g_0+g_1,h_0+h_1}.\n\\end{align}\nFor $q\\neq r$, this is a nontrivial 2-cocycle for $\\ZZ{p}\\times\\ZZ{p}$\\cite{propitius,DEWILDPROPITIUS1997297}, and therefore these annuli form a nontrivial twisted group algebra. There is an algebra isomorphism from this annulus algebra to the $p^2$ dimensional Pauli algebra. This isomorphism is defined by\n\\begin{align}\nM_{g,h}\\mapsto X^{(q-r)g}Z^h,\\label{eqn:tubetopauli}\n\\end{align}\nwhere $X$ and $Z$ are Pauli matrices obeying $ZX=\\omega XZ$. The Pauli matrices span the full $p\\times p$ matrix algebra. Up to isomorphism, there is a unique primitive idempotent of this algebra \n\\begin{align}\nP&=\\begin{pmatrix}\n1&0&\\cdots&0\\\\\n0&0&\\cdots&0\\\\\n\\vdots&\\vdots&\\ddots&\\vdots\\\\\n0&0&\\cdots&0\n\\end{pmatrix}\\\\&=\\frac{1}{p}\\sum_g Z^g.\n\\end{align}\nFinally, we use the algebra isomorphism  Eqn.~\\ref{eqn:tubetopauli} to obtain the idempotent for the single defect\n\\begin{align}\n\\defect{F_q}{F_r}{}{}{}&=\\frac{1}{p}\\sum_g\n\\begin{array}{c}\n\\includeTikz{FqFrtube_example_4}{\n\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\\annparamst{$*$}{$*$}{}{};\n\t\\annst{}{$g$};\n\t\\end{tikzpicture}\n}\n\\end{array}.\n\\end{align}\n\n\\end{exmp}\n\nOther idempotents can be found using the techniques outlined here. A full set of representative idempotents is provided in Table~\\ref{tab:idempotents}.\n\n\n\n\\section{Horizontal fusion outcomes} \\label{sec:horizontal_fusion_table}\n\\vspace*{12mm}\n\\begin{table}[h]\n\t\\begin{minipage}[t][.8\\textheight][c]{\\textwidth}\n\t\t\\renewcommand{\\arraystretch}{1.5}\n\t\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{T}{T}{c}{d}{} $&$ \\defect{T}{L}{c}{}{} $&$ \\defect{T}{R}{c}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{X_{n}}{c}{}{} $&$ \\defect{T}{F_{t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\delta_{\\nu -\\mu}^{b+c} \\defect{T}{T}{a}{d}{\\mu,\\nu} $&$ \\defect{T}{T}{a}{c}{\\mu} $&$ \\delta_{\\nu -\\mu}^{b+c} \\defect{T}{R}{a}{}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\defect{T}{T}{a}{n^{-1}(b+c+\\mu )}{\\mu} $&$ \\defect{T}{R}{a}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{L}{a}{}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{T}{L}{d}{}{\\mu,\\nu} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{T}{F_{0}}{}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\defect{T}{L}{n^{-1}(a+c+\\mu )}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{R}{a}{}{} $&$ \\defect{T}{T}{a}{d}{\\mu} $&$ \\defect{T}{T}{a}{c}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\defect{T}{R}{a}{}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{L}{d}{}{\\mu} $&$ \\defect{T}{L}{c}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\defect{T}{T}{a+l (c+\\mu )}{d}{\\mu} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\defect{T}{R}{a+l (c+\\mu )}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\defect{T}{X_{l n}}{a+l (c+\\mu )}{}{\\mu} $&$ \\defect{T}{F_{l^{-1}t}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ \\defect{T}{L}{d}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{\\mu} $&$ \\defect{T}{F_{n r}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{\\mu} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{L}{T}{a}{}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{L}{T}{d}{}{\\mu,\\nu} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{L}{R}{}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\defect{L}{T}{n^{-1}(a+c+\\mu )}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{L}{L}{d}{x}{\\mu,\\nu} $&$ \\defect{L}{L}{c}{x}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{L}{F_{0}}{x}{}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\defect{L}{L}{n^{-1}(a+c+\\mu )}{x}{\\mu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{T}{d}{}{\\mu} $&$ \\defect{L}{T}{c}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{L}{d}{x}{\\mu} $&$ \\defect{L}{L}{c}{x}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ \\defect{L}{T}{d}{}{\\mu} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{\\mu} $&$ \\defect{L}{X_{l n}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\defect{L}{L}{d}{x+r (c+\\mu )}{\\mu} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\defect{L}{F_{0}}{x+r (c+\\mu )}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\defect{L}{F_{n r}}{x+r (c+\\mu )}{}{\\mu} $&$ \\defect{L}{X_{r^{-1}t}}{}{}{\\mu} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{R}{T}{a}{}{} $&$ \\defect{T}{T}{a}{d}{\\nu} $&$ p\\cdot \\defect{T}{T}{a}{c}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ p\\cdot \\defect{T}{R}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{L}{}{}{} $&$ \\defect{T}{L}{d}{}{\\nu} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\defect{T}{T}{a}{d}{} $&$ \\defect{T}{T}{a}{c}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\defect{T}{L}{d}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{d}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\defect{T}{L}{d}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{L}{T}{d}{}{\\nu} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{L}{L}{d}{x}{\\nu} $&$ p\\cdot \\defect{L}{L}{c}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\defect{L}{T}{d}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\defect{L}{L}{d}{x}{} $&$ \\defect{L}{L}{c}{x}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\defect{L}{T}{d}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{d}{\\beta}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{T}{T}{a+k (c-\\nu )}{d}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{R}{a+k (c-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{(k n)^{-1}(a+c k-\\alpha ) }{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\defect{T}{L}{d}{}{\\nu} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{d}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{T}{T}{\\alpha}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\defect{T}{L}{d}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{T}{T}{a+c k}{d}{} $&$ \\defect{T}{L}{c}{}{} $&$ \\defect{T}{R}{a+c k}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{X_{k n}}{a+c k}{}{} $&$ \\defect{T}{F_{k^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{d}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\defect{T}{L}{d}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\defect{L}{T}{d}{}{\\nu} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{L}{L}{d}{c q-\\nu  q+x}{\\nu} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{F_{0}}{c q-\\nu  q+x}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{c q-n \\alpha  q+x}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\defect{L}{T}{d}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{d}{\\beta}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{L}{L}{\\alpha}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\defect{L}{T}{d}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{L}{L}{d}{c q+x}{} $&$ \\defect{L}{T}{c}{}{} $&$ \\defect{L}{F_{0}}{c q+x}{}{} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{F_{n q}}{c q+x}{}{} $&$ \\defect{L}{X_{q^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{d}{\\beta}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t\\end{tabular}\n\t\t\t\\end{minipage}\n\t\t\t\\vspace*{16mm}\n\t\t\t\\caption{Defect fusion table (part a). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\t\t\\vspace*{-30mm}\n\t\t\t\\label{tab:horizontal_table_1}\n\t\t\t\\end{table}\n\n\\begin{table} \n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{L}{T}{c}{}{} $&$ \\defect{L}{L}{c}{z}{} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{F_{0}}{z}{}{} $&$ \\defect{L}{X_{n}}{}{}{} $&$ \\defect{L}{F_{t}}{z}{}{} $\\\\\\toprule[1pt]\n\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\defect{T}{T}{a}{c}{\\nu} $&$ \\defect{T}{T}{a}{c}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{R}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{L}{a}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ \\defect{T}{L}{c}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{R}{a}{}{} $&$ p\\cdot \\defect{T}{T}{a}{c}{} $&$ \\defect{T}{T}{a}{c}{\\nu} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{L}{T}{a}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ \\defect{L}{T}{c}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\defect{L}{L}{c}{x}{\\nu} $&$ \\defect{L}{L}{c}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{R}{}{}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ p\\cdot \\defect{L}{L}{c}{x}{} $&$ \\defect{L}{L}{c}{x}{\\nu} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{R}{T}{a}{}{} $&$ \\defect{T}{T}{a}{c}{\\mu,\\nu} $&$ \\defect{T}{T}{a}{c}{\\mu} $&$ \\defect{T}{R}{a}{}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{\\mu} $&$ \\defect{T}{R}{a}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{L}{}{}{} $&$ \\defect{T}{L}{c}{}{\\mu,\\nu} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\defect{T}{T}{a}{c}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{T}{T}{a}{c}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{T}{R}{a}{}{\\mu,\\nu} $&$ \\defect{T}{R}{a}{}{\\mu} $&$ \\defect{T}{T}{a}{t^{-1}(x+z+\\mu )}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{T}{L}{c}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{T}{F_{0}}{}{}{\\mu,\\nu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\defect{T}{L}{t^{-1}(x+z+\\mu )}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{\\mu} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{\\mu} $&$ \\defect{T}{F_{l^{-1}t}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\defect{T}{L}{c}{}{\\mu} $&$ \\defect{T}{T}{r^{-1}(x+z+\\mu )}{c}{\\mu} $&$ \\defect{T}{F_{0}}{}{}{\\mu} $&$ \\defect{T}{R}{r^{-1}(x+z+\\mu )}{}{\\mu} $&$ \\defect{T}{F_{n r}}{}{}{\\mu} $&$ \\defect{T}{X_{r^{-1}t}}{r^{-1}(x+z+\\mu )}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{L}{T}{c}{}{\\mu,\\nu} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{L}{L}{c}{x}{\\mu,\\nu} $&$ \\defect{L}{L}{c}{x}{\\mu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{\\mu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{L}{T}{c}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{L}{R}{}{}{\\mu,\\nu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\defect{L}{T}{t^{-1}(x+z+\\mu )}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\defect{L}{L}{c}{x}{\\mu} $&$ \\delta_{\\nu -\\mu}^{y+z} \\defect{L}{L}{c}{x}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{y+z} \\defect{L}{F_{0}}{x}{}{\\mu,\\nu} $&$ \\defect{L}{F_{0}}{x}{}{\\mu} $&$ \\defect{L}{L}{t^{-1}(y+z+\\mu )}{x}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\defect{L}{L}{c}{l^{-1}(l x+z+\\mu )}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\defect{L}{F_{0}}{l^{-1}(l x+z+\\mu )}{}{\\mu} $&$ \\defect{L}{X_{l n}}{}{}{\\mu} $&$ \\defect{L}{F_{l^{-1}t}}{l^{-1}(l x+z+\\mu )}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{\\mu} $&$ \\defect{L}{T}{c}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{\\mu} $&$ \\defect{L}{R}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{\\mu} $&$ \\defect{L}{X_{r^{-1}t}}{}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ \\defect{L}{T}{c}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{L}{L}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\defect{L}{T}{c}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{L}{c}{k^{-1}(k x+z-\\nu )}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{F_{0}}{k^{-1}(k x+z-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{k^{-1}(k x+z-t \\alpha )}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{L}{T}{c}{}{} $&$ \\defect{L}{L}{c}{k^{-1}(x+z)}{} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{F_{0}}{k^{-1}(x+z)}{}{} $&$ \\defect{L}{X_{k n}}{}{}{} $&$ \\defect{L}{F_{k^{-1}t}}{k^{-1}(x+z)}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ p\\cdot \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\oplus_{\\beta} \\defect{L}{L}{c}{\\beta}{} $&$ \\defect{L}{T}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{T}{T}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ \\defect{T}{L}{c}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{T}{q^{-1}(x+z-\\nu )}{c}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{R}{q^{-1}(x+z-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{t^{-1}(x+z-q \\alpha )}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\defect{T}{L}{c}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{T}{L}{c}{}{} $&$ \\defect{T}{T}{q^{-1}(y+z)}{c}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{R}{q^{-1}(y+z)}{}{} $&$ \\defect{T}{F_{n q}}{}{}{} $&$ \\defect{T}{X_{q^{-1}t}}{q^{-1}(y+z)}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ p\\cdot \\defect{T}{L}{c}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{c}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n\t\\caption{Defect fusion table (part b). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\\label{tab:horizontal_table_2}\n\\end{table}\n\n\\begin{table} \n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{R}{T}{c}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{R}{c}{z}{} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{X_{n}}{}{}{} $&$ \\defect{R}{F_{t}}{z}{}{} $\\\\\\toprule[1pt]\n\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\delta_{\\nu -\\mu}^{b+c} \\defect{R}{T}{a}{}{\\mu,\\nu} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{b+c} \\defect{R}{R}{a}{z}{\\mu,\\nu} $&$ \\defect{R}{R}{a}{z}{\\mu} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\defect{R}{R}{a}{z+t (b+\\mu )}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{L}{a}{}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{R}{L}{}{}{\\mu,\\nu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{R}{F_{0}}{z}{}{\\mu,\\nu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\defect{R}{F_{0}}{z+t (a+\\mu )}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{R}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\defect{R}{T}{a}{}{\\mu,\\nu} $&$ \\defect{R}{R}{a}{z}{\\mu} $&$ \\defect{R}{R}{a}{z}{\\mu,\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{\\mu} $&$ \\defect{R}{T}{a}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu,\\nu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu,\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\defect{R}{T}{a+l (c+\\mu )}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\defect{R}{R}{a+l (c+\\mu )}{z}{\\mu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\defect{R}{X_{l n}}{}{}{\\mu} $&$ \\defect{R}{F_{l^{-1}t}}{\\mu  t+l^{-1}(a t)+z}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{\\mu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{\\mu} $&$ \\defect{R}{X_{r^{-1}t}}{}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{L}{T}{a}{}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{F_{0}}{T}{}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{F_{0}}{R}{z}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\defect{F_{0}}{R}{z+t (a+\\mu )}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{F_{0}}{L}{x}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{a+c} \\defect{F_{0}}{F_{0}}{x}{z}{\\mu,\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\mu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\defect{F_{0}}{F_{0}}{x}{z+t (a+\\mu )}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{R}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu,\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\mu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\mu,\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{\\mu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{\\mu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{\\mu} $&$ \\defect{F_{0}}{F_{l^{-1}t}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\defect{F_{0}}{L}{x+r (c+\\mu )}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\defect{F_{0}}{F_{0}}{x+r (c+\\mu )}{z}{\\mu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\defect{F_{0}}{F_{n r}}{}{}{\\mu} $&$ \\defect{F_{0}}{X_{r^{-1}t}}{x+r \\left(t^{-1}z+\\mu \\right)}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\defect{R}{R}{a}{z}{\\nu} $&$ p\\cdot \\defect{R}{R}{a}{z}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ \\defect{R}{R}{a}{z}{} $&$ \\defect{R}{R}{a}{z}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{T}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ p\\cdot \\defect{R}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\nu} $&$ p\\cdot \\defect{F_{0}}{F_{0}}{x}{z}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{L}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{R}{T}{a+k (c-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{R}{a+k (c-\\nu )}{z}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z+k^{-1}(t (a-\\alpha ))}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{\\nu} $&$ \\oplus_{\\alpha,\\beta} \\defect{R}{R}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{R}{T}{a+c k}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{R}{a+c k}{z}{} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{X_{k n}}{}{}{} $&$ \\defect{R}{F_{k^{-1}t}}{k^{-1}(a t)+z}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ p\\cdot \\defect{R}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{c q-\\nu  q+x}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{0}}{c q-\\nu  q+x}{z}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z+q^{-1}(t (x-\\alpha ))}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{\\nu} $&$ \\oplus_{\\alpha,\\beta} \\defect{F_{0}}{F_{0}}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{F_{0}}{L}{c q+x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{F_{0}}{c q+x}{z}{} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{F_{n q}}{}{}{} $&$ \\defect{F_{0}}{X_{q^{-1}t}}{x+t^{-1}(q z)}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n\t\\caption{Defect fusion table (part c). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\\label{tab:horizontal_table_3}\n\\end{table}\n\n\\begin{table} \n\n\t\\renewcommand{\\arraystretch}{1.5}\n\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{L}{z}{}{} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{F_{0}}{z}{w}{} $&$ \\defect{F_{0}}{X_{n}}{z}{}{} $&$ \\defect{F_{0}}{F_{t}}{}{}{} $\\\\\\toprule[1pt]\n\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{R}{a}{z}{\\nu} $&$ \\defect{R}{R}{a}{w}{} $&$ \\defect{R}{T}{a}{}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{L}{a}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ \\defect{R}{F_{0}}{w}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{R}{a}{}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ p\\cdot \\defect{R}{R}{a}{z}{} $&$ \\defect{R}{R}{a}{w}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{F_{0}}{w}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\defect{R}{F_{0}}{w}{}{} $&$ \\defect{R}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{L}{T}{a}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ \\defect{F_{0}}{R}{w}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{w}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{R}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{R}{w}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{F_{0}}{x}{z}{} $&$ \\defect{F_{0}}{F_{0}}{x}{w}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\defect{F_{0}}{R}{w}{}{} $&$ \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\mu,\\nu} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\defect{R}{R}{a}{z}{\\mu,\\nu} $&$ \\defect{R}{R}{a}{w}{\\mu} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\mu,\\nu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu,\\nu} $&$ \\defect{R}{F_{0}}{w}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\defect{R}{T}{a}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{R}{T}{a}{}{\\mu,\\nu} $&$ \\defect{R}{R}{a}{z}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{R}{R}{a}{w}{\\mu,\\nu} $&$ \\defect{R}{R}{a}{n (x+z+\\mu )}{\\mu} $&$ \\defect{R}{T}{a}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{R}{L}{}{}{\\mu,\\nu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{R}{F_{0}}{w}{}{\\mu,\\nu} $&$ \\defect{R}{F_{0}}{n (x+z+\\mu )}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{\\mu} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{\\mu} $&$ \\defect{R}{F_{0}}{w}{}{\\mu} $&$ \\defect{R}{X_{l n}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\defect{R}{L}{}{}{\\mu} $&$ \\defect{R}{T}{r^{-1}(x+z+\\mu )}{}{\\mu} $&$ \\defect{R}{F_{0}}{z}{}{\\mu} $&$ \\defect{R}{R}{r^{-1}(x+z+\\mu )}{w}{\\mu} $&$ \\defect{R}{F_{n r}}{n (x+z+\\mu )}{}{\\mu} $&$ \\defect{R}{X_{r^{-1}t}}{}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{R}{w}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\mu,\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{w}{\\mu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{F_{0}}{T}{}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{x+z} \\defect{F_{0}}{R}{w}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{R}{n (x+z+\\mu )}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $&$ \\delta_{\\nu -\\mu}^{y+z} \\defect{F_{0}}{L}{x}{}{\\mu,\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{z}{\\mu} $&$ \\delta_{\\nu -\\mu}^{y+z} \\defect{F_{0}}{F_{0}}{x}{w}{\\mu,\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{n (y+z+\\mu )}{\\mu} $&$ \\defect{F_{0}}{L}{x}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\defect{F_{0}}{L}{l^{-1}(l x+z+\\mu )}{}{\\mu} $&$ \\defect{F_{0}}{R}{z}{}{\\mu} $&$ \\defect{F_{0}}{F_{0}}{l^{-1}(l x+z+\\mu )}{w}{\\mu} $&$ \\defect{F_{0}}{X_{l n}}{l^{-1}(l x+z+\\mu )}{}{\\mu} $&$ \\defect{F_{0}}{F_{l^{-1}t}}{}{}{\\mu} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{\\mu} $&$ \\defect{F_{0}}{T}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{\\mu} $&$ \\defect{F_{0}}{R}{w}{}{\\mu} $&$ \\defect{F_{0}}{F_{n r}}{}{}{\\mu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{\\mu} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{z}{}{\\nu} $&$ \\defect{F_{0}}{R}{w}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{F_{0}}{F_{0}}{\\alpha}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{R}{w}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{L}{k^{-1}(k x+z-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\defect{F_{0}}{F_{0}}{k^{-1}(k x+z-\\nu )}{w}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x+(k n)^{-1}(n z-\\beta)}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{L}{k^{-1}(x+z)}{}{} $&$ \\defect{F_{0}}{R}{z}{}{} $&$ \\defect{F_{0}}{F_{0}}{k^{-1}(x+z)}{w}{} $&$ \\defect{F_{0}}{X_{k n}}{k^{-1}(x+z)}{}{} $&$ \\defect{F_{0}}{F_{k^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{R}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z}{} $&$ \\defect{F_{0}}{R}{w}{}{} $&$ \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{R}{R}{\\alpha}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{z}{}{\\nu} $&$ \\defect{R}{F_{0}}{w}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{T}{q^{-1}(x+z-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\defect{R}{R}{q^{-1}(x+z-\\nu )}{w}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{n (x+z-q \\alpha )}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{F_{0}}{w}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z}{} $&$ \\defect{R}{F_{0}}{w}{}{} $&$ \\defect{R}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{T}{q^{-1}(y+z)}{}{} $&$ \\defect{R}{F_{0}}{z}{}{} $&$ \\defect{R}{R}{q^{-1}(y+z)}{w}{} $&$ \\defect{R}{F_{n q}}{n (y+z)}{}{} $&$ \\defect{R}{X_{q^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{F_{0}}{z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{w}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\\toprule[1pt]\n\t\\end{tabular}\n\n\t\\caption{Defect fusion table (part d). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\\label{tab:horizontal_table_4}\n\\end{table}\n\n\\begin{table}  \n\t\\resizebox{\\textwidth}{!}{\n\t\t\\renewcommand{\\arraystretch}{1.5}\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{X_{m}}{T}{c}{}{} $&$ \\defect{X_{m}}{L}{}{}{} $&$ \\defect{X_{m}}{R}{}{}{} $&$ \\defect{X_{m}}{F_{0}}{z}{}{} $&$ \\defect{X_{m}}{X_{m}}{c}{z}{} $&$ \\defect{X_{m}}{X_{n}}{}{}{} $&$ \\defect{X_{m}}{F_{t}}{z}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\defect{T}{T}{a}{m^{-1}(b+c-\\nu )}{\\nu} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\defect{T}{T}{a}{m^{-1}(b+c)}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{R}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{L}{a}{}{} $&$ \\defect{T}{L}{m^{-1}(a+c-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{L}{m^{-1}(a+c)}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{\\nu} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{(l m)^{-1}(a+c l-\\alpha ) }{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{X_{l m}}{a+c l}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{T}{T}{\\alpha}{\\beta}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{m r}}{}{}{} $&$ p\\cdot \\defect{T}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{L}{T}{a}{}{} $&$ \\defect{L}{T}{m^{-1}(a+c-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{T}{m^{-1}(a+c)}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\defect{L}{L}{m^{-1}(a+c-\\nu )}{x}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{L}{m^{-1}(a+c)}{x}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{\\nu} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{L}{L}{\\alpha}{\\beta}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{X_{l m}}{}{}{} $&$ p\\cdot \\defect{L}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{c r-m \\alpha  r+x}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{F_{m r}}{c r+x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{T}{a}{}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{R}{a}{m (x+z-\\nu )}{\\nu} $&$ \\defect{R}{R}{a}{m x+z}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{T}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{F_{0}}{m (x+z-\\nu )}{}{\\nu} $&$ \\defect{R}{F_{0}}{m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{R}{R}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{X_{l m}}{}{}{} $&$ p\\cdot \\defect{R}{X_{l n}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{m (x+z-r \\alpha )}{} $&$ \\defect{R}{F_{m r}}{m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{R}{m (x+z-\\nu )}{}{\\nu} $&$ \\defect{F_{0}}{R}{m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{F_{0}}{x}{m (y+z-\\nu )}{\\nu} $&$ \\defect{F_{0}}{F_{0}}{x}{m y+z}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{L}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x+(l m)^{-1}(m z-\\beta)}{\\beta}{} $&$ \\defect{F_{0}}{X_{l m}}{x+(l m)^{-1}z }{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{F_{0}}{F_{0}}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{m r}}{}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{X_{k m}}{T}{a+k (c-\\nu )}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $&$ \\defect{X_{k m}}{R}{}{}{\\nu} $&$ \\defect{X_{k m}}{R}{}{}{} $&$ \\defect{X_{k m}}{T}{a+c k}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $&$ \\defect{X_{k m}}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\defect{X_{k m}}{L}{}{}{\\nu} $&$ p\\cdot \\defect{X_{k m}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{k m}}{L}{}{}{} $&$ p\\cdot \\defect{X_{k m}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{\\nu} $&$ p\\cdot \\defect{X_{k m}}{R}{}{}{} $&$ \\defect{X_{k m}}{R}{}{}{\\nu} $&$ \\defect{X_{k m}}{R}{}{}{} $&$ p\\cdot \\defect{X_{k m}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\defect{X_{k m}}{L}{}{}{} $&$ \\defect{X_{k m}}{L}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{k m}}{F_{0}}{k^{-1}(k x+z-\\nu )}{}{\\nu} $&$ \\defect{X_{k m}}{F_{0}}{x+(k m)^{-1}z }{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{k m}}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{X_{k m}}{T}{a+c k}{}{} $&$ \\defect{X_{k m}}{L}{}{}{} $&$ \\defect{X_{k m}}{R}{}{}{} $&$ \\defect{X_{k m}}{F_{0}}{k^{-1}(x+z)}{}{} $&$ \\defect{X_{k m}}{X_{k m}}{a+c k}{m x+z}{} $&$ \\defect{X_{k m}}{X_{k n}}{}{}{} $&$ \\defect{X_{k m}}{F_{k^{-1}t}}{k^{-1}(a t)+m x+z}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $&$ p\\cdot \\defect{X_{k m}}{L}{}{}{} $&$ p\\cdot \\defect{X_{k m}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{k m}}{X_{l m}}{}{}{} $&$\\begin{cases}\\oplus_{\\alpha,\\beta} \\defect{X_{k m}}{X_{k m}}{\\alpha}{\\beta}{} & k m=l n \\\\ p\\cdot \\defect{X_{k m}}{X_{l n}}{}{}{} &\\text{otherwise}\\end{cases}$&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\defect{X_{k m}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{k m}}{R}{}{}{} $&$ \\defect{X_{k m}}{F_{m r}}{c k m r+m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{k m}}{F_{n r}}{\\alpha}{}{} $&$\\begin{cases} \\oplus_{\\alpha} \\defect{X_{k m}}{X_{k m}}{\\alpha}{m x+z-m r \\alpha}{} &k m=r^{-1}t\\\\ \\defect{X_{k m}}{X_{r^{-1}t}}{}{}{} &\\text{otherwise}\\end{cases}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\defect{F_{m q}}{T}{}{}{\\nu} $&$ p\\cdot \\defect{F_{m q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $&$ \\defect{F_{m q}}{T}{}{}{} $&$ p\\cdot \\defect{F_{m q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{F_{m q}}{L}{c q-\\nu  q+x}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $&$ \\defect{F_{m q}}{F_{0}}{}{}{\\nu} $&$ \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\defect{F_{m q}}{L}{c q+x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $&$ \\defect{F_{m q}}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\defect{F_{m q}}{T}{}{}{} $&$ \\defect{F_{m q}}{T}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $&$ \\defect{F_{m q}}{R}{m (x+z-\\nu )}{}{\\nu} $&$ \\defect{F_{m q}}{R}{m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $&$ \\defect{F_{m q}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{\\nu} $&$ p\\cdot \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\defect{F_{m q}}{F_{0}}{}{}{\\nu} $&$ \\defect{F_{m q}}{F_{0}}{}{}{} $&$ p\\cdot \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\defect{F_{m q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $&$ \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\defect{F_{m q}}{X_{l m}}{c l m q+m x+z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{X_{l n}}{\\alpha}{}{} $&$\\begin{cases}\\oplus_{\\alpha} \\defect{F_{m q}}{F_{m q}}{\\alpha}{m x+z-l m \\alpha}{}& q m=l^{-1}t\\\\ \\defect{F_{m q}}{F_{l^{-1}t}}{}{}{} &\\text{otherwise}\\end{cases}$\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{F_{m q}}{L}{c q+x}{}{} $&$ \\defect{F_{m q}}{T}{}{}{} $&$ \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\defect{F_{m q}}{R}{m (y+z)}{}{} $&$ \\defect{F_{m q}}{F_{m q}}{c q+x}{m y+z}{} $&$ \\defect{F_{m q}}{F_{n q}}{}{}{} $&$ \\defect{F_{m q}}{X_{q^{-1}t}}{q^{-1}(t x)+m y+z}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{L}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{m q}}{T}{}{}{} $&$ p\\cdot \\defect{F_{m q}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{m q}}{R}{\\alpha}{}{} $&$ \\defect{F_{m q}}{F_{m r}}{}{}{} $&$\\begin{cases}\\oplus_{\\alpha,\\beta} \\defect{F_{m q}}{F_{m q}}{\\alpha}{\\beta}{} &q m=r n\\\\p\\cdot \\defect{F_{m q}}{F_{n r}}{}{}{}& \\text{otherwise}\\end{cases}$&$ \\oplus_{\\alpha} \\defect{F_{m q}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n\t\\caption{Defect fusion table (part e). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\\label{tab:horizontal_table_5}\n\\end{table}\n\n\\begin{table} \n\t\\resizebox{\\textwidth}{!}{\n\t\t\\renewcommand{\\arraystretch}{1.5}\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\color[gray]{.8}\\vrule}c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]&$ \\defect{F_{s}}{T}{}{}{} $&$ \\defect{F_{s}}{L}{z}{}{} $&$ \\defect{F_{s}}{R}{z}{}{} $&$ \\defect{F_{s}}{F_{0}}{}{}{} $&$ \\defect{F_{s}}{X_{n}}{z}{}{} $&$ \\defect{F_{s}}{F_{s}}{z}{w}{} $&$ \\defect{F_{s}}{F_{t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{T}{T}{a}{b}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{R}{a}{b s-\\nu  s+z}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{T}{a}{}{} $&$ \\defect{R}{R}{a}{b s+w}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{L}{a}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{a s-\\nu  s+z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\defect{R}{F_{0}}{a s+w}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{R}{a}{}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $&$ \\defect{R}{T}{a}{}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{R}{R}{a}{\\beta}{} $&$ \\defect{R}{T}{a}{}{} $&$ p\\cdot \\defect{R}{T}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{0}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\defect{R}{L}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{X_{l}}{a}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{R}{\\alpha}{z+l^{-1}(s (a-\\alpha ))}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\defect{R}{X_{l n}}{}{}{} $&$ \\defect{R}{F_{l^{-1}s}}{l^{-1}(a s)+w}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{T}{F_{r}}{}{}{} $&$ p\\cdot \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{R}{R}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{R}{F_{n r}}{\\alpha}{}{} $&$ \\defect{R}{X_{r^{-1}s}}{}{}{} $&$ p\\cdot \\defect{R}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{L}{T}{a}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{a s-\\nu  s+z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{R}{a s+w}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{L}{a}{x}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ \\defect{F_{0}}{L}{x}{}{}$ &$ \\defect{F_{0}}{F_{0}}{x}{a s-\\nu  s+z}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{F_{0}}{x}{a s+w}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{R}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\defect{F_{0}}{T}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{0}}{x}{}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $&$ \\defect{F_{0}}{L}{x}{}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{F_{0}}{F_{0}}{x}{\\beta}{} $&$ \\defect{F_{0}}{L}{x}{}{} $&$ p\\cdot \\defect{F_{0}}{L}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{X_{l}}{}{}{} $&$ p\\cdot \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{F_{0}}{F_{0}}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{l n}}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{l^{-1}s}}{}{}{} $&$ p\\cdot \\defect{F_{0}}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{L}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{L}{\\alpha}{}{} $&$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{F_{0}}{\\alpha}{z+r^{-1}(s (x-\\alpha ))}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{R}{\\alpha}{}{} $&$ \\defect{F_{0}}{F_{n r}}{}{}{} $&$ \\defect{F_{0}}{X_{r^{-1}s}}{s^{-1}(r w)+x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{0}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{R}{T}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{\\nu} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ p\\cdot \\defect{T}{R}{a}{}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{R}{a}{}{} $&$ p\\cdot \\defect{T}{R}{a}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{R}{a}{x}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $&$ \\defect{T}{T}{a}{s^{-1}(x+z-\\nu )}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\defect{T}{R}{a}{}{\\nu} $&$ \\defect{T}{R}{a}{}{} $&$ \\defect{T}{T}{a}{s^{-1}(x+z)}{} $&$ \\oplus_{\\beta} \\defect{T}{T}{a}{\\beta}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\defect{T}{L}{s^{-1}(x+z-\\nu )}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{F_{0}}{}{}{\\nu} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\defect{T}{L}{s^{-1}(x+z)}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{X_{l}}{}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{T}{T}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{l n}}{\\alpha}{}{} $&$ \\defect{T}{F_{l^{-1}s}}{}{}{} $&$ p\\cdot \\defect{T}{F_{l^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{R}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{T}{\\alpha}{s^{-1}(x+z-r \\alpha )}{} $&$ \\defect{T}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{R}{\\alpha}{}{} $&$ \\defect{T}{F_{n r}}{}{}{} $&$ \\defect{T}{X_{r^{-1}s}}{r^{-1}(x+z)}{}{} $&$ \\oplus_{\\alpha} \\defect{T}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{0}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{R}{}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{L}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{\\nu} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ p\\cdot \\defect{L}{F_{0}}{x}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{R}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\defect{L}{T}{s^{-1}(x+z-\\nu )}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{R}{}{}{\\nu} $&$ \\defect{L}{R}{}{}{} $&$ \\defect{L}{T}{s^{-1}(x+z)}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{0}}{x}{y}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $&$ \\defect{L}{L}{s^{-1}(y+z-\\nu )}{x}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{F_{0}}{x}{}{\\nu} $&$ \\defect{L}{F_{0}}{x}{}{} $&$ \\defect{L}{L}{s^{-1}(y+z)}{x}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{x}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{X_{l}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{L}{\\alpha}{l^{-1}(l x+z-s \\alpha )}{} $&$ \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ \\defect{L}{X_{l n}}{}{}{} $&$ \\defect{L}{F_{l^{-1}s}}{x+l^{-1}z}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{0}}{F_{r}}{}{}{} $&$ \\oplus_{\\alpha,\\beta} \\defect{L}{L}{\\alpha}{\\beta}{} $&$ \\oplus_{\\alpha} \\defect{L}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{L}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{L}{F_{n r}}{\\alpha}{}{} $&$ \\defect{L}{X_{r^{-1}s}}{}{}{} $&$ p\\cdot \\defect{L}{X_{r^{-1}t}}{}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{X_{k}}{T}{a}{}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{\\nu} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\defect{F_{k^{-1}s}}{R}{-\\nu  s+k^{-1}(a s)+z}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\defect{F_{k^{-1}s}}{R}{k^{-1}(a s)+w}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{\\nu} $&$ p\\cdot \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ p\\cdot \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{R}{}{}{} $&$ p\\cdot \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ p\\cdot \\defect{F_{k^{-1}s}}{T}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{0}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{L}{k^{-1}(k x+z-\\nu )}{}{\\nu} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{\\nu} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\defect{F_{k^{-1}s}}{L}{x+k^{-1}z}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{k}}{a}{x}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\defect{F_{k^{-1}s}}{L}{k^{-1}(x+z)}{}{} $&$ \\defect{F_{k^{-1}s}}{R}{k^{-1}(a s)+z}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\defect{F_{k^{-1}s}}{X_{k n}}{k^{-1}(a s)+n x+z}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{k^{-1}s}}{k^{-1}(x+z)}{k^{-1}(a s)+w}{} $&$ \\defect{F_{k^{-1}s}}{F_{k^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{X_{l}}{}{}{} $&$ p\\cdot \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $&$ p\\cdot \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{X_{l n}}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{l^{-1}s}}{}{}{} $&$\\begin{cases} \\oplus_{\\alpha,\\beta} \\defect{F_{k^{-1}s}}{F_{k^{-1}s}}{\\alpha}{\\beta}{} & k^{-1} s=l^{-1}t\\\\p\\cdot \\defect{F_{k^{-1}s}}{F_{l^{-1}t}}{}{}{} &\\text{otherwise}\\end{cases}$\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{X_{k}}{F_{r}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{L}{\\alpha}{}{} $&$ \\defect{F_{k^{-1}s}}{T}{}{}{} $&$ \\defect{F_{k^{-1}s}}{F_{0}}{}{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{R}{\\alpha}{}{} $&$\\begin{cases} \\oplus_{\\alpha} \\defect{F_{n r}}{F_{n r}}{\\alpha}{n x+z-k n \\alpha}{} & k^{-1} s=r n \\\\ \\defect{F_{k^{-1}s}}{F_{n r}}{}{}{} &\\text{otherwise}\\end{cases}$&$ \\defect{F_{k^{-1}s}}{X_{r^{-1}s}}{w+(k r)^{-1}(s (x+z)) }{}{} $&$ \\oplus_{\\alpha} \\defect{F_{k^{-1}s}}{X_{r^{-1}t}}{\\alpha}{}{} $\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$ \\defect{F_{q}}{T}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{\\nu} $&$ p\\cdot \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ p\\cdot \\defect{X_{q^{-1}s}}{R}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{L}{x}{}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{\\nu} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\defect{X_{q^{-1}s}}{F_{0}}{x+s^{-1}(q z)-q \\nu}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\defect{X_{q^{-1}s}}{F_{0}}{s^{-1}(q w)+x}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{R}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{T}{q^{-1}(x+z-\\nu )}{}{\\nu} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{\\nu} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\defect{X_{q^{-1}s}}{T}{q^{-1}(x+z)}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{0}}{}{}{} $&$ p\\cdot \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{\\nu} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ p\\cdot \\defect{X_{q^{-1}s}}{L}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{X_{l}}{x}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $&$\\begin{cases} \\oplus_{\\alpha} \\defect{X_{l n}}{X_{l n}}{\\alpha}{n x+z-n q \\alpha}{} & q^{-1} s=l n \\\\ \\defect{X_{q^{-1}s}}{X_{l n}}{}{}{} &\\text{otherwise}\\end{cases}$&$ \\defect{X_{q^{-1}s}}{F_{l^{-1}s}}{w+(l q)^{-1}(s (x+z)) }{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{l^{-1}t}}{\\alpha}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{q}}{x}{y}{} $&$ \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\defect{X_{q^{-1}s}}{T}{q^{-1}(y+z)}{}{} $&$ \\defect{X_{q^{-1}s}}{F_{0}}{x+s^{-1}(q z)}{}{} $&$ \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\defect{X_{q^{-1}s}}{F_{n q}}{q^{-1}(s x)+n y+z}{}{} $&$ \\defect{X_{q^{-1}s}}{X_{q^{-1}s}}{q^{-1}(y+z)}{w+q^{-1}(s x)}{} $&$ \\defect{X_{q^{-1}s}}{X_{q^{-1}t}}{}{}{} $\\\\\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t$ \\defect{F_{q}}{F_{r}}{}{}{} $&$ p\\cdot \\defect{X_{q^{-1}s}}{L}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{T}{\\alpha}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{0}}{\\alpha}{}{} $&$ p\\cdot \\defect{X_{q^{-1}s}}{R}{}{}{} $&$ \\oplus_{\\alpha} \\defect{X_{q^{-1}s}}{F_{n r}}{\\alpha}{}{} $&$ \\defect{X_{q^{-1}s}}{X_{r^{-1}s}}{}{}{} $&$\\begin{cases} \\oplus_{\\alpha,\\beta} \\defect{X_{r^{-1}t}}{X_{r^{-1}t}}{\\alpha}{\\beta}{} & q^{-1} s=r^{-1}t \\\\ p\\cdot \\defect{X_{q^{-1}s}}{X_{r^{-1}t}}{}{}{} &\\text{otherwise}\\end{cases}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n\t\\caption{Defect fusion table (part f). $\\mu$ ($\\nu$) indexes degeneracy in the bottom (top) domain wall fusion.}\n\t\\label{tab:horizontal_table_6}\n\\end{table}\n\n\\setcounter{table}{\\value{curtable}}\n\\renewcommand{\\thetable}{\\Roman{table}}\n\n\\section{The Vertical Fusion Algorithm} \\label{sec:vertical_fusion_algorithm}\n\nIn this section, we explain the vertical defect fusion algorithm. The structure of this section parallels Section~\\ref{sec:hotizontal_defect_fusion_algorithm}. Since there is no domain wall fusion, vertical fusion is simpler than horizontal fusion. No inflation is required, so the algorithm proceeds in three key steps:\n\\begin{enumerate} \n\t\\item Determine idempotents corresponding to source defects using Table~\\ref{tab:idempotents}.\n\t\\item Determine idempotent for target defect using Tables~\\ref{tab:zptable} and \\ref{tab:idempotents}.\n\t\\item Find a nonzero pants diagram that absorbs the source idempotents on the legs and (inflated) target on the waist.\n\\end{enumerate}\n\n\n\nWe shall explain how the algorithm works, step by step, using an example.\n\n\\begin{exmp}[$\\defect{R}{R}{a}{x}{} \\circ \\defect{R}{F_s}{z}{}{}$]\nConsider the vertical defect fusion\n\\begin{align}\n  \\defect{R}{R}{a}{x}{} \\circ \\defect{R}{F_s}{z}{}{}.\n\\end{align}\nThe convention for the $\\circ$ operator is that the left argument is the bottom defect and the right argument is the top defect.\n\\subsubsection{Writing down the defect idempotents to compose}\nThe first step in the procedure is to look up the idempotents that we are composing. These are found in Table~\\ref{tab:idempotents}. In our current example, we have\n\\begin{align}\n\\defect{R}{F_r}{z}{}{}=\\frac{1}{p}\\sum_h\\omega^{hz}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RFr_idempotent_1}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-h$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}, \\quad\n\\defect{R}{R}{a}{x}{}=\\frac{1}{p}\\sum_g\\omega^{gx}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RR_idempotent_1}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamss{$0$}{$a$}{}{};\n\t\t\t\t\t\\annss{}{$-g$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}.\n\\end{align}\n\\subsubsection{Writing down the target idempotent}\nNext, we need to write down the target defect idempotent. These can be found in Table~\\ref{tab:idempotents}.\n\\begin{align}\n\\defect{R}{F_r}{\\zeta}{}{}=\\frac{1}{p}\\sum_{\\gamma}\\omega^{\\gamma\\zeta}\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{RFr_idempotent_2}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\annparamst{$0$}{$*$}{}{};\n\t\t\t\t\t\\annst{}{$-\\gamma$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}.\n        \\end{align}\n\\subsubsection{Decorating the pants}\nAt this point we have extracted all the data we need from the tables. Now we need to find all the pants diagrams which absorb our source defect idempotents on the legs and our target defect idempotent on the waist. In this case, the, the pants are oriented perpendicular to the horizontal case. The most general pair of pants for any 3 domain walls $\\mathcal{M},\\mathcal{N},\\mathcal{P}$ is\n\\begin{align}\n\\begin{array}{c}\n\t\t\\includeTikz{generalpants_vert_fusion_algo}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t\\vpantsparams{$m$}{$n$}{$p$};\n\t\t\t\\vpantsstp{$k_2$}{$k_3$}{$k_0$}{$k_1$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nWe use the following equation to bring every vertical pair of pants into a standard form\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{generalpants_vert}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t\\vpantsparams{$m$}{$n$}{$p$};\n\t\t\t\\vgeneralpantsstp{$g_0$}{$g_1$}{$g_2$}{$g_3$}{$h_0$}{$h_1$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t&=\\frac{\\Omega_M(h_0^{-1},g_1^{-1})\\Omega_N(h_0^{-1},g_1)}{\\Omega_P(h_0,g_3)\\Omega_N(h_0,g_3^{-1})}\n\t\\begin{array}{c}\n\t\t\\includeTikz{generalpants_canonical_vert}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.9}]\n\t\t\t\\vpantsparams{$m$}{$n$}{$p$};\n\t\t\t\\vpantsstp{$h_0g_0$}{$g_1h_1$}{$h_0g_2$}{$g_3h_1$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nWhen we insert our source defect idempotents on the legs and our target defect idempotent on the waist, we get\n\\begin{align}\\frac{1}{p^3}\\sum_{g,h,\\gamma}\\omega^{hz+gx+\\gamma\\zeta+sk_0h}\n\\begin{array}{c}\n\t\t\\includeTikz{vert_fusion_algo_pants_2}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.7}]\n\t\t\t\\vpantsparams{$0$}{$0$}{$*$};\n\t\t\t\\vpantsstp{$k_2$}{$k_3{-}g{-}\\gamma$}{$k_0$}{$k_1{-}h{-}\\gamma$};\n\t\t\t\\node[orange,inline text] at (0,-.55) {$a$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nThe variables $k_1$ and $k_2$ can be converted into a global phase by the translations $h \\to h + k_1, g \\to g + k_3$. For the object labels to match up, we need $k_0 = a$ and $k_2 = 0$. Now, if we make the change of variables,\n\\begin{align}\n  c &= g + \\gamma \\\\\n  b &= h + \\gamma\n\\end{align}\nwe get\n\\begin{align}\\frac{1}{p^3}\\sum_{c,b,\\gamma}\\omega^{(b-\\gamma)z+(c-\\gamma)x+\\gamma\\zeta+sa(b-\\gamma)}\n\\begin{array}{c}\n\t\t\\includeTikz{vert_fusion_algo_pants_3}{\n\t\t\t\\begin{tikzpicture}[scale=.9,,every node\/.style={scale=.7}]\n\t\t\t\\vpantsparams{$0$}{$0$}{$*$};\n\t\t\t\\vpantsstp{}{$-c$}{$a$}{$-b$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nFor this to be non zero, we need to have $\\zeta=x+z+sa$,\nso\n\\begin{align}\n \\defect{R}{R}{a}{x}{}  \\circ  \\defect{R}{F_s}{z}{}{}= \\defect{R}{F_s}{x+z+sa}{}{}\n\\end{align}\n\n\\end{exmp}\n\\section{Introduction}\n\nTopological phases are promising candidate materials for robust encoding, storage and manipulation of quantum information\\cite{Dennis2002,MR1951039,Brown2014,Terhal2015}. Formed by locally interacting degrees of freedom, these quantum systems have emergent global properties that are protected against the presence of environmental noise. In addition to the bulk properties, inclusion of defects has been shown to improve the power of these materials from a quantum computational perspective\\cite{Dennis2002,0610082,Bombin2007a,Bombin2010,Brown2013a,Pastawski2014,Yoshida2015a,1606.07116,Brown2016,IrisCong1,IrisCong2,PhysRevB.96.195129,Yoshida2017,SETPaper,Brown2018}. It is therefore important to understand the full theory, including defects. In this paper, we study non-chiral, 2-dimensional, long-range-entangled topological phases with defects.\n\nA defect of a topological phase is a region of positive codimension which differs from the ground state. For example, in a 2-dimensional topological phase, domain walls are codimension 1 defects and anyonic excitations are codimension 2 defects. Much work has been done on defects in topological phases, for example \\onlinecites{Dennis2002,0610082,Bombin2007a,Bombin2010,MR2942952,FUCHS2002353,MR3370609,Kong2013,Brown2013a,MR3063919,Barkeshli2013,Barkeshli2014,Pastawski2014,Yoshida2015a,1606.07116,Brown2016,IrisCong1,IrisCong2,PhysRevB.96.195129,Yoshida2017,PhysRevB.96.125104,SETPaper,Bridgeman2017,Brown2018,1809.00245}. In previous work, the term defect frequently refers to a 0-dimensional interface between two invertible domain walls. To avoid confusion, we shall use the term {\\em binary interface defect} to refer to \\emph{any} 0-dimensional interface between two, not necessarily invertible, domain walls.\n\nThis work builds on our previous paper \\onlinecite{1806.01279}, in which we computed the fusions of all domain walls in Kitaev's $\\vvec{\\ZZ{p}}$ models\\cite{MR1951039} (with $p$ prime). In this paper, we compute all possible fusions between all possible binary interface defects in the Kitaev $\\vvec{\\ZZ{p}}$ model. The tools from both \\onlinecite{1806.01279} and the present work can be adapted to more general Levin-Wen models. In the physics literature, defect fusion is often synonymous with symmetry gauging. We compute the fusions even when no gauging exists. \n\nIn \\onlinecite{MR1951039}, Kitaev defined a 2D lattice model associated to any finite group $G$ with particle-like low energy excitations (known as \\emph{anyons}) parameterized by the simple representations of the Drinfeld double of $G$. These Kitaev models are some of the most well known examples of topological phases, and are of great experimental interest\\cite{chow2014implementing,Gambetta1}.\n\nThese models were generalized in \\onlinecite{Levin2005} to allow any fusion category $\\mathcal{C}$ as input. The low energy excitations of these \\emph{Levin-Wen} models are particle-like and parameterized by simple object of the Drinfeld center $Z(\\mathcal{C})$. When $\\mathcal{C} = \\vvec{G}$, the Levin-Wen model can be transformed into the Kitaev model associated to $G$ with a finite depth quantum circuit, so they describe the same phase of matter. Indeed, the category of representations of the Drinfeld double of $G$ is equivalent to the Drinfeld center of $\\vvec{G}$.\n\nThere are many interesting examples of fusion categories, so the Levin-Wen construction gives us many interesting 2D lattice models. Unfortunately they are too complicated to simulate or study using conventional lattice quantum field theoretic techniques. Moreover, for most interesting fusion categories, the data required to write down the Hamiltonian in the Levin-Wen construction is not known explicitly. For this reason, we need alternative tools to study these models.\n\nThe renormalization invariant properties of a topological phase is described by a topological quantum field theory (TQFT). In mathematics, a TQFT is a functor from a bordism category into a linear algebraic category. It would be counter-productive for us to give a precise definition here. The reader interested in details is encouraged to consult the recent survey \\onlinecite{MR3674995} which includes complete definitions. In \\onlinecite{MR1357878}, Barrett and Westbury described how to construct a (2+1)D TQFT from a fusion category, generalizing the Turaev-Viro construction from \\onlinecite{MR1191386}. It is well understood that the TQFT associated to a fusion category captures the renormalization invariant properties of the corresponding Levin-Wen model. In this paper, we use TQFTs as studied in mathematics to compute renormalization invariant properties of Levin-Wen models which are intractable from the lattice theoretic perspective.\n\nOur work builds on the work of Morrison and Walker in \\onlinecite{MR2978449}. If $\\mathcal{C}$ is a fusion category, morphisms in $\\mathcal{C}$ can be described by 2-dimensional string diagrams up to isotopy. If $\\Sigma$ is a 2-manifold with boundary, the TQFT associated to $\\mathcal{C}$ sends $\\Sigma$ to the vector space of string diagrams from $\\mathcal{C}$ drawn on $\\Sigma$ modulo local relations. These vector spaces are called Skein modules. The string diagrams are allowed to terminate on boundary components. When $\\Sigma$ has a boundary, the vector space is graded by the object labels on the boundary components, and the graded pieces often assemble into an algebraic object. The process of drawing string diagrams from $\\mathcal{C}$ on $\\Sigma$ also extends to modules over $\\mathcal{C}$. Morphisms in these module categories can also be described using string diagrams, and drawing string diagrams from module categories on $\\Sigma$ allows us to use diagrammatic techniques to study defects of codimension 1 and 2 in the corresponding topological phase. The TQFT assigns a category to the annulus, which can be additionally decorated with bimodule strings\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{generic_morphism_in_tube_category}{\n\t\t\t\\begin{tikzpicture}[scale=.7,,every node\/.style={scale=.6}]\n\t\t\t\t\\annparamst{$m$}{$n$}{$m'$}{$n'$};\n\t\t\t\t\\annst{$g$}{$h$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array} : (m,n) \\to (m',n').\\label{eqn:generic_morphism_in_tube_category}\n\\end{align}\nThe representations of this category classify codimension 2 defects (binary interface defects in our language). We refer the reader to the recent survey \\onlinecite{1607.05747} for more details. \nIn \\onlinecite{MR2978449}, representations of this annular category are called sphere modules. Indeed, when $\\mathcal{C} = \\vvec{G}$ and both of the modules are $\\vvec{G}$, then this category is Morita equivalent to the Drinfeld double of $G$. The representations (defects) then correspond to the anyonic excitations of the model. In \\onlinecite{MR2942952}, Kitaev and Kong explain that fusion category bimodules correspond to domain walls and bimodule functors correspond to codimension 2 defects in the Levin-Wen models. In \\onlinecite{SETPaper}, annular categories, called \\emph{dube algebras} by Williamson, Bultinck and Verstraete, are used to study defects interfacing invertible domain walls.\n\nMany of the defect fusions which we compute have multiplicity, which correspond to multiple fusion channels. These multiplicities are somewhat mysterious from the physical perspective.\n\nFrom a mathematical perspective, we are computing decomposition rules for relative tensor product and composition of bimodule functors. In an upcoming paper \\footnote{D.~Barter, J.~C.~Bridgeman, C.~Jones, \\emph{in preparation}}, we will provide a rigorous proof of this fact using a robust theory of skeletalization of fusion categories and their bimodules.\n\n\\subsection*{What is being computed in this work}\n\nSuppose that $\\mathcal{A},\\,\\mathcal{B}$ are 2-dimensional phases of matter and $\\mathcal{M},\\,\\mathcal{N}$ are domain walls between the two phases. A defect $\\alpha$ interfaces between two domain walls\n\\begin{align}\n\\begin{array}{c}\n\\includeTikz{general_defect_picture}\n{\n\t\\begin{tikzpicture}\n\t\\fill[red!10](-1.5,-1) rectangle (0,1);\n\t\\fill[darkgreen!10](1.5,-1) rectangle (0,1);\n\t\\draw[thick,purple] (0,-1)--(0,0);\n\t\\draw[thick,orange](0,0)--(0,1);\n\t\\draw[thick,orange,fill=white] (0,0) ellipse (.2 and .2);\n\t\\draw[thick,purple,dash pattern = on 2 off 2,dash phase=0] (0,0) ellipse (.2 and .2);\n\t\\node[] at (0,0) {$\\alpha$};\n\t\\node at (-1,0) {$\\mathcal{A}$};\n\t\\node at (1,0) {$\\mathcal{B}$};\n  \\node at (0,1.4) {$\\mathcal{N}$};\n  \\node at (0,-1.4) {$\\mathcal{M}$};\n\t\\end{tikzpicture}\t\n}\n\\end{array}.\n\\end{align}\nIn this paper, we are interested in defects when $\\mathcal{A}$ and $\\mathcal{B}$ are the phase associated to the fusion category $\\vvec{\\ZZ{p}}$, and all possible ways these defects can be fused. \nThere are two ways in which defects can fuse:\n\\begin{itemize}\n\t\\item Horizontally, where the supporting domain walls also fuse\n\t\\begin{align} \\label{fig:horizontal_defect_fusion}\n\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{setup_A}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\fill[red!10](-1.5,-1) rectangle (-.4,1);\n\t\t\t\t\\fill[blue!10](-.4,-1) rectangle (.4,1);\n\t\t\t\t\\fill[darkgreen!10](1.5,-1) rectangle (.4,1);\n\t\t\t\t\\draw[thick,red] (-.4,-1)--(-.4,0);\n\t\t\t\t\\draw[thick,blue](-.4,0)--(-.4,1);\n\t\t\t\t\\draw[thick,red,fill=white] (-.4,0) ellipse (.2 and .2);\n\t\t\t\t\\node[fill=white,circle,inner sep=0pt] at (-.4,0) {$\\alpha$};\n\t\t\t\t\\draw[thick,blue,dash pattern = on 2 off 2,dash phase=0] (-.4,0) circle (.2);\n\t\t\t\t%\n\t\t\t\t\\draw[thick,darkgreen] (.4,-1)--(.4,0);\n\t\t\t\t\\draw[thick,darkred](.4,0)--(.4,1);\n\t\t\t\t\\draw[thick,darkgreen,fill=white] (.4,0) circle (.2);\n\t\t\t\t\\node[fill=white,circle,inner sep=0pt] at (.4,0) {$\\beta$};\n\t\t\t\t\\draw[thick,darkred,dash pattern = on 2 off 2,dash phase=0] (.4,0) circle (.2);\n\t\t\t\t\\node at (-1,.5) {$\\mathcal{A}$};\n\t\t\t\t\\node at (0,.5) {$\\mathcal{B}$};\n\t\t\t\t\\node at (1,.5) {$\\mathcal{C}$};\n\t\t\t\t\\node[inline text,below] at (-.4,-1) {$\\mathcal{M}$};\n\t\t\t\t\\node[inline text,below] at (.4,-1) {$\\mathcal{N}$};\n\t\t\t\t\\node[inline text,above] at (-.4,1) {$\\mathcal{P}$};\n\t\t\t\t\\node[inline text,above] at (.4,1) {$\\mathcal{Q}$};\n\t\t\t\t\\end{tikzpicture}\t\n\t\t\t}\n\t\t\\end{array}\n\t\t\\to\n\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{setup_B}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\fill[red!10](-1.5,-1) rectangle (-.15,1);\n\t\t\t\t\\fill[blue!10](-.15,-1) rectangle (.15,1);\n\t\t\t\t\\fill[darkgreen!10](1.5,-1) rectangle (.15,1);\n\t\t\t\t\\draw[thick,red] (-.15,-1)--(-.15,0);\n\t\t\t\t\\draw[thick,blue](-.15,0)--(-.15,1);\n\t\t\t\t\\draw[thick,darkgreen] (.15,-1)--(.15,0);\n\t\t\t\t\\draw[thick,darkred](.15,0)--(.15,1);\n\t\t\t\t\\draw[thick,red,fill=white] (0,0) ellipse (.5 and .3);\n\t\t\t\t\\draw[thick,blue,dash pattern = on 5 off 15] (0,0) ellipse (.5 and .3);\n\t\t\t\t\\draw[thick,darkgreen,dash pattern = on 5 off 15,dash phase=10] (0,0) ellipse (.5 and .3);\n\t\t\t\t\\draw[thick,darkred,dash pattern = on 5 off 15,dash phase=5] (0,0) ellipse (.5 and .3);\n\t\t\t\t\\node[] at (0,0) {$\\alpha\\otimes\\beta$};\n\t\t\t\t\\node at (-1,.5) {$\\mathcal{A}$};\n\t\t\t\t\\node at (0,.5) {$\\mathcal{B}$};\n\t\t\t\t\\node at (1,.5) {$\\mathcal{C}$};\n\t\t\t\t\\node[inline text,below] at (0,-1) {$\\mathcal{M}\\otimes\\mathcal{N}$};\n\t\t\t\t\\node[inline text,above] at (0,1) {$\\mathcal{P}\\otimes\\mathcal{Q}$};\n\t\t\t\t\\end{tikzpicture}\t\n\t\t\t}\n\t\t\\end{array}\n\t\t\\to\n\t\t\\sum_{\\mathcal{R},\\mathcal{S},\\gamma}\n\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{setup_C}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\fill[red!10](-1.5,-1) rectangle (0,1);\n\t\t\t\t\\fill[darkgreen!10](1.5,-1) rectangle (0,1);\n\t\t\t\t\\draw[thick,purple] (0,-1)--(0,0);\n\t\t\t\t\\draw[thick,orange](0,0)--(0,1);\n\t\t\t\t\\draw[thick,orange,fill=white] (0,0) ellipse (.2 and .2);\n\t\t\t\t\\draw[thick,purple,dash pattern = on 2 off 2,dash phase=0] (0,0) ellipse (.2 and .2);\n\t\t\t\t\\node[] at (0,0) {$\\gamma$};\n\t\t\t\t\\node at (-1,.5) {$\\mathcal{A}$};\n\t\t\t\t\\node at (1,.5) {$\\mathcal{C}$};\n\t\t\t\t\\node[inline text,below] at (0,-1) {$\\mathcal{R}$};\n\t\t\t\t\\node[inline text,above] at (0,1) {$\\mathcal{S}$};\n\t\t\t\t\\end{tikzpicture}\t\n\t\t\t}\n\t\t\\end{array}.\n\t\\end{align}\n\t\\\\\n\\item Vertically, where the defects fuse along a common domain wall\n\\begin{align}\n\t\\begin{array}{c} \\label{fig:vertical_defect_fusion}\n\t\t\\includeTikz{setup_A_vert}\n\t\t{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10](-1,-1) rectangle (0,1);\n\t\t\t\\fill[darkgreen!10](0,-1) rectangle (1,1);\n\t\t\t\\draw[thick,red] (0,-1)--(0,-.33);\n\t\t\t\\draw[thick,blue](0,-.33)--(0,.33);\n\t\t\t\\draw[thick,darkgreen](0,.33)--(0,1);\n\t\t\t\\draw[thick,red,fill=white] (0,-.33) circle (.2);\n\t\t\t\\node at (0,-.33) {$\\alpha$};\n\t\t\t\\draw[thick,blue,dashed] (0,-.33) circle (.2);\n\t\t\t\\draw[thick,darkgreen,fill=white] (0,.33) circle (.2);\n\t\t\t\\node at (0,.33) {$\\beta$};\n\t\t\t\\draw[thick,darkred,dashed] (0,.33) circle (.2);\n\t\t\t\\node at (-.75,.5) {$\\mathcal{A}$};\n\t\t\t\\node at (.75,.5) {$\\mathcal{B}$};\n\t\t\t\\end{tikzpicture}\t\n\t\t}\n\t\\end{array}\n\t\\to\n\t\\begin{array}{c}\n\t\t\\includeTikz{setup_B_vert}\n\t\t{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10](-1,-1) rectangle (0,1);\n\t\t\t\\fill[darkgreen!10](0,-1) rectangle (1,1);\n\t\t\t\\draw[thick,red] (0,-1)--(0,0);\n\t\t\t\\draw[thick,darkgreen] (0,0)--(0,1);\n\t\t\t\\draw[thick,red,fill=white] (0,0) ellipse (.5 and .3);\n\t\t\t\\draw[thick,darkgreen,dash pattern = on 5 off 5] (0,0) ellipse (.5 and .3);\n\t\t\t\\node[] at (0,0) {$\\alpha\\circ\\beta$};\n\t\t\t\\node at (-.75,.5) {$\\mathcal{A}$};\n\t\t\t\\node at (.75,.5) {$\\mathcal{B}$};\n\t\t\t\\end{tikzpicture}\t\n\t\t}\n\t\\end{array}\n\t\\to\n\t\\sum_{\\gamma}\n\t\\begin{array}{c}\n\t\t\\includeTikz{setup_C_vert}\n\t\t{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10](-1,-1) rectangle (0,1);\n\t\t\t\\fill[darkgreen!10](0,-1) rectangle (1,1);\n\t\t\t\\draw[thick,red] (0,-1)--(0,0);\n\t\t\t\\draw[thick,darkgreen] (0,0)--(0,1);\n\t\t\t\\draw[thick,red,fill=white] (0,0) ellipse (.2 and .2);\n\t\t\t\\draw[thick,darkgreen,dash pattern = on 2 off 2,dash phase=0] (0,0) ellipse (.2 and .2);\n\t\t\t\\node[] at (0,0) {$\\gamma$};\n\t\t\t\\node at (-.75,.5) {$\\mathcal{A}$};\n\t\t\t\\node at (.75,.5) {$\\mathcal{B}$};\n\t\t\t\\end{tikzpicture}\t\n\t\t}\n\t\\end{array}.\n\\end{align}\n\\end{itemize}\nIn this work, we present algorithms that can be used to compute both the horizontal and vertical fusions between all binary interface defects.\n\\subsection*{Structure of the paper}\nThe paper is structured as follows. In Section~\\ref{sec:constructing_idempotents}, we describe how to compute idempotent representations of all the binary interface defects in Kitaev's $\\ZZ{p}$ model. For a semi-simple category, the Karoubi envelope agrees with the category of representations, so these idempotents parameterize representations of the annulus categories. In Section~\\ref{sec:hotizontal_defect_fusion_algorithm}, we describe the procedure used to compute the horizontal fusion of binary interface defects. Rather than give a formal algorithm, we describe how to proceed in a sufficiently general example. In Section~\\ref{sec:horizontal_defect_fusion}, we demonstrate some of the horizontal fusion computations to elucidate some of the complications. In Section~\\ref{sec:vertical_fusion_algorithm}, we outline the procedure used to compute vertical fusion. In Section~\\ref{sec:vertical_defect_fusion}, we include some example vertical fusion computations. In Section~\\ref{sec:physical_interpretations}, we give physical interpretations for all the binary interface defects. The physical interpretations can be used to reproduce all the horizontal and vertical fusion tables, except for the multiplicities, which are still somewhat mysterious from the physical perspective. In Section~\\ref{sec:natural_transformations}, we explain how natural transformations between bimodule categories fit into our framework. This will be expanded on in future work.\n\nIn Appendix~\\ref{sec:idempotents_table}, we tabulate the idempotent representations of all the binary interface defects in Kitaev's $\\ZZ{p}$ model. In Appendix~\\ref{sec:inflations}, we tabulate all the inflation data required to compute the horizontal fusions. This data was computed as an intermediate step in \\onlinecite{1806.01279}. In Appendix~\\ref{sec:horizontal_fusion_table}, we tabulate the horizontal fusions and in Appendix~\\ref{sec:vertical_fusion_table}, we tabulate the vertical fusions. \n\n\\section{Physical interpretation of defects} \\label{sec:physical_interpretations}\n\\begin{table}\n\t\\resizebox{.9\\textwidth}{!}{\n\t\t\\begin{tabular}{!{\\vrule width 1pt}c!{\\vrule width 1pt}c|c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\t\t\\rowcolor[gray]{.9}[\\tabcolsep]Bimodule label & Domain wall & Action on particles\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$T$&$\\begin{array}{c}\\includeTikz{T}{\n\t\t\t\t\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\t\\fill[red!10](1,-1) rectangle (.25,1);\n\t\t\t\t\\draw[thick](-.25,-1)--(-.25,1);\n\t\t\t\t\\draw[thick](.25,-1)--(.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw (.1,-.9+.2*#3)--(.25,-.9+.2*#3);};\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\t\\end{array}$&Condenses $e$ on both sides\\\\\n\t\t\n\t\t\t$L$&$\\begin{array}{c}\\includeTikz{L}{\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\t\\fill[red!10](1,-1) rectangle (.25,1);\n\t\t\t\t\\draw[thick](-.25,-1)--(-.25,1);\n\t\t\t\t\\draw[thick](.25,-1)--(.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw (.1,-.9+.2*#3)--(.25,-.9+.2*#3);};\n\t\t\t\t\\end{tikzpicture}}\\end{array}$&Condenses $m$ on left and $e$ on right\\\\\n\t\t\n\t\t\t$R$&$\\begin{array}{c}\\includeTikz{R}{\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\t\\fill[red!10](1,-1) rectangle (.25,1);\n\t\t\t\t\\draw[thick](-.25,-1)--(-.25,1);\n\t\t\t\t\\draw[thick](.25,-1)--(.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{tikzpicture}}\\end{array}$&Condenses $e$ on left and $m$ on right\\\\\n\t\t\n\t\t\t$F_0$&$\\begin{array}{c}\\includeTikz{F_0}{\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\t\\fill[red!10](1,-1) rectangle (.25,1);\n\t\t\t\t\\draw[thick](-.25,-1)--(-.25,1);\n\t\t\t\t\\draw[thick](.25,-1)--(.25,1);\n\t\t\t\t\\end{tikzpicture}}\\end{array}$&Condenses $m$ on both sides\\\\\n\t\t\t\\toprule[1pt]\n\t\t\t$X_k$&$\\begin{array}{c}\\includeTikz{Xk}{\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (1,1);\n\t\t\t\t\\draw[ultra thick,dashed] (0,-1)--(0,1);\n\t\t\t\t\\end{tikzpicture}}\\end{array}$&$X_k:e^am^b\\mapsto e^{ka}m^{k^{-1} b}$ (moving left to right), where $k^{-1}$ is taken multiplicatively modulo $p$\\\\\n\t\t\n\t\t\t$F_{q}=F_1 X_q$&$\\begin{array}{c}\\includeTikz{Fq}{\\begin{tikzpicture}[yscale=.3]\n\t\t\t\t\\draw[white](0,-1.1)--(0,1.21);\n\t\t\t\t\\fill[red!10](-1,-1) rectangle (1,1);\n\t\t\t\t\\draw[ultra thick,dotted] (0,-1)--(0,1);\\end{tikzpicture}}\\end{array}$&\n\t\t\t$F_1:e^am^b\\mapsto e^{b}m^{a}$\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n\t\\caption{Domain walls on the lattice corresponding to bimodules. Reproduced from \\onlinecite{1806.01279}.}\\label{tab:domainwallinterp}\n\\end{table}\n\nAssociated to a fusion category $\\mathcal{C}$ is a topological phase described by its Drinfeld center $Z(\\mathcal{C})$. The Levin-Wen procedure\\cite{Levin2005} can be used to construct a lattice model which realizes the topological phase in its low energy space. In \\onlinecite{1806.01279}, we showed how the bimodule categories for $\\vvec{\\ZZ{p}}$ can be interpreted in terms of boundaries and domain walls in the physical theory. This data is reproduced in Table~\\ref{tab:domainwallinterp} for completeness. \n\nIn this section, we discuss a physical interpretation for all defects in Table~\\ref{tab:idempotents}. We will also show how the fusion rules in Tables~\\ref{tab:horizontal_table_1}-\\ref{tab:horizontal_table_6} and \\ref{tab:vertical_fusion_tables} can be obtained from the physical theory up to multiplicity. The multiplicities remain mysterious from the physical perspective, but they can be computed from the Frobenius-Perron dimensions of the defects.\n\nThe simplest defect to interpret is $\\defect{X_1}{X_1}{a}{x}{}$. By studying Table~\\ref{tab:zptable} we observe that $X_1$ is the `identity' domain wall, corresponding to $\\vvec{\\ZZ{p}}$ as a bimodule over itself. The only defects of such a domain wall are the anyons of the theory. Therefore\n\\begin{align}\n\\defect{X_1}{X_1}{a}{x}{}\\mapsto e^xm^a,\n\\end{align}\nwhere the $p^2$ particles of the (domain wall free) theory associated to $\\vvec{\\ZZ{p}}$ are usually denoted $e^xm^a$, with $a,x\\in \\ZZ{p}$\\cite{MR1951039}. \n\nFor other defects, the physical interpretation is found by studying how anyons can be introduced to obtain distinct states. For example, consider the defects $\\defect{X_k}{X_l}{\\cdot}{}{}$. If an anyon $m^a$ is introduced on the left of the wall and $e^x$ is introduced on the right, they can be split and passed through the upper and lower walls to find the equivalent states\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XX_phys_1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\\fill[red!10] (-1.25,-1) rectangle (1.25,1);\n\t\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_k$};\n\t\t\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_l$};\n\t\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{x}$};\n\t\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{a}$};\n\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XX_phys_2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.25,-1) rectangle (1.25,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_k$};\n\t\t\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_l$};\n\t\t\t\\fill[red] (-.25,.5) circle (.1) node[left,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\\fill[red] (-.25,-.5) circle (.1) node[left,inline text,text=black] {$m^{a-\\alpha}$};\n\t\t\t\\fill[blue] (.25,.5) circle (.1) node[right,inline text,text=black] {$e^{\\beta}$};\n\t\t\t\\fill[blue] (.25,-.5) circle (.1) node[right,inline text,text=black] {$e^{x-\\beta}$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XX_phys_3}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2.5,-1) rectangle (2.5,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_k$};\n\t\t\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_l$};\n\t\t\t\\draw[thick,red,>=stealth,->] (-.25,.5)--(.25,0)->(-.15,-.4);\n\t\t\t\\draw[thick,blue,>=stealth,->] (.25,.5)--(-.25,0)->(.15,-.4);\n\t\t\t\\fill[red] (-.25,-.5) circle (.1) node[left,inline text,text=black] {$m^{a+(kl^{-1}{-}1)\\alpha}$};\n\t\t\t\\fill[blue] (.25,-.5) circle (.1) node[right,inline text,text=black] {$e^{x+(kl^{-1}{-}1)\\beta}$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\\label{eqn:XkXldefectstates}\n\\end{align}\nIf $k=l$, then all such states are distinct, so there are $p^2$ defects, corresponding to a basic defect with anyons pushed onto it. For $k\\neq l$, we are free to choose $\\alpha$ and $\\beta$ in Eqn.~\\ref{eqn:XkXldefectstates}, so all states are equivalent, and there is a unique defect.\n\nA very similar computation can be performed for other domain wall interfaces. In the case of $\\defect{F_q}{X_l}{\\cdot}{}{}$, the computation is\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{FX_phys_1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.25,-1) rectangle (1.25,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,0);\n\t\t\t\\draw[ultra thick,dotted](0,0)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$F_q$};\n\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_l$};\n\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{x}$};\n\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{a}$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{FX_phys_2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.25,-1) rectangle (1.25,1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$F_q$};\n\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_l$};\n\t\t\t\\draw[thick,red](-.25,0)--(0,-.25);\\draw[thick,blue,>=stealth,->] (0,-.25)--(.15,-.1);\n\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{x+qa}$};\n\t\t\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,0);\n\t\t\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,-1);\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array},\\label{eqn:FqXldefectstates}\n\\end{align}\nwith no further equivalence possible. Therefore, there are $p$ distinct states corresponding to a base defect with a number of $e$ particles absorbed from the right. We remark that, just as in the choice of idempotents, there is a choice of labeling. We could have instead chosen to label by some combination of $e$ and $m$ on the left and right. This corresponds to permuting the labels.\n\nWhen there is a boundary present, particles can be discarded (condensed) into the boundary. In the case of a $\\defect{L}{T}{\\cdot}{}{}$ defect, we have\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{LT_phys_1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-1.25,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-1.25,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[red] (-.5,-.5) circle (.1) node[left,inline text,text=black] {$m^a$};\n\t\t\t\t\\fill[red] (.5,-.5) circle (.1) node[right,inline text,text=black] {$m^b$};\n\t\t\t\t\\fill[blue] (-.5,.5) circle (.1) node[left,inline text,text=black] {$e^x$};\n\t\t\t\t\\fill[blue] (.5,.5) circle (.1) node[right,inline text,text=black] {$e^y$};\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{LT_phys_2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-1.25,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-1.25,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[red] (.5,-.5) circle (.1) node[right,inline text,text=black] {$m^b$};\n\t\t\t\t\\draw[thick,red,>=stealth,->] (-.5,-.5)--(-.25,-.5);\n\t\t\t\t\\draw[thick,blue,>=stealth,->] (-.5,.5)--(-.25,.5);\n\t\t\t\t\\draw[thick,blue,>=stealth,->] (.5,.5)--(.25,.5);\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array},\\label{eqn:LTdefectstates}\n\\end{align}\nso there are $p$ distinct defects, corresponding to pinning a number of $m$ defects to the right hand boundary.\n\nThe full set of defect interpretations are listed in Table~\\ref{tab:defectinterps}. \nWe remark that in the case $\\ZZ{2}$, many of the defects studied here are the subject of previous work, for example \n\\begin{align}\n\t\\defect{X_1}{F_1}{x}{}{},\\defect{F_1}{X_1}{x}{}{}\\mapsto \\sigma_x\n\\end{align}\nwas referred to as a `twist' in \\onlinecite{Bombin2010}, and defects involving rough\/smooth interfaces in Table~\\ref{tab:defectinterps} were named `corners' in \\onlinecite{Brown2016}.\n\n\\begin{table}\n\t\\resizebox{.9\\textwidth}{!}{\n\t\t\\begin{tabular}{!{\\vrule width 1pt}>{\\columncolor[gray]{.9}[\\tabcolsep]}c!{\\vrule width 1pt}c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c !{\\color[gray]{.8}\\vrule} c!{\\vrule width 1pt}}\n\t\t\t\\toprule[1pt]\n\t\t\\rowcolor[gray]{.9}[\\tabcolsep]\t&$T$&$L$&$R$&$F_0$&$X_l$&$F_r$\\\\\n\t\t\\toprule[1pt]\n\t\t\t$T$&$\\defect{T}{T}{a}{b}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TT_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^a$};\t\t\n\t\t\t\t\\fill[red] (.25,0) circle (.1) node[right,inline text,text=black] {$m^b$};\t\t\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{T}{L}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TL_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[red] (.25,0) circle (.1) node[right,inline text,text=black] {$m^a$};\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{T}{R}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TR_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^a$};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{T}{F_0}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TF0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{T}{X_l}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TXl_idempotent_physical_A}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[red] (-.05,-.2) circle (.1) node[left,inline text,text=black] {$m^{a}$};\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}\n\t\t\t$\n\t\t\t&$\\defect{T}{F_r}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{TFr_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$L$&\n\t\t\t$\\defect{L}{T}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LT_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[red] (.25,0) circle (.1) node[right,inline text,text=black] {$m^a$};\n\t\t\t\t\\end{tikzpicture}\t\n\t\t\t}\n\t\t\t\\end{array}$\n\t\t\t&$\\defect{L}{L}{a}{x}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LL_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[blue] (-.25,0) circle (.1) node[left,inline text,text=black] {$e^x$};\t\t\n\t\t\t\t\\fill[red] (.25,0) circle (.1) node[right,inline text,text=black] {$m^a$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{L}{R}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LR_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&\n\t\t\t$\\defect{L}{F_0}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LF0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[blue] (-.25,0) circle (.1) node[left,inline text,text=black] {$e^x$};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&\n\t\t\t$\\defect{L}{X_l}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LXl_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&\n\t\t\t$\\defect{L}{F_r}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{LFr_idempotent_physical_A}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[blue] (-.05,-.2) circle (.1) node[left,inline text,text=black] {$e^{x}$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}\n\t\t\t$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$R$&$\\defect{R}{T}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RT_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^a$};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{R}{L}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RL_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{R}{R}{a}{x}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RR_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^a$};\t\t\n\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{R}{F_0}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RF0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{R}{X_l}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RXl_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{R}{F_r}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{RFr_idempotent_physical_A}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[blue] (.05,-.2) circle (.1) node[right,inline text,text=black] {$e^{x}$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}\n\t\t\t$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$F_0$&$\\defect{F_0}{T}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0T_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_0}{L}{x}{{}}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0L_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[blue] (-.25,0) circle (.1) node[left,inline text,text=black] {$e^x$};\t\t\n\t\t\t\t\\fill[black] (.25,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_0}{R}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0R_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\foreach #3 in {5,...,9}{\\draw[ultra thick] (-.1,-.9+.2*#3)--(-.25,-.9+.2*#3);};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[black] (-.25,0) circle (.1) node[left,inline text,text=black] {};\t\t\n\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_0}{F_0}{{(x,y)}}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0F0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\t\\fill[red!10](-.5,-1) rectangle (-.25,1);\n\t\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\t\\end{scope}\n\t\t\t\t\\fill[blue] (-.25,0) circle (.1) node[left,inline text,text=black] {$e^x$};\t\t\n\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^y$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_0}{X_l}{{x}}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0Xl_idempotent_physical_A}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,1);\n\t\t\t\t\\fill[blue] (-.05,-.2) circle (.1) node[left,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}\n\t\t\t$&$\\defect{F_0}{F_r}{}{}{}\n\t\t\t=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{F0Fr_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1)--(-.25,-1)--(0,0)--(.25,-1)--(.5,-1)--(.5,1)--(-.5,1);\n\t\t\t\t\\draw[ultra thick](-.25,-1)--(0,0)--(.25,-1);\n\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,1);\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}\n\t\t\t$\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$X_k$&$\\defect{X_k}{T}{a}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkT_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[red] (-.05,.2) circle (.1) node[left,inline text,text=black] {$m^a$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{X_k}{L}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkL_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{X_k}{R}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkR_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{X_k}{F_0}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkF0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\fill[blue] (-.05,.2) circle (.1) node[left,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t$\\defect{X_k}{X_k}{a}{x}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkXk_idempotent_physical}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\t\t\\begin{scope}\n\t\t\t\t\t\\clip (0,-1) rectangle (-1,1);\n\t\t\t\t\t\\fill[red] (-.05,0) circle (.1) node[left,inline text,text=black] {$m^a$};\n\t\t\t\t\t\\end{scope}\n\t\t\t\t\t\\begin{scope}\n\t\t\t\t\t\\clip (0,-1) rectangle (1,1);\n\t\t\t\t\t\\fill[blue] (.05,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\t\\end{scope}\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\t\\\\\n\t\t\t\t\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t\t$\\defect{X_k}{X_l}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{XkXl_idempotent_physical}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[left,inline text,text=black]  at (0,-.5) {$X_k$};\n\t\t\t\t\t\\node[left,inline text,text=black]  at (0,.5) {$X_l$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\\end{tabular}\n\t\t\t&\t\n\t\t\t$\\defect{X_k}{F_r}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkFr_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,-1)--(0,0);\n\t\t\t\t\\draw[ultra thick,dotted](0,1)--(0,0);\n\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\clip (0,-1) rectangle (-1,1);\n\t\t\t\t\\fill[black] (-.05,0) circle (.1) node[left,inline text,text=black] {$$};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\clip (0,-1) rectangle (1,1);\n\t\t\t\t\\fill[blue] (.05,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$\n\t\t\t\\\\\n\t\t\t\\greycline{2-7}\n\t\t\t$F_q$&$\\defect{F_q}{T}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{FqT_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dotted](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_q}{L}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{FqL_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[xscale=-1,yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[blue] (-.05,.2) circle (.1) node[left,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_q}{R}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{FqR_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\foreach #3 in {0,...,3}{\\draw[yscale=-1,ultra thick,shift={(-.25+.05*#3,-.9+.2*#3)}] (0,0)--(.15,-.0375);};\n\t\t\t\t\\fill[blue] (.05,.2) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&$\\defect{F_q}{F_0}{}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{FqF0_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,1)--(-.25,1)--(0,0)--(.25,1)--(.5,1)--(.5,-1)--(-.5,-1);\n\t\t\t\t\\draw[ultra thick](-.25,1)--(0,0)--(.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,0)--(0,-1);\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$&\n\t\t\t$\\defect{F_q}{X_l}{x}{}{}=\n\t\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{FqXl_idempotent_physical}\n\t\t\t{\n\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\\draw[ultra thick,dotted](0,-1)--(0,0);\n\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,0);\n\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\clip (0,-1) rectangle (-1,1);\n\t\t\t\t\\fill[black] (-.05,0) circle (.1) node[left,inline text,text=black] {$$};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\begin{scope}\n\t\t\t\t\\clip (0,-1) rectangle (1,1);\n\t\t\t\t\\fill[blue] (.05,0) circle (.1) node[right,inline text,text=black] {$e^x$};\n\t\t\t\t\\end{scope}\n\t\t\t\t\\end{tikzpicture}\n\t\t\t}\n\t\t\t\\end{array}$\n\t\t\t&\n\t\t\t\\begin{tabular}{c}\n\t\t\t\t$\\defect{F_q}{F_q}{x}{y}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqFq_idempotent_physical}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\t\\draw[ultra thick,dotted](0,-1)--(0,0);\n\t\t\t\t\t\\draw[ultra thick,dotted](0,1)--(0,0);\n\t\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\begin{scope}\n\t\t\t\t\t\\clip (0,-1) rectangle (-1,1);\n\t\t\t\t\t\\fill[blue] (-.05,0) circle (.1) node[left,inline text,text=black] {$e^{x}$};\n\t\t\t\t\t\\end{scope}\n\t\t\t\t\t\\begin{scope}\n\t\t\t\t\t\\clip (0,-1) rectangle (1,1);\n\t\t\t\t\t\\fill[blue] (.05,0) circle (.1) node[right,inline text,text=black] {$e^y$};\n\t\t\t\t\t\\end{scope}\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\\\\\n\t\t\t\t\\arrayrulecolor[gray]{.8}\\hline\\arrayrulecolor{black}\n\t\t\t\t$\\defect{F_q}{F_r}{}{}{}=\n\t\t\t\t\\begin{array}{c}\n\t\t\t\t\\includeTikz{FqFr_idempotent_physical}\n\t\t\t\t{\n\t\t\t\t\t\\begin{tikzpicture}[scale=.7,every node\/.style={scale=.7}]\n\t\t\t\t\t\\fill[red!10] (-.5,-1) rectangle (.5,1);\n\t\t\t\t\t\\draw[ultra thick,dotted](0,-1)--(0,0);\n\t\t\t\t\t\\draw[ultra thick,dotted](0,1)--(0,0);\n\t\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[left,inline text,text=black]  at (0,-.5) {$F_q$};\n\t\t\t\t\t\\node[left,inline text,text=black]  at (0,.5) {$F_r$};\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t}\n\t\t\t\t\\end{array}$\n\t\t\t\\end{tabular}\n\t\t\t\\\\\n\t\t\t\\toprule[1pt]\n\t\t\\end{tabular}\n\t}\n\t\\caption{Physical interpretation of all defects.}\\label{tab:defectinterps}\n\\end{table}\n\n\\subsection{Fusing defects: Horizontal}\nThe physical interpretations from Table~\\ref{tab:defectinterps} can be used to compute the fusion rules. In this section, we will illustrate how this is done using a few examples.\n\n\\begin{exmp}[$\\defect{X_k}{X_k}{}{}{} \\otimes \\defect{X_l}{X_l}{}{}{}$]\n\nConsider the fusion $\\defect{X_k}{X_k}{a}{x}{}\\otimes \\defect{X_l}{X_l}{c}{z}{}$. In the physical theory, we begin by clearing any anyons occurring on the central region\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXkXlXl_phys1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\\fill[red!10] (-2,-1) rectangle (2,1);\n\t\t\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$X_k$};\n\t\t\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_l$};\n\t\t\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$X_l$};\n\t\t\t\t\t\\fill[blue] (-.5,.25) circle (.1) node[right,inline text,text=black] {$e^{x}$};\n\t\t\t\t\t\\fill[red] (-1,0) circle (.1) node[left,inline text,text=black] {$m^{a}$};\n\t\t\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{z}$};\n\t\t\t\t\t\\fill[red] (.5,-.25) circle (.1) node[left,inline text,text=black] {$m^{c}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXkXlXl_phys2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2.1,-1) rectangle (2.1,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$X_k$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_l$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$X_l$};\n\t\t\t\\fill[red] (-1,0) circle (.1) node[left,inline text,text=black] {$m^{a+kc}$};\n\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{z+lx}$};\n\\draw[thick,blue,>=stealth,->] (-.5,.25)->(.9,.1);\n\\draw[thick,red,>=stealth,->] (.5,-.25)->(-.9,-.1);\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXkXlXl_phys3}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.5,-1) rectangle (1.5,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_{kl}$};\n\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_{kl}$};\n\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{a+kc}$};\n\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{z+lx}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nThis recovers the fusion computed using the annular algebra.\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{X_k}{X_l}{}{}{} \\otimes \\defect{X_m}{X_n}{}{}{}$]\nA more interesting computation arises when we allow the upper and lower domain walls to differ\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXlXmXn_phys1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1,-1) rectangle (1,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$X_l$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$X_n$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXlXmXn_phys2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1,-1) rectangle (1,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$X_l$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$X_n$};\n\t\t\t\t\t\t\\fill[blue] (.5,.25) circle (.1) node[left,inline text,text=black] {$e^{\\beta}$};\n\t\t\t\t\t\t\\fill[blue] (.5,-.25) circle (.1) node[left,inline text,text=black] {$e^{-\\beta}$};\n\t\t\t\t\t\t\\fill[red] (-.5,.7) circle (.1) node[right,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\t\t\t\\fill[red] (-.5,-.7) circle (.1) node[right,inline text,text=black] {$m^{-\\alpha}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXlXmXn_phys3}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2.5,-1) rectangle (2.5,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$X_l$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$X_n$};\n\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{\\beta(n-m)}$};\n\t\t\t\\fill[red] (-1,0) circle (.1) node[left,inline text,text=black] {$m^{\\alpha(l-k)}$};\n\t\t\t\\draw[thick,blue,>=stealth,->] (.5,.25)->(.9,.1);\\draw[thick,blue,>=stealth,->] (.5,-.25)->(.9,-.1);\n\t\t\t\\draw[thick,red,>=stealth,->] (-.5,.7)->(-.9,.1);\\draw[thick,red,>=stealth,->] (-.5,-.7)->(-.9,-.1);\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkXlXmXn_phys4}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.75,-1) rectangle (1.75,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_{km}$};\n\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_{ln}$};\n\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{\\beta(n-m)}$};\n\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{\\alpha(l-k)}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nIf $km\\neq ln$, then from Eqn.~\\ref{eqn:XkXldefectstates}, this is equivalent to the base defect for all choices $\\alpha,\\beta$. In the case where $km=ln$, these states are distinct for each $\\alpha, \\beta$, giving the fusion rule\n\\begin{align}\n\t\\defect{X_k}{X_l}{}{}{}\\otimes \\defect{X_m}{X_n}{}{}{}&=\n\t\\begin{cases}\n\t\tp\\cdot \\defect{X_{km}}{X_{ln}}{}{}{}&km\\neq ln\\\\\n\t\t\\oplus_{\\alpha,\\beta} \\defect{X_{km}}{X_{km}}{\\alpha}{\\beta}{}&km=ln\n\t\\end{cases}.\n\\end{align}\nThe coefficient $p$ is not obtained from this computation, but can be calculated using the Frobenius-Perron dimensions computed in Section~\\ref{S:FPd}.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\protect \\defect{X_k}{F_r}{}{}{} \\otimes \\protect\\defect{X_m}{F_t}{}{}{}$]\n\nConsider the fusion process\n\\begin{align}\n\t\\defect{X_k}{F_r}{x}{}{}\\otimes\\defect{X_m}{F_t}{z}{}{}.\n\\end{align}\nIn the physical theory, this is computed using\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkFrXmFt_phys1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1,-1) rectangle (1.5,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$F_r$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$F_t$};\n\t\t\t\\fill[blue] (-.5,0) circle (.1) node[right,inline text,text=black] {$e^{x}$};\n\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{z}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkFrXmFt_phys2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1,-1) rectangle (1.5,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$F_r$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$F_t$};\n\t\t\t\\fill[blue] (-.5,.25) circle (.1) node[right,inline text,text=black] {$e^{\\beta}$};\n\t\t\t\\fill[blue] (.5,-.25) circle (.1) node[left,inline text,text=black] {$e^{x-\\beta}$};\n\t\t\t\\fill[red] (.5,.7) circle (.1) node[left,inline text,text=black] {$m^{-\\alpha}$};\n\t\t\t\\fill[red] (-.5,-.7) circle (.1) node[right,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{z}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkFrXmFt_phys3}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2.75,-1) rectangle (3.25,1);\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[black] (-.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\fill[black] (.75,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (-.75,-1) {$X_k$};\n\t\t\t\\node[above,inline text,text=black]  at (-.75,1) {$F_r$};\n\t\t\t\\node[below,inline text,text=black]  at (.75,-1) {$X_m$};\n\t\t\t\\node[above,inline text,text=black]  at (.75,1) {$F_t$};\n\t\t\t\\fill[blue] (1,0) circle (.1) node[right,inline text,text=black] {$e^{z-t\\alpha+m(x-\\beta)}$};\n\t\t\t\\draw[thick,blue] (-.5,.25)--(-.75,.165);\\draw[thick,red,>=stealth,->] (-.75,.165)->(-.9,.1);\n\t\t\t\\draw[thick,red,>=stealth,->] (-.5,-.7)->(-.9,-.1);\n\t\t\t%\n\t\t\t\\draw[thick,blue,>=stealth,->] (.5,-.25)->(.9,-.1);\n\t\t\t\\draw[thick,red] (.5,.7)--(.75,.3);\\draw[thick,blue,>=stealth,->] (.75,.3)->(.9,.1);\n\t\t\t%\n\t\t\t\\draw[ultra thick,dashed](-.75,1)--(-.75,-1);\n\t\t\t\\draw[ultra thick,dashed](.75,1)--(.75,-1);\n\t\t\t\\fill[red] (-1,0) circle (.1) node[left,inline text,text=black] {$m^{k\\alpha+r^{-1}\\beta}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n\t=\n\t\\begin{array}{c}\n\t\t\\includeTikz{XkFrXmFt_phys4}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1.25,-1) rectangle (1.25,1);\n\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_{km}$};\n\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_{r^{-1}t}$};\n\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{\\beta^\\prime}$};\n\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{\\alpha^\\prime}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array},\n\\end{align}\nwhere $\\alpha^\\prime=k\\alpha+r^{-1}\\beta$ and $\\beta^\\prime=z+mx-mr\\alpha^\\prime+(mrk-t)\\alpha$. If $km\\neq r^{-1}t$, these states are all equivalent to the base defect following Eqn.~\\ref{eqn:XkXldefectstates}. In the case $km=r^{-1}t$, we have\n\\begin{align}\n\t\t\\begin{array}{c}\n\t\t\t\\includeTikz{XkFrXmFt_phys5}{\n\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\fill[red!10] (-1.25,-1) rectangle (2.25,1);\n\t\t\t\t\\draw[ultra thick,dashed](0,1)--(0,-1);\n\t\t\t\t\\fill[black] (0,0) circle (.1) node[right,inline text,text=black] {};\n\t\t\t\t\\node[below,inline text,text=black]  at (0,-1) {$X_{km}$};\n\t\t\t\t\\node[above,inline text,text=black]  at (0,1) {$X_{r^{-1}t}$};\n\t\t\t\t\\fill[blue] (.25,0) circle (.1) node[right,inline text,text=black] {$e^{z+mx-mr\\alpha}$};\n\t\t\t\t\\fill[red] (-.25,0) circle (.1) node[left,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\t\\end{tikzpicture}}\n\t\t\\end{array},\n\\end{align}\nfor any $\\alpha$. The fusion rule is therefore\n\\begin{align}\n\t\\defect{X_k}{F_r}{x}{}{}\\otimes \\defect{X_m}{Ft}{z}{}{}&=\n\t\\begin{cases}\n\t\t\\defect{X_{km}}{X_{r^{-1}t}}{}{}{}&km\\neq r^{-1}t\\\\\n\t\t\\oplus_{\\alpha} \\defect{X_{km}}{X_{km}}{\\alpha}{z+m(x-r\\alpha)}{}&km=r^{-1}t\n\t\\end{cases},\n\\end{align}\nwhere the Frobenius-Perron dimensions (Section~\\ref{S:FPd}) can be used to check that no multiplicity is required.\n\n\\end{exmp}\n\n\\begin{exmp}[$\\defect{T}{T}{}{}{} \\otimes \\defect{T}{T}{}{}{}$]\n\nMultiplicity in the domain wall fusion (Table~\\ref{tab:zptable}) has an effect on the defect fusion. For the fusion\n\\begin{align}\n\t\\defect{T}{T}{a}{b}{}\\otimes \\defect{T}{T}{c}{d}{},\n\\end{align}\nthe physical pre-fusion state is\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{DTTabxDTTcd_interp_1}\n\t\t{\n\t\t\t\\begin{tikzpicture}[yscale=.75]\n\t\t\t\\draw[thick,white](1,0)--(1,1) node[above,inline text,pos=1]{\\footnotesize$m^{b+c}$};\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[red,text=black] (-.25,0) circle (.1) node[left,inline text] {$m^a$};\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^b$};\n\t\t\t\\begin{scope}[shift={(2,0)}]\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[red,text=black] (-.25,0) circle (.1) node[left,inline text] {$m^c$};\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^d$};\n\t\t\t\\end{scope}\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\nThe central strip supports a $p$ dimensional vector space. The qudit state can be read out by exchanging an $e$ particle between the boundaries. The state is changed by inserting an $m$ line vertically. Suppose the strip is in the state $m^\\mu$. To perform the fusion, we push the inner $m$ particles away from the fusion region\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{DTTabxDTTcd_interp_2}\n\t\t{\n\t\t\t\\begin{tikzpicture}[yscale=.75]\n\t\t\t\\filldraw[red!10](.25,-1)--(1.75,-1)--(1,0)--cycle;\n\t\t\t\\filldraw[yscale=-1,red!10](.25,-1)--(1.75,-1)--(1,0)--cycle;\n\t\t\t\\draw[thick,red](1,0)--(1,1) node[above,inline text,pos=1]{\\footnotesize$m^{b+c}$};\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[red,text=black] (-.25,0) circle (.1) node[left,inline text] {$m^a$};\n\t\t\t\\begin{scope}[shift={(2,0)}]\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^d$};\n\t\t\t\\end{scope}\n\t\t\t\\draw[ultra thick] (.25,-1) --(1.75,1) (.25,1) --(1.75,-1);\n\t\t\t\\foreach #3 in {0,...,8}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick,shift={(.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t\t\\draw [yscale=-1,ultra thick,shift={(.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t\t\\draw [xscale=-1,ultra thick,shift={(-2+.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t\t\\draw [xscale=-1,yscale=-1,ultra thick,shift={(-2+.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t}\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\nAfter fusing the central strip, there are still $p$ states. These are now understood as a subspace of a 2 qudit system, with one supported on each of the upper and lower regions. The subspace is spanned by states of the form $\\ket{m^\\mu}\\otimes\\ket{m^{\\mu+b+c}}$. The fusion outcome is therefore\n\\begin{align}\n\t\\left[ \\defect{T}{T}{a}{b}{}\\otimes \\defect{T}{T}{c}{d}{} \\right]_{\\mu,\\nu} = \\delta_{\\nu-\\mu}^{b+c} \\cdot \\defect{T}{T}{a}{d}{}.\n\\end{align}\n\\end{exmp}\n\n\n\\begin{exmp}[$\\defect{L}{T}{}{}{} \\otimes \\defect{L}{T}{}{}{}$]\nAs a final example, we show how the fusion\n\\begin{align}\n\t\\defect{L}{T}{a}{}{}\\otimes \\defect{L}{T}{b}{}{}\n\\end{align}\ncan be computed.\n\nPhysically, this process is represented by\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{DLTaDLTb_interp_1}\n\t\t{\n\t\t\t\\begin{tikzpicture}[yscale=.75]\n\t\t\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {7,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[black,text=black] (-.25,0) circle (.075);\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^a$};\n\t\t\t\\begin{scope}[shift={(2,0)}]\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {7,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[black,text=black] (-.25,0) circle (.075);\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^b$};\n\t\t\t\\end{scope}\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\nThe central strip supports a $p$ dimensional vector space. The qudit state can be read out by exchanging an $e$ particle between the boundaries in the upper region. The state is changed by inserting an $m$ line vertically. To perform the fusion, we must push the $m$ particles away from the fusion region\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{DLTaDLTb_interp_2}\n\t\t{\n\t\t\t\\begin{tikzpicture}[yscale=.75]\n\t\t\t\\filldraw[red!10](.25,-1)--(1.75,-1)--(1,0)--cycle;\n\t\t\t\\filldraw[yscale=-1,red!10](.25,-1)--(1.75,-1)--(1,0)--cycle;\n\t\t\t\\draw[thick,red](1,0)--(1,1) node[above,inline text,pos=1]{\\footnotesize$m^a$};\n\t\t\t\\begin{scope}[xscale=1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {7,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[black,text=black] (-.25,0) circle (.075);\n\t\t\t\\begin{scope}[shift={(2,0)}]\n\t\t\t\\begin{scope}[xscale=-1]\n\t\t\t\\fill[red!10](-1,-1) rectangle (-.25,1);\n\t\t\t\\draw[ultra thick] (-.25,-1)--(-.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (-.25,-1+.1+#3*.15)--(-.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\end{scope}\n\t\t\t\\fill[red,text=black] (.25,0) circle (.1) node[right,inline text] {$m^b$};\n\t\t\t\\end{scope}\n\t\t\t\\draw[ultra thick] (.25,-1) --(1.75,1) (.25,1) --(1.75,-1);\n\t\t\t\\foreach #3 in {0,...,8}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick,shift={(.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t\t\\draw [yscale=-1,ultra thick,shift={(.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t\t\\draw [xscale=-1,yscale=-1,ultra thick,shift={(-2+.075+3\/40*#3,.1+.1*#3)}] (.25,-1)--(.25-.075,-1+.1);\n\t\t\t}\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{array}.\n\\end{align}\nAfter fusing the central strip, there are still $p$ states, fully supported on the upper region. The fusion outcome is therefore\n\\begin{align}\n\t\\left[\\defect{L}{T}{a}{}{}\\times \\defect{L}{T}{b}{}{}\\right]_\\mu=\\defect{L}{T}{b}{}{}\n\\end{align}\n\n\\end{exmp}\n\n\\subsection{Fusing defects: Vertical}\nSince the vertical fusions are much simpler than the horizontal ones, we will provide a single example to illustrate how the physical interpretation can be used for the fusion calculation. \n\\begin{exmp}[$\\defect{L}{F_r}{}{}{}\\circ \\defect{F_r}{L}{}{}{}$]\nConsider the fusion\n\\begin{align}\n\t\\defect{L}{F_r}{x}{}{}\\circ \\defect{F_r}{L}{z}{}{}. \n\\end{align}\nPhysically, the fusion is\n\\begin{align}\n\t\\begin{array}{c}\n\t\t\\includeTikz{vertfusionexample_phys_1}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-1,-1)rectangle(1,1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,-1)--(0,-.25)--(.25,-1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,1)--(0,.25)--(.25,1);\n\t\t\t\\draw[thick,dotted](0,-.25)--(0,.25);\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,-1+.12*#3+.075)}] (0,0)--(-.075,-.025);};\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,1-.12*#3-.075)}] (0,0)--(-.075,.025);};\n\t\t\t\\fill[blue] (-.25,-.25) circle (.1) node[left,inline text,text=black] {$e^{x}$};\n\t\t\t\\fill[blue] (-.25,.25) circle (.1) node[left,inline text,text=black] {$e^{z}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n=\n\t\\begin{array}{c}\n\t\t\\includeTikz{vertfusionexample_phys_2}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2,-1)rectangle(1,1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,-1)--(0,-.25)--(.25,-1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,1)--(0,.25)--(.25,1);\n\t\t\t\\draw[thick,dotted](0,-.25)--(0,.25);\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,-1+.12*#3+.075)}] (0,0)--(-.075,-.025);};\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,1-.12*#3-.075)}] (0,0)--(-.075,.025);};\n\t\t\t\\fill[blue] (-.25,-.25) circle (.1) node[left,inline text,text=black] {$e^{x+z-r\\alpha}$};\n\t\t\t\\fill[blue] (-.25,.25) circle (.1) node[left,inline text,text=black] {$e^{r\\alpha}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n=\n\t\\begin{array}{c}\n\t\t\\includeTikz{vertfusionexample_phys_3}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2,-1)rectangle(1,1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,-1)--(0,-.25)--(.25,-1);\n\t\t\t\\filldraw[white,draw=black,ultra thick] (-.25,1)--(0,.25)--(.25,1);\n\t\t\t\\draw[thick,dotted](0,-.25)--(0,.25);\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,-1+.12*#3+.075)}] (0,0)--(-.075,-.025);};\n\t\t\t\\foreach #3 in {0,...,4}{\\draw[ultra thick,shift={(.25-.04*#3-.025,1-.12*#3-.075)}] (0,0)--(-.075,.025);};\n\t\t\t\\fill[blue] (-.25,-.25) circle (.1) node[left,inline text,text=black] {$e^{x+z-r\\alpha}$};\n\t\t\t\\fill[red] (.25,-.25) circle (.1) node[right,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\\draw[thick,blue](-.25,.25)--(0,0);\n\t\t\t\\draw[thick,red,>=stealth,->] (0,0)--(.15,-.15);\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}\n=\n\t\\begin{array}{c}\n\t\t\\includeTikz{vertfusionexample_phys_4}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\fill[red!10] (-2,-1)rectangle(1.25,1);\n\t\t\t\\filldraw[white] (-.25,-1) rectangle (.25,1);\n\t\t\t\\draw[ultra thick](-.25,-1)--(-.25,1);\\draw[ultra thick](.25,-1)--(.25,1);\n\t\t\t\\foreach #3 in {0,...,12}\n\t\t\t{\n\t\t\t\t\\draw [ultra thick] (.25,-1+.1+#3*.15)--(.1,-1+.1+#3*.15);\n\t\t\t}\n\t\t\t\\fill[blue] (-.5,0) circle (.1) node[left,inline text,text=black] {$e^{x+z-r\\alpha}$};\n\t\t\t\\fill[red] (.5,0) circle (.1) node[right,inline text,text=black] {$m^{\\alpha}$};\n\t\t\t\\end{tikzpicture}}\n\t\\end{array}.\n\\end{align}\nSince any $\\alpha$ could have been chosen for this calculation, the fusion rule is\n\\begin{align}\n\t\\defect{L}{F_r}{x}{}{}\\circ\\defect{F_r}{L}{z}{}{}=\\oplus_\\alpha \\defect{L}{L}{\\alpha}{x+z-r\\alpha}{}.\n\\end{align}\n\\end{exmp}\n\n\\subsection{Frobenius-Perron Dimension}\\label{S:FPd}\nThe Frobenius-Perron dimension (FPd) of the defects can be computed using the fusion table. For defects $a$, $b$ and $c$, the FPd obeys \n\\begin{align}\nd_a d_b=\\sum_c N_{a,b}^c d_c,\n\\end{align}\nwhere $N_{a,b}^c$ is the multiplicity of the fusion. The FPd of the defects of $\\vvec{\\ZZ{p}}$ do not depend on the defect label, only on the domain walls involved. The FPds are \n\\begin{align}\n\\dim\\left(\\defect{X_k}{X_l}{\\bullet}{}{}\\right)&=\\dim\\left(\\defect{F_k}{F_l}{\\bullet}{}{}\\right)=\\begin{cases}1&k=l\\\\p&k\\neq l\\end{cases},\\\\\n\\dim\\left(\\defect{X_k}{F_r}{\\bullet}{}{}\\right)&=\\dim\\left(\\defect{F_r}{X_k}{\\bullet}{}{}\\right)=\\sqrt{p},\\\\\n\\dim\\left(\\defect{W_1}{W_2}{\\bullet}{}{}\\right)&=0&\\text{for all other defects}.\n\\end{align}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nVeering triangulations were introduced by the first author in \\cite{Ago11} to study mapping tori of pseudo-Anosov homeomorphisms. In that setting, these are ideal triangulations of the punctured mapping tori that encode a folding sequence of train tracks associated to the pseudo-Anosov homeomorphism. In \\cite{Ago11}, veering triangulations were used to prove a quantitative refinement of a result obtained initially by Farb, Leininger, and Margalit (\\cite{FLM11}), which states that given a bound on the normalized dilatation of the pseudo-Anosov monodromy, there are only finitely many homeomorphism types of such punctured mapping tori. However, the proof presented in \\cite{Ago11} contained a gap: an estimate by Ham and Song (\\cite{HS07}) was applied to a matrix which records how weights on the branches of the train tracks distribute under the folding moves, but the estimate only applies to irreducible matrices, and there exists examples for which this fails to be the case. \n\nMeanwhile, since \\cite{Ago11} appeared, the study of veering triangulations has developed in many other directions, see for example \\cite{Gue16}, \\cite{HIS16}, \\cite{HRST11}, \\cite{Lan18}. In particular, Gu\\'eritaud reconstructed the veering triangulations for punctured mapping tori in \\cite{Gue16} in terms of the pseudo-Anosov suspension flow. Among other things, this shows that the veering triangulation is canonically associated to each fibered face of the Thurston unit ball in $H_2$, instead of individual fiberings in the interior of the face. Generalizing this, the first author and Gu\\'{e}ritaud showed that if a 3-manifold has a pseudo-Anosov flow without perfect fits, then the manifold obtained by drilling out the singular orbits of the flow admits a veering triangulation (see \\cite{LMT21}). \n\nRecently, Schleimer and Segerman proved the converse of this: if a 3-manifold admits a veering triangulation, then appropriate Dehn fillings of it carry pseudo-Anosov flows without perfect fits. Their construction involves isotoping the stable and unstable branched surfaces in order to form a dynamic pair in the sense of Mosher (\\cite{Mos96}). The isotopy is in turn constructed by analyzing how the branched surfaces behave with respect to a canonical decomposition of the 3-manifold into veering solid tori. Details of this construction will appear in \\cite{SS22}. We remark that Schleimer and Segerman's construction is in fact part of a big program showing that veering triangulations and pseudo-Anosov flows without perfect fits are in correspondence with each other. In particular, their construction is inverse to that of Agol and Gu\\'eritaud in both directions, in a suitable sense. The first three parts of their program are \\cite{SS18}, \\cite{SS21}, and \\cite{SS19}. Also see the introduction of \\cite{SS19} for an outline of the whole program. \n\nThis paper was born out of an attempt to reprove Schleimer and Segerman's construction using more direct means. There is a standard way of constructing pseudo-Anosov flows on a 3-manifold starting from an embedded graph with desirable properties. See the concept of templates, introduced by Birman and Williams in \\cite{BW83a} and \\cite{BW83b} (under the name of `knot-holders'), and the concept of dynamic pairs, introduced by Mosher in \\cite{Mos96}. The general strategy is to thicken up the graph using flow boxes, then collapse along the complement to obtain a flow on the whole 3-manifold. Given a veering triangulation, there is a natural candidate to apply this strategy to: the flow graph. (See \\Cref{sec:veertri} for definitions of objects associated to veering triangulations.) However, the fact that the flow graph might not be strongly connected presents difficulties in both constructing the flow and analyzing it.\n\nThis turns out to be exactly the same issue underlying the gap in \\cite{Ago11}. In this paper, we explain a way of addressing this. We show that the problematic infinitesimal components must arise in a certain form, which have to do with subsets of the veering triangulation which we call `walls'. By throwing out these components, we obtain the reduced flow graph, which share many of the same features as the flow graph.\n\nBy analyzing this reduced flow graph, we are able to tackle the proof in \\cite{Ago11}. The explicit quantitative bound we get is the following:\n\\begin{thm:boundveertet}\n\nIf $M$ is the punctured mapping torus of a pseudo-Anosov homeomorphism $\\phi: S_{g,n} \\to S_{g,n}$, where the normalized dilatation $\\lambda(\\phi)^{2g-2+\\frac{2}{3}n} \\leq P$, then $M$ has a veering triangulation with at most $\\frac{P^9-1}{2} (\\frac{2 \\log P^9}{\\log(2 P^{-9}+1)}-1)$ tetrahedra.\n\\end{thm:boundveertet}\n\nNote that the bound we obtain is worse than that stated in \\cite{Ago11} by an exponent of $2+\\epsilon$, but there might be room for improvement.\n\nTo reprove Schleimer and Segerman's construction, we apply the general strategy outlined above to the reduced flow graph: we thicken up the graph by flow boxes, and collapse along its complement, with the help of the unstable branched surface. When the filling slopes satisfy a necessary condition, it is not difficult to see that the resulting flow is pseudo-Anosov. Furthermore, we can leverage the structure of the unstable branched surface to show that the pseudo-Anosov flow obtained has no perfect fits relative to the `filled' orbits. The precise result is as below, see \\Cref{sec:veertri,sec:pAflow} for relevant definitions.\n\n\\begin{thm:vtpAflow}\nSuppose $M$ admits a veering triangulation. Let $l=(l_i)$ denote the collection of ladderpole curves on each boundary component. Then $M(s)$ admits a transitive pseudo-Anosov flow $\\phi$ if $|\\langle s,l \\rangle| \\geq 2$. \n\nFurthermore, there are closed orbits $c_i$ isotopic to the cores of the filling solid tori, such that each $c_i$ is $|\\langle s_i,l_i \\rangle|$-pronged, and $\\phi$ is without perfect fits relative to $\\{c_i\\}$.\n\\end{thm:vtpAflow}\n\nWe remark that there are 2 notions of pseudo-Anosov flows common in the 3-manifold topology literature: topological pseudo-Anosov flows and smooth pseudo-Anosov flows. \\Cref{thm:vtpAflow} holds for both notions.\n\nWe also remark that, in contrast to the work by Schleimer and Segerman, it is not clear whether our construction provides a correspondence between veering triangulations and pseudo-Anosov flows. We pose this as a question more carefully in \\Cref{sec:questions}.\n\nThe results we discuss in this paper run parallel to those in \\cite{LMT21}. The walls in this paper are closely related to what are called AB regions in \\cite{LMT21}. In the setting of \\cite{LMT21}, one starts with a pseudo-Anosov flow without perfect fits and builds a veering triangulation transverse to the flow. Then it turns out that the flow graph encodes the orbits of the flow, with AB cycles governing the extent of over- and under-counting. In this paper we go in the opposite direction, starting with a veering triangulation and constructing a pseudo-Anosov flow without perfect fits. We end up with a similar conclusion: the flow graph encodes the orbits of the flow, with infinitesimal cycles of walls governing the extent of over- and under-counting. We will explain these connections between this paper and \\cite{LMT21} where relevant.\n\nHere is an outline of this paper. In \\Cref{sec:veertri}, we review the notion of veering triangulations and related constructions. In \\Cref{sec:infcomp}, we study the infinitesimal components of the flow graph and show that they must arise from walls. We also show how to define the reduced flow graph by throwing away these infinitesimal components. In \\Cref{sec:finiteness}, we analyze the reduced flow graph of layered veering triangulations to fix the gap in \\cite{Ago11} and prove \\Cref{thm:boundveertet}. In \\Cref{sec:pAflow}, we reprove Schleimer and Segerman's result (\\Cref{thm:vtpAflow}) using the reduced flow graph. Finally in \\Cref{sec:questions}, we discuss some questions coming out this paper. We remark that \\Cref{sec:finiteness,sec:pAflow} are independent of each other, and the reader can jump to any of them after reading \\Cref{sec:infcomp}.\n\n{\\bf Acknowledgments.} The first author discovered the gap in \\cite{Ago11} with the help of the veering triangulation census. We thank Andreas Giannopolous, Saul Schleimer, and Henry Segerman for making the data in the census available. We also thank Saul Schleimer and Henry Segerman for sharing with us an early version of \\cite{SS22}. We thank Michael Landry for the support and encouragement throughout this project. We thank Rafael Potrie on MathOverflow for pointing us in the direction of Mario Shannon's thesis, and Mario Shannon for guiding us through the material in \\Cref{subsec:smooth}. We thank Anna Parlak and Samuel Taylor for comments on an early version of this paper.\n\nThe first author was supported by MSRI and the Simonyi Professorship at IAS during part of this project. Both authors are partially supported by the Simons Foundation.\n\n{\\bf Notational conventions.} Throughout this paper, \n\\begin{itemize}\n    \\item $\\overline{M}$ will be an oriented compact 3-manifold with torus boundary components, while $M$ will be the interior of such a manifold. We will sometimes conflate a torus end of $M$ with the corresponding boundary component of $M$.\n    \\item $X \\backslash \\backslash Y$ will denote the metric completion of $X \\backslash Y$ with respect to the induced path metric from $X$. In addition, we will call the components of $X \\backslash \\backslash Y$ the complementary regions of $X$ in $Y$.\n    \\item $\\widetilde{X}$ will mean a universal cover of $X$, unless otherwise stated.\n\\end{itemize}\n\n\\section{Background: veering triangulations} \\label{sec:veertri}\n\nWe recall the definition of a veering triangulation.\n\n\\begin{defn} \\label{defn:tautstructure}\nAn \\textit{ideal tetrahedron} is a tetrahedon with its 4 vertices removed. The removed vertices are called the \\textit{ideal vertices}. An \\textit{ideal triangulation} of $M$ is a decomposition of $M$ into ideal tetrahedra glued along pairs of faces.\n\nA \\textit{taut structure} on an ideal triangulation is a labelling of the dihedral angles by $0$ or $\\pi$, such that \n\\begin{enumerate}\n    \\item Each tetrahedron has exactly two dihedral angles labelled $\\pi$, and they are opposite to each other.\n    \\item The angle sum around each edge in the triangulation is $2\\pi$.\n\\end{enumerate}\n\nIntuitively this means that there is a degenerate geometric structure on the triangulation where every tetrahedron is flat.\n\nA \\textit{transverse taut structure} is a taut structure along with a coorientation on each face, such that for any edge labelled $0$ in a tetrahedron, exactly one of the faces adjacent to it is cooriented out of the tetrahedron.\n\nA \\textit{transverse taut ideal triangulation} is an ideal triangulation with a transverse taut structure.\n\\end{defn}\n\nWe will always take the convention that the face coorientations are pointing upwards in our figures and descriptions.\n\n\\begin{defn} \\label{defn:veertri}\nA \\textit{veering structure} on a transverse taut ideal triangulation is a coloring of the edges by red or blue, so that looking at each flat tetrahedron from above, when we go through the 4 outer edges counter-clockwise, starting from an endpoint of the inner edge in front, the edges are colored red, blue, red, blue in that order. We call such a tetrahedron a \\textit{veering tetrahedron}.\n\nA \\textit{veering triangulation} is a transverse taut ideal triangulation with a veering structure.\n\\end{defn}\n\n\\Cref{fig:veertet} shows a veering tetrahedron in a veering triangulation.\n\n\\begin{figure} \n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{4cm}{\\input{veertet.pdf_tex}}\n    \\caption{A tetrahedron in a transverse veering triangulation. There are no restrictions on the colors of the top and bottom edges.} \n    \\label{fig:veertet}\n\\end{figure}\n\nFrom now on $\\Delta$ will denote a veering triangulation on $M$.\n\n\\begin{rmk} \\label{rmk:transverse}\nSome authors call the notion we have just defined a transverse veering triangulation, and call the version of this notion without face coorientations veering triangulations instead. These two versions do not differ by much, since one can always take a double cover to make the faces coorientable. However, we will just be studying the version with face coorientations in this paper. Without the face coorientations, our main object of study, the flow graph (\\Cref{defn:flowgraph}), cannot be defined.\n\\end{rmk}\n\nWe recall some combinatorial facts and constructions for veering triangulations.\n\n\\begin{defn} \\label{defn:fantet}\nA veering tetrahedron in $\\Delta$ is called a \\textit{toggle tetrahedron} if the colors on its top and bottom edges differ. It is called a \\textit{red\/blue fan tetrahedron} if both its top and bottom edges are red\/blue respectively.\n\nNote that some authors call toggle and fan tetrahedra hinge and non-hinge respectively.\n\\end{defn}\n\n\\begin{prop}(\\cite[Observation 2.6]{FG13}) \\label{prop:fantet}\nEvery edge $e$ in $\\Delta$ has one tetrahedron above it, one tetrahedron below it, and two stacks of tetrahedra, in between the tetrahedra above and below, on either of its sides.\n\nEach stack must be nonempty. Suppose $e$ is red (blue, respectively). If there is exactly one tetrahedron in one stack, then that tetrahedron is a red (blue, respectively) fan tetrahedron. If there are $n>1$ tetrahedron in one stack, then going from top to bottom in that stack, the tetrahedra are: one toggle tetrahedron, $n-2$ blue (red, respectively) fan tetrahedra, and one toggle tetrahedron. \n\\end{prop}\n\n\\begin{defn} \\label{defn:branchsurf}\nFor each tetrahedron of $\\Delta$, define a branched surface inside it by placing a quadrilateral with vertices on the top and bottom edges and the two side edges of the same color as the top edge, then adding a triangular sector for each side edge of the opposite color to the top edge, with a vertex on that side edge and attached to the quadrilateral along an arc going between the two faces adjacent to that side edge. We also require that the arcs of attachment for the two triangular sectors intersect only once on the quadrilateral. See \\Cref{fig:branchsurf} top left. These branched surfaces in each tetrahedron can arranged to match up across faces, thus glue up to a branched surface in $M$, which we call the \\textit{unstable branched surface} $B$.\n\nThe intersection of the unstable branched surface with the faces of $\\Delta$ is called the \\textit{unstable train track}. Notice that as one goes from the top faces to the bottom faces of each veering tetrahedron, the unstable train track undergoes a \\textit{folding move}. See \\Cref{fig:branchsurf} bottom. \n\nThe branch locus of $B$ is a union of circles, smoothly carried by the branch locus and intersecting transversely at \\textit{double points} of the branch locus. We call these circles the \\textit{components} of the branch locus. Orient the circles so that they intersect the 2-skeleton of the veering triangulation negatively, i.e. they disagree with the coorientation of the faces whenever they meet. This orientation has the special property that at double points of the branch locus, it always points from the side with more sectors to the side with less sectors.\n\\end{defn}\n\nNote that the unstable branched surface, when considered as a cell complex by taking the 0-cells to be the double points of its branch locus, the 1-skeleton to be the branch locus, and the 2-cells to be the sectors, is dual to the veering triangulation. As such, it makes sense for us to talk about, for example, the sector of $B$ dual to a given edge of $\\Delta$.\n\nAlso, notice that each sector of $B$ is of the form of a diamond. Each sector has two upper edges and at least two lower edges, at least one on either side; one top vertex, two side vertices, some vertices between the lower edges on either side, and one bottom vertex.\n\n\\begin{figure} \n    \\centering\n    \\resizebox{!}{8cm}{\\input{branchsurf.pdf_tex}}\n    \\caption{The unstable branched surface meets the faces of a veering tetrahedron in the unstable train track, which undergoes a folding move as one goes from the top faces to the bottom faces. The flow graph can be embedded in the unstable branched surface.}\n    \\label{fig:branchsurf}\n\\end{figure}\n\n\\begin{defn} \\label{defn:flowgraph}\n\nLet $\\Delta$ be a veering triangulation. The \\textit{flow graph} $\\Phi$ is a directed graph with vertex set $V(\\Phi)$ equals to the set of edges of $\\Delta$, and edge set $E(\\Phi)$ defined by adding 3 edges per tetrahedron, going from the top edge and the two side edges of opposite color to the top edge, into the bottom edge.\n\n\\end{defn}\n\nWe note that the flow graph is also defined in \\cite{LMT20}. However the reader is cautioned that there the flow graph is oriented in the opposite direction compared to here.\n\nThe flow graph $\\Phi$ can be embedded in the unstable branched surface $B$ in the following way. Place each vertex of $\\Phi$ at the top vertex of the sector dual to the corresponding edge of $\\Delta$, and place the edges of $\\Phi$ exiting that vertex in the interior of the sector, as shown in \\Cref{fig:branchsurf} top right. The flow graph inherits the structure of a (non-generic) oriented train track from this embedding, by making the edges meeting a vertex tangent to a vertical line. We will consider the flow graph to be a subset of $B$ in this way from now on.\n\nThe ideal triangulation $\\Delta$ on $M$ induces a triangulation $\\partial \\Delta$ on the torus boundary components of $\\overline{M}$, by considering them as links of the ideal vertices. The branch locus of the unstable branched surface inside each tetrahedron opens up towards two opposite ideal vertices, hence each side determines an oriented interval between two edges of a face in $\\partial \\Delta$. Join these intervals end-to-end. Since each face in $\\partial \\Delta$ contains at most one such interval, the paths must close up to form oriented parallel loops. Also note that since every edge in $\\Delta$ is the top edge of some tetrahedron, each boundary component of $\\overline{M}$ will receive at least one loop.\n\n\\begin{defn} \\label{defn:ladderpole}\nThe homology classes of these oriented parallel loops on each boundary component of $\\overline{M}$ are called the \\textit{ladderpole curves}. The slopes they determine on each boundary component of $\\overline{M}$ are called the \\textit{ladderpole slopes}.\n\\end{defn}\n\nWe end this section by analyzing the complementary regions of $B$ in $M$ and the complementary regions of $\\Phi$ in $B$.\n\n\\begin{prop} \\label{prop:complbranchsurf}\nThe components of $M \\backslash \\backslash B$ are (once-punctured cusped polygons)$\\times S^1$, where each cusp$\\times S^1$ represents the ladderpole slope on the corresponding boundary component of $\\overline{M}$, and the sum over all cusp circles in each component represents the ladderpole curve on the corresponding boundary component of $\\overline{M}$.\n\\end{prop}\n\n\\begin{proof}\nLet $T$ be such a component. $T$ is a neighborhood of a torus end of $M$. The part of $T$ inside each tetrahedron is homeomorphic to a product, hence the product structures glue up to give a homeomorphism $T \\cong T^2 \\times [0,\\infty)$. $\\partial T$ inherits the branch locus of $B$ as cusp circles, with these representing the ladderpole classes and slopes as described in the statement by definition. An identification of $\\partial T$ with $T^2$ sending these cusp circles to longitudes extend to an identification of $T$ with (once-punctured cusped $n$-gon)$\\times S^1$, where $n$ is the number of cusp circles. \n\\end{proof}\n\n\\begin{lemma} \\label{lemma:flowgraphcompl}\nThe components of $B \\backslash \\backslash \\Phi$ are annuli or Mobius bands with tongues, i.e. they can be obtained by attaching triangular sectors (`tongues') along arcs to a smooth annulus or Mobius band. (This terminology is borrowed from \\cite{Mos96}.)\n\nMoreover, the arcs of attachment of the triangular sectors zig-zag along the annulus\/Mobius band. More precisely, the arcs on the annulus\/Mobius band lift to $y=\\pm x + 2i, i \\in \\mathbb{Z}$ in the universal cover $[0,1] \\times \\mathbb{R}$. These arcs are subintervals of the branch locus of $B$ and are oriented downwards (i.e. decreasing $y$ in the above model). See \\Cref{fig:flowgraphcompl} bottom.\n\\end{lemma}\n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{12cm}{\\input{flowgraphcompl.pdf_tex}}\n    \\caption{The complementary regions of $\\Phi$ in $B$ are annuli\/Mobius bands with tongues.}\n    \\label{fig:flowgraphcompl}\n\\end{figure}\n\n\\begin{proof}\nThe topology of the complementary regions is unchanged if we thicken up $\\Phi$ to be a regular neighborhood of itself in $B$ within each sector. Hence we can take a neighborhood of $\\Phi$ in each sector of $B$ as in \\Cref{fig:flowgraphcompl} top, where the black lines in the figure are where the faces of $\\Delta$ meet the sector.\n\nWith this adjustment in place, it is straightforward to analyze the portion of each complementary region within a veering tetrahedron. There are two of these for each tetrahedron, corresponding to the two side edges of the same color as the top edge. We show the form of these components in \\Cref{fig:flowgraphcompl} middle. In particular they intersect the faces of the tetrahedron in train track switches. \n\nComponents of $B \\backslash \\backslash \\Phi$ can be obtained by gluing these pieces along the train track switches on the top and bottom, and the result of the gluing must be an annulus or Mobius band with tongues with the arcs of attachments of the tongues as described.\n\\end{proof}\n\n\\section{Infinitesimal components of the flow graph} \\label{sec:infcomp}\n\n\\begin{defn} \\label{defn:PF}\nA directed graph $G$ is said to be \\textit{strongly connected} if for every ordered pair of vertices $(v,w)$ there a directed edge path going from $v$ to $w$.\n\nThe \\textit{adjacency matrix} of $G$ is defined to be the matrix $A \\in Hom(\\mathbb{R}^{V(G)}, \\mathbb{R}^{V(G)})$ with entries $A_{wv}$=number of edges going from $v$ to $w$. It is easy to see that $G$ being strongly connected is equivalent to its adjacency matrix $A$ being \\textit{irreducible}, i.e. for every $v,w$, the entry $(A^n)_{wv}$ is positive for some $n>0$.\n\\end{defn}\n\nFor reasons explained in the introduction, it would be convenient to have a strongly connected flow graph associated to a given veering triangulation. However, a brief search through the veering triangulation census (\\cite{GSS}) gives examples which this property fails, for instance: \\texttt{eLAkbbcdddhwqj\\_2102} (which is layered) and \\texttt{fLAMcaccdeejsnaxk\\_20010} (which is non-layered). See \\cite{GSS} for what these codes mean. For completeness, we also note that there are examples which this property holds: \\texttt{cPcbbbdxm\\_10} (which is layered), and \\texttt{gLLAQbddeeffennmann\\_011200} (which is non-layered). As we shall see later, further examples can be constructed by taking covers.\n\nOur goal in this section is to analyze how this property can fail and explain how to prune the flow graph to make a version of this property hold.\n\n\\subsection{Strongly connected components} \\label{subsec:PFcomp}\n\nWe first set up some notation for discussing strongly connectedness.\n\n\\begin{defn} \\label{defn:PFcomp}\nLet $G$ be a directed graph, let $v,w$ be vertices of $G$. Write $v \\gtrsim w$ if there is a directed edge path going from $v$ to $w$. Write $v \\sim w$ if $v \\gtrsim w$ and $v \\lesssim w$. Note that $\\sim$ is an equivalence relation, call equivalence classes of $\\sim$ \\textit{strongly connected components}.\n\nConstruct a directed graph $G\/{\\sim}$ with vertex set equals to the equivalence classes $[v]$ of $\\sim$, and an edge from $[v]$ to $[w]$ if $v \\gtrsim w$ and $[v] \\neq [w]$. Call a strongly connected component $[v]$ of $G$ an \\textit{infinitesimal component} if $[v]$ has an incoming edge in $G\/{\\sim}$.\n\nA subset $V$ of $V(G)$ is called a \\textit{minimal set} if it has the property that if $v \\in V$ and $v \\gtrsim w$ then $w \\in V$.\n\\end{defn}\n\nIt is easy to see that a finite directed graph is a disjoint union of strongly connected graphs if and only if it has no infinitesimal components. Similarly, a finite directed graph is strongly connected if and only if it has no proper minimal sets. In this sense, infinitesimal components, or minimal sets in general, are the obstruction to strong connectedness. Hence our approach to understanding the failure of strongly connectedness of a given flow graph is to analyze its infinitesimal components and minimal sets.\n\n\\subsection{Walls} \\label{subsec:wall}\n\nIn this section, we introduce the concept of walls of a veering triangulation. These present one way in which infinitesimal components of flow graphs can arise. What we will show in \\Cref{subsec:infcompproof} is that these account for all infinitesimal components of flow graphs. \n\n\\begin{defn} \\label{defn:wall}\nLet $\\Delta$ be a veering triangulation on a 3-manifold $M$. Suppose there are tetrahedra $t_{i,j}, 1 \\leq i \\leq w+1, j \\in \\mathbb{Z}\/h$, where $w \\geq 2$, such that for $2 \\leq i \\leq w$,\n\n\\begin{enumerate}\n    \\item The bottom edge of $t_{i,j}$ is the top edge of $t_{i,j+1}$\n    \\item If $i$ is odd, the bottom edge of $t_{i,j}$ are side edges of $t_{i-1,j}$ and $t_{i+1,j}$ and no other tetrahedra in $\\Delta$. If $i$ is even, the bottom edge of $t_{i,j}$ are side edges of $t_{i-1,j+1}$ and $t_{i+1,j+1}$ and no other tetrahedra in $\\Delta$.\n\\end{enumerate}\n\nThen we call the collection $\\{t_{i,j}\\}$ a wall, and call $w$ the \\textit{width} of the wall. We emphasize that in this definition, $t_{i,j}$ for different $i,j$ may not be different tetrahedra in $\\Delta$.\n\nNotice that for $w\\geq 3$, it is possible to extract a proper subcollection of $\\{t_{i,j}\\}$ which will form a wall of smaller width. A wall will be called \\textit{maximal} if the collection of tetrahedra $\\{t_{i,j}\\}$ cannot be enlarged into a wall of larger width. Similarly, one can always replace $h$ with a multiple of $h$ by renaming the tetrahedra appropriately. In the sequel we will implicitly assume that $h$ is the minimum possible value that satisfies the definition. \n\\end{defn}\n\nSuppose we have a wall $\\{t_{i,j}\\}$ of a veering triangulation $\\Delta$. Notice that for $2 \\leq i \\leq w$, by \\Cref{prop:fantet} and \\Cref{defn:flowgraph}, the bottom edge of $t_{i,j}$ has only one outgoing edge in the flow graph $\\Phi$, namely the edge going to the bottom edge of $t_{i,j+1}$. Hence there is a cycle $c_i$ in the flow graph passing through the top\/bottom edges of $t_{i,j}$, for which there are edges entering $c_i$ but no edges exiting $c_i$. In other words, $c_i$ is an infinitesimal component of the flow graph for each $2 \\leq i \\leq w$. We call these $c_i$ the \\textit{infinitesimal cycles} of the wall.\n\nMeanwhile for each $j$ there is an edge of $\\Phi$ going from the bottom edge of $t_{1,j}$ to the bottom edge of $t_{1,j+1}$, since the former is either the top edge of $t_{1,j+1}$ or a side edge of opposite color as the top edge of $t_{1,j+1}$. Let $c_1$ be the cycle formed by these edges. Similarly, there is a cycle $c_{w+1}$ passing through the bottom edges of $t_{w+1,j}$. We call $c_1$ and $c_{w+1}$ the \\textit{boundary cycles} of the wall.\n\nHere is a convenient way to visualize a wall, at least for those with width $w \\geq 3$. Inside each tetrahedron, there is a unique quadrilateral carried by the unstable branched surface with boundary lying on the unstable train track. If $w \\geq 3$, by applying \\Cref{prop:fantet} to the bottom edges of $t_{i,j}$ for $2 \\leq i \\leq w$, we can see that the tetrahedra $t_{i,j}$ are all fan tetrahedra, and they are either all blue fan tetrahedra or all red fan tetrahedra. Hence the quadrilaterals that lie inside them are adjacent to each other in $B$, and their union forms a surface homeomorphic to an annulus or a Mobius band which is carried by $B$. The quadrilaterals form a tiling of the annulus or Mobius band, with 4 quadrilaterals meeting at each vertex, resembling a tiled wall. In fact, this is the reason why we have chosen to call the collection of tetrahedra a wall. In particular, we note that in this case (1) in the definition holds for $i=1,w+1$ as well, and so the infinitesimal and boundary cycles $c_i$ lie within the quadrilateral tiling as vertical loops.\n\nIf $w=2$, we can try to repeat the above argument, but the picture is not as clean. By the same argument as above, $t_{1,j}$ and $t_{3,j}$ are fan tetrahedra, but the bottom edge of $t_{1,j}$ may be different from the top edge of $t_{1,j+1}$, and similarly for $t_{3,j}$. As a result, the union of quadrilaterals inside the tetrahedra do not nicely tile up an annulus or Mobius band, but instead at some vertices some quadrilaterals from the bottom might `peel away'.\n\nWe remark that the quadrilaterals we considered above are among the ones considered in \\cite{HRST11}, and this idea of looking at how quadrilaterals tile up also appeared there.\n\nIn \\Cref{fig:width4wall}, we present 3 ways of illustrating a wall of width $4$ in order to aid the reader's intuition. The first picture shows the quadrilateral tiling as mentioned above (\\Cref{fig:width4wall} top left). The second picture shows a layered view in terms of a folding sequence of the unstable train track on the faces of tetrahedra in the wall (\\Cref{fig:width4wall} right). The third picture shows the portion of the unstable branched surface in a small neighborhood of the wall (\\Cref{fig:width4wall} bottom left). The flow graph contains vertical lines as subgraphs in the picture. These are the infinitesimal and boundary cycles of the wall.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{10cm}{\\input{width4wall.pdf_tex}}\n    \\caption{Illustrating a wall of width $4$ from three viewpoints. Top left: tiling with quadrilaterals. Right: folding sequence of the unstable train track. Bottom left: unstable branched surface in a small neighborhood.}\n    \\label{fig:width4wall}\n\\end{figure}\n\nIn \\Cref{fig:width2wall}, we also demonstrate the same 3 viewpoints for a wall of width $2$, since as pointed out before, the combinatorics for width $2$ walls are slightly more general than that for higher width walls. \n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{12cm}{\\input{width2wall.pdf_tex}}\n    \\caption{Illustrating a wall of width $2$ from the same three viewpoints as \\Cref{fig:width4wall}.}\n    \\label{fig:width2wall}\n\\end{figure}\n\nWe caution the reader that $c_i$ for different $i$ may be the same cycle in $\\Phi$. In fact, call a wall \\textit{twisted} if $t_{i,j}=t_{w+2-i,j+h'}$ for some $h'$, and call a wall \\textit{untwisted} otherwise. For a twisted wall, $c_i$ and $c_{w+2-i}$ are the same cycle for each $i$; for a untwisted wall, $c_i$ are distinct cycles. Equivalently, a wall is twisted if and only if the surface tiled by the quadrilaterals is homeomorphic to a Mobius band.\n\nAnother point of caution is that two distinct maximal walls $\\{t_{i,j}\\}, \\{t'_{i,j}\\}$ can share a tetrahedron, say $t_{i,j}=t'_{i',j'}$. By maximality, this is only possible if $i=1$ or $w+1$ and $i'=1$ or $w'+1$, where $w$ and $w'$ are the widths of the walls respectively. Intuitively, this can be visualized as two walls touching along their boundary cycles within a tetrahedron. In fact, this behaviour can happen within a single wall as well, that is, a tetrahedron can appear as $t_{1,j}$ or as $t_{w+1,j}$ for more than one value of $j$. However, a tetrahedron can only appear at most twice in walls, corresponding to the $2$ side edges of the same color as the top edge. In particular a vertex of $\\Phi$ can lie in at most $2$ boundary cycles.\n\n\\subsection{Classification of infinitesimal components} \\label{subsec:infcompproof}\n\nWe begin our analysis of infinitesimal components of flow graphs. Throughout this subsection, fix a minimal set $V$ of the flow graph $\\Phi$ associated to a veering triangulation $\\Delta$. Recall that vertices of $\\Phi$ are edges of $\\Delta$, so it makes sense to say whether $e$ is in $V$ for an edge $e$ of $\\Delta$.\n\nEach branch of the unstable train track $\\tau$ in a face of $\\Delta$ is dual to some edge of that face. Let $\\tau'$ be the union of those branches dual to edges of $\\Delta$ that lie in $V$. Up to rotation, there are 5 configurations for the portion of $\\tau'$ lying on a face of $\\Delta$. We show and name these 5 types of faces in \\Cref{fig:facetype}.\n\n\\begin{figure}\n    \\centering\n        \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{8cm}{\\input{facetype.pdf_tex}}\n    \\caption{The 5 possible types of faces, labelled by letters b,i,e,c,f.}\n    \\label{fig:facetype}\n\\end{figure}\n\nWe also consider the configurations for the portion of $\\tau'$ lying on the boundary of a tetrahedron. Since $V$ is minimal, and since the flow graph $\\Phi$ contains the cycles formed by edges going from the top edge to the bottom edge of each tetrahedron, the top edge of a tetrahedron lies in $V$ if and only if the bottom edge lies in $V$. Again by minimality, if a side edge of opposite color to the top edge is in $V$, then the bottom edge is in $V$, hence the top edge will be in $V$ as reasoned above. From these restrictions, we can enumerate 13 configurations for the portion of $\\tau'$ on the boundary of a tetrahedron, up to rotation and reflection. We show and name these 13 types of tetrahedra in \\Cref{fig:tettype}.\n\n\\begin{figure}\n    \\centering\n        \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{18cm}{\\input{tettype.pdf_tex}}\n    \\caption{The 13 possible types of tetrahedra, labelled by Roman numerals 0-XII. We drew the tetrahedra by their top faces followed by their bottom faces. The tetrahedron of type (VI)-(X) inside the box are eliminated by \\Cref{lemma:tettypeelim}.}\n    \\label{fig:tettype}\n\\end{figure}\n\n\\begin{lemma} \\label{lemma:tettypeelim}\nTetrahedra of type (VI)-(X) will not appear.\n\\end{lemma}\n\n\\begin{proof}\nWe use a double-counting argument. $$\\sum \\text{\\# type (c) upper faces} - \\text{\\# type (c) lower faces} =0$$ where the sum is over all tetrahedra of $\\Delta$, since each face belongs to the upper face of exactly one tetrahedron and the lower face of exactly one tetrahedron. By inspection of \\Cref{fig:tettype}, the number of type (c) upper faces is always greater or equal to the number of type (c) lower faces for each tetrahedron; for type (VI).(VIII),(IX),(X) tetrahedra, strict inequality holds, so these in fact cannot occur. \n\nA similar argument using type (i) faces eliminates the possiblity of type (VII) tetrahedra (now that type (VIII) tetrahedra are eliminated).\n\\end{proof}\n\nWe now analyze what happens if we have a type (i) face. Let $f$ be such a face, and $e$ be the unique edge of $f$ that lies in $V$. \n\n$e$ is the top edge of one tetrahedron, the bottom edge of one tetrahedron, and the side edges of two stacks of tetrahedra, one stack on each side of $e$. The face $f$ determines a side of $e$, and we claim that the stack of tetrahedra on that side consists of one tetrahedron only. \n\nSuppose otherwise, then the top edge of the tetrahedron on the bottom of the stack is of opposite color to $e$ (by \\Cref{prop:fantet}). Looking at \\Cref{fig:tettype}, we claim that this tetrahedron must be of type (XI) or (XII). This is because these are the only types of tetrahedron with a side edge in $V$ that is of opposite color to the top edge. As a consequence, the face of this tetrahedron which meet $e$ must be of type (f). But again by looking at \\Cref{fig:tettype}, the face $f$ being of type (i) forces the face below it and sharing the edge $e$ to be of type (i) too (and the tetrahedron between them being of type (I) or (II)). In fact, the same is true for the face above $f$ and sharing the edge $e$. Hence inducting upwards and downwards, the faces on the side of $f$ that are adjacent to $e$ must all be of type (i). This gives us a contradiction at the bottom of the stack of tetrahedra.\n\nThe tetrahedra on the top and bottom of $e$ have to be of type (III) or (IV). In fact, they are either both of type (III) or both of type (IV), since by the reasoning above, a type (i) face on a side of $e$ forces all of the faces in that stack with edge $e$ to be of type (i).\n\nIf the tetrahedra on the top and bottom of $e$ are of type (III), the stack on the side of $e$ opposite to $f$ also consists of one tetrahedron only by the reasoning two paragraphs above. In particular, in this case, $e$ is adjacent to $4$ (possibly non-distinct) tetrahedra. Note that there is still the freedom of the side tetrahedra being of type (I) or (II).\n\nAt this point, we recall the quadrilaterals that we considered in \\Cref{subsec:wall}. By looking at the surface formed by the union of the quadrilaterals in the tetrahedra we have considered so far, this gives us a good way of keeping track of our argument. For example, in the situation of the paragraph above, we have $4$ tetrahedra that share an edge $e$, hence we have 4 quadrilaterals sharing a vertex at $e$, see \\Cref{fig:wallarg1} left.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{4cm}{\\input{wallarg1.pdf_tex}}\n    \\caption{The types of the tetrahedra around $e$ split into three cases. The Roman numeral inside a quadrilateral is the type of the tetrahedron the quadrilateral belongs to.}\n    \\label{fig:wallarg1}\n\\end{figure}\n\nNow, our analysis can be extended in the vertical direction. The top edge of the tetrahedron above $e$ is adjacent to a type (i) face. So we can repeat our arguments on there. This gives a group of 4 tetrahedra centered around that edge. In terms of the quadrilaterals, we are extending the tiling vertically. See \\Cref{fig:wallarg2} left. A caveat of this picture, however, is that the top and bottom vertices of the quadrilaterals representing type (I) and (II) tetrahedra may not match up, since they might be representing different edges of the adjacent type (III) tetrahedron. We denote this by making the quadrilaterals peel away slightly. Since there are only finitely many tetrahedra, the vertical extension must loop back on itself eventually, giving us a width $2$ wall whose infinitesimal cycle lies in $V$.\n\nWe still need to tackle the case when the tetrahedra on the top and bottom of $e$ are of type (IV), we claim that the stack on the side of $e$ opposite to $f$ also consists of one tetrahedron only. Suppose otherwise, then arguing as before, the tetrahedron on the bottom of the stack is of type (XI) or (XII) and its faces that meet $e$ are of type (f). But we know that the face at the bottom of the stack that meets $e$ is of type (e), so this is not the case. \n\nLooking at \\Cref{fig:tettype}, this single tetrahedron must be of type (IV) or (V). If the side tetrahedron is of type (IV), it has two type (i) faces, and the tetrahedra sharing those type (i) faces must be of type (I) or (II). In particular, we get a cluster of 6 tetrahedra. The corresponding 6 quadrilateral tile up a region as shown in \\Cref{fig:wallarg1} center.\n\nNow repeat the argument vertically as in the last case. We obtain a width $3$ wall eventually, for which the infinitesimal cycles lie in $V$.\n\nFinally, we tackle the case where $e$ has type (IV) tetrahedra above and below and a type (V) tetrahedron on the side opposite to $f$. See \\Cref{fig:wallarg1} right. Let $e'$ be the top edge of this type (V) tetrahedron. By repeating our arguments up to this point vertically, we know that the tetrahedron on top of $e'$ is of type (IV) or (V). It is not of type (IV) otherwise $e'$ has a side where all the faces are of type (i), contradicting the tetrahedron of type (V) below it. So the sides of $e'$ have type (e) faces on the top and bottom of their stacks of tetrahedra. We have argued that this implies the stacks of tetrahdra on the sides of $e'$ have one tetrahedron each, and that these side tetrahedra are of type (IV) or (V). The same argument also applies to the bottom edge of the original type (V) tetrahedron. In terms of our picture with quadrilaterals, this extends the tiling horizontally.\n\nNow continue the analysis inductively, both horizontally and vertically. Again, the vertical extension must loop back on itself at some point. This implies that the horizontal extension has to stop at some point, otherwise it will loop back on itself and we will get a Klein bottle or a torus tiled by quadrilaterals from type (V) tetrahedra, which contradicts $\\Delta$ having a strict angle structure (see \\cite[Theorem 1.5 and Corollary 3.11]{HRST11}). The horizontal extension stops by hitting a column of type (IV) tetrahedra, after which the tiling is `capped off' by type (I) or (II) tetrahedra. See \\Cref{fig:wallarg2} right. Hence in every case, we get a wall with infinitesimal cycles contained in $V$.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{6cm}{\\input{wallarg2.pdf_tex}}\n    \\caption{The patterns of quadrilaterals that can occur in our proof. Left: a width $2$ wall. Right: a width $\\geq 3$ wall.}\n    \\label{fig:wallarg2}\n\\end{figure}\n\nWe claim that if $V$ is a proper subset of $V(\\Phi)$, it consists solely of these infinitesimal cycles of walls. Suppose otherwise, then since these infinitesimal cycles have no incoming edges from vertices in $V$, we can remove all of them from $V$ and we would still have a non-empty minimal set. Rename $V$ as this new minimal set, and reclassify the faces and tetrahedra of $\\Delta$ according to \\Cref{fig:facetype,fig:tettype} with respect to this new minimal set.\n\nAfter this reclassification, there are no type (i) faces anymore, since those must be contained in walls by our analysis above, and we have eliminated all strongly connected components of $V$ within walls. In particular, we can only have tetrahedra of type (0), (V), (XI), and (XII).\n\nSuppose we have a type (e) face $f$. Let $e$ be the unique edge of $f$ not lying in $V$. Consider the stack of tetrahedra on the side of $e$ determined by $f$. Looking at \\Cref{fig:tettype}, with the limited types of tetrahedra we can have, as we move downwards the stack starting from $f$, we can only get to faces of type (e). So the stack must end on a type (e) face, but then the tetrahedron directly below this face, which is also the tetrahedron below $e$, cannot be any of the remaining types in \\Cref{fig:tettype}. This contradiction shows that there are no type (e) faces anymore.\n\nAnd so the only types of tetrahedra we can have are type (0) and (XII). In this case, it is clear that a tetrahedron sharing a face with a type (0) or a type (XII) tetrahedron must be of type (0) or type (XII) itself, respectively. So by a connectedness argument, $V$ is either empty or the entirety of $V(\\Phi)$. This proves the claim and concludes our analysis of minimal sets of $\\Phi$, which we summarize as:\n\n\\begin{thm} \\label{thm:infinitesimal}\nFor a given veering triangulation $\\Delta$, the infinitesimal components of its flow graph $\\Phi$ are exactly the infinitesimal cycles of walls. Furthermore, $\\Phi\/{\\sim}$ is a rooted height 1 tree, i.e. it has a unique vertex from which there is an edge from it to any other vertex.\n\\end{thm}\n\n\\begin{proof}\nWe showed that any proper minimal set of $\\Phi$ is a disjoint union of infinitesimal cycles of walls. This implies that every proper minimal set of $\\Phi\/{\\sim}$ is a disjoint union of vertices. Furthermore, each infinitesimal cycle of a wall has incoming edges, hence each vertex of $\\Phi\/{\\sim}$ which is an infinitesimal component has an incoming edge. This is enough to imply the second statement of the theorem.\n\\end{proof}\n\nWe use this theorem to justify our statement earlier that for a finite cover $\\widetilde{\\Delta}$ of a veering triangulation $\\Delta$, $\\widetilde{\\Delta}$ has a strongly connected flow graph if and only if $\\Delta$ has a strongly connected flow graph. The flow graph of $\\widetilde{\\Delta}$ is a covering of that of $\\Delta$, and so the foward implication is easy. For the converse, if the flow graph of $\\widetilde{\\Delta}$ is not strongly connected, it contains infinitesimal cycles of a wall. Such a wall projects down to a wall of $\\Delta$ by \\Cref{defn:wall}, thus $\\Delta$ contains infinitesimal cycles as well.\n\n\\begin{rmk} \\label{rmk:LMTABwalls}\nWith this theorem, we can also discuss how walls are related to the material in \\cite{LMT21}.\n\nWe first recall some terminology from \\cite{LMT21}. The \\textit{dual graph} to a veering triangulation $\\Delta$ is defined to be the 1-skeleton, i.e. the branch locus, of the unstable branched surface $B$, with the edges oriented by the coorientations on $\\Delta^{(2)}$. An \\textit{AB cycle} is then defined to be a loop carried by the dual graph which only makes \\textit{anti-branching turns}, i.e. does not follow components of the branch locus at each vertex of $B$. The lift of an AB cycle to the universal cover $\\widetilde{M}$ determines a properly embedded plane carried by $\\widetilde{B}$, by taking the union over all sectors lying below the lifted AB cycle. Such a plane is called a \\textit{dynamic plane}. The total number of lifted AB cycles in a dynamic plane is defined to be the \\textit{width} of the dynamic plane. If there are $2$ or more such lifted AB cycles, the region bounded inbetween them is called an \\textit{AB region}. For more details, we refer the reader to \\cite[Section 2.2, 3.1]{LMT21}.\n\nNow one can count that there are $\\lceil \\frac{w}{2} \\rceil$ or $w$ adjacent AB cycles within a wall of width $w$, depending on whether the wall is twisted or not. These lift to $w$ lifted AB cycles in a dynamic plane, which bound AB regions inbetween. Conversely, if in the universal cover $\\widetilde{B}$ carries a dynamic plane containing an AB region, then the portion of the flow graph within the AB region carries a line which only has incoming edges, hence quotients down to an infinitesimal cycle of a wall.\n\\end{rmk}\n\n\\subsection{Reduced flow graph}\n\nGiven \\Cref{thm:infinitesimal}, a way of arranging for strong connectedness is to simply  throw away all the infinitesimal cycles.\n\n\\begin{defn} \\label{defn:reducedflowgraph}\n\nThe \\textit{reduced flow graph} $\\Phi_{red}$ of a veering triangulation is the flow graph with all infinitesimal cycles in walls and the edges that enter the cycles removed.\n\n\\end{defn}\n\nThus by \\Cref{thm:infinitesimal}, $\\Phi_{red}$ is strongly connected. Because of the simple nature of infinitesimal cycles, $\\Phi_{red}$ inherits some of the properties of $\\Phi$. We end this section with two examples of this, which will be useful in \\Cref{sec:pAflow}.\n\n\\begin{lemma} \\label{lemma:flowgraphsorbits}\nFor every cycle $c$ of $\\Phi$, $c$ or $c^2$ is homotopic to a cycle of $\\Phi_{red}$ in $M$.\n\\end{lemma}\n\n\\begin{proof}\nThe cycles of $\\Phi$ are those of $\\Phi_{red}$ and the infinitesimal cycles of walls in $\\Phi$. Each infinitesimal cycle of a wall is homotopic (isotopic, even) to a boundary cycle in $M$, unless the wall has even width and is twisted, in which case one has to double the infinitesimal cycle $c_{\\frac{w}{2}}$ before it is homotopic to a boundary cycle in $M$.\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:redflowgraphcompl}\nThe components of $B \\backslash \\backslash \\Phi_{red}$ are annulus or Mobius bands with tongues.\n\nMoreover, the arcs of attachment of the triangular sectors criss-cross along the annulus\/Mobius band. More precisely, the arcs on the annulus\/Mobius band lift to $y=\\pm x + \\frac{2i}{w}, i \\in \\mathbb{Z}$ in the universal cover $[0,1] \\times \\mathbb{R}$, for some $w \\geq 1$. These arcs are subintervals of the branch locus of $B$ and are oriented downwards (i.e. decreasing $y$ in the above model). See \\Cref{fig:redflowgraphcompl}.\n\\end{lemma}\n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{12cm}{\\input{redflowgraphcompl.pdf_tex}}\n    \\caption{The complementary regions of $\\Phi_{red}$ in $B$ are annuli\/Mobius bands with tongues. The attaching arcs form a criss-cross pattern on the annulus\/Mobius band. Here we show the situation in the neighborhood of the width $4$ wall from \\Cref{fig:width4wall} (top) and the width $2$ wall from \\Cref{fig:width2wall} (bottom).}\n    \\label{fig:redflowgraphcompl}\n\\end{figure}\n\n\\begin{proof}\nBy \\Cref{lemma:flowgraphcompl}, it suffices to look at the components that contains infinitesimal cycles of some wall. For these, the statement of the lemma is clear from \\Cref{fig:redflowgraphcompl}, where we illustrate the case for $w=2$ and $w \\geq 3$ respectively. We remark that $w$ in the statement will in fact be the width of the corresponding wall. \n\\end{proof}\n\n\\begin{rmk} \\label{rmk:stablegluesides}\nWe remark that if $w \\geq 3$ for a component of $B \\backslash \\backslash \\Phi_{red}$ as in the lemma, the tongues attached along the $y=x+\\frac{2i}{N+1}$ arcs must all lie on one side and the tongues attached along the $y=-x+\\frac{2i}{N+1}$ arcs must all lie on the opposite side, in the universal cover. In contrast, the sides of attachment do not have fixed patterns for $w=1,2$.\n\\end{rmk}\n\n\\section{Finiteness of layered veering triangulations} \\label{sec:finiteness}\n\nIn this section, we fix the gap in the proof of \\cite[Theorem 6.2]{Ago11}. For completeness, we explain the setup of the original theorem then proceed to show a way of fixing its proof, albeit with a weaker bound.\n\nWe first have to explain layered veering triangulations.\n\nLet $\\phi:S_{g,n} \\to S_{g,n}$ be a pseudo-Anosov homeomorphism on a finite type surface with $\\chi(S_{g,n})=2-2g-n<0$. $S^\\circ$ will denote the surface obtained by removing the singularities of the stable and unstable foliations for $\\phi$, and $\\phi^\\circ$ will denote the restriction of $\\phi$ to $S^\\circ$. Write $T(\\phi^\\circ)$ for the mapping torus of $\\phi^\\circ$.\n\nIn \\cite{Ago11}, it is shown that there exists a periodic folding sequence of train tracks $\\tau_0 \\rightsquigarrow ...  \\rightsquigarrow \\tau_N$, i.e. train tracks such that $\\tau_{i+1}$ is obtained from $\\tau_i$ using a folding move and $\\phi(\\tau_N)=\\tau_0$, on $S^\\circ$. The sequence of ideal triangulations $\\delta_i$ of $S^\\circ$ dual to $\\tau_i$ are then related to one another by diagonal switches in quadrilaterals, and $\\phi^\\circ$ sends $\\delta_N$ to $\\delta_0$. \n\nA veering triangulation can be constructed in the following way: Start with $\\delta_0$ on $S^\\circ$ and attach a veering tetrahedron to the bottom that effects the diagonal switch from $\\delta_0$ to $\\delta_1$. The bottom boundary of the complex can be identified with $(S^\\circ,\\delta_1)$. We inductively add veering tetrahedra to the bottom, until the bottom boundary of the complex can be identified with $(S^\\circ,\\delta_N)$. Finally, we glue this bottom boundary to the top boundary to get a veering triangulation $\\Delta$ of $T(\\phi^\\circ)$, using the fact that $\\phi^\\circ$ sends $\\delta_N$ to $\\delta_0$. \n\nWe construct a directed graph according to this description. Start with a set of $e$ vertices given by the set of edges in $\\delta_0$. Notice that $\\delta_1$ and $\\delta_0$ differ by one edge exactly, say $\\delta_1 \\backslash e_1=\\delta_0 \\backslash e_0$. Add a vertex corresponding to $e_1$, and add three edges going from the three elements of $E(\\delta_0)$ dual to the three branches of $\\tau_0$ which fold onto the branch of $\\tau_1$ dual to $e_1$. One of these edges goes from $e_0$ to $e_1$, call this edge \\textit{vertical}, and call the other two edges \\textit{slanted}. Inductively, for each $i$, add a vertex that corresponds to the new edge in $\\delta_i$ and add three edges according to the folding move $\\tau_i \\rightsquigarrow \\tau_{i+1}$, one vertical and two slanted. Call the resulting directed graph the \\textit{cut open flow graph} $\\Phi \\backslash \\backslash S$. See \\Cref{fig:cutopenflowgraph}.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{6cm}{\\input{cutopenflowgraph.pdf_tex}}\n    \\caption{The cut open flow graph for a layered veering triangulation. There is a natural identification between layers $E(\\delta_i)$. The cut open reduced flow graph for the veering cellulation can be obtained by deleting some columns.}\n    \\label{fig:cutopenflowgraph}\n\\end{figure}\n\nThe flow graph of $\\Delta$ can be obtained from $\\Phi \\backslash \\backslash S$ by identifying the vertices at the bottom corresponding to $E(\\delta_N)$ to those at the top for $E(\\delta_0)$ according to how $\\phi^\\circ$ sends $\\delta_N$ to $\\delta_0$. \n\nNow define another directed graph $G$ by setting the set of vertices to be the set of edges in $\\delta_0$, and placing an edge from $i$ to $j$ for every directed edge path in the cut open flow graph $\\Phi \\backslash \\backslash S$ which starts at $i$ and ends at $(\\phi^\\circ)^{-1}(j)$. Let $A$ be the adjacency matrix of $G$. \n\nWe note that $A$ can also be defined as the transition matrix describing how weights on the branches of $\\tau_0$ distribute under the sequence of folding moves $\\tau_0 \\rightsquigarrow ... \\rightsquigarrow \\tau_N$ and the return map $\\phi^\\circ$.\n\nMeanwhile, let $\\lambda=\\lambda(\\phi)>1$ be the dilatation of $\\phi$. The train tracks $\\tau_i$ in fact carry the unstable measured lamination of $\\phi$, hence the transverse measure on the leaves of the foliation collapse down to weights on the branches of $\\tau_0$. These in turn define an eigenvector $(w_i)$ of $A$ with eigenvalue $\\lambda$. \n\nThe assertion of \\cite[Theorem 6.2]{Ago11} is that if $\\lambda^{2g-2+\\frac{2}{3}n} \\leq P$, then the veering triangulation $\\Delta$ has at most $\\frac{P^9-1}{2}$ tetrahedra.\n\nWe recall the proof presented in \\cite{Ago11}, but phrased in the language here. We first claim that $G$ has at least $2N+e$ edges. This is because for every slanted edge in $\\Phi \\backslash \\backslash S$, we can take an edge path from $E(\\delta_0)$ to $E(\\delta_N)$ by concatenating paths of vertical edges to the back and front of the slanted edge. Also, for every vertex $i$ in $E(\\delta_0)$, there is a edge path from $i$ to $E(\\delta_N)$ that consists entirely of vertical edges. Since there are $2N$ slanted edges and $e$ vertices in $E(\\delta_0)$, we find $2N+e$ edges in $G$.\n\nAlso notice that by an index calculation, we have $e \\leq 9(2g-2+\\frac{2}{3}n)$. See \\cite[Lemma 6.1]{Ago11} for details of this.\n\nNow assume for the moment that $G$ is strongly connected. Then \\cite{Ago11} proceeds by using an estimate of Ham and Song (\\cite{HS07}). We repeat Ham and Song's argument here since it is a good warm-up for the proof which we will present later. Fix a vertex $i$ of $G$ and fix a spanning tree $T$ of $G$ rooted at $i$, i.e. $T$ is a subgraph of $G$ which is a tree, and for every vertex $j$ of $G$ there is a edge path from $j$ to $i$ within $T$. Then there are at least $2N+1$ edges in $G \\backslash T$, and for every edge $d$ in $G \\backslash T$, we can find a path in $G$ ending at $i$ and of length $e$ by appending to the path travelling across $d$ then to $i$ within $T$ an arbitrary path at the front. These paths will be distinct, hence using the fact that $(w_i)$ is an eigenvector of $A^e$ of eigenvalue $\\lambda^e$, and summing over the paths, we have $\\lambda^e w_i \\geq \\Sigma w_j \\geq (2N+1) \\min_j w_j$. Taking the minimum over the left hand side, $\\lambda^e \\min w_i \\geq (2N+1) \\min w_j$, hence $N \\leq \\frac{\\lambda^e-1}{2} \\leq \\frac{P^9-1}{2}$.\n\nThe problem with this argument however, is that $G$ is not always strongly connected.\n\nIn fact, $G$ is strongly connected if and only if the flow graph $\\Phi$ is strongly connected. For if $\\Phi$ is strongly connected, then for every pair of vertices $(i,j)$ in $E(\\delta_0)$, their images can be connected by an edge path $\\alpha$ in $\\Phi$. The preimage of such an edge path under $\\Phi \\backslash \\backslash S \\to \\Phi$ is a collection of paths $\\alpha_1,...,\\alpha_s$, for which the ending point of $\\alpha_i$, which lies in $E(\\delta_N)$, is sent by $\\phi$ to the starting point of $\\alpha_{i+1}$, which lies in $E(\\delta_0)$. Hence the $\\alpha_i$ define an edge path $\\alpha'$ in $G$ connecting $i$ to $j$.\n\nConversely, if $\\Phi$ is not strongly connected, then we can find a infinitesimal cycle $c$ of a wall. As above, $c$ lifts to a collection of paths $c_1,...,c_s$ in $\\Phi \\backslash \\backslash S$ and determines a cycle $c'$ in $G$. In fact, $c_i$ will consist entirely of vertical edges by the definition of the infinitesimal cycles of a wall. The vertices of $G$ that lie in $c'$ are exactly those elements of $E(\\delta_0)$ that have image in $c$. Moreover, $c'$ has no outgoing edges in $G'$, for otherwise there is an outgoing path of $c$ in $\\Phi$.\n\nThis motivates us to consider instead the full subgraph of $G$ obtained by restricting to the set of vertices that have image in the reduced flow graph $\\Phi_{red}$ in $\\Phi$. We call this subgraph $G_{red}$. By the argument above, $G_{red}$ can also be obtained by removing $c'$ for all infinitesimal cycles $c$ of $\\Phi$. \n\nWe also set the \\textit{cut open reduced flow graph}, $\\Phi_{red} \\backslash \\backslash S$, to be the preimage of $\\Phi_{red} \\subset \\Phi$ in $\\Phi \\backslash \\backslash S$. Let $E_{red}(\\delta_i)$ be the subset of $E(\\delta_i) \\subset \\Phi \\backslash \\backslash S$ that lie in $\\Phi_{red} \\backslash \\backslash S$. $G_{red}$ can be obtained by gluing $E_{red}(\\delta_N)$ at the bottom of $\\Phi_{red} \\backslash \\backslash S$ to $E_{red}(\\delta_0)$ at the top. Hence arguing as above, we can see that $G_{red}$ is strongly connected.\n\nThe strategy now is to apply Ham and Song's argument on $G_{red}$ to bound the number of vertices in $\\Phi_{red}$, then bound the number of vertices in $\\Phi \\backslash \\Phi_{red}$. \n\nWe first set up some notation. Let $N'$ be the number of vertices in $\\Phi_{red}$. Let $e'$ be the number of elements in $E_{red}(\\delta_0)$.\n\nWe claim that $G_{red}$ has at least $2N'+e'$ edges. This follows from the same argument as for $G$: we can find an edge path in $\\Phi_{red} \\backslash \\backslash S$ from $E_{red}(\\delta_0)$ to $E_{red}(\\delta_N)$ for every slanted edge in $\\Phi_{red} \\backslash \\backslash S$ and for every vertex in $E_{red}(\\delta_0)$.\n\nMeanwhile, recall that $G \\backslash G_{red}$ is the union of $c'$ for the infinitesimal cycles $c$ of $\\Phi$, where each $c'$ has no outgoing edges. Hence the adjacency matrix $A_{red}$ of $G_{red}$ can be expressed as a block of $A$ with zeros to the left and right. As a consequence, $(w_i)_{i \\in G_{red}}$ is an eigenvector of $A_{red}$ with eigenvalue $\\lambda$. \n\nHence repeating Ham and Song's argument, we have $N' \\leq \\frac{\\lambda^{e'}-1}{2}$.\n\nNow suppose we have a width $w$ wall. For simplicity, first suppose that the wall is untwisted. Then there are $w-1$ infinitesimal cycles and $2$ boundary cycles of the wall, all of the same length $h$. The $(w-1)h$ vertices in the infinitesimal cycles are discarded in $\\Phi_{red}$, but the $2h$ vertices in the boundary cycles remain. Similarly, if the wall is twisted, there are $(w-1)h$ vertices discarded while $2h$ vertices remain, for an appropriate $h$. Meanwhile, as discussed in \\Cref{subsec:wall}, each vertex of $\\Phi_{red}$ can appear at most twice in the collection of boundary cycles. Hence we conclude that $N \\leq N'W$ where $W$ is the maximum width of a wall in the veering triangulation $\\Delta$. Thus it remains to bound $W$.\n\nTo that end, let $c_2,...,c_W$ be the infinitesimal cycles in a width $W$ wall, and let $c_1,c_{W+1}$ be the boundary cycles of the wall. Again, for simplicity, first suppose that the wall is untwisted. Then there are corresponding disjoint cycles $c'_1,...,c'_{W+1}$ in $G$. The length of each cycle $c'_i$ is given by the number of paths in the lift of $c_i$ in the cut open flow graph. But this number also equals to the intersection number of $c_i$ with $S^\\circ$ in $T(\\phi^\\circ)$. Since the $c_i$ are parallel to each other, this number is of the same value $L$ for each $i=1,...,W+1$ and hence at most $\\frac{e}{W+1}$ by the pigeonhole principle. Similarly, if the wall is twisted, the infinitesimal and boundary cycles determine cycles in $G$ of lengths at most $\\frac{2e}{W+1}$.\n\nNow pick a vertex of $\\Phi_{red}$ that lies in $c_1$. One of its preimages in $\\Phi_{red} \\backslash \\backslash S$ has three incoming edges. $c_1$ passes through one of these edges, and for each of the remaining two, we can construct a path from $E_{red}(\\delta_0)$ to $E_{red}(\\delta_N)$ by concatenating vertical edges to the back and front of the edge. These two paths determines two edges that enter $c'_1$ at a vertex $i$ in $G_{red}$. Since $G_{red}$ is strongly connected, we can locate two edge paths in $G_{red}$ of length $L$ that end at the two edges respectively. Suppose these two paths have start at vertices $j$ and $k$. Then using the fact that $(w_i)$ is an eigenvector of $A^L_{red}$ with eigenvalue $\\lambda^L$, we have $\\lambda^L w_i \\geq w_i + w_j + w_k$, which implies \n\n$$w_i \\geq \\frac{2}{\\lambda^L-1} \\min_j w_j$$\n\nBut for every two vertices $i,j$ in $G_{red}$, by strong connectedness, there is a edge path of length $\\leq e'$ from $i$ to $j$, hence $w_i \\geq \\lambda^{-e'} w_j$. So $\\frac{max_j w_j}{max_i w_i} \\leq \\lambda^{e'}$. Applying this to the above inequality.\n\n$$ \\lambda^{e'} \\geq \\frac{2}{\\lambda^L-1} $$\n\n$$ \\lambda^L \\geq 2 \\lambda^{-e'}+1 $$\n\nHence,\n\n$$ \\frac{2e}{W+1} \\geq L \\geq \\frac{\\log(2 \\lambda^{-e'}+1)}{\\log \\lambda} $$\n\n$$ W \\leq \\frac{2 \\log \\lambda^e}{\\log(2 \\lambda^{-e'}+1)} -1 $$\n\nPutting everything together, \n\n\\begin{align*}\nN \\leq& \\frac{\\lambda^{e'}-1}{2} (\\frac{2 \\log \\lambda^e}{\\log(2 \\lambda^{-e'}+1)}-1)\\\\\n\\leq& \\frac{\\lambda^e-1}{2} (\\frac{2 \\log \\lambda^e}{\\log(2 \\lambda^{-e}+1)}-1)\\\\\n\\leq& \\frac{P^9-1}{2} (\\frac{2 \\log P^9}{\\log(2 P^{-9}+1)}-1)\n\\end{align*}\n\nWe record this as a theorem.\n\n\\begin{thm} \\label{thm:boundveertet}\nIf $M$ is the punctured mapping torus of a pseudo-Anosov homeomorphism $\\phi: S_{g,n} \\to S_{g,n}$, where the normalized dilatation $\\lambda(\\phi)^{2g-2+\\frac{2}{3}n} \\leq P$, then $M$ has a veering triangulation with at most $\\frac{P^9-1}{2} (\\frac{2 \\log P^9}{\\log(2 P^{-9}+1)}-1)$ tetrahedra.\n\\end{thm}\n\n\\begin{rmk} \\label{rmk:boundcompare}\n$\\frac{P^9-1}{2} (\\frac{2 \\log P^9}{\\log(2 P^{-9}+1)}-1)$ is asymptotically $\\frac{9}{2} P^{18}\\log P$ as $P \\to \\infty$, so we have worsened the exponent on the bound in \\cite[Theorem 6.2]{Ago11} by a factor of $2+\\epsilon$. \n\nWe remark that the original bound of $\\frac{P^9-1}{2}$ still holds for veering triangulations that have strongly connected flow graph. However, it is not clear if this is the case for `most' $\\phi$ or if there is a way to tell if this is the case just from $\\phi$.\n\nIt also seems likely that there is room for improvement for our bound in the general case. For example, it should be possible to obtain better bounds on $W$ by finding more paths that enter $c'_1$, which should be easy when $L$ is large. When $W$ or $L$ is large, it should also be possible to bound $e'$ more effectively than just using $e$, which is what we have done here. See \\Cref{sec:questions} for another discussion on how one might improve the bound.\n\\end{rmk}\n\n\\section{Pseudo-Anosov flows} \\label{sec:pAflow}\n\nIn this section we will reprove Schleimer and Segerman's result that a veering triangulation induces a pseudo-Anosov flow without perfect fits on suitable Dehn fillings. We use the following notations to simplify the statement. If $s=(s_i)$ is a collection of slopes on each boundary component of $\\overline{M}$, we write $M(s)$ to mean the closed 3-manifold obtained by Dehn filling $\\overline{M}$ along the slopes recorded by $s$. If $a=(a_i)$ and $b=(b_i)$ are collections of slopes on each boundary component of $\\overline{M}$, then by $|\\langle a,b \\rangle| \\geq n$ we mean that the geometric intersection numbers between $a_i$ and $b_i$ on each boundary component is at least $n$.\n\n\\begin{thm} \\label{thm:vtpAflow}\nSuppose $M$ admits a veering triangulation. Let $l=(l_i)$ denote the collection of ladderpole curves on each boundary component. Then $M(s)$ admits a transitive pseudo-Anosov flow $\\phi$ if $|\\langle s,l \\rangle| \\geq 2$. \n\nFurthermore, there are closed orbits $c_i$ isotopic to the cores of the filling solid tori, such that each $c_i$ is $|\\langle s_i,l_i \\rangle|$-pronged, and $\\phi$ is without perfect fits relative to $\\{c_i\\}$.\n\\end{thm}\n\nWe will recall the definitions of pseudo-Anosov flows, transitivity, and no perfect fits in \\Cref{subsec:pAproof}. \n\nA subtle point of the theorem is that there are actually two common notions of pseudo-Anosov flows on 3-manifolds in the literature, which we differentiate by calling them topological pseudo-Anosov flows and smooth pseudo-Anosov flows. \\Cref{thm:vtpAflow} holds for both notions, due to \\Cref{thm:top2smooth}. We will explain this technicality more in \\Cref{subsec:pAproof,subsec:smooth}.\n\nThe proof can be outlined as follows. We first thicken up the reduced flow graph $\\Phi_{red}$ in $M$ to $N(\\Phi_{red})$ by replacing its edges with flow boxes. This set can be considered as a subset of a neighborhood of the unstable branched surface, $N(B)$, naturally. By understanding the complement of $\\Phi_{red}$ in $B$, we are able to glue faces of $N(\\Phi_{red})$ across its complementary regions in $N(B)$. Similarly, by understanding the complement of $B$ in $M$, we are able to glue faces of $N(B)$ across its complementary regions in $M(s)$. These gluings preserve the (singular) 1-dimensional foliation on the flow boxes, hence that descends down to a (honest) 1-dimensional foliation on $M(s)$, which can be parametrized into a topological flow, for which we show is pseudo-Anosov, transitive, and without perfect fits (relative to the orbits $\\{c_i\\}$).\n\n\\begin{rmk}\nIt is not too difficult to show that a veering triangulation induces a (topological) pseudo-Anosov flow on $M(s)$ for $|\\langle s,l \\rangle| \\geq 2$, using the tool of dynamic pairs developed by Mosher in \\cite{Mos96}. Specifically, one can apply the proof of \\cite[Proposition 2.6.2]{Mos96} to $(B, \\Phi)$ to produce a dynamic pair in $M(s)$, which by \\cite[Theorem 3.4.1]{Mos96} gives rise to a pseudo-Anosov flow. The more challenging part however, at least from this approach, is to show that such a pseudo-Anosov flow is transitive and has no perfect fits. \n\nIn our proof, we use a lot of the same ideas as \\cite{Mos96}, but most notably we skip over constructing the `stable branched surface' in a dynamic pair, and instead construct a pseudo-Anosov flow directly from the `unstable branched surface' $B$ and the `dynamic train track' $\\Phi_{red}$, to use the terminology from \\cite{Mos96}. This allows us to analyze the pseudo-Anosov flow using the special properties of $\\Phi_{red}$ and $B$, proving transitivity and no perfect fits.\n\\end{rmk}\n\n\\subsection{Thickening up $\\Phi_{red}$}\n\nWe know that $\\Phi_{red}$ is strongly connected by construction. This is equivalent to its adjacency matrix $A \\in Hom(\\mathbb{R}^{V(\\Phi_{red})}, \\mathbb{R}^{V(\\Phi_{red})})$ being irreducible. As such, by the Perron-Frobenius Theorem, $A$ has a positive eigenvector $(w_v)$ with eigenvalue $\\lambda \\geq 1$. We know that $\\lambda>1$ since each vertex of $\\Phi_{red}$ has three incoming edges. Meanwhile, $A$ being irreducible implies $A^T$ is irreducible as well, and so the latter has a positive eigenvector $(w'_v)$ with the same eigenvalue $\\lambda >1$.\n\nWe remark in passing that each vertex of $\\Phi_{red}$ having $3$ incoming edges in fact implies that $\\lambda=3$ and $w_v=1$ for all $v$. This knowledge, however, will play no role in the construction at all. We simply wish to point out that the value of $\\lambda$ has nothing to do with the dilatation factor of the monodromy when $\\Delta$ is layered.\n\nRecall that $\\Phi_{red}$ naturally sits on $B$ inside $M$. We will thicken up $\\Phi_{red}$ by replacing each edge $e$ of $\\Phi_{red}$ going from $v$ to $w$ by a flow box $$Z_e \\cong \\{(s,u,t) \\in \\mathbb{R}^3: |s|\\leq w_v \\lambda^{t-1}, |u| \\leq w'_w \\lambda^{-t}, t \\in [0,1] \\}$$ See \\Cref{fig:flowbox}. There are two 2-dimensional foliations on $Z_e$: the first one by leaves of the form $\\{u=u_0 \\lambda^{-t} \\}_{u_0}$, which we will call the stable foliation, and the second one by leaves of the form $\\{s=s_0 \\lambda^{t-1}\\}_{s_0}$, which we will call the unstable foliation. There is also an oriented 1-dimensional foliation on $Z_e$ by curves $\\{(s_0\\lambda^{t-1}, u_0\\lambda^{-t},t): t \\in [0,1]\\}_{s_0,u_0}$, oriented by decreasing $t$. Notice that the leaves of the stable and unstable foliations intersect transversely along leaves of the oriented 1-dimensional foliation.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{16pt}{16pt}\\selectfont\n    \\resizebox{!}{4cm}{\\input{flowbox.pdf_tex}}\n    \\caption{A flow box.}\n    \\label{fig:flowbox}\n\\end{figure}\n\nGive $Z_e$ its Euclidean metric induced from $\\mathbb{R}^3$. There is a natural way of gluing up the $Z_e$ across their top and bottom faces at the vertices of $\\Phi_{red}$, preserving the metric on those faces, by definition of $w_v$ and $w'_v$. A priori there is a freedom for $Z_e$ to twist along $e$, we get rid of this by requiring that the framing on $Z_e$ induced from its $u$ coordinate match up with the framing on $e$ induced from $B$. See \\Cref{fig:thicken}. We call the resulting set $N(\\Phi_{red})$, since it is a regular neighborhood of $\\Phi_{red}$ in $M$.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{14pt}{14pt}\\selectfont\n    \\resizebox{!}{8cm}{\\input{thicken.pdf_tex}}\n    \\caption{Thickening up $\\Phi_{red}$ to $N(\\Phi_{red})$, which naturally embeds in $N(B)$.}\n    \\label{fig:thicken}\n\\end{figure}\n\nNote that the oriented 1-dimensional foliations on $Z_e$ piece together to give a decomposition of $N(\\Phi_{red})$. This decomposition is almost an oriented 1-dimensional foliation, except the `leaves' are oriented 1-manifolds possibly with train track singularities branching off in forward and backward directions, arising from the multiple outgoing and incoming edges at the vertices. Despite this, for convenience we will still refer to this decomposition as an oriented 1-dimensional foliation. \n\nSimilarly, the stable and unstable foliations on $Z_e$ each piece together to give a decomposition of $N(\\Phi_{red})$ which is almost a 2-dimensional foliation, except that the `leaves' have branching. We will refer to these as the stable and unstable foliations on $N(\\Phi_{red})$ respectively. \n\nDenote $\\partial^s Z_e=\\{u= \\pm w'_w \\lambda^{-t}\\}$ and $\\partial^u Z_e=\\{s= \\pm w_v \\lambda^{t-1}\\}$. We will call $\\partial^s N(\\Phi_{red}) := \\bigcup_e \\partial^s Z_e$ the \\textit{stable boundary} of $N(\\Phi_{red})$ and $\\partial^u N(\\Phi_{red}) := \\bigcup_e \\partial^u Z_e$ the \\textit{unstable boundary} of $N(\\Phi_{red})$. We will also call the collection of the closures of intervals where the interior of the top and bottom faces of $Z_e$ meets $\\partial^s N(\\Phi_{red})$ or $\\partial^u N(\\Phi_{red})$ the branch locus. These are exactly the places where the leaves of the oriented 1-dimensional foliation on $N(\\Phi_{red})$ have branching.\n\nNow consider an $I$-fibered neighborhood $N(B)$ of $B$ in the cusped model. This means that $N(B)$ is a closed regular neighborhood of $B$, with a map $M \\to M$ restricting to a projection on $N(B) \\to B$ with $I$-fibers and a homeomorphism outside of $N(B)$, and so that the boundary of $N(B)$ can be decomposed into surfaces with boundary, the interior on which the projection is a local homeomorphism, and the boundary curves correspond to the components of the branch locus of $B$. We will often conflate a component of the branch locus of $B$ with the corresponding circle on $\\partial N(B)$. We will also call the collection of circles in the latter setting the branch locus of $N(B)$. \n\n$N(\\Phi_{red})$ can be arranged to be a subset of $N(B)$, in such a way that $\\partial^u N(\\Phi_{red}) \\subset \\partial N(B)$, the branch locus of $N(\\Phi_{red})$ is a subset of the branch locus of $N(B)$, and $N(\\Phi_{red})$ is saturated with respect to the $I$-fibering of $N(B)$.\n\nNote that we do not require the intervals $[-w_v \\lambda^{t_0-1}, w_v \\lambda^{t_0-1}] \\times \\{ u_0 \\} \\times \\{ t_0 \\} \\subset Z_e \\subset N(\\Phi_{red})$ to coincide with the $I$-fibers of $N(B)$. Indeed, this is impossible since the branch locus of $N(\\Phi_{red})$ are parallel along such intervals, while branch locus of $N(B)$ project to transverse curves on $B$. One can arrange for this property by `splitting' $N(\\Phi_{red})$ slightly, but since this does not aid our construction, we will not do so. \n\n\\subsection{Gluing along the stable boundary} \\label{subsec:stableglue}\n\nThe next step is to glue $N(\\Phi_{red})$ along its stable boundary across its complementary regions in $N(B)$, so that the foliations on $N(\\Phi_{red})$ descend to respective foliations on $N(B)$. To perform the gluing, we have to understand the complementary regions in question. \n\nBy \\Cref{lemma:redflowgraphcompl}, the complementary regions of $\\Phi_{red}$ in $B$ are annuli or Mobius bands with tongues. The complementary regions of $N(\\Phi_{red})$ in $N(B)$ are $I$-fibered neighborhoods of these, see \\Cref{fig:stableglue1}. Fix one of these components $K$. The boundary of $K$ is naturally divided into $\\partial^s K \\cup \\partial^u K$, where $\\partial^s K$ is identified to $\\partial^s N(\\Phi_{red})$ in $M$ and $\\partial^u K \\subset \\partial N(B)$. In particular $\\partial^s K$ inherits the oriented 1-dimensional foliation on $\\partial^s N(\\Phi_{red})$. Furthermore, if we call the $I$-fibers over the tips of the tongues of $B \\backslash \\backslash \\Phi_{red}$ the \\textit{cusps} of $K$, $\\partial^s K$ can be divided along the cusps into two or one components (depending on whether the corresponding component of $B \\backslash \\backslash \\Phi_{red}$ is an annulus or a Mobius band with tongues respectively). Call these components the \\textit{stable faces} of $K$. \n\nMeanwhile, recall that $N(\\Phi_{red})$, hence $\\partial^s N(\\Phi_{red})$, inherits the Euclidean metric of the flow box $Z_e$. The oriented 1-dimensional foliation contracts the metric in the transverse direction. Hence the oriented foliation on each stable face of $K$ has exactly one $S^1$ leaf, and all other leaves enter through the cusps of $K$ and spiral into the $S^1$ leaf. \n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{8cm}{\\input{stableglue1.pdf_tex}}\n    \\caption{Complementary regions of $N(\\Phi_{red})$ in $N(B)$. We construct an $I$-fibering conjugating the flows on the stable face, and collapse along the $I$-fibers.}\n    \\label{fig:stableglue1}\n\\end{figure}\n\nMore rigorously, parametrize the oriented 1-dimensional foliation on $\\partial^s N(\\Phi_{red})$ in some way, for example according to the $t$ coordinate of each $Z_e$, so that we can consider it as a forward semiflow. Take the intervals $[-w_v, w_v] \\times \\{ \\pm w'_w \\lambda^{-1} \\} \\times \\{ 1 \\} \\subset Z_e \\subset N(\\Phi_{red})$, with the Euclidean metrics, as sections of the flow, and note that the first return map contracts the metric by $\\lambda^{-1}$, then apply Banach fixed point theorem. \n\nWe claim that there is an $I$-fibering of $K$, transverse to the stable faces and parallel to $\\partial^u K$, for which the induced homeomorphism between the stable faces preserves the oriented 1-dimensional foliations on them, i.e. send leaves to leaves in an orientation preserving way. \n\nSuppose first that the corresponding component of $B \\backslash \\backslash \\Phi_{red}$ is an annulus with tongues. Choose short local sections to the foliation near the unique $S^1$ leaf on each stable face of $K$, call them $I, I'$. The return maps of the oriented foliations on $I, I'$, which we call $h, h'$, are \\textit{contracting} by our analysis above, i.e. $h^k(I)$ is a strictly decreasing collection of subintervals with $\\bigcap_{k=1}^\\infty h^k(I)$ being a point, and similarly for $h'^k(I')$. Note that these points could lie on endpoints of $I,I'$ if one side of the annulus has no tongues attached.\n\nStart by defining the $I$-fibering on the cusps of $K$, where the $I$-fibering is uniquely determined but degenerate, in the sense that the $I$-fibers are just points. Then extend the $I$-fibering to the triangular sectors of $\\partial^u K$, so that the base of the triangles, which are among the branch locus of $N(B)$, are $I$-fibers.\n\nNow extend the $I$-fibering across subintervals of the leaves of the foliation on the stable faces which start on the branch locus and end at the interior of $I$ or $I'$. Then extend the $I$-fibering across subintervals of the leaves which start on the interior of the cusps of $K$ and end at the interior of $I$ or $I'$, using the fact that the union of these subintervals is a finite union of rectangles foliated as products.\n\nFinally, apply the lemma below by taking $f$ to be the homeomorphism induced by portion of the $I$-fibering already defined. Then use the extended $f$ given by the lemma to construct the $I$-fibering between $I$ and $I'$, and complete the construction of the $I$-fibering by extending along the leaves as they go around the stable faces. See \\Cref{fig:stableglue2} for a graphical summary of these steps.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{16pt}{16pt}\\selectfont\n    \\resizebox{!}{6cm}{\\input{stableglue2.pdf_tex}}\n    \\caption{The steps for constructing the $I$-fibering on each component of $N(B) \\backslash \\backslash N(\\Phi_{red})$.}\n    \\label{fig:stableglue2}\n\\end{figure}\n\n\\begin{lemma} \\label{lemma:expandextendbb}\nLet $h:I \\to I, h': I' \\to I'$ be injective contracting maps on intervals. Let $f:I \\backslash h(I) \\to I' \\backslash h'(I')$ be a given homeomorphism. Then there is a unique way of extending $f$ to a homeomorphism $I \\to I'$ so that $h' f=f h$.\n\\end{lemma}\n\\begin{proof}\nWrite $I^{(k)}=h^k(I), I'^{(k)}=h'^k(I')$. By assumption, $I^{(k)}$ is a shrinking subinterval of $I$ which contracts to a point $s$ and $I'^{(k)}$ is a shrinking subinterval of $I'$ which contracts to a point $s'$. Extend $f$ by defining it to be $h'^k f (h^k)^{-1}$ on $I^{(k)} \\backslash I^{(k+1)}$. Finally define $f(s)=s'$.\n\\end{proof}\n\nIf the corresponding component of $B \\backslash \\backslash \\Phi_{red}$ is a Mobius band with tongues instead, we choose two local sections $J_1, J_2$ on the one stable face near the $S^1$ leaf, so that we have return maps $J_1 \\to J_2, J_2 \\to J_1$ for the oriented foliation. (We can pick $J_1$ first, then pick $J_2$ near $J_1$ using the fact that the return map $J_1 \\to J_1$ is contracting.) Let $s_1, s_2$ be where the $S^1$ leaf meets $J_1, J_2$. $s_1, s_2$ have to be in the interior of $J_1, J_2$ in this setting. Let $I_i$ be the subinterval of $J_i$ to the left of $s_i$, $I'_i$ be the subinterval of $J_{i+1}$ to the right of $s_{i+1}$ (indices taken mod $2$, and left\/right measured relative to the direction of the flow and a fixed orientation on the stable face). First construct the $I$-fibering on subintervals of leaves that start on cusps of $K$ and end at the interior of $J_1$ or $J_2$, then apply the following generalized lemma (with $m=2$) to construct the $I$-fibering across a neighborhood of the periodic trajectory as above.\n\n\\begin{lemma} \\label{lemma:expandextend}\nLet $h_i:I_i \\to I_{i+1}, h'_i:I'_i\\to I'_{i+1}$ be injective contracting maps on intervals $I_1,...,I_m,I'_1,...,I'_m$ (indices taken mod $m$). Here, contracting means \\{$h_{mk+i-1} \\cdots h_i (I_i)\\}_k$ is a decreasing collection of subintervals and $\\bigcap_{k=1}^\\infty h_{mk+i-1} \\cdots h_i (I_i)$ is a single point for each $i$, and similarly for $h'_i$. Let $f_i:I_i \\backslash h_{i-1}(I_{i-1}) \\to I'_i \\backslash h'_{i-1}(I'_{i-1})$ be given homeomorphisms. Then there is a unique way of extending $f_i$ to homeomorphisms $I_i \\to I'_i$ so that $h'_i f_i=f_{i+1} h_i$ for all $i$.\n\\end{lemma}\n\n\\begin{proof}\nThis can be proved as in the previous lemma (which is the `baby case' when $m=1$), with more annoying bookkeeping. We remark that we stated this lemma for all $m \\geq 1$ because we will need this generality in the next subsection.\n\\end{proof}\n\n\\begin{rmk} \\label{rmk:kopell}\nThe homeomorphism across the stable faces induced by such an $I$-fibering cannot be made to preserve the Euclidean structure on $\\partial^s N(\\Phi_{red})$ in general. In particular, the homeomorphism will not, in general, take an interval of the form $[-w_v \\lambda^{t_0-1}, w_v \\lambda^{t_0-1}] \\times \\{ \\pm w'_w \\lambda^{-t_0} \\} \\times \\{ t_0 \\} \\subset Z_e \\subset N(\\Phi_{red})$ to another interval of this form.\n\nAlso, the homeomorphism cannot be chosen to be a diffeomorphism in general (with respect to the smooth structures induced by the Euclidean structures). This arises from the fact that one cannot replace `homeomorphisms' by `diffeomorphisms' in \\Cref{lemma:expandextend} in general by Kopell's Lemma (\\cite{Kop70}). \n\\end{rmk}\n\nNow collapse across all such components $K$ along the $I$-fiberings. This collapses $N(\\Phi_{red})$ onto $N(B)$. The oriented 1-dimensional foliation on $N(\\Phi_{red})$ descends to a decomposition of $N(B)$. The `leaves' of which are still not all 1-manifolds, but now we only have branching in the backwards direction, since all the forward branching occurs along some cusp of some $K$, and we have collapsed all forward trajectories starting from those points. As before, we will still call this decomposition an oriented 1-dimensional foliation for convenience of notation.\n\nSimilarly, the stable and unstable foliations on $N(\\Phi_{red})$ descend to decompositions of $N(B)$, which we call the stable and unstable foliations on $N(B)$ respectively. The leaves of the stable foliation on $N(B)$ are in fact 2-manifolds by the same argument as the last paragraph, but those of the unstable foliation on $N(B)$ have branching. \n\nWhen restricted to the unstable boundary, the collapse takes $\\partial^u N(\\Phi_{red})$ onto $\\partial N(B)$. In particular, we can recover the oriented 1-dimensional foliation restricted to $\\partial N(B)$ in the following way: take the image of $\\Phi_{red}$ on the boundary of $M \\backslash \\backslash B$, thicken it up within $\\partial (M \\backslash \\backslash B)$ by replacing each edge by a flow box $\\{ (u,t): |u| \\leq w'_w \\lambda^{-t}, t \\in [0,1]\\}$ foliated by $\\{(u_0 \\lambda^{-t}, t): t \\in [0,1] \\}_{u_0}$ and piecing together their top and bottom edges at the vertices of $\\Phi_{red}$. The $I$-fiberings on the components $K$ constructed above, when restricted to $\\partial^u K$, will determine $I$-fiberings of the complementary regions, and we collapse along these $I$-fibers to get the oriented foliation on $\\partial N(B)$.\n\nBefore we move on, we will construct some local sections to the foliation which will be used to prove no perfect fits in \\Cref{subsec:pAproof}. Fix a component $c$ of the branch locus of $B$. Consider the subset of double points of the branch locus of $B$ that lie on $c$. These are each dual to some veering tetrahedron in $\\Delta$. Let $V_c$ be the subset of vertices of $\\Phi_{red}$ which are the bottom edges of one of these veering tetrahedra. Cyclically order $V_c=\\{v_1,...,v_s\\}$ according to the order in which $c$ meets the corresponding double points. \n\nMeanwhile, for a vertex $v$ of $\\Phi_{red}$, let $R_v$ be the union of the top faces of $Z_e$ as $e$ varies over the edges of $\\Phi_{red}$ that exit $v$. Equivalently, this is also the union of the bottom faces of $Z_e$ as $e$ varies over the edges of $\\Phi_{red}$ that enter $v$. These are embedded rectangles transverse to the foliation on $N(\\Phi_{red})$.\n\nFor a collection $V_c=\\{v_1,..,v_s\\}$ as above, each $R_{v_i} \\subset N(\\Phi_{red})$ contains an subinterval of $c$. Here we remind the reader that we are using the same name for a component of the branch locus of $B$ and the corresponding component of the branch locus of $N(B)$. Let $\\partial^- R_{v_i}$ be the side of $R_{v_i}$ which $c$ enters, and let $\\partial^+ R_{v_i}$ be the side of $R_{v_i}$ which $c$ exits. Note that $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$ lie on stable faces of the same component of $N(B) \\backslash \\backslash N(\\Phi_{red})$ (indices taken mod $s$). We drew one instance of these as red intervals in \\Cref{fig:stableglue1}.\n\nAfter collapsing along the components of $N(B) \\backslash \\backslash N(\\Phi_{red})$, the image of the union $\\bigcup_{i=1}^s R_{v_i}$ in $N(B)$ contains $c$, by our choice of the $I$-fiberings. The images of $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$ may not match up away from $c$. However we claim that we can at least find surfaces $Q_i$ which are unions of finite subintervals of leaves of the 1-dimensional foliation on $N(B)$ which connect the image of $\\partial^- R_{v_i}$ to a subinterval of that of $\\partial^+ R_{v_{i+1}}$. See \\Cref{fig:stableglue3} top.\n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{8cm}{\\input{stableglue3.pdf_tex}}\n    \\caption{The surfaces $P_c$ are unions of images of $R_{v_i}$ and connecting surfaces $Q_i$.}\n    \\label{fig:stableglue3}\n\\end{figure}\n\nLet $K$ be the component of $N(B) \\backslash \\backslash N(\\Phi_{red})$ which contains both $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$. To construct $Q_i$, consider the universal cover $\\widetilde{K}$ of $K$. Note that there is a notion of height on $\\widetilde{K}$, with respect to the $y$-coordinate in the model described in \\Cref{lemma:redflowgraphcompl}, as opposed to $K$ where that coordinate is circular. The oriented 1-dimensional foliations on the stable faces of $K$ lift to oriented 1-dimensional foliations on the stable faces of $\\widetilde{K}$. Similarly, the $I$-fibering on $K$ lifts to an $I$-fibering on $\\widetilde{K}$. A component of the branch locus of $K$ is a subinterval of $c$ connecting up $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$. Lift $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$ together with the connecting subinterval to $\\widetilde{K}$. We abuse notation and still call the lifts of $\\partial^+ R_{v_i}$ and $\\partial^- R_{v_{i+1}}$ by the same name. By the criss-cross property stated in \\Cref{lemma:redflowgraphcompl}, the components of $\\partial^u \\widetilde{K}$ containing endpoints of $\\partial^+ R_{v_i}$ do not lie below the components that contain endpoints of $\\partial^- R_{v_{i+1}}$. In other words, if we transfer $\\partial^+ R_{v_i}$ to the other stable face of $\\widetilde{K}$ using the homeomorphism induced by the $I$-fibering, and call the image $\\partial^{++} R_{v_i}$ temporarily, then the leaves of the 1-dimensional foliation passing through $\\partial^{++} R_{v_i}$ will also pass through $\\partial^- R_{v_{i+1}}$. Hence we can take the union of subintervals of leaves going between $\\partial^{++} R_{v_i}$ and $\\partial^- R_{v_{i+1}}$, project down to $K$ then collapse to $N(B)$ to get $Q_i$.\n\nLet $P_c$ be the union of the images of $R_{v_i}$ and $Q_i$. $P_c$ is an immersed annulus with corners in $N(B)$. $P_c$ contains $c$ and intersects the same leaves as the images of $R_{v_i}$. $P_c$ is not strictly speaking a local section, since the $Q_i$ are tangent along the flow. One can perturb $P_c$ so that it becomes transverse to the flow everywhere, but since we ultimately only use the structure of $P_c$ after projecting to the orbit space, this is not necessary for our arguments. We will point out more features of $P_c$ in \\Cref{subsec:pAproof}.\n\n\\subsection{Gluing along the unstable boundary} \\label{subsec:unstableglue}\n\nWe will follow the same strategy to glue $N(B)$ along its boundary across its complementary regions in $M(s)$, so that the foliations on $N(B)$ descend to respective foliations on $M(s)$, for suitable $s$.\n\n\\begin{lemma} \\label{lemma:unstableglue}\nThe components of $M \\backslash \\backslash B$ are (once-punctured cusped polygons)$\\times S^1$. Fix one of these components $T$, $\\partial T$ is naturally decomposed into a number of annulus faces meeting along cusp circles. $\\Phi_{red}$ on each annulus face consists of one or two oriented circles along with some branches attached inductively which exit through the cusp circles. Each oriented circle of $\\Phi_{red} \\cap \\partial T$ has isotopy class equal to the ladderpole slope, in particular they are parallel. See \\Cref{fig:unstableglue} left. \n\\end{lemma}\n\n\\begin{proof}\nThe first sentence is \\Cref{prop:complbranchsurf}, and the second sentence follows easily. For the third sentence, note that $\\Phi_{red}$ on each annulus face $A$ of $\\partial T$ is an oriented train track with branches exiting through the boundary and with only diverging switches. This forces $\\Phi_{red} \\cap A$ to be a union of parallel circles with branches attached inductively which exit through the cusp circles. There cannot be more than 2 circles on each annulus or else the inner circles will have no branches attached. These will then contradict $\\Phi_{red}$ being strongly connected, since these circles can have no outgoing edges. Meanwhile, each cusp circle meets $\\Phi_{red}$, otherwise there will be a component of $B \\backslash \\backslash \\Phi_{red}$ carrying a circle in its branch locus, contradicting \\Cref{lemma:redflowgraphcompl}. These two observations imply the third sentence.\n\nTo show the fourth sentence, it suffices to show that the oriented circles of $\\Phi_{red} \\cap \\partial T$ are oriented coherently. We will show this using \\cite[Lemma 5.4]{LMT20}. (Again, we caution the reader that the flow graph in \\cite{LMT20} is oriented in the opposite direction compared to this paper.) Their lemma implies that each cycle of $\\Phi \\cap \\partial T$ has slope equal to the ladderpole slope. Since $\\Phi_{red}$ is a subgraph of $\\Phi$, the same statement holds for $\\Phi_{red} \\cap \\partial T$. We remark that the third sentence of our lemma follows from their lemma as well.\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{6cm}{\\input{unstableglue.pdf_tex}}\n    \\caption{Collapsing across complementary regions of $M(s) \\backslash \\backslash N(B)$.}\n    \\label{fig:unstableglue}\n\\end{figure}\n\nRecall that we can recover the oriented 1-dimensional foliation on the boundary of each component of $M \\backslash \\backslash N(B)$ from the portion of $\\Phi_{red}$ on the boundary of the corresponding component of $M \\backslash \\backslash B$, as described in the last subsection. From the lemma, it follows that the foliation on each annulus will consist of a band of $S^1$ leaves, with all other leaves spiralling out of the band and exiting through the cusp circles. Using the transverse measure $du$ on the rectangles $\\{ (u,t): |u| \\leq w'_w \\lambda^{-t}, t \\in [0,1] \\}$, one can see that the band of $S^1$ leaves actually just consists of one leaf. For if the band has nonzero width, pick a local section to the foliation within the band and observe that the backward return map of the foliation has to be contracting, providing a contradiction. See \\Cref{fig:unstableglue} middle.\n\nNow fix a component $T$ of $M \\backslash \\backslash N(B)$, fix an identification $T \\cong$ (punctured cusped $k$-gon)$\\times S^1$ and take the orientation on the punctured cusped $k$-gon to be the one induced from that of $T \\subset M$ and the ladderpole slope. Label the annulus faces of $\\partial T$ as $A^{(1)},...,A^{(k)}$ cyclically according to the orientations we have chosen. Suppose a slope $s$ is given on the torus end of $T$, whose geometric intersection number with the ladderpole class $l$ is at least $1$. We want to produce a `pronged $I$-fibering' of $T(s)$ preserving the leaves on $A^{(j)}$. By this we mean a decomposition of $T(s)$ into intervals and $|\\langle s,l \\rangle|$-prongs, so that the endpoints of the prongs lie along the unique $S^1$ leaves on the $A^{(j)}$, and so that the homeomorphisms on the halves of $A^{(j)}$ induced from the interval fibers preserve the leaves of the oriented 1-dimensional foliations.\n\nLet $m$ be the meridian of $T$ in the description $T \\cong$ (punctured cusped $k$-gon)$\\times S^1$, i.e. the isotopy class of $\\partial$(punctured cusped $k$-gon)$\\times \\{t_0\\}$. We pick $\\frac{|\\langle s,l \\rangle|}{k}$ local sections $J^{(j)}_1,...,J^{(j)}_{\\frac{|\\langle s,l \\rangle|}{k}}$ near the $S^1$ leaf on $A^{(j)}$, so that we have backward return maps $J^{(j)}_i \\to J^{(j)}_{i+1}$ (sub-indices taken mod $\\frac{|\\langle s,l \\rangle|}{k}$) of the oriented foliation. Let $s^{(j)}_i$ be where $J^{(j)}_i$ meets the $S^1$ leaf on $A^{(j)}$. First define the fibering on the cusp circles of $T(s)$, where it is uniquely determined and degenerate, then extend the fibering over subintervals of leaves that start at the interior of $J^{(j)}_i$ and end on the cusp circles, using the fact that the union of these trajectories are unions of rectangles foliated as products. Finally apply \\Cref{lemma:expandextend} to $I_i=$ subinterval of $J^{(j)}_i$ to the left of $s^{(j)}_i$ and $I'_i=$ subinterval of $J^{(j+1)}_{i\\pm \\langle s,m \\rangle}$ to the right of $s^{(j+1)}_{i\\pm \\langle s,m \\rangle}$ for each $j$ (mod $k$), where the $\\pm$ is taken to be the sign of $\\langle s,l \\rangle$, to produce the pronged $I$-fibering in the remaining subset of $T(s)$. \n\nNow given a collection of slopes $s$ on each torus end of $M$ with $|\\langle s,l \\rangle| \\geq 1$, collapse along the fibers of the pronged $I$-fibering in each corresponding component of $M(s) \\backslash \\backslash N(B)$. See \\Cref{fig:unstableglue} right for an illustration of a collapsed component of $M(s) \\backslash \\backslash N(B)$ with $|\\langle s,l \\rangle|=3$. The oriented 1-dimensional foliation on $N(B)$ descends to a decomposition of $M(s)$, which is a genuine oriented 1-dimensional foliation, since all the branching of the leaves in $N(B)$ occurs along the branch locus of $N(B)$, and we have collapsed all backward trajectories ending at those points. In particular this oriented 1-dimensional foliation can be continuously parametrized into a topological flow. \n\nFor each torus end of $M$, call the image of the $S^1$ leaves on $A^{(j)}$ after collapsing the \\textit{core orbit} of the corresponding end.\n\n\\subsection{Showing pseudo-Anosovity and no perfect fits} \\label{subsec:pAproof}\n\nWe have constructed our flow at this point. It remains to check that it satisfies the properties described in \\Cref{thm:vtpAflow}. We first recall the relevant definitions.\n\n\\begin{defn} \\label{defn:phorbit}\nConsider the map $\\begin{pmatrix} \\lambda^{-1} & 0\\\\ 0 & \\lambda \\end{pmatrix}: \\mathbb{R}^2 \\to \\mathbb{R}^2$. This preserves the foliations of $\\mathbb{R}^2$ by horizontal and vertical lines. Let $\\phi_{n,0,\\lambda}:\\mathbb{R}^2 \\to \\mathbb{R}^2$ be the lift of this map over $z \\mapsto z^{\\frac{n}{2}}$ that preserves the lifts of the quadrants. (When $n$ is odd, one has to choose a branch of $z \\mapsto z^{\\frac{n}{2}}$ but it is easy to see that the result is independent of the choice.) Let $\\phi_{n,k, \\lambda}: \\mathbb{R}^2 \\to \\mathbb{R}^2$ be the composition of $\\phi_{n,0,\\lambda}$ and rotating by $\\frac{2\\pi k}{n}$ anticlockwise. Also pull back the foliations of $\\mathbb{R}^2$ by horizontal and vertical lines. The resulting two singular foliations are preserved by $\\phi_{n,k}$, call them $l^s, l^u$ respectively. Let $\\Phi_{n,k,\\lambda}$ be the mapping torus of $\\phi_{n,k, \\lambda}$, $\\Lambda^s, \\Lambda^u$ be suspensions of $l^s, l^u$ respectively, and consider the suspension flow on $\\Phi_{n,k, \\lambda}$. Call the suspension of the origin the \\textit{pseudo-hyperbolic orbit} of $\\Phi_{n,k,\\lambda}$. The local behaviour of the flow near the pseudo-hyperbolic orbit serves as the local model for singular orbits in a pseudo-Anosov flow.\n\\end{defn}\n\n\\begin{defn} \\label{defn:pAflow}\nA \\textit{smooth pseudo-Anosov flow} on a closed smooth 3-manifold $N$ is a continuous flow $\\phi^t$ satisfying\n\\begin{itemize}\n    \\item There is a finite collection of closed orbits $\\{\\gamma_1, ..., \\gamma_s \\}$, called the \\textit{singular orbits} such that $\\phi^t$ is smooth away from the singular orbits.\n    \\item There is a path metric $d$ on $N$, which is induced from a Riemannian metric $g$ away from the singular orbits.\n    \\item Away from the singular orbits, there is a splitting of the tangent bundle into three $\\phi^t$-invariant line bundles $TM=E^s \\oplus E^u \\oplus T\\phi^t$ such that $$|d\\phi^t(v)| < C \\lambda^{-t} |v|$$ for every $v \\in E^s, t>0$, and $$|d\\phi^t(v)| < C \\lambda^t |v|$$ for every $v \\in E^u, t<0$, for some $C, \\lambda>1$.\n    \\item Each singular orbit $\\gamma_i$ has a neighborhood $N_i$ and a map $f_i$ sending $N_i$ to a neighborhood of the pseudo-hyperbolic orbit in $\\Phi_{n_i, k_i, \\lambda}$, for some $n_i \\geq 3$, such that $f_i$ is bi-Lipschitz on $N_i$ and smooth away from $\\gamma_i$, preserves the orbits, and sends $E^s, E^u$ to line bundles tangent to $\\Lambda^s, \\Lambda^u$ respectively. In this case, we say that $\\gamma_i$ is \\textit{$n_i$-pronged}.\n\\end{itemize}\n\nA \\textit{topological pseudo-Anosov flow} on a closed 3-manifold $N$ is a continuous flow $\\phi^t$ satisfying:\n\\begin{itemize}\n    \\item There is a finite collection of closed orbits $\\{\\gamma_1, ..., \\gamma_s \\}$, called the \\textit{singular orbits}, and two singular 2-dimensional foliations $\\Lambda^s, \\Lambda^u$, called the stable and unstable foliations respectively, which are non-singular away from the singular orbits.\n    \\item Away from the singular orbits, every point has a neighborhood which is a \\textit{flow box}, i.e. a set of the form $I_s \\times I_u \\times I_t$ such that the flow lines are the lines with constant $s$ and $u$, the stable and unstable foliations are the foliations by planes with constant $u$ or $s$ coordinate respectively.\n    \\item There is a \\textit{Markov partition}, i.e. there is a finite collection of flow boxes $\\{ I^{(i)}_s \\times I^{(i)}_u \\times I_t \\}_i$ covering $N$ with disjoint interiors, such that $$(I^{(i)}_s \\times I^{(i)}_u \\times \\{1\\}) \\cap (I^{(j)}_s \\times I^{(j)}_u \\times \\{0\\}) = \\bigcup_k J^{(ij.k)}_s \\times I^{(i)}_u \\times \\{1\\} = \\bigcup_k I^{(j)}_s \\times J^{(ji.k)}_u \\times \\{0\\}$$ for some finite collection of subintervals $J^{(ij,k)}_s \\subset I^{(i)}_s$ and $J^{(ji,k)}_u \\subset J^{(j)}_u$.\n    \\item Pick a path metric $d$ on $M$. For every $p,q$ on the same stable leaf, there exists an orientation-preserving homeomorphism $T:(-\\infty,\\infty) \\to (-\\infty,\\infty)$ such that $$\\lim_{t \\to \\infty} d_N(\\phi^t(p),\\phi^{T(t)}(q))=0$$ Respectively, for every $p,q$ on the same unstable leaf, there exists an orientation-preserving homeomorphism $T:(-\\infty,\\infty) \\to (-\\infty,\\infty)$ such that $$\\lim_{t \\to -\\infty} d_N(\\phi^t(p),\\phi^{T(t)}(q))=0$$ \n    \\item Each singular orbit $\\gamma_i$ has a neighborhood $N_i$ and a continuous map $f_i$ sending $N_i$ to a neighborhood of the pseudo-hyperbolic orbit in $\\Phi_{n_i, k_i, \\lambda}$, for some $n_i \\geq 3, \\lambda>1$, such that $f_i$ preserves the orbits and $\\Lambda^s, \\Lambda^u$ on the two sets. In this case, we say that $\\gamma_i$ is \\textit{$n_i$-pronged}.\n\\end{itemize}\n\\end{defn}\n\n\\begin{defn}\nA (smooth\/topological) pseudo-Anosov flow is \\textit{transitive} if it has a dense orbit.\n\\end{defn}\n\nIt has long been a folklore fact that the notions of smooth and topological pseudo-Anosov flows are essentially equivalent in the case when the flows are transitive. The easier direction is that a transitive smooth pseudo-Anosov flow is a topological pseudo-Anosov flow. Indeed, the only nontrivial facts to check are that $E^s \\oplus T\\phi^t$ and $E^u \\oplus T\\phi^t$ are integrable (away from singular orbits), which follows from stable manifold theory (see for example, \\cite[Chapter 17.4]{KH95}), and that there exists a Markov partition, which follows from the arguments in \\cite[Section 2]{Rat73}. For the other direction, we have\n\n\\begin{thm} \\label{thm:top2smooth}\nGiven a transitive topological pseudo-Anosov flow $\\phi^t$ on a closed 3-manifold $N$, there exists a homeomorphism $F:N \\to N$ and a smooth pseudo-Anosov flow $\\hat{\\phi^t}$ (with respect to some smooth structure on $N$), such that $F$ maps the trajectories of $\\phi^t$ to that of $\\hat{\\phi^t}$ (preserving their orientations), i.e. $\\phi^t$ is \\textit{$C^0$-orbit equivalent} to $\\hat{\\phi^t}$.\n\\end{thm}\n\nThis theorem has been proven recently by Shannon in \\cite{Sha21} in the case of transitive Anosov flows, i.e. when there are no singular orbits. His methods in fact generalize immediately to the general case of transitive pseudo-Anosov flows. We will explain more carefully how to apply his arguments to prove \\Cref{thm:top2smooth} in \\Cref{subsec:smooth}. We remark that it is still open whether \\Cref{thm:top2smooth} is true without the hypothesis of transitivity. Also see \\cite[Section 3.1]{Mos96} for a related discussion. \n\nBefore we recall the definition of no perfect fits, we need a nontrivial fact: Suppose a closed 3-manifold $N$ admits a (smooth\/topological) pseudo-Anosov flow. Lift everything up to the universal cover $\\widetilde{N}$. It is shown in \\cite[Proposition 4.1]{FM01} that the orbit space $\\mathcal{O}$ of the flow on $\\widetilde{N}$ is homeomorphic to $\\mathbb{R}^2$, and inherits two (possibly singular) 1-dimensional foliations $\\mathcal{O}^s, \\mathcal{O}^u$.\n\n\\begin{rmk}\n\\cite{FM01} only deals with smooth pseudo-Anosov flows, but given \\Cref{thm:top2smooth}, the facts stated above holds for topological pseudo-Anosov flows as well. Alternatively, the proof of \\cite[Proposition 4.1]{FM01} is valid verbatim for topological pseudo-Anosov flows.\n\\end{rmk}\n\nWe will in fact generalize the definition of no perfect fits slightly, compared to the usual definition found in, for example, \\cite{Fen12}.\n\n\\begin{defn} \\label{defn:perfectfit}\n\nLet $\\phi$ be a pseudo-Anosov flow on a closed 3-manifold $N$, and let $\\{c_1,...,c_k\\}$ be a collection of closed orbits of $\\phi$. Lift these up to a flow $\\widetilde{\\phi}$ on the universal cover $\\widetilde{N}$ together with a collection of orbits $\\{ \\widetilde{c_i} \\}$ which are the preimages of $\\{c_i\\}$.\n\nA \\textit{perfect fit rectangle} is a rectangle-with-one-ideal-vertex properly embedded in $\\mathcal{O}$ such that 2 opposite sides of the rectangle lie along leaves of $\\mathcal{O}^s$ and the remaining 2 opposite sides lie along leaves of $\\mathcal{O}^u$, and such that the restrictions of $\\mathcal{O}^s$ and $\\mathcal{O}^u$ to the rectangle foliate it as a product, i.e. conjugate to the foliations of $[0,1]^2 \\backslash \\{(1,1)\\}$ by vertical and horizontal lines. See \\Cref{fig:perfectfitdefn}. \n\nThe collection of orbits $\\{ \\widetilde{c_i} \\}$ determines a set $\\mathcal{C}$ in $\\mathcal{O}$. We will say that $\\phi$ has no perfect fits relative to $\\{c_1,...,c_k \\}$ if there are no perfect fit rectangles in $\\mathcal{O}$ disjoint from $\\mathcal{C}$.\n\n\\end{defn}\n\n\\begin{figure}\n    \\centering\n    \\fontsize{18pt}{18pt}\\selectfont\n    \\resizebox{!}{4cm}{\\input{perfectfitdefn.pdf_tex}}\n    \\caption{A perfect fit rectangle.}\n    \\label{fig:perfectfitdefn}\n\\end{figure}\n\nGiven a perfect fit rectangle with sides $F,H$ along leaves of $\\mathcal{O}^s$, sides $G,K$ along leaves of $\\mathcal{O}^u$, and sides $F,G$ adjacent to the ideal vertex, notice that we can always choose a smaller perfect fit rectangle with sides $F,G,H',K'$ where $H'$ is closer to $F$ than $H$ and $K'$ is closer to $G$ than $K$. We will frequently use this observation when analyzing perfect fit rectangles.\n\nA pseudo-Anosov flow is without perfect fits according to the definition in \\cite{Fen12} if and only if it is without perfect fits relative to $\\varnothing$ in \\Cref{defn:perfectfit}. This is in turn equivalent to the pseudo-Anosov flow having no perfect fits relative to the set of its singular orbits, since by definition a perfect fit rectangle cannot contain a singular point in its interior, and we can always choose a smaller perfect fit rectangle that does not contain any singular points on the boundary as well.\n\nIt is immediate from their definitions that the property of transitivity and no perfect fits relative to a collection of orbits is preserved under $C^0$-orbit equivalence. Hence in view of \\Cref{thm:top2smooth}, it suffices for us to show \\Cref{thm:vtpAflow} with `pseudo-Anosov flow' taken to mean topological pseudo-Anosov flow.\n\nWe first verify that our flow on $M(s)$ is a topological pseudo-Anosov flow. Take the collection of singular orbits to be the core orbits of the ends of $M$ with $|\\langle s,l \\rangle| \\geq 3$. Take $\\Lambda^s$ and $\\Lambda^u$ to be the stable and unstable foliations on $M(s)$ respectively.\n\nOne can construct flow box neighborhoods (away from the singular orbits) in $M(s)$ by piecing together those in $N(\\Phi_{red})$.\n\nWe will call the images of the flow boxes $Z_e$ in $M(s)$ after collapsing $N(B) \\backslash \\backslash N(\\Phi_{red})$ and $M(s) \\backslash \\backslash N(B)$ by the same name. These form a Markov partition for the flow by definition. Take the metric $d_{M(s)}$ on $M(s)$ to be induced from the Euclidean metrics on $Z_e$. \n\nFor the fourth item in the definition, it suffices to prove it when $p,q$ are close to each other, so we can assume that $p,q$ lie in the same $Z_e$. In that case, something stronger is in fact true: if $p,q$ lie on the same stable leaf, $$d_{M(s)}(\\phi^t(p),\\phi^t(q))<C \\lambda^{-t}$$ when the orbits through $p$ and $q$ are parametrized so that they go through each $Z_e$ in unit time (where $C$ depends on $p,q$). Similarly if $p,q$ lie on the same unstable leaf.\n\nFor the final item in the definition, the neighborhoods $N_i$ can be constructed by piecing up the flow box neighborhoods in $N(\\Phi_{red})$ again. We then have to find conjugations $f_i$ between $N_i$ and neighborhoods of the pseudo-hyperbolic orbits in $\\Phi_{n_i,k_i,\\lambda}$ for $n_i=|\\langle s_i,l_i \\rangle|$. These can be constructed by choosing local sections to the flow in $N_i$ and in $\\Phi_{n_i,k_i,\\lambda}$, then applying \\Cref{lemma:expandextend} to the prongs that are the intersections of these local sections with the singular stable and unstable leaves, taking $f$ there to be an arbitrary homeomorphism. This constructs $f_i$ on the singular stable and unstable leaves on the local sections, then $f_i$ is uniquely determined on the local sections (after possibly shrinking them), by requiring that it preserves the stable and unstable foliations. Then we complete the construction of $f_i$ by following the trajectories as they go around $N_i$. Observe that $|\\langle s, l \\rangle| \\geq 2$ is required here, otherwise we will have 1-pronged singular orbits. (Meanwhile `2-pronged singular orbits' are just non-singular orbits.)\n\nWe then show that the flow is transitive. Pick an infinite path in $\\Phi_{red}$ which contains all finite paths in $\\Phi_{red}$ as sub-paths. Such a path can be constructed using the fact that $\\Phi_{red}$ is strongly connected. Write the path as $v_1 \\overset{e_1}{\\to} v_2 \\overset{e_2}{\\to} ...$. We claim that there is an infinite flow line in $M(s)$ passing through $Z_{e_1},Z_{e_2},...$ in that order. To show this claim, consider the subset in $Z_{e_1}$ of points whose forward trajectories flow through $Z_{e_1},...,Z_{e_k}$. These subsets will be nonempty decreasing sub-flow boxes of $Z_{e_1}$, hence taking the intersection as $k \\to \\infty$, we get points in $Z_{e_1}$ whose flow lines have the required property. Pick one these flow lines and denote it as $l$. Now for any point $y \\in M(s)$, suppose the forward and backward trajectories of $y$ pass through $Z_{d_0}, Z_{d_1}, Z_{d_2}...$ and $Z_{d_0}, Z_{d_{-1}}, Z_{d_{-2}},...$ respectively. Note that these sequences may not be unique, due to the fact that leaves of the 1-dimensional foliation on $N(\\Phi_{red})$ have forward and backward branching, but we can always just make a choice. Now fix some $N>0$, the flow line $l$ passes through $Z_{d_{-N}},...,Z_{d_N}$ in that order at some point, by construction. Since both $l$ and the backward flow line through $y$ pass through $Z_{d_{-N}},...,Z_{d_0}$, they intersect $Z_{d_0}$ in flow lines not more than $w_{v_{-N}} \\lambda^{-N}$ apart in the $s$ coordinate. Similarly, looking in the forward direction, $l$ and the flow line through $y$ are not more than $w'_{w_N} \\lambda^{-N}$ apart in the $u$ coordinate within $Z_{d_0}$. Hence we conclude that $l$ passes through a $(\\max w_v+\\max w'_v) \\lambda^{-N}$ neighborhood of $y$. Letting $N \\to \\infty$, this proves transitivity.\n\nFinally we prove that the flow has no perfect fits relative to the core orbits, which we denote as $c_i$. To do this we will make use of the surfaces $P_c$ constructed in \\Cref{subsec:unstableglue}. There, we defined these as subsets of $N(B)$, but here we will consider their images in $M(s)$ after collapsing $M \\backslash \\backslash N(B)$ and still refer to them by the same name. Consider the lifts of $P_c$ to the universal cover $\\widetilde{M(s)}$. We denote these as $\\widetilde{P_{\\widetilde{c}}}$, where $\\widetilde{c}$ ranges over components of the branch locus of $\\widetilde{B}$ that are lifts of $c$, which we will conflate with the corresponding components of the branch locus of $\\widetilde{N(B)}$. After collapsing $\\widetilde{M(s)} \\backslash \\backslash \\widetilde{N(B)}$, each $\\widetilde{c} \\subset \\widetilde{M(s)}$ lies on a leaf of $\\widetilde{\\Lambda^u}$ and is transverse to the flow, meeting exactly those flow lines which spiral out of the lift of the core orbit produced by collapsing the component of $\\widetilde{M(s)} \\backslash \\backslash \\widetilde{N(B)}$ which $\\widetilde{c}$ opens up towards. Moreover, as one goes along $\\widetilde{c}$ in its orientation lifted from $c$, we meet flow lines that spiral closer to the lift of the core orbit. Hence when we project $\\widetilde{c}$ to $\\mathcal{O}$, we get an open half leaf of $\\mathcal{O}^u$, oriented away from infinity. This projection of $\\widetilde{c}$ is a homeomorphism, and so we will call the image $\\widetilde{c}$ as well for convenience.\n\nMeanwhile, each $\\widetilde{P_{\\widetilde{c}}}$ projected to $\\mathcal{O}$ will be a surface containing $\\widetilde{c}$. More precisely, recall that $P_c$ is the union of the images of $R_{v_i}$ and $Q_i$. Let $\\widetilde{R_{v_i}} \\subset \\widetilde{\\Phi_{red}}$ be the lifts of $R_{v_i}$ that meet $\\widetilde{c}$ and $\\widetilde{Q_i} \\subset \\widetilde{N(B)}$ the lifts of $Q_i$ that meet $\\widetilde{c}$. Then $\\widetilde{P_c}$ is the union of the images of $\\widetilde{R_{v_i}}$ and $\\widetilde{Q_i}$ after collapsing $\\widetilde{N(B)} \\backslash \\backslash \\widetilde{N(\\Phi_{red})}$ and $\\widetilde{M(s)} \\backslash \\backslash \\widetilde{N(B)}$. The image of each $\\widetilde{R_{v_i}}$ will be projected homeomorphically to a rectangle in $\\mathcal{O}$ along $\\widetilde{c}$ since these are transverse to the flow, while the image of each $\\widetilde{Q_i}$ is tangent to the flow hence will be projected into sides of the rectangles. By construction, the rectangles will get wider (with respect to $\\mathcal{O}^u$) as one goes along $\\widetilde{c}$. Hence images of these $\\widetilde{P_{\\widetilde{c}}}$ in $\\mathcal{O}$ will be half infinite strips with ridges protruding towards infinity, see \\Cref{fig:stableglue3}. We will basically argue that if there is a perfect fit rectangle in $\\mathcal{O}$ disjoint from the images of $\\widetilde{c_i}$, then we can reduce it to sit inside one of these $\\widetilde{P_{\\widetilde{c}}}$ with the ideal vertex facing away from infinity, hence by the shape of these surfaces, the ideal vertex must actually close up, giving us a contradiction. For convenience, in the argument we will write $\\widetilde{P_{\\widetilde{c}}}$ and $\\widetilde{R_{v_i}}$ for their images in $\\mathcal{O}$ as well.\n\nSo suppose we have a perfect fit rectangle in $\\mathcal{O}$ disjoint from the images of $\\widetilde{c_i}$ with sides $F,H$ along leaves of $\\mathcal{O}^s$, sides $G,K$ along leaves of $\\mathcal{O}^u$, and sides $F,G$ adjacent to the ideal vertex. Lift $H$ to $\\widetilde{M(s)}$. By flowing forward (and possibly rechoosing $K$ to be closer to $G$) we can assume that the lift of $H$ is short enough to be contained within some $\\widetilde{Z_e}$. In particular we can take the lift of $H$ to lie on the image of some $\\widetilde{R_v}$. If the lift of $H$ does not meet the image of the branch locus of $\\widetilde{N(B)}$ in its interior, i.e. it is fully contained in the bottom face of some $\\widetilde{Z_e}$, then we can rechoose the lift of $H$ to be on the top face of $\\widetilde{Z_e}$. When we do so the length of the lift (with respect to the Euclidean metric on the $\\widetilde{Z_e}$'s) is increased by a factor of $\\lambda$, so we cannot continue this operation indefinitely. In other words, we can choose a lift of $H$ which lies on the image of some $\\widetilde{R_v}$ and whose interior meets the image of the branch locus of $\\widetilde{N(B)}$.\n\nRecall that the components of the branch locus of $\\widetilde{N(B)}$ carry orientations induced from those of $B$. We split into two cases depending on how the lift of $H$ meets the image of the branch locus of $\\widetilde{N(B)}$ with respect to this orientation. Case 1 is if the lift of $H$ meets the image of a component $\\widetilde{c}$ of the branch locus oriented in the same direction as $K$ leaving $H$. In this case, by moving $K$ closer to $G$, we can suppose that $K$ lies within $\\widetilde{c}$ on $\\mathcal{O}$. This uses the assumption that the perfect fit rectangle is disjoint from the images of $\\widetilde{c_i}$, otherwise $K$ could go beyond $\\widetilde{c}$ on the leaf of $\\mathcal{O}^s$. Now by construction, $\\widetilde{P_{\\widetilde{c}}}$ in $\\mathcal{O}$ contains both $H$ and $K$. Also, $H$ and $K$ open up away from infinity, towards the end without ridges, hence $F$ and $G$ are forced to meet within $\\widetilde{P_{\\widetilde{c}}}$, giving us a contradiction. See \\Cref{fig:perfectfitarg1}.\n\n\\begin{figure}\n    \\centering\n    \\resizebox{!}{4cm}{\\input{perfectfitarg1.pdf_tex}}\n    \\caption{In case 1, $H$ and $K$ open up towards the end of $\\widetilde{P_{\\widetilde{c}}}$ without ridges.}\n    \\label{fig:perfectfitarg1}\n\\end{figure}\n\nCase 2 is if the lift of $H$ meets a component $\\widetilde{c}$ of the branch locus oriented in the different direction as $K$ leaving $H$. As above, we can assume that $K$ lies within $\\tilde{c}$ on $\\mathcal{O}$. Recall that $\\widetilde{P_{\\widetilde{c}}}$ consists of a union of rectangles $\\widetilde{R_{v_i}}$ along $\\tilde{c}$, and $H$ is contained in one of these rectangles. Since $K$ is a finite interval, it only goes through finitely many $\\widetilde{R_{v_i}}$, call this finite number the length of $K$. Notice that it is sometimes possible to reduce the length of $K$ by moving $H$ closer to $F$ and from $\\widetilde{R_{v_{i+1}}}$ to $\\widetilde{R_{v_i}}$ while keeping it inside $\\widetilde{P_{\\widetilde{c}}}$. In any case, we can assume that the length of $K$ is the minimum possible value among all perfect fit rectangles that are disjoint from the images of $\\widetilde{c_i}$ in $\\mathcal{O}$.\n\nIf the length of $K$ is 1, then we have a perfect fit rectangle with $H$ and $K$ within a single rectangle $\\widetilde{R_v}$, which is impossible. \n\nHence the remaining case is if we cannot move $H$ from $\\widetilde{R_{v_{i+1}}}$ to $\\widetilde{R_{v_i}}$ for some consecutive rectangles $\\widetilde{R_{v_i}}, \\widetilde{R_{v_{i+1}}}$ along $\\widetilde{c}$. Let $\\partial^+ \\widetilde{R_{v_i}}$ be the side of $\\widetilde{R_{v_i}}$ which $\\widetilde{c}$ exits, and $\\partial^- \\widetilde{R_{v_{i+1}}}$ be the side of $\\widetilde{R_{v_{i+1}}}$ which $\\widetilde{c}$ enters. (This is consistent with the notation used in \\Cref{subsec:stableglue}.) We can at least move $H$ onto the image of $\\partial^- \\widetilde{R_{v_{i+1}}}$ in $\\mathcal{O}$ in this case. Then we can pick a lift of $H$ which lies in the image of the stable face of some component $\\widetilde{K}$ of $\\widetilde{N(B)} \\backslash \\backslash \\widetilde{N(\\Phi_{red})}$ which contains $\\partial^- \\widetilde{R_{v_{i+1}}}$. The assumption that we cannot move $H$ to $\\widetilde{R_{v_i}}$ means that an endpoint of $\\partial^+ \\widetilde{R_{v_i}}$ lies on a component of $\\partial^u \\widetilde{K}$ which meets the other stable face at a point whose image in $\\widetilde{M(s)}$ has forward trajectory meeting $H$ in its interior.\n\nBut by the criss-cross property established in \\Cref{lemma:redflowgraphcompl}, since $\\partial^+ \\widetilde{R_{v_i}}$ and $\\partial^- \\widetilde{R_{v_{i+1}}}$ both meet $\\widetilde{c}$, each endpoint of $\\partial^+ \\widetilde{R_{v_i}}$ either lies on a component of $\\partial^u \\widetilde{K}$ which contains an endpoint of $\\partial^- \\widetilde{R_{v_{i+1}}}$, or lies on a component of $\\partial^u \\widetilde{K}$ whose branch locus is oriented in the different direction as $\\widetilde{c}$ (i.e. if $\\widetilde{c}$ is one of the $y=x+c$ lines in the model in \\Cref{lemma:redflowgraphcompl}, then the branch locus is one of the $y=-x+c$ lines, and vice versa). We are in the latter scenario here for at least one of the endpoints, and in that case the branch locus meets the stable face of $\\widetilde{K}$ containing $\\partial^- \\widetilde{R_{v_{i+1}}}$ at a point on or above $\\partial^- \\widetilde{R_{v_{i+1}}}$. If the point is on $\\partial^- \\widetilde{R_{v_{i+1}}}$ (which must be the case if $w=1$, in the notation of \\Cref{lemma:redflowgraphcompl}) then we are in case 1 since this branch locus will meet the lift of $H$ and is oriented in the same direction as $K$ leaving $H$. If the point is strictly above $\\partial^- \\widetilde{R_{v_{i+1}}}$, then up to moving $K$ closer to $G$, we can rechoose a lift of $H$ on the image of a higher $\\widetilde{R_v}$ so that it meets a component of the branch locus of $\\widetilde{N(B)}$ oriented in the different direction as $\\widetilde{c}$, hence the same direction as $K$ leaving $H$, see \\Cref{fig:perfectfitarg2}. Then we have reduced to case 1 again. Hence in any case, we see that the perfect fit rectangle cannot exist.\n\n\\begin{figure}\n    \\centering\n    \\fontsize{16pt}{16pt}\\selectfont\n    \\resizebox{!}{12cm}{\\input{perfectfitarg2.pdf_tex}}\n    \\caption{In case 2, up to possibly rechoosing a lift of $H$ that is higher up, we can assume that the lift of $H$ meets a component of the branch locus of $\\widetilde{N(B)}$ oriented in the same direction as $K$ leaving $H$.}\n    \\label{fig:perfectfitarg2}\n\\end{figure}\n\n\\subsection{Proof of \\Cref{thm:top2smooth}} \\label{subsec:smooth}\n\nBy \\cite{Bru95}, $\\phi^t$ admits a Birkhoff section. Recall that this means there is an embedding of an oriented surface with boundary $\\Sigma \\hookrightarrow N$, such that $\\partial \\Sigma$ is a union of orbits, $int \\Sigma$ is positively transverse to $N$, and every point intersects $\\Sigma$ in finite forward time. We briefly sketch an argument here.\n\nFor every point $x \\in N$, take a local section to the flow near $x$. The local section is divided into $2n$ regions by the stable and unstable leaves containing $x$ (where $n=2$ unless $x$ lies on a singular orbit). Name them $A_1,B_1,...,A_n,B_n$ in a cyclic order. It is standard that the set of periodic orbits in a transitive pseudo-Anosov flow is dense. For example, this can proved using a symbolic dynamics argument similar to how we proved transitivity before. Use this fact to find non-singular periodic orbits $p_i$ passing through $A_i$. We can further assume that $p_i$ are orientation preserving paths on the stable and unstable leaves they lie on, since the set of orientation preserving periodic orbits is dense as well. Encode the closed orbits $p_i$ by periodic symbolic sequences using a Markov partition, then concatenate these sequences to get a long periodic sequence which corresponds to a closed orbit $q$ in $N$ `shadowing' $p_1,...,p_n$, in that cyclic order. Now construct a $2n$-gon on the local section with vertices at where $p_1,q,p_2,q,...,p_n,q$ meet the local section. Here the multiple listings of $q$ refer to the multiple times $q$ meets the local section as it shadows the $p_i$'s. Connect up the edges $[q,p_k]$ and $[p_k,q]$ on the local section for each $k$ by following the flow. After perturbation, we get an immersed local Birkhoff section. See \\Cref{fig:birkhoffsection}. Since $N$ is compact, we can take the union of finitely many such surfaces and have all points meet the union in finite forward time. Then we can perform some surgeries along the self-intersections to get a genuine Birkhoff section $\\Sigma$. \n\n\\begin{figure}\n    \\centering\n    \\fontsize{12pt}{12pt}\\selectfont\n    \\resizebox{!}{8cm}{\\input{birkhoffsection.pdf_tex}}\n    \\caption{Left: pick a long periodic orbit $q$ which weaves around the shorter periodic orbits $p_1,...,p_n$. Right: take a $2n$-gon transverse to the flow with vertices at $p_1,q,...,p_n,q$ and connect up its sides to produce an immersed local Birkhoff section.}\n    \\label{fig:birkhoffsection}\n\\end{figure}\n\nOur argument here is a generalization of Fried's argument in \\cite{Fri83} to pseudo-Anosov flows. We remark that there is an alternative argument to construct $q$ in \\cite{Bru95} without using symbolic dynamics.\n\nLet $\\Sigma^\\circ$ be $\\Sigma$ without its boundary components, and let $\\overline{\\Sigma}$ be $\\Sigma$ with all its boundary components collapsed to points. The first return map on $\\Sigma$ restricts to a pseudo-Anosov homeomorphism $h^\\circ$ on $\\Sigma^\\circ$, with the intersections of $\\Lambda^s, \\Lambda^u$ with $\\Sigma^\\circ$ acting as the stable and unstable foliations. This in turns induces a pseudo-Anosov homeomorphism $\\overline{h}$ on $\\overline{\\Sigma}$. (Note that a map obtained this way on a general Birkhoff section may have 1-pronged singularities, but this is eliminated by requiring that the components of $\\partial \\Sigma$ be orientation preserving curves on the stable and unstable leaves.) By classical results, there is a smooth structure on $\\overline{\\Sigma}$ such that $\\overline{h}$ is smooth away from the singular points. Hence the mapping torus of $(\\bar{\\Sigma}, \\bar{h})$ carries a smooth structure so that the suspension flow is smooth away from the singular points. Moreover, for each closed orbit of the suspension flow, there is a neighborhood diffeomorphic to a neighborhood of the pseudo-hyperbolic orbit of some $\\Phi_{n,k,\\lambda}$ in \\Cref{defn:phorbit}. In particular this is true for the closed orbits which are suspensions of points of $\\bar{\\Sigma} \\backslash \\Sigma^\\circ$. We call these closed orbits the surgery orbits for ease of notation.\n\nMeanwhile, note that $N \\backslash \\partial \\Sigma$ with the restricted $\\phi^t$ flow is $C^0$-orbit equivalent to the suspension flow on the mapping torus $(\\Sigma^\\circ, h^\\circ)$. In particular, there is a vector field on $N \\backslash \\partial \\Sigma$ (with respect to some smooth structure) whose trajectories are equal to the flow lines of $\\phi^t$. Using the orbit equivalence, we can transfer neighborhoods of the surgery orbits, minus the surgery orbits themselves, to neighborhoods of the ends of $N \\backslash \\partial \\Sigma$, then fill back in the components of $\\partial \\Sigma$ to get neighborhoods of $\\partial \\Sigma$ in $N$. It remains to perform Dehn surgery on these neighborhoods, which we will do by cutting out small special shaped sub-neighborhoods and gluing similarly shaped regions back in, and arguing that the result is orbit equivalent to $\\phi^t$ on the original manifold $N$. The first step is a restatement of the arguments in \\cite{Goo83}, while the second step relies on \\cite[Theorem 3.9]{Sha21}. \n\nWithin each such neighborhood in $N$, excise a small cross-shaped neighborhood of $\\partial \\Sigma$. These neighborhoods are defined in \\cite[Section 5.2]{Sha21}, and in this setting, located using the orbit equivalence map between $N \\backslash \\partial \\Sigma$ and the mapping torus of $(\\Sigma^\\circ, h^\\circ)$. The feature of these neighborhoods is that their boundaries have alternating annulus faces, the flow transverse to half of the faces and tangent to the remaining faces. Then we glue some cross-shaped neighborhoods back in by matching up the vector fields at the boundary, so that the glued vector field determines a smooth flow (away from the singular orbits). See \\cite[Section 5.3]{Sha21} for an explicit description of this gluing map. The key observation that makes this work is that there is a freedom in this gluing: one can do Dehn twists suitably along annulus faces which the flow is transverse to, and still have the vector fields match up. So far there is no difference between the Anosov case and the general pseudo-Anosov case. \n\nThen one uses a cone field argument to find the stable and unstable line bundles $E^s, E^u$ and show that the glued up flow is pseudo-Anosov, provided that the neighborhood we excised is small enough. This is done in the Anosov case in \\cite[Section 5.4]{Sha21}, and is in fact more simple in the pseudo-Anosov case, since one only needs to construct $E^s, E^u$ away from the singular orbits. On the singular orbits, the definition only requires an orbit equivalence between a neighborhood in $N$ and a neighborhood in $\\Phi_{n,k,\\lambda}$, and these are provided by the orbit equivalence between $N \\backslash \\partial \\Sigma$ and the mapping torus of $(\\Sigma^\\circ, h^\\circ)$, since by construction the orbits of $\\partial \\Sigma$ are nonsingular. As remarked above, this method of constructing (pseudo-)Anosov flows on Dehn surgeries appeared back in \\cite{Goo83}. \n\nFinally, we construct a Birkhoff section on this glued flow by taking the portion of the original Birkhoff section $\\Sigma$ outside of the excised cross-shaped neighborhood, and extending it inside the glued cross-shaped neighborhood by taking a union of straight lines towards the pseudo-hyperbolic orbit. Call this surface $\\Sigma'$. The computation in \\cite[Section 5.5]{Sha21} shows that this indeed defines a Birkhoff section, and that the return map $h'$ is pseudo-Anosov. Moreover, $h$ and $h'$ induce the same automorphism on $\\pi_1(\\Sigma) \\cong \\pi_1(\\Sigma')$, preserving the peripheral subgroups. Hence using \\cite[Theorem 3.9]{Sha21}, which applies to pseudo-Anosov flows as well, as stated there, we get a $C^0$-orbit equivalence between $\\phi^t$ on $N$ and the smooth pseudo-Anosov glued flow. \n\n\\subsection{Properties of the construction} \\label{subsec:prop}\n\nWe point out two properties of the pseudo-Anosov flows we constructed. We will phrase our proofs in terms of the initial topological pseudo-Anosov flow we constructed, but thanks to \\Cref{thm:top2smooth}, one can just take an orbit equivalence map to prove the properties for the smooth pseudo-Anosov flow as well.\n\n\\begin{prop} \\label{prop:unstablelamination}\nThe unstable lamination of the pseudo-Anosov flow, obtained by blowing air into the singular leaves of the unstable foliation, is carried by $B \\subset M(s)$.\n\\end{prop}\n\n\\begin{proof}\nThe unstable lamination can also be obtained by taking the unstable foliation on $N(B)$ and blowing air into the non-manifold leaves. There is a projection $N(B) \\to B$ by definition, and this projects the lamination to $B$.\n\\end{proof}\n\n\\begin{defn}\nSuppose a pseudo-Anosov flow $\\phi$ has a Markov partition as in \\Cref{defn:pAflow}. Define a directed graph $G$ by letting the set of vertices be the flow boxes, and putting an edge from $(I^{(j)}_s \\times I^{(j)}_u \\times [0,1]_t)$ to $(I^{(i)}_s \\times I^{(i)}_u \\times [0,1]_t)$ for every $J^{(ij,k)}_s$. \n\nNotice that $G$ has a natural embedding in $N$ by placing the vertices in the interior of the corresponding flow box and placing the edges through the corresponding intersections $J^{(ij,k)}_s \\times I^{(i)}_u \\times \\{1\\}$. $G$ together with this embedding in $N$ is said to \\textit{encode the Markov partition}. \n\\end{defn}\n\n\\begin{prop} \\label{prop:Markovpart}\nThe pseudo-Anosov flow constructed in \\Cref{thm:vtpAflow} admits a Markov partition encoded by the reduced flow graph $\\Phi_{red}$.\n\\end{prop}\n\n\\begin{proof}\nThe flow boxes $Z_e$ form a Markov partition by construction, but notice that the graph that encodes this Markov partition is not $\\Phi_{red}$. Indeed, the flow boxes $Z_e$ correspond to edges, not vertices, of $\\Phi_{red}$, hence the graph encoding this Markov partition is instead a `dual' of $\\Phi_{red}$. \n\nWhat we will do is extract from this an alternate Markov partition that is encoded by $\\Phi_{red}$. For each vertex $v$ of $\\Phi_{red}$, recall the rectangle $R_v$ considered in \\Cref{subsec:stableglue}. Let $Z'_v$ be the closure of the union of trajectories in $M(s)$ that start in the image of the interior of $R_v$ and end when it meets the image of the interior of another (possibly the same) $R_{v'}$. These $Z'_v$ cover $M(s)$ with disjoint interiors, but they might not be flow boxes. Instead, they are in general homeomorphic to $\\{(s,u,t): s \\in I_s, u \\in I_u, t \\in [f(s,u),1]\\}$ with a function $f$ with discontinuities along finitely many lines $u=u_0$. Consider the total number of such intervals of discontinuities at the bottom of these $Z'_v$. If this number is $0$, then we have a genuine flow box decomposition which gives a Markov partition encoded by $\\Phi_{red}$. \n\nIf this number is positive, we claim that we can modify the $Z'_v$ to reduce it. Fix an interval of discontinuity $J$ of $Z'_v$. Suppose the two sides to it on the bottom of the $Z'_v$ lie on the image of $R_{v_1}$ and $R_{v_2}$. We can modify $R_{v_1}$ so that it matches up with $R_{v_2}$, elminating the discontinuity across $J$. See \\Cref{fig:markovpartarg}. One might worry that this produces new intervals of discontinuity, but we argue that this will not happen. This is because this operation only possibly affects the intervals on the images of the boundaries of the $R_{v_i}$ that are adjacent to $J$, but for one such interval $K$, $J \\cap K$ is a point that lies on the image of a component $c$ of the branch locus of $N(B)$. In particular, the two $R_{v_i}$ match up along $c$ in $N(B)$, hence in $M(s)$ already. So we can in fact modify $R_{v_1}$ while fixing $K$. Doing this inductively, we can reduce the number of intervals of discontinuity to $0$.\n\\end{proof}\n\n\\begin{figure}\n    \\centering\n    \\fontsize{12pt}{12pt}\\selectfont\n    \\resizebox{!}{8cm}{\\input{markovpartarg.pdf_tex}}\n    \\caption{Modifying $R_v$ to eliminate the intervals of discontinuity, so that we obtain a decomposition by flow boxes.}\n    \\label{fig:markovpartarg}\n\\end{figure}\n\n\\begin{cor} \\label{cor:symbdyn}\nGiven a cycle $l$ of $\\Phi$ in $M(s)$, there is a periodic orbit $c$ of the flow such that $l$ is homotopic to $c$. Conversely, given a periodic orbit $c$ of the flow, there exists a cycle $l$ of $\\Phi$ such that $l$ is homotopic to some multiple $c^k$, $k \\geq 1$.\n\\end{cor}\n\n\\begin{proof}\nThis is mostly a straightforward consequence of \\Cref{prop:Markovpart} and \\Cref{lemma:flowgraphsorbits}.\n\nGiven a cycle $l$ of $\\Phi$, we first assume that there is a homotopic cycle of $\\Phi_{red}$, which we also write as $l$. Then we can proceed as in the proof of transitivity to construct orbits of the flow passing through the sequence of flow boxes corresponding to the sequence of vertices of $\\Phi_{red}$ passed through by $l'$. A similar argument as in that proof shows that such an orbit will be unique, hence the orbit $c$ will be periodic, since the sequence of flow boxes is periodic. A homotopy between $l$ and $c$ can now be constructed using a straight line homotopy within each flow box.\n\nWe caution that $c$ may not be a primitive closed orbit. Indeed, the above construction in fact gives a $S^1$ leaf $c'$ of the singular 1-dimensional foliation on $N(\\Phi_{red})$. But when we quotient $N(\\Phi_{red})$ to $M(s)$, $c'$ may cover an orbit multiple times.\n\nIf $l$ is not homotopic to a cycle of $\\Phi_{red}$, then by the proof of \\Cref{lemma:flowgraphsorbits}, $l$ must be an infinitesimal cycle within a twisted wall. Let $l'$ be the boundary cycle of that wall. Applying the argument above for this $l'$, we find a circular leaf $c'$ in $N(\\Phi_{red})$ lying on the stable boundary $\\partial_s N(\\Phi_{red})$ which is homotopic to $l'$ in $M(s)$. When we quotient $N(\\Phi_{red})$ to $N(B)$, $c'$ double covers its image $c''$, which is hence homotopic to $l$. The image of $c''$ after quotienting to $M(s)$ will be the desired periodic orbit $c$ in this case. Again, we remark that $c$ may not be primitive.\n\nConversely, given a periodic orbit $c$ of the flow on $M(s)$, consider its preimage in $N(\\Phi_{red})$. This will be a disjoint union of circular leaves which cover $c$ possibly multiple times. Pick one of these circles $c'$. The periodic sequence of flow boxes which it meets will determine a cycle $l$ of $\\Phi_{red} \\subset \\Phi$. A homotopy between $l$ and $c$ can now be constructed using a straight line homotopy within each box as before.\n\\end{proof}\n\nCombining \\cite[Theorem A]{Lan20} and \\cite[Proposition 5.7]{LMT20}, it is known that when $|\\langle s,l \\rangle| \\geq 3$, the homology classes of the cycles of $\\Phi$ in $M(s)$ span a cone dual to a face of the Thurston unit ball, namely the face determined by the negative of the Euler class of the veering triangulation $$-e(\\Delta)=\\sum \\frac{1}{2}(\\text{\\# prongs}(\\gamma)-2) \\cdot [\\gamma] \\in H_1(M(s))$$ Hence \\Cref{cor:symbdyn} implies that the homology classes of the periodic orbits of the pseudo-Anosov flow in \\Cref{thm:vtpAflow} span the same dual cone. \n\n\\begin{rmk} \\label{rmk:LMTorbits}\n\n\\Cref{cor:symbdyn} and its proof also allows one to count the number of periodic orbits of the pseudo-Anosov flow using $\\Phi$. The precise statements one can get are essentially the same as those shown in \\cite[Theorem 6.1]{LMT21}, but again we remind the reader that our setting in this paper is opposite to that in \\cite{LMT21}.\n\nWe elaborate on this a little more. The proof of \\Cref{cor:symbdyn} shows that for each cycle $l$ of $\\Phi$, there is a unique primitive periodic orbit $c$ of the flow homotopic to $l$, unless if $l$ determines a circular leaf that lies on the boundary of $N(\\Phi_{red})$ and multiple covers a periodic orbit of the flow under the quotient maps. If such a circular leaf lies on $\\partial^s N(\\Phi_{red}) \\backslash \\backslash \\partial^u N(\\Phi_{red})$, then $l$ lies on the boundary of a component of $B \\backslash \\backslash \\Phi_{red}$, and if the circular leaf multiple covers a periodic orbit, the component must be a Mobius band with tongues. If such a circular leaf lies on $\\partial^u N(\\Phi_{red})$, then $l$ can be homotoped into a torus end of $M$.\n\nConversely, the proofs of \\Cref{cor:symbdyn} and \\Cref{lemma:flowgraphsorbits} show that for each primitive periodic orbit $c$ of the flow, there is a unique cycle $l$ of $\\Phi$ homotopic to $c$, unless the preimage of $c$ in $N(\\Phi_{red})$ multiple covers $c$ or if $l$ is homotopic to a boundary cycle of a wall. The former can only happen when if the preimage lies in the boundary of $N(\\Phi_{red})$. If the preimage lies in $\\partial^s N(\\Phi_{red}) \\backslash \\backslash \\partial^u N(\\Phi_{red})$, then $c$ is homotopic to the core of a component of $B \\backslash \\backslash \\Phi_{red}$ which is a Mobius band with tongues as above. If the preimage lies in $\\partial^u N(\\Phi_{red})$ then $c$ is a core orbit. In the latter case, $l$ is homotopic to an AB cycle, by the discussion in \\Cref{rmk:LMTABwalls}.\n\\end{rmk}\n\n\\section{Discussion and further questions} \\label{sec:questions}\n\nWe conclude with some questions coming out of this paper.\n\nIn \\Cref{sec:infcomp}, we analyzed \\emph{how} the flow graph of a veering triangulation can fail to be strongly connected. An equally interesting question is \\emph{when} does the flow graph fail to be strongly connected. For example, for layered veering triangulations, one could ask if there is a topological criterion on the monodromy that determines whether the flow graph is strongly connected. Another interesting question is if having a strongly connected flow graph is a generic property among all veering triangulations.\n\nWe already pointed out in \\Cref{sec:finiteness} that in our argument for \\Cref{thm:boundveertet}, there is a lot of room for improvement for the bounds we used. A more interesting approach to try to obtain better bounds overall is to study the flow graphs themselves. It is shown in \\cite{McM15} how to recover the dilatation of the pseudo-Anosov monodromy from the Teichm\\\"uller polynomial of the associated fibered face. The Teichm\\\"uller polynomial is in turn related to the veering polynomial of the layered veering triangulation, defined in \\cite{LMT20} as the inclusion of the Perron polynomial of the flow graph in the 3-manifold. Now, the combinatorial rigidity of veering triangulations might impose restrictions on the graph isomorphism type of flow graphs, which might in turn have consequences for the behavior of their Perron polynomials, possibly giving some bounds on dilatations. We remark that Parlak has computed some data for veering polynomials of veering triangulations in the census (see \\cite{Par21}). This data might be helpful for extrapolating patterns for the approach above.\n\nOne motivation for improving the bound is to use \\Cref{thm:boundveertet} to solve the minimal dilatation problem by computation. Starting with an upper bound, we can use \\Cref{thm:boundveertet} to reduce the search to a finite list. However, the best currently known upper bound (for every surface at once) is $P=(2+\\sqrt 3)^2$ (obtained in \\cite{HK06} and \\cite{Min06}), and plugging this into our bound, we have to look at all 3-manifolds triangulated by at most $9.2 \\times 10^{21}$ veering tetrahedra. In comparison, the veering triangulation census only covers triangulations up to 16 veering tetrahedra for now, so huge advancements would have to be made before this approach is computationally feasible.\n\nFinally, there are plenty of natural questions one could ask about our construction of the psuedo-Anosov flow in \\Cref{sec:pAflow}.\n\nWe have been using the unstable branched surface $B$ throughout this paper, but there is also a stable branched surface associated to a veering triangulation, which can be defined as the unstable branched surface associated to the veering triangulation with reversed coorientation on the faces. It is natural to ask whether the stable lamination of our pseudo-Anosov flow is carried by the stable branched surface. \n\nAlso, one can repeat our entire construction with the stable branched surface replacing the unstable branched surface. Then one can ask whether the pseudo-Anosov flows obtained using the stable and unstable branched surfaces are $C^0$-orbit equivalent. A related development is that Parlak (\\cite{Par21}) has found veering triangulations whose veering polynomial is different from that of the veering triangulation with reversed coorientation on the faces.\n\nWe mentioned in the introduction that the construction of a pseudo-Anosov flow without perfect fits from a veering triangulation was first done by Schleimer and Segerman. It is an interesting question to ask whether the flow constructed by our approach ends up being equivalent to the one constructed by Schleimer and Segerman.\n\nOne could also ask if our construction is inverse, in either direction, to the construction by Agol and Gu\\'eritaud, which produces a veering triangulation out of a pseudo-Anosov flow without perfect fits. More precisely, if one starts with a pseudo-Anosov flow without perfect fits relative to orbits $\\{c_i\\}$ on a closed orientable 3-manifold $N$ and constructs a veering triangulation on $N$ with $\\{c_i\\}$ drilled out, then constructs a pseudo-Anosov flow by filling in slopes that recover $N$, is the final pseudo-Anosov flow $C^0$-orbit equivalent to the initial pseudo-Anosov flow? Conversely, if one starts with a veering triangulation and constructs a pseudo-Anosov flow by filling along slopes $s$ with $| \\langle s,l \\rangle | \\geq 2$, then constructs a veering triangulation with the core orbits drilled out, is the final veering triangulation isomorphic to the initial veering triangulation? If one could show that our construction is equivalent to that of Schleimer and Segerman, then the answer to both questions would be `yes', since as mentioned in the introduction, this is a feature of their construction.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMultiferroics (MFs) are materials in\nwhich different ferroic orders  such as ferromagnetism,\nferroelectricity and\/or  ferroelasticity may coexist in a single\ncompound.\\cite{Khomskii} They have attracted much attention for\ntheir potential applications in memory devices and other electronic\ncomponents, due to the intriguing\n possibility of controlling magnetism by an applied\n  electric field, and\nviceversa (magnetoelectric effect).\\cite{MF1,MF2,MF3,bfo}\n\nMultiferroics are compounds where electron correlations are rather\nimportant, and where the electron charge shows atomic-like features,\nsuch as strong space localization, poorly dispersed band energies,\nand large on-site Coulomb repulsion.\\cite{MF4,MF2,MF3,MF7} For these\nsystems, there are well-known deficiencies of local-spin-density\napproximation (LSDA) or spin-polarized\ngeneralized-gradient-approximation (SGGA) to\ndensity-functional-theory (DFT). Among them, we recall the\n  underestimation of the band-gap magnitude for most insulating\n materials.\\cite{failures,MetallicGe,hexYMnO3}\nPart of these failures can be traced back\nto  the self-interaction error in  approximate\ndensity functionals: the electron charge experiences a spurious interaction\nwith the Coulomb and exchange-correlation potential generated by itself.\\cite{ReviewGW1,ReviewGW2}\n\nThe LSDA+$U$\\cite{ldau1,ldau2,ldau3} and the self-interaction  correction (SIC) schemes\\cite{sic1,sic2,sic3}  can overcome some of the deficiencies of LSDA. However, LSDA+$U$ suffers\nambiguities in the choice of the $U$ parameter \nand needs a choice regarding which orbitals to treat within a Hubbard-like approach. For simple materials, a self-consistent evaluation of the $U$ parameter can be obtained, although this method is not widely used.\\cite{ldau.kresse} For BiFeO$_{3}$, the value U$_{eff}$=3.8 eV has been recently calculated.\\cite{Ueff}\n\n\nSIC-schemes are not commonly available in electronic structure codes\nfor extended solid state systems.  The implementation of a fully self-consistent\nSIC-LSDA approach for extended systems was\ndone by Svane and co-workers.\\cite{Svane} Since then,\nother approaches have been implemented (for a review, see Refs.\\cite{sic2,AtomicSic}). SIC-schemes  suffer from the ``nonvariationality-problem'' of the energy functional which makes forces and stress calculation not commonly available.\\cite{Stengel} \n\nIn the last few years, hybrid Hartree-Fock density functionals\\cite{hybrid2,hybrid3,hybrid4,hybrid5} have been\nwidely used in solid state physics,\\cite{hybrid5,hybrid6,hybrid7,hybrid7.5} ranging from simple semiconductor systems,\\cite{hybrid7.5} to transition metals, lanthanides, actinides,\\cite{itinerant1,itinerant2,itinerant3,itinerant4} molecules at surfaces,\\cite{Stroppa1,Stroppa2} diluted magnetic semiconductors,\\cite{Stroppa3} carbon nanostructures.\\cite{hybrid11,hybrid12} For  a recent review see\nRef.\\cite{ScuseriaReview}.  Hybrid functionals \nmix  the exact nonlocal exchange\nof Hartree-Fock theory\\cite{hybrid2,hybrid3,hybrid4,hybrid5} with the density functional exchange.\nThe Heyd-Scuseria-Ernzerhof screened hybrid functional (HSE)\\cite{hsedef,hsedef1}\n is well suited for extended solid state systems. \n\nThere are very few studies  dealing with ferroelectric oxides  and even less\nwith multiferroics. Wahl \\textit{ et al.} re-investigated the well-known SrTiO$_{3}$ and BaTiO$_{3}$ \\cite{Roman}\n using   HSE and semilocal functionals (LDA,PBE,\\cite{pbe} PBEsol\\footnote{PBEsol\\cite{PBEsol} is a variant of the PBE exchange functional which improves equilibrium properties of densely-packed solids and their surfaces.}).\nBilc \\textit{et al.}\nstudied in great details BaTiO$_{3}$ as well as  PbTiO$_{3}$ using the B1-WC\n hybrid-functional and concluded that the latter gives\n an accurate description of both the structural\nand electronic properties.\\cite{Bilc}\nGoffinet \\textit{et al.} extended the analysis to the prototypical multiferroic bismuth ferrite   showing that hybrid-functionals, specifically the B1-WC functional, open\nnew perspectives for a better first-principles description\nof multiferroics.\\cite{Goffinet} In passing, we recall that the WC gradient corrected functional\\cite{WC} is very similar to PBEsol and the hybrid B1-WC\\cite{B1WC}  functional mixes the WC\nfunctional with 16\\% nonlocal exchange. The hybrid HSE functional mixes 25\\% nonlocal exchange with the PBE functional and the mix is performed only on the short range component of the Coulomb interaction\n(for further details, we refer to Ref.\\ \\onlinecite{Roman}). However, which functional to prefer for simple ferroelectric compounds is still an open issue.\\cite{Roman}\n\n\nSo far, a good performance of HSE or B1-WC functionals has\nbeen recognized\nfor \\textit{proper}   multiferroics where the ferroelectric polarization\n is of displacive type.\n On the other hand, magnetically\ndriven multiferroics, also known as \\textit{improper} multiferroics,\nare largely unexplored  using hybrid functionals. The purpose of this work is to extend the previous hybrid\ndensity functional studies from prototypical ferroelectric oxides\n(SrTiO$_{3}$, BaTiO$_{3}$, PbTiO$_{3}$)\\cite{Roman,Bilc} or simple multiferroic system (BiFeO$_{3}$)\nto more complicated and exotic multiferroic compounds, such as HoMnO$_{3}$.\n\n First of all, we focus on  BiFeO$_{3}$,\nalready investigated using the B1-WC functional, but not yet  using HSE.\nIn this way, we are able to compare two different, although similar, approaches for BiFeO$_{3}$. Most importantly,\nwe consider another prototypical case\n of \\textit{improper} multiferroic,\nnamely HoMnO$_{3}$, which has   recently attracted\nmuch attention.\\cite{HMO1,HMO2}  We will show that important differences compared to standard DFT approaches\narise when a proper description of correlated electrons, such as that given by HSE, is taken into account.\n \nOur study suggests that HSE functional improves the description compared to standard DFT approaches for multiferroic\nsystems.\n\nThe  paper is organized as follows. Details of the computational setups\nare given in   Sect.\\ \\ref{Computational}. \nAn extended discussion of the\nstructural, electronic, magnetic and ferroelectric properties of BiFeO$_{3}$ is reported in Sect.\\ \\ref{BFO}. Sect.\\ \\ref{HMOSect}\nis devoted to HoMnO$_{3}$ focussing on the paraelectric AFM-A (Sect.\\ \\ref{A-HMO}) and ferroelectric AFM-E (Sect.\\ \\ref{E-HMO}) phases.\nFinally, in Sect.\\ \\ref{conclusions}, we draw our conclusions.\n\n\\section{Computational details}\\label{Computational}\nAll the calculations presented in this study are performed\nby using the latest version of the \\textit{Vienna ab initio simulation package}\n(VASP 5.2).\\cite{vasp1}\n For BiFeO$_{3}$, all the results are obtained using the projector-augmented plane-wave method\\cite{paw1,paw2} by explicitly treating 15 valence electrons for Bi (5$d^{10}$6$s^{2}$6$p^{3}$), 14 for Fe (3p$^{6}$3$d^{6}$4$s^{2}$), and 6 for oxygen (2$s^{2}$2$p^{4}$). We used a 6$\\times$6$\\times$6 Monkhorst-Pack {\\bf k}-mesh for the\nBrillouin-zone integration and 400-eV energy cutoff. Tests using a\n8$\\times$8$\\times$8 mesh as well as 600 eV cutoff\ndid not give significant differences in the calculated properties.\n Brillouin zone integrations are performed with a Gaussian broadening of 0.1 eV during all relaxations.\nThe experimental unit cell for the $R3c$ (ferroelectric phase)\nwas used as an input in the full-optimization procedure.\n For this phase as well as for  the paraelectric one (see below), we used the rhombohedral setting. The geometries were relaxed until\nall force components were less than\n0.01 eV\/\\AA\\ and the stress tensor components less than 50 meV\/cell. The spin configuration was fixed in order to reproduce the G-type antiferromagnetic state of BiFeO$_{3}$ and the spin-orbit coupling was neglected. For the paraelectric phase,\nwe used the non-polar R$\\bar{3}$c LiNbO$_{3}$ phase.\\cite{paraelectric}\nWe compute the \\textit{difference} of electric polarization, \\textit{i.e.} $\\Delta P=P^{FE}-P^{PE}=\n(P^{FE}_{ion}+P^{FE}_{ele})-(P^{PE}_{ion}+P^{PE}_{ele})=\n\\Delta P_{ion}+\\Delta P_{ele}$, where $FE$, $PE$, $ion$ and $ele$ denote ferroelectric, paraelectric, ionic and electronic contribution, respectively.\nFor the paraelectric phase, we used the same lattice constant and rhombohedral angle of the ferroelectric one.  Note that, although counterintuitively,\n$P^{PE}_{ele}$ may be different from zero, as explained in Ref.\\ \\onlinecite{Neaton}.  $P^{FE,PE}_{ion}$  is calculated  by summing the position of each ion in the unit cell\n times the  number of its valence electrons.\nThe electronic contribution is obtained by using the Berry phase formalism,\nwithin the ``modern'' theory of polarization.\\cite{berry1,berry2,berry3}\n\nConcerning the HSE calculations, due to the high computational load,\\footnote{To have an idea of the increased computational cost involved\n in the HSE calculation,\nwe note that, by considering the same computational setup for BiFeO$_{3}$, \neach electronic minimization step takes about 50 times more CPU time than PBE or PBE+U. \nThis means that if a PBE (or PBE+U) self-consistent calculation takes 10 minutes, \nthe HSE will take about 9 hours.} we always used the 400 eV and 6$\\times$6$\\times$6 {\\bf k}-point mesh. The Fock exchange was sampled using the twofold reduced \\textbf{k}-point grid (using the full grid, gives however negligible changes in the computed properties).\nFinally, we performed G$_{0}$W$_{0}$ calculations\\cite{GW,GW1,GW2,GW3,GW4,GW5}\n on top of the HSE electronic and ionic structure, which usually represent a good starting point for a perturbative quasiparticle excitation energies.\\cite{GW1} We also included vertex correction in W via an effective nonlocal exchange correlation kernel.\\cite{GW4}\n\nFor orthorhombic HoMnO$_{3}$, the $Pnma$ symmetry is chosen with the $b$ basis vector as the longest one. The paraelectric phase was simulated\nusing 20-atoms cell in the AFM-A spin configuration showing ferromagnetic (FM) (AFM) intraplanar (interplanar) coupling; for the ferroelectric one\nwe used a 40-atoms cell (doubling the previous cell along the $a$ axis) in the AFM-E spin configuration (\\textit{i.e.} in-plane FM zigzag chains anti-ferromagnetically coupled to the neighboring chains with the interplanar coupling also AFM).\nThe energy cutoff was set to 300 eV and the Brillouin zone mesh was fixed to 4$\\times$2$\\times$4 and 2$\\times$2$\\times$4 grid for the AFM-A and AFM-E phase respectively. Ho $4f$ electrons were assumed as frozen in the core.\nThe experimental lattice constants were used for all the calculations but the internal positions were relaxed.\nFor the HSE calculations, the Fock operator was evaluated\non the down-folded \\textbf{k}-point mesh. In order to assess the relative stability of the two magnetic phases,\nwe used the same simulation cell containing 40 atoms for both phases, increasing the cutoff to 400 eV and using a 4$\\times$2$\\times$4 {\\bf k}-point grid.\n\n\n\\section{BiFeO$_{3}$: results and discussions}\\label{BFO}\n\\subsection{Structural properties}\nThe ferroelectric structure is represented by a distorted\ndouble perovskite structure with $R3c$ symmetry (N. 161, point group C$_{3v}$)\nas reported by Kubel and Schmid.\\cite{Kubel} The paraelectric phase has $R\\overline{3}c$\nsymmetry (N. 167, point group D$_{3d}$). Both phases are shown in Fig.\\ref{bfo.1}.\nIn Table\\ \\ref{tab1} we report relevant properties such as the  structural parameters, the Fe magnetic moment\n and the energy gap calculated using the PBE and HSE functionals. We also report the values using the B1-WC functional taken from Ref.\\cite{Goffinet}\n\nFirst of all, HSE reduces the lattice parameter $a_ {rh}$\nwith respect to PBE, giving a much better agreement\nwith the experimental value: the error decreases\nfrom $\\sim$1 \\%(PBE) to $\\sim$0.3 \\%(HSE).\nAs a consequence,\n the unit cell volume $V$ also shrinks,\ngetting closer to the experimental value.\nThe rhombohedral angle, $\\alpha_{rh}$,  is almost insensitive to the applied functional.  \nThus, the inclusion of  Fock exchange makes the structure more compact, \\emph{i.e.}  \nthe lattice constant decreases.\n Note that the B1-WC functional gives too small lattice constant and too small equilibrium volume as compared to HSE, worsening the comparison with the experiments.\nThere is a very good agreement between the\nrelaxed coordinates of the\nWyckoff positions and the experimental ones\nusing HSE, while the PBE as well as  the B1-WC functional give slightly worse  results (\nthe only exception being the $x$ component of the oxygen atoms in the $6b$ site symmetry).\nIn the experimental structure, the BiO$_{6}$ cage is strongly distorted with three coplanar   nearest\nneighbors (NNs) lying above Bi along [111] at 2.270 \\AA \\ (d$^{s}_{Bi-O}$, $s$ refers to short) and three\n NNs sitting below at 2.509 \\AA \\ (d$^{l}_{Bi-O}$, $l$ refers to long).\nFrom Table\\ \\ref{tab1}, we see that the theoretical\n NNs distances  compare well with   experiments, with errors from\n $\\sim$ 1 \\% to  $\\sim$4 \\% (PBE), from $\\sim$ 3 \\% to $\\sim$ 5 \\%\n(B1-WC) and  $\\sim$ 2 \\% (HSE).\nThe O-$\\widehat{Fe}$-O bond angle would be\n180$^{\\circ}$ in the ideal\ncubic perovskite. In this system, it buckles to an experimental\nvalue of 165.03$^{\\circ}$. The HSE value (164.56$^{\\circ}$) is close to  PBE, and in both cases, they are slightly underestimated with respect to experiment. The B1-WC angle, on the other hand, is clearly underestimated.\nOverall, the predicted HSE values clearly are in much better  agreement with experiments than those calculated using the PBE or B1-WC functional.\n\n\\subsection{Electronic and magnetic properties}\nLet us consider now the magnetic and  electronic properties.\nAs shown  in Table\\ \\ref{tab1},\nthe calculated local moments are generally very similar for all the functionals,\nand close to the experimental value. In particular,\nthe HSE local moment is slightly larger than   PBE , suggesting\n a more localized picture of the spin-polarized electrons.\nThe calculated  PBE (HSE) electronic energy gap is\n 1.0 (3.4) eV. The expected band-gap opening   using hybrid\nfunctionals can be understood as follows:\nthe exact exchange  acts on occupied states only, correcting them for the self-interaction, thus\nshifting downwards the occupied valence bands. In turn, this has a clear interpretation: within the Hartree-Fock approximation for the ground state\nof an N electron system, the potential felt by each of the N electrons in the ground state is that due to N-1 other electrons, \\textit{i.e.} they feel a more attractive ionic potential. On the other hand, for unoccupied states, the potential is that due to the N occupied orbitals, so these orbitals effectively experience  a potential from one more electron, the latter screens the ionic potential which in turn becomes less attractive. Therefore, the unoccupied states are shifted upwards, opening the gap. \n\nAs for the experimental energy gap for BiFeO$_{3}$, the situation is not clear.  There have been several measurements of the band gap using UV-visible absorption spectroscopy and \nellipsometry on polycrystalline BFO films, epitaxial BFO films grown by pulsed-laser deposition, nanowires, nanotubes, and bulk single crystals. Reported band-gap values vary from 2.5 to 2.8 eV.\\cite{gap1,gap2,gap3,gap4,Kanai,Gao}\nAn estimate gives $\\sim$ 2.5 eV from the optical absorption\nspectra by Kanai \\textit{et al.}\\cite{Kanai} and Gao\\textit{ et al. }\\cite{Gao}. From  the theoretical side, there is a spread of values: a small gap of 0.30-0.77 eV using LSDA,\\cite{Neaton}\n or from 0.3 to 1.9 eV using ``LDA+U'', depending on the value of\nU\\cite{Neaton}; 0.8-1.0 eV using PBE (WC) GGA functional;\\cite{Bilc}\n3.0-3.6 eV using B1-WC and B3LYP hybrid functionals.\\cite{Bilc}\nThus, a parameter-free \n theoretical reference  value\nis  clearly needed. The most accurate (but expensive) method is the GW approximation.\\cite{GW} Here, we provide for the first\ntime, the value of the BFO energy gap based on the GW method.\n\nFirst of all,  at the PBE level, we estimate an energy gap of 1.0 eV. When introducing the exact exchange (HSE),  the gap opens up to E$_{g}$=3.4  eV.\nUpon inclusion of many-body effects (G$_{0}$W$_{0}$),\n it  opens even more (E$_{g}$=3.8 eV).\n Finally, when including vertex corrections, we find that   the gap reduces to 3.3 eV. Remarkably, vertex corrections almost confirm the HSE band gap.\nThis is perfectly in line with recent works\\cite{ScuseriaReview} where it is argued  that HSE band gaps\nrepresent a  very accurate estimate due to\npartial inclusion of the  derivative discontinuity of the exchange-correlation functional.\\cite{failures} Clearly, effects beyond bare DFT are important in this compound.\nOur results show that hybrid-functional calculations\ngive already a very good estimate at a lower computational cost compared to GW. In this respect, we mention the very recent experimental study based on resonant\nsoft x-ray emission spectroscopy\\cite{Guo-exp-gap} where  the\nband gap corresponding to the energy separation between the top of the O $2p$\nvalence band and the bottom of the Fe $d$ conduction band is 1.3 eV.\nThe discrepancy between theory and experiment may be due to the presence\nof defects in the experimental sample  as well as\n to the resolution involved in photoemission spectra. We hope to stimulate further experimental work to test our first-principles prediction of the\nenergy gap  of this important multiferroic material.\n\nIn Fig.\\ \\ref{fig1} we show the Density of States (DOS) for the optimized\natomic structure.\n Let's focus on the PBE DOS. The lowest states at $\\sim$ $-$ 10 eV\nare Bi $s$  hybridized with O $p$ states (blue curve).\n Above $-$ 6 eV  there are hybridized O $p$ and Fe $d$ states. The Fe $d$ states extend  in the conduction band as well; the Bi $p$ states can be found above 4 eV.  As for the HSE DOS, we see that  the conduction bands are shifted upwards, opening the valence-conduction gap.\n There is  a change in the spectral distribution  above $-$ 8 eV: a valley appears around $-$ 6 eV  and the lower shoulder of the peak increases its spectral weight. It is easy to trace back the above changes to modifications of the  majority Fe $d$ states, as shown in the panel beneath: while in PBE the band states in the vicinity of the top of the valence band have predominantly $d$ character, in HSE the spectral weight of the Fe $d$ states is concentrated far away from the top of the valence band. This can be interpreted as a change from a more itinerant picture to a more  localized description of the Fe $d$ states  going from PBE to HSE.\nIn Fig.\\ \\ref{fig1}, we also include the spectral distribution of the Fe $d$  states derived from a recent experimental work\\cite{Guo-exp-gap} (see dotted lines): the position of the main HSE Fe $d$ peak almost perfectly matches the experimental PDOS, although the bandwidth of the calculated DOS is different because of energy resolution,  etc. Indeed, if we include the Fe $d$ DOS calculated using G$_{0}$W$_{0}+$ vertex corrections,\nthe agreement between the theoretical and experimental peak position becomes excellent. \nNote that the HSE and G$_{0}$W$_{0}+$Vertex peak position are very close to each other,\n confirming the accurate HSE description of the BiFeO$_{3}$ electronic structure. \nIn passing we note that while DFT+$U$ gives a better estimate \nof the Fe $d$ peak position,\\cite{Neaton}\nthe energy gap is still underestimated with respect our GW calculation.  \n \nWe previously mentioned that HSE may change the ionic\/covalent character in this compound.\nTo support this, in Fig.\\ \\ref{fig2} we show the difference between the PBE and HSE charge density, $\\Delta \\rho=\\rho^{PBE}-\\rho^{HSE}$ calculated at fixed geometry.\nIn a purely ionic description, all the valence electrons would be located on the oxygens,  acting as ``electron sinks'', and the cations would donate their nominal valence charge.\nThe more the electrons populate the anions, or conversely, the more the electrons depopulate the cations, the more the picture shows an ionic character. Fig.\\ \\ref{fig2} confirms the trend discussed before: upon adding  a fraction of exact exchange to the PBE functional, the electronic charge at the cations decreases,\n \\textit{i.e.} $\\Delta \\rho$ is positive (grey areas in Fig.\\ \\ref{fig2}), while the electronic charge at the anions  increases \\textit{i.e.} $\\Delta \\rho$\nis negative (yellow areas in Fig.\\ \\ref{fig2}). Thus, the introduction of exact-exchange generates a flux of charge from the cations towards the anions, clearly shown in Fig.\\ \\ref{fig2}, increasing the ionicity of the compound.\n\nIn order to discuss more quantitatively these effects,\nwe perform a Bader analysis\nof the electronic charge.\\cite{Bader1,Bader2,Bader3}\nThe atom in molecules (AIM) theory is a well established analysis tool for studying the topology of the electron density and suitable for\n discussing the ionic\/covalent character of  a compound. The charge ($Q_{B}$) enclosed within the Bader ($V_{B}$) volume is a good approximation to the total electronic charge of an atom.  In Table\\ \\ref{bader} we report $Q_{B}$ and $V_{B}$ calculated for Bi, Fe, and O at a fixed geometric\nstructure, \\textit{i.e.} HSE geometry. This is needed in order to avoid\ndifferent volumes for the normalization of the charge in the unit cell and for highlighting the electronic structure modifications due to the exact exchange. Furthermore, we consider only the valence charge\nfor our analysis  (although one should formally include also the core charge, we do not expect variations as far as the trends are concerned).\n Let us first consider the cations:\nthe Bader charge and volume are larger in PBE than in HSE. For the anions, the opposite holds true.\nThis is not unexpected and in agreement with intuition: upon introducing Fock exchange, the system\nevolves towards a more ionic picture, through a  flux of charge from cations towards anions,\nwhich reduces (increases) the ``size'' of the cations (anions) when going from the PBE to the HSE solution.\nFinally, we note that a different degree of ionicity modifies   the calculated equilibrium lattice parameter: in a partially covalent material, such as BiFeO$_{3}$,\\cite{Eriksson} the increased ionicity changes the different net charges generating a higher Madelung field,\n which is an important contribution to the bonding in the solid, and contracts the equilibrium structure.\\cite{cora1,cora2}\n\n\\subsection{Ferroelectric properties}\n\n\nLet's finally focus on the electric polarization. In Table\\ \\ref{tab4}, we report the ionic and electronic contributions to the difference of ferroelectric polarization  between the polar ($R3c$) and non-polar ($R\\overline{3}c$) both in PBE and HSE. In order to disentangle the purely electronic effects from the ionic ones\nupon introduction of Fock exchange, \nwe report also the PBE(HSE) electronic contribution calculated at fixed HSE(PBE) geometry.\n\nAs a general comment, we note that a large polarization of $\\sim$ 100 $\\mu C \/cm^{2}$ along (111) for bulk BFO has been reported experimentally by new measurements on high-quality single crystals.\\cite{BFOexp}  in good agreement with our calculated values.\nIn what follows, we will mainly focus on the differences between PBE and HSE calculations. Note that the unit cell volume\n  is different for PBE and HSE, as shown in Table\\ \\ref{tab1}.\n\nFirst, we note that the polarization calculated according\nto the point charge model (P$_{pcm}$) is closer to P$_{tot}$ at the\nHSE than at the PBE level.\n This confirms that the HSE description of BFO points towards an  ionic picture,\n \\emph{i.e.} by decreasing the covalency effects.  \nThe calculated total polarization P$_{tot}$ is $\\sim$ 105 $\\mu C\/cm^2$ using PBE and $\\sim$ 110 $\\mu C\/cm^2$ using HSE, \\textit{i.e.} HSE predicts an increase of \\textit{total} electric polarization. The occurrence of ferroelectricity \n in BiFeO$_{3}$ is usually discussed in terms of ``polarizable lone pair'' carried by the Bismuth atom. This has a physical interpretation in terms of cross gap hybridization between occupied O $2p$ states and unoccupied Bi $6p$ states.\\cite{Singh1,Singh2,Singh3,Singh4,Eriksson}. Intuitively, \n the larger  the energy gap, the lower the polarization should be. Accordingly, one might expect   HSE  to reduce the polarization compared to PBE because of the larger energy gap. We will show below that this is not  in contraddiction with the results of\nTable\\ \\ref{tab1}. In fact, let's consider P$_{tot}$ calculated at the \\textit{same} atomic structure \n(for example at the PBE relaxed structure of the paraelectric and ferroelectric phases)\nbut using both PBE and HSE. We denote the former as  P$_{tot}^{PBE}$, the latter as P$_{tot}^{HSE(PBE)}$. Note that the ionic contributions is of course the same for both cases. From Table\\ \\ref{tab1}, we have P$_{tot}^{PBE}$=105.6 $\\mu C\/cm^2$  and  P$_{tot}^{HSE(PBE)}$=103.2  $\\mu C\/cm^2$, \\textit{i.e.} a decrease of total polarization is found in going from PBE to HSE for the same ionic structure. A similar behavior is found in opposite conditions: for the HSE ionic structure, P$_{tot}^{HSE}$=110.3  $\\mu C\/cm^2$ and P$_{tot}^{PBE(HSE)}$=112.6  $\\mu C\/cm^2$.\nThus, keeping the same volume and including Fock exchange, the polarization reduces as   expected.\n On the other hand, when we evaluate \nthe \\textit{total} polarization at the appropriate equilibrium and relaxed structures using PBE and HSE , the ionic contribution also varies and one loses a direct connection between the increase of the \nenergy gap  and the decrease of total polarization. In our case, the total polarization, when evaluated at the appropriate equilibrium volume for each functional, increases from PBE to HSE. This clearly points out a strong volume-dependence of the polarization, therefore calling for a correct \nestimate  of the volume (as provided by HSE).\n\n\n\n\n\n\n\n\n\\section{HoMnO$_{3}$:results and discussions}\n\\label{HMOSect}\n\\subsection{Paraelectric AFM-A phase} \\label{A-HMO}\n\\subsubsection{Structural properties}\nAn extended review of the main properties of orthorhombic HoMnO$_{3}$\n  within a standard PBE approach\ncan be found in Ref. \\cite{HMO4} where it is also shown that the inclusion of the $U$ correction\nworsens the structural properties. Therefore, in this paper,\nwe will focus on the comparison between the predictions of HSE with respect to  PBE results. \nIn Fig.\\ \\ref{HMO} we show the perspective view of HoMnO$_{3}$ and the\nparaelectric (AFM-A) and ferroelectric (AFM-E) spin configurations in the $c-a$ plane.\n\nIn Table\\ \\ref{tab.afma} we report the optimized structural\nparameters in the AFM-A magnetic configuration, calculated\nusing  PBE and HSE.\nThe in-plane\nshort ($s$) and long ($l$) Mn-oxygen bond-lengths\nget closer to experimental values using HSE; on the other hand,\nthe  out-of-plane length is slightly overestimated with respect to\nthe PBE and the experimental value. In order to quantify   structural distortions,\n  the  Jahn-Teller (JT) distortion vector $\\textbf{Q}=[Q_{1},Q_{2}]=[\\sqrt{l-s},\\sqrt{\\frac{2}{3}}(2m-l-s)]$\n  is often introduced.\n From Table\\ \\ref{tab.afma}, it is\nthus clear that the HSE  functional improves the JT distortion\n upon the PBE description: the magnitude of $\\textbf{Q}$ is Q=0.55 \\AA \\ and\n0.61  \\AA \\ using PBE and HSE, respectively, whereas the experimental value is 0.59 \\AA.\n As far as the structural angles are concerned, we first notice that the GdFeO$_{3}$-like tilting ($\\alpha$) in the Mn-O$_{6}$ octahedron is slightly overestimated using the hybrid functional:\n the deviation from the experimental angle is 3.7 (PBE) and 5.0 \\% (HSE) with the Mn-O-Mn in-plane  angles calculated using HSE  slightly reduced\nwith respect to  PBE. We note, however, that the experimental uncertainty    on the angles may be up to $\\sim$ 1$^{\\circ}$, \\cite{HMO3}\ndue to synthesis problems of ortho-HoMnO$_{3}$.\n\\cite{HMO2,HMO3} The octahedral tilting is  related to \n the ionic size of the rare-earth ion:\\cite{Tilting,Tilt2} the tilting increases when the radius of the rare-earth atom decreases (for example, from La to Lu in the manganites series).\\cite{HMO4} In this respect, the tendency towards a larger octahedral tilting, upon inclusion of exact-exchange-functional, goes hand in hand with the  reduced ionic size of Ho ion when going from PBE to HSE. As in the previous case,\nwe performed a Bader analysis of the valence charge distribution. Results are shown in Table\\ \\ref{bader}: as expected, the ''size`` of the Ho ion is reduced within  HSE. \n According to\nour previous discussion, the ionic\/covalent character of the charge density is modified by HSE\nin favor of a more ionic picture. This will have important consequences for the\nelectronic polarization, as shown below.\n\n\\subsubsection{Electronic properties}\n\nIn Fig.\\ \\ref{HMO.bande} we show the band structure for the AFM-A phase as calculated using standard PBE (left panel) as well as HSE (right panel) along the main  symmetry lines.\nThe PBE band structure shows a small gap equal to $\\sim$ 0.2 eV. The bands below $\\sim$ $-$2  eV are mainly oxygen $p$ states\nand those  2-3 eV below (above) the Fermi level are mainly spin-up (spin-down)\nMn $d$ states. There is also a considerable weight of the Mn $d$ states \nin the oxygen bands near the top of the valence band.\\footnote{Although the set of ``$t_{2g}$'' or ``$e_{g}$''\norbitals is well defined in a local coordinate frame centered on each Mn ion, this is not any more true when using\nthe standard orthorhombic system as a global coordinate frame, due to different tilting angles and distortions on neighboring MnO$_6$ cages. Thus, our discussion for the Mn $d$ states has only a qualitative meaning.}\n \n\nThe group of bands between $-$ 1 and $-$ 2 eV are mainly $d_{xy}$,$d_{yz}$ with some small weight of $d_{z^{2}}$.\nThe  two  bands just below the Fermi energy are  mainly $d_{x^{2}-y^{2}}$-like with some $d_{xz}$ weight; at   $Y$,$S$,$Z$   they become degenerate.\nHigher in energy, between 0 and 1 eV,\nthere are two  more bands showing a similar behavior, \\textit{i.e.} degenerate at $Y$,$S$,$Z$, almost degenerate along $Z$-$R$ and with a similar overall band dispersion. Even  higher in energy,\nthe are the  Mn minority states.\nFrom Fig.\\ \\ref{HMO.bande}, we can extract the JT splitting ($\\Delta_{JT}$ of the ``$e_{g}$'' states), the CF splitting ($\\Delta_{CF}$ between ``$e_{g}$'' and ``$t_{2g}$''), and the exchange splitting ($\\Delta_{EX}$ between majority and minority spin states, evaluated at the $S$ point, for simplicity). We thus have\n $\\Delta_{JT}$=1.05, $\\Delta_{CF}$=2.34, and $\\Delta_{EX}$=2.75 eV.\nThe comparison between the PBE and HSE band structure highlights some\ndifferences.  First we note that the oxygen bands are slightly shifted to lower binding energy together with the Mn $d_{xy}$,$d_{yz}$ bands. On the other hand, the average position of occupied Mn $d_{xz}$, $d_{x^{2}-y^{2}}$  bands remain almost\nunchanged, but the band-width increases. The latter effect is mainly   shown by the lowest  of the two bands, \\textit{i.e.} the $d_{xy}$-like band. As   well known,\\cite{MartinBook}\nHartree-Fock hamiltonians naturally leads to larger band-width and down-shift of electronic states, even for\nsimple homogeneous systems. Thus, the larger band-widths obtained by HSE can not be simply  connected\nto a stronger hybridization, because it is an intrinsic feature of Fock exchange for every electronic state.\nIndeed, we will show below that the $d-p$ hybridization is expected to decrease using HSE.\n \n$\\Delta_{JT}$ is evaluated as  3.4 eV, larger than in the PBE case, suggesting a stronger local distortion related to the  Jahn-Teller instability. The increase of the JT  splitting is linked to the increase of the energy gap as well, which is now  $\\sim$ 2.7 eV. The exchange splitting, $\\Delta_{EX}$=4.6 eV, is also larger along with an  increased  local Mn moments with respect to PBE.\nNote, that the enhancement of Jahn-Teller distortion by HSE is not unexpected. In fact,\nit was previously suggested that the HSE functional is able to reveal the Jahn-Teller effect for Mn$^{+4}$ through a \\textit{symmetry broken solution}\ngiving rise to an orbitally ordered state and consequent Jahn-Teller distortion.\\cite{Stroppa3}\nThe larger $\\Delta_{JT}$ is mainly driven by a purely electronic effect due to the inclusion of  Fock-exchange. Infact, by calculating the PBE self-consistent charge density on top of the HSE ionic\nstructure, $\\Delta_{JT}$ becomes $\\sim$ 1.2 eV, \\textit{i.e.} nearly equal to the previous PBE case.\nThus, we expect  HSE to cause a rearrangement of the charge density that will \\textit{reduce} the\nelectronic contribution to the electronic polarization when considering the polar phase.  This  can be understood as follows.\nThe increase of the Jahn-Teller splitting goes hand in hand  with the increase of the energy gap:  the larger  the  gap, the smaller  the  dielectric constant is, \\textit{i.e.} the smaller the screening is. Now, let us consider the fixed ionic configuration of the paraelectric phase: the charge in the non-centrosymmetric \\textit{spin arrangement}   can be thought as a ``small'' perturbation of the centrosymmetric one   upon the application of the  ``internal'' electric field.\nThe electronic charge will  respond to such a field, and  each electronic state will change assuming a polarized configuration. If the gap is large, the ``electric field'' will hardly mix the electronic states in the valence band with the electronic states in the conduction band, since in second order perturbation theory approach the denominator will be of order of the band gap energy,\n so that  the electrons don't polarize much, \\textit{i.e.} the charge distribution becomes more ``rigid''. In conclusion, we expect  the electronic contribution to the electric polarization to decrease upon introduction of Fock exchange.\nThis will be confirmed by our calculations. Note that the above reasoning is not appropriate for BFO where the polarization is mainly due to  ionic displacements.\n\n\n\\subsection{AFM-E phase}\\label{E-HMO}\n\\subsubsection{Structural properties}\nLet us focus on the AFM-E phase, where the resulting symmetry is lowered by\nthe spin configuration with respect to the AFM-A spin arrangements by removing\n the inversion symmetry.\nIn Fig.\\ \\ref{chain}, we show the relevant structural internal parameters, for the relaxed PBE and HSE structure,  by considering the Mn-O-Mn-O-Mn chain (compare Fig.\\ \\ref{chain} and Fig.\\ \\ref{HMO}).\n The Mn-O $short$  bond-lengths do not show significant differences\nbetween the parallel and antiparallel spin configuration  in PBE as well in HSE. On the other hand, the long Mn-O bond lengths are mostly affected:\ntheir difference, $l^{p}$ - $l^{ap}$ in PBE is about 0.07 \\AA \\ and decreases to $\\sim$ 0.02 \\AA \\ upon introduction of exact exchange.\nAt the same time, the angle changes: $\\alpha^{p}$ decreases while $\\alpha^{ap}$\nremains almost equal to the PBE value.\n  The results can be interpreted as follows: bare PBE is expected to overestimate hybridization effects between oxygen $p$-states and Mn $d$-states, therefore inducing a stronger rearrangement of ionic positions consistent with a ``softer\" structure when moving from, say a centrosymmetric A-type to a ferroelectric E-type phase. Viceversa, upon introduction of correlation effects, the reduced hybridization is expected to lead to a more ``rigid\"  ionic arrangement. Indeed, this is evident when comparing\nthe difference between   $\\alpha^{p}$ - $\\alpha^{ap}$, which drastically reduces upon introduction of HSE with respect to PBE. We recall that, ultimately, it is this difference that gives rise to the ionic polarization, as clearly shown in Fig. 2 b) of Ref.\\cite{HMO1}. We can therefore anticipate that a reduction of the polarization will occur upon introduction of HSE, as discussed in detail below.\nWhat is worthwhile noting is that the Mn-Mn distances in PBE dramatically depends on their having parallel ($d_{Mn-Mn}^p$ = 3.98 \\AA) or antiparallel spins ($d_{Mn-Mn}^{ap}$ = 3.87 \\AA), so that\n$d_{Mn-Mn}^p - d_{Mn-Mn}^{ap}$ = 0.11 \\AA. However, this dependence is smoothed upon introduction of HSE, so that the difference in Mn-Mn distance strongly reduces to  $d_{Mn-Mn}^p - d_{Mn-Mn}^{ap}$ = 0.03 \\AA. In general, the marked (weak) dependence of the structural properties within PBE (HSE) is consistent with a strong (small) efficiency of the double--exchange mechanism, which ultimately relies on the $p-d$ hybridization and hopping integral.\n\n\\subsubsection{Electronic and magnetic properties}\n The band-structure of the AFM-E is quite similar to the A-type and is therefore not shown. However, there are some small differences which we comment on.\n As expected, the increase of the number of the AFM bonds\nof each Mn with its four neighbors  associated with the change of the magnetic state going from AFM-A to AFM-E type results\nin a narrowing of all bands. This is  further enhanced by HSE due to the  reduced hopping upon introduction of exact exchange, as expected. Furthermore, the increase of the\nband gap is facilitated by the interplay of the crystal distortion,\nwhich is generally enhanced by HSE,  with\nthe AFM arrangement of spins. As expected, the energy gap\nis the largest in the AFM-E-HSE band structure, being now $\\sim$ 3 eV.\nThe $\\Delta_{JT}$ evaluated at $S$ point, is also the largest in this case,\nbeing $\\sim$ 3.7 eV.  \nBefore turning our attention to the electronic polarization,\nwe discuss the magnetic properties. First of all,\nwe found that the AFM-E is more stable than the AFM-A by $\\sim$ 4 meV\/cell in the HSE\nformalism. Note that this value has been obtained using the same simulation cell for both phases, therefore reducing the influence of numerical errors. Although the relative stability \nis still comparable with the numerical accuracy, it is indeed consistent   with experiments.\\cite{HMO2,HMO3} In AFM-E-PBE, the Mn moment is 3.4 $\\mu_{B}$ which induces a small spin-polarization on the oxygen equal to $\\pm$ 0.04 $\\mu_{B}$. In AFM-E-HSE, the Mn moment slightly increases to 3.7 $\\mu_{B}$\nand the oxygen moment slightly decreases to $\\pm$ 0.01 $\\mu_{B}$:\nthe increased localization of the Mn $d$ states correlates with the increased\nManganese spin moment and goes hand by hand with the  decreased $p$-$d$ hybridization and a decreased induced spin moment on oxygens.\n\n\\subsubsection{Ferroelectric properties}\nFinally, we calculated the ferroelectric polarization by considering the AFM-A as the reference paraelectric structure.\nThe results show that the polarization (both the electronic and ionic terms) strongly reduces upon introduction of HSE. However, it is remarkable that the total P is still of the order of 2 $\\mu C\/cm^2$:\n this confirms HoMnO$_3$ as the magnetically-induced ferroelectric having the highest \n polarization predicted so far. Our estimate is in very good agreement with model Hamiltonian\n calculations.\\cite{dagotto}  The comparison between\ntheory and experiments as far as the electric polarization is concerned is still a matter of debate. Whereas earlier studies predicted negligible values for polycrystalline HoMnO$_{3}$ samples,\\cite{Lorenz}  more recent studies for TmMnO$_{3}$\nin the E-type (where the exchange-striction mechanism is exactly the same as in HoMnO$_{3}$) point to a polarization which could exceed 1 $\\mu C\/cm^2$,\\cite{Yu}\nin excellent agreement with our predicted HSE value.\n\nThe reasons why we expect a reduction upon introduction of HSE have been already discussed in previous paragraphs and can be traced back to the reduced $p-d$ hybridization. As in the case of BFO, we disentangle the structural and electronic effects, by using the HSE (PBE) geometry with the PBE (HSE) functional \n(cfr.  Table\\ \\ref{tab4}). What we infer from these ``ad-hoc\"-built systems is that the use of\n HSE dramatically reduces the electronic contribution (cfr $P_{ele}$ in PBE and HSE(PBE)), {\\em i.e.} reduced by $\\sim$ 2.5 $\\mu C\/cm^2$. Less important, though still appreciable, seem to be the ionic displacements: their dipole moment is reduced by $\\sim$ 1.5 $\\mu C\/cm^2$ when comparing $P_{tot}$ in HSE and HSE(PBE). This is consistent with what shown approximately by P$_{pcm}$.\n\n\n\n\\section{Conclusions}\\label{conclusions}\nIn this work, we have revised the two workhorse materials of the exponentially growing\nfield of multiferroics, namely BiFeO$_{3}$ for proper MFs and HoMnO$_{3}$\nfor improper MFs by using the  screened hybrid functional (HSE).\n\nFrom our study,  several important points emerge.\nFor BFO:\ni)  the structural, electronic and magnetic properties well agree  with experiments;\nii) the ferroelectric polarization agrees with reported values in the literature;\niii) even if PBE allows the description of ferroelectric properties by\n opening an energy gap, it is by no means  satisfactory in correctly describing\n all the properties on the same footing. On the other hand, HSE improves the  PBE and LDA+$U$ description; this is clearly shown by benchmark\ncalculations using the most advanced and accurate\nstate-of-the art GW+vertex corrections (which basically confirm the HSE results);\niv) the previous comment, and very recent studies\\cite{ScuseriaReview}\n suggests that optical properties, so far not investigated at all by ab--initio calculations for BFO, can be properly addressed within HSE.\nv) finally, we note that the electronic polarization \\textit{increases}  upon introduction of exact exchange. \nFor HMO, we note that:\ni) the HSE results are in good agreement with experiments when available;\nii) the  Jahn-Teller effect is correctly described in agreement with experiment;\niii) despite a reduction of the polarization value with respect to PBE, HoMnO$_{3}$ still shows the highest $P$ predicted among magnetically-induced ferroelectrics.\n\nWe have shown that\n introduction of ''correlation`` effects may both enhance the polarization\nor reduce it: the former effect will most\nlikely occur for proper MFs, and the latter\nfor improper MFs, \\textit{e.g.} magnetically driven.\nNote that an increase of HSE polarization with respect to LDA, for example,\n is also found by Wahl \\textit{et al.}\\cite{Roman} for BaTiO$_{3}$, a standard ferroelectric compound. \nAlso for BiFeO$_{3}$, an increase of polarization using DFT+$U$ \nhas been noticed.\\cite{Eriksson,Neaton,BFO.LDAU} The increase of ferroelectric polarization\nwhen including a fraction of exact exchange and using the theoretical equilibrium \nvolume has been reported also for simple ferrolectric compound such as KNbO$_{3}$.\\cite{cora1}\n \n\n\nOne final comment is in order: although the HSE results certainly point\ntowards a truly realistic description,\nit is still possible that, to some extent,\n the good performances of HSE may be material-dependent, \\textit{i.e.} the universal 1\/4 fraction of the exact exchange may be not appropriate for some  specific material. What is certainly true is that\nthe predictive capability of HSE, combined with\n its nowadays affordable computational costs, make the functional an attractive\n choice for the study of a wide range of materials,\n from well-behaved insulators to doped semiconductors\n exhibiting magnetic ordering, multifunctional complex oxides  of interest for many industrial applications therefore representing a very good starting point for materials design.\n\n\n\\acknowledgments\nThe research leading to these results has received funding from the European Research Council under the European Community,  7th Framework Programme - FP7 (2007-2013)\/ERC Grant Agreement n. 203523. A.S. would like to thank  G. Kresse for kind assistance for the GW calculations and M. Marsman for useful discussions. Furthermore,\nA.S. thanks  L. Kronik (Weizmann Institute) for helpful comments. \nThe authors acknowledge kind hospitality at the S$^3$ CNR-INFM National Center in Modena after the catastrophic earthquake  of April 6$^{th}$ 2009 in L'Aquila.\nThe computational support by Caspur Supercomputing Center in Roma and technical assistance by \nDr. L. Ferraro is gratefully acknowledged. Figures have been done by using the VESTA package.\\cite{vesta}\n\\newpage\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDisentangling the highly complex and densely grown nerve fiber network in the brain is key to understanding its function and to developing treatments for neurodegenerative diseases. Especially the detailed reconstruction of crossing, long-range nerve fiber pathways in densely packed white matter regions poses a major challenge for many neuroimaging techniques. Diffusion magnetic resonance imaging allows to measure the spatial orientations of crossing nerve fibers, but only with resolutions down to a few hundred micrometers in post-mortem human brains \\cite{calabrese2018,roebroeck2018}, which is not sufficient to resolve individual nerve fibers with diameters in the order of 1\\,\\textmu m \\cite{liewald2014}. \nThe microscopy technique \\textit{3D Polarized Light Imaging (3D-PLI)}, on the other hand, determines the three-dimensional course of nerve fiber pathways in whole histological brain sections with in-plane resolutions of 1.3 \\textmu m, but yields only a single fiber orientation vector for each measured tissue voxel \\cite{MAxer2011_1,MAxer2011_2,dohmen2015,menzel2015}, leaving uncertainties if the brain section (with a thickness between 20\\,\\textmu m and 100\\,\\textmu m) contains several crossing nerve fibers.\n\nRecent studies \\cite{menzel2020,menzel2020-BOEx} have shown that nerve fiber crossings can be visualized with scattered light: When shining light in the optical regime through unstained, histological brain sections and studying the spatial distribution of scattered light behind the sample (\\textit{scattering pattern}), we obtain valuable information about the tissue substructure of the illuminated region, such as the individual directions of crossing nerve fibers.\n\nOne possibility to measure these scattering patterns is to illuminate the sample by a non-focused laser beam, place the camera in the back-focal plane of the lens, and measure the Fourier transform of the image plane (\\textit{coherent Fourier scatterometry} \\cite{menzel2020-BOEx}). However, the technique demands raster-scanning of the brain section, and the minimum diameter of the laser beam ($>$ 100 \\textmu m) limits the spatial resolution. \n\nThe recently developed neuroimaging technique \\textit{Scattered Light Imaging (SLI)} uses a reverse setup: Instead of measuring the scattering patterns for each brain region separately, the whole brain section is illuminated from different angles and the transmitted light is measured under normal incidence \\cite{menzel2021}. In contrast to coherent Fourier scatterometry, SLI can be performed with commercial LED light sources and reconstructs crossing nerve fiber directions for whole brain tissue samples with micrometer resolution. So far, SLI measurements have been performed with a fixed polar angle of illumination of about $\\theta \\approx 47^{\\circ}$ and azimuthal steps of $\\Delta\\phi=15^{\\circ}$ (\\textit{angular SLI measurement} \\cite{menzel2021}). Each pixel in the resulting image series is associated with a light intensity profile that shows peaks at different positions, revealing the directions of crossing nerve fibers. The software \\textit{SLIX (Scattered Light Imaging ToolboX} \\cite{slix}) enables the generation of human-readable parameter maps, showing e.\\,g.\\ the individual orientations of several crossing nerve fibers. However, because the accuracy is limited by the azimuthal step size, the light intensity profiles contain less information than the complete scattering patterns, and the fiber orientations cannot be reliably determined at the borders of the image due to asymmetric illumination.\n\nHere, we present SLI scatterometry, which allows measurements of full scattering patterns for each image pixel at once: Making use of an LED display with individually controllable LEDs (instead of the masked light source used in previous angular SLI measurements), we realize scatterometry measurements for whole brain tissue samples, providing detailed information about the tissue substructure and enabling an exact determination of the individual (crossing) nerve fiber directions. \nTo validate our results, we compare the measured scattering patterns to previous measurements with coherent Fourier scatterometry \\cite{menzel2020-BOEx} and angular SLI measurements \\cite{menzel2021}, see Sec.\\  \\ref{sec:comparison}. Finally, we present SLI scatterometry measurements of a human brain section with 3 \\textmu m in-plane resolution (Sec.\\ \\ref{sec:human-section}), and demonstrate that our technique enables new insights into the nerve fiber architecture of human brain tissue structures. \n\n\n\n\\section{Materials and Methods}\n\n\\subsection{Preparation of Brain Tissue}\n\\label{sec:methods-preparation}\n\nThe studies were performed on various post-mortem brain sections.\nFor the comparison in Sec.\\ \\ref{sec:comparison}, the exact same tissue samples were used as in \\cite{menzel2020-BOEx} and \\cite{menzel2021}: Two 60\\,\\textmu m thin, coronal sections of a vervet monkey brain (sections no.\\ 493 and 512), and three 30\\,\\textmu m thin sections of human optic tracts that were manually placed on top of each other to obtain a model of three crossing nerve fiber bundles (sections no.\\ 32\/33), cf.\\ Tab.\\ F1 in \\cite{menzel2021}. \nThe vervet monkey brain was obtained from a healthy adult male (2.4 years old) in accordance with the Wake Forest Institutional Animal Care and Use Committeee (IACUC \\#A11-219). Euthanasia procedures conformed to the AVMA Guidelines for the Euthanasia of Animals. The optic tracts were extracted from the optic chiasm of a human brain (female body donor, 74 years old, without known neurological\/psychiatric disorders) and cut along the fiber tracts of the visual pathway. \nThe coronal section of the human brain hemisphere shown in Sec.\\ \\ref{sec:human-section} is 50\\,\\textmu m thin and was obtained from a female body donor (89 years old, without known neurological\/psychiatric disorders).\nThe human brains were provided by the Netherlands Brain Bank, in the Netherlands Institute for Neuroscience, Amsterdam. A written informed consent of the subjects is available.\n\nWithin 24 hours after death, the brains were removed from the skull and fixed in a buffered solution of 4\\,\\% formaldehyde in which they remained for several weeks. Subsequently, they were cryoprotected in a solution of 20\\,\\% glycerin, deeply frozen, cut with a large-scale cryostat microtome (\\textit{Polycut CM 3500, Leica Microsystems}, Germany), mounted on glass slides, and stored at -80\\,$^{\\circ}$C. Prior to imaging, the brain sections were thawed, embedded in 20\\,\\% glycerin solution, and cover-slipped.\n\n\n\\subsection{Measurement Setup}\n\\label{sec:methods-setup}\n\nFigure \\ref{fig:setup}A shows the setup for the SLI scatterometry measurements. \nIt consists of an LED display (\\textit{INFiLED s1.8 LE Indoor LED Cabinet}) comprising $256 \\times 256$ individually controllable RGB-LEDs with a pixel pitch of 1.8\\,mm and a sustained brightness of 1000 cd\/m$^2$, a specimen stage, and a camera (\\textit{BASLER acA5472-17uc}) with $5472 \\times 3648$ pixels and $13.1 \\times 8.8$\\,mm$^2$ sensor size. \nThe distance between sample and camera objective was chosen to be about $L=40$\\,cm, the distance between light source and sample $H=13$\\,cm. The width of the LED display and the distance between light source and sample determine the maximum possible angle of illumination: $\\theta = \\arctan\\left(\\frac{256\\,\\cdot\\,1.8\\,\\text{mm} \/ 2}{13\\,\\text{cm}}\\right) \\approx 60.6^{\\circ}$.\nThe measurements were performed using an objective lens (\\textit{Rodenstock Apo-Rodagon-D120}) with a focal length of 120\\,mm and a full working distance of 24.3\\,cm, yielding an object-space resolution of 3.0\\,\\textmu m\/px and a field of view of 16.1 $\\times$ 11.0\\,mm$^2$. \nTo avoid detection of ambient light and suppress internal reflexes, the light path between sample and camera objective was sheltered by a light-absorbing conic tube (cf.\\ Fig.\\ \\ref{fig:setup}A).\nIn addition, a light-absorbing mask was placed on top of the specimen stage to cover everything outside the field of view and suppress reflexes in the camera objective (not shown in the figure).\nDuring a measurement, a square of $n \\times n$ illuminated RGB-LEDs (white light) was moved over the LED display ($m \\times m$ kernels) and an image was taken for every position of the square, yielding a series of $m \\times m$ images. Depending on the size of the illuminated square, the measurements were performed for different times of illumination (from 100\\,ms to 10\\,s) and different camera gain factors (from 1 to 27).\nTo reduce noise, up to four shots were recorded for the same position of illuminated LEDs and averaged.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig1_Setup.pdf}\n\\end{center}\n\\caption{\\textbf{(A)} Setup for SLI scatterometry measurement. \\textbf{(B)} Example of measured scattering pattern (obtained from an SLI scatterometry measurement with $2\\times2$ illuminated LEDs and $81 \\times 81$ kernels, shown in Fig.\\ \\ref{fig:chiasm}C(ii)) with corresponding azimuthal line profile (integrated from the center to the outer border, evaluated in steps of $\\Delta\\phi=1^{\\circ}$).}\n\\label{fig:setup}\n\\end{figure}\n\n\n\\subsection{Generation of Scattering Patterns and SLI Profiles}\n\\label{sec:methods-scatteringpatterns}\n\nFor every image pixel in the resulting series of $m \\times m$ images, a scattering pattern with $m \\times m$ pixels can be generated (cf.\\ Fig.\\ \\ref{fig:setup}B, top): The pixel value in the upper left corner of the scattering pattern, for example, shows the intensity value of the respective image pixel in the first recorded image (obtained when illuminating the sample with $n \\times n$ LEDs in the upper left corner of the display).\nIn this way, the scattering pattern shows the distribution of scattered light for the respective image pixel. \nWhen comparing the scattering patterns to measurements with coherent Fourier scatterometry, it should be noted that they do not show the distribution of scattered light on a hemisphere projected onto the xy-plane (as in \\cite{menzel2020-BOEx}, Fig.\\ 9a), but the distribution of scattered light on a plane (gnomonic projection). Therefore, the distance between the rings denoting steps of $\\Delta\\theta=10^{\\circ}$ increases with increasing $\\theta$ (cf.\\ Fig.\\ \\ref{fig:chiasm}C).\n\nAs every image pixel represents a different position in the sample, and hence a different position with respect to the center of the LED display, the center of the scattering patterns (i.\\,e.\\ the region of maximum brightness where unscattered light falls straight into the camera) varies between image pixels.\nIn order to evaluate the scattering patterns independent of their center position, the region of maximum brightness (centroid of pixels with maximum intensity) was determined for each scattering pattern. (As the scattering patterns are not always radially symmetric, we cannot simply use the centroid of the scattering patterns.) Subsequently, the scattering patterns were cropped to the maximum possible circle around the center (cf.\\ Fig.\\ \\ref{fig:setup}B, dashed circle). \n\nTo quantify the distribution of scattered light and compare the resulting scattering patterns to previous results from angular SLI measurements \\cite{menzel2021}, \\textit{SLI profiles} (polar integrals $I(\\phi)$, as in \\cite{menzel2020-BOEx}) were computed: For this purpose, the intensity values were integrated from the center to the outer circle of the (centered) scattering pattern in one pixel steps for a defined azimuthal angle $\\phi$, starting on top and moving clock-wise in defined steps $\\Delta\\phi$, using bilinear interpolation to compute the intensity value at the respective position. Figure \\ref{fig:setup}B shows an example of such an SLI profile (integrated intensity values $I(\\phi)$ plotted against $\\phi$, for $\\Delta\\phi = 1^{\\circ}$).\nNote that, when comparing the SLI profiles to line profiles obtained from coherent Fourier scatterometry, it is only possible to compare the general form of the line profiles and the number\/location of the peaks due to differences in the measurement techniques.\n\n\n\\subsection{Smoothing of SLI Profiles}\n\\label{sec:methods-smoothing}\n\nWhile the line profiles obtained from previous angular SLI measurements (with $\\Delta\\phi=15^{\\circ}$ steps) are highly discretized, the line profiles obtained from coherent Fourier and SLI scatterometry measurements allow for much smaller azimuthal steps (e.\\,g.\\ $\\Delta\\phi = 1^{\\circ}$) when using interpolation.\nTo make these line profiles analyzable with the software SLIX (originally designed to deal with highly discretized SLI profiles, see \\cite{slix}), smoothing was applied to suppress high frequency components that represent details of the underlying fibers which are not relevant when characterizing the overall fiber structure \\cite{menzel2020-BOEx}.\n\nThe SLI line profiles were smoothed using a Fourier low pass filter with a \\textit{cutoff frequency} (determining the percentage of passing frequencies) and a \\textit{window width} (controlling the sharpness of the low pass filter):\n\\begin{equation}\n\\text{FFT} \\left\\lbrace 1 - \\left[0.5 + 0.5 \\cdot \\tanh \\left( \\frac{ \\vert \\text{frequencies \/ max. frequency} \\vert - \\text{cutoff frequency}}{ \\text{window width}} \\right) \\right] \\right\\rbrace.\n\\end{equation}\n\nTo identify the optimum parameters for the filter, line profiles from SLI scatterometry measurements of regions with known anatomical fiber structures (parallel in-plane fibers, out-of-plane fibers, two and three in-plane crossing fiber bundles) were filtered using different cutoff frequencies and window widths, and the fraction of correctly determined regions (i.\\,e.\\ two approx.\\ $180^{\\circ}$-separated distinct peaks for in-plane parallel fibers, four distinct peaks for two in-plane crossing fiber bundles, etc.) was determined for each pair of parameters. \nFor each region, 1600 line profiles from SLI scatterometry measurements were constructed and evaluated as a black box.\n\nFigure \\ref{fig:optimization} shows the optimization of the filter parameters. \nFor azimuthal steps of $\\Delta\\phi = 1^{\\circ}$ and $5^{\\circ}$, the Fourier low pass filter was applied to all line profiles and optimized by shifting both the cutoff frequency and the window width of the filter for each iteration. \nFor the cutoff frequency, steps of $2\\,\\%$ were selected. \nThe window width was incremented in steps of $0.025$ for a range between $0.00$ and $0.25$. \nThe exact implementation of the Fourier low pass filter is described in the software repository of SLIX, which is publicly available (\\url{https:\/\/github.com\/3d-pli\/SLIX}).\n\nFor each region, the optimization algorithm yields a matrix with the fraction of line profiles for which the number of significant peaks was correctly determined by SLIX (\\textit{detection rate}).  \nEach fiber arrangement has its own optimum set of parameters: For example, two or three crossing fibers (expected to generate line profiles with four or six peaks) need a higher cutoff frequency, i.\\,e.\\ higher passing frequencies, than parallel in\/out-of-plane fibers (expected to generate one or two peaks). \nAlso, crossing regions, which were less correctly determined in previous angular SLI measurements \\cite{menzel2021}, may yield lower values in the corresponding matrix than their in\/out-of-plane counterparts. \nTo ensure that each type of fiber region is equally considered in the selection of optimum filter parameters, all matrices were normalized and summed up to identify the best choice of parameters.\n\nThe results are shown in Fig.\\ \\ref{fig:optimization}C for both azimuthal steps. \nRegions in yellow have higher detection rates than blue areas. \nThe parameter combination with the highest value, indicated by the asterisk, was chosen as optimum filter parameters for the given azimuthal step: Line profiles with $\\Delta\\phi = 1^{\\circ}$ were filtered with a cutoff frequency of $4\\,\\%$ and a window width of 0.125 (used in Figs.\\ \\ref{fig:optimization}B, \\ref{fig:chiasm}D, \\ref{fig:vervet}E). Line profiles with $\\Delta\\phi = 5^{\\circ}$ were filtered with a cutoff frequency of $40\\,\\%$ and a window width of 0.225 (used in Fig.\\ \\ref{fig:human}B).  \nThe effect of the Fourier low pass filter can be seen in Fig.\\ \\ref{fig:optimization}D where the detection rate (correct peak detection) of unfiltered $15^{\\circ}$-line profiles (used in Figs.\\ \\ref{fig:chiasm}A\/B and \\ref{fig:vervet}A,B) is compared to the detection rate of filtered (smoothed) $1^{\\circ}$- and $5^{\\circ}$-line profiles: the filtered line profiles obtained from SLI measurements with small azimuthal steps yield better detection rates than those obtained from the highly discretized SLI measurements.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig2_Optimization.pdf}\n\\end{center}\n\\caption{Optimized smoothing of SLI profiles. \\textbf{(A)} Averaged scattered light intensity of a coronal vervet monkey brain section (section 493, top) and three crossing sections of human optic tracts (sections 32\/33, bottom). The white rectangles mark the evaluated regions with anatomically known nerve fiber structures: (I) in-plane, (II) out-of-plane, (III) two times crossing, (IV) three times crossing fibers. Relevant anatomical structures are labeled: corona radiata (cr), corpus callosum (cc), cingulum (cg), fornix (f). \\textbf{(B)} Examples of original (blue) and smoothed (orange) azimuthal line profiles (SLI profiles) with $\\Delta\\phi = 1^{\\circ}$-steps, obtained from scattering patterns measured at locations indicated by the yellow asterisks in A. The SLI scatterometry measurements were performed with one illuminated LED and 10\\,sec illumination. For the vervet brain section, the measurement was performed 16 months after tissue embedding with $64 \\times 64$ kernels and a gain factor of 27. For the three sections of optic tracts, the measurement was performed 20 months after tissue embedding with $50 \\times 50$ kernels and a gain factor of 10. \\textbf{(C)} Detection rate of correctly determined regions, i.\\,e.\\ correctly determined number of peaks (average over the four selected types) for different parameters of the Fourier low pass filter (different cutoff frequencies and windows widths) applied to the SLI profiles generated with $\\Delta\\phi=1^{\\circ}$-steps (top) and $5^{\\circ}$-steps (bottom). The magenta asterisks mark the set of parameters (shown in magenta numbers) for which the maximum detection rate is reached: 83.4\\,\\% for $1^{\\circ}$-steps and 80.8\\,\\% for $5^{\\circ}$-steps. \\textbf{(D)} Detection rates of correctly determined regions evaluated separately for the different regions in A, for SLI profiles with $15^{\\circ}$-steps (without smoothing) as well as for SLI profiles with $1^{\\circ}$- and $5^{\\circ}$-steps using the optimum smoothing parameters (magenta numbers in C).} \n\\label{fig:optimization}\n\\end{figure}\n\n\n\\subsection{Visualization of Nerve Fiber Directions}\n\\label{sec:methods-visualization}\n\nThe smoothed SLI profiles were analyzed with SLIX \\cite{slix} to determine the positions of the peaks and compute the corresponding nerve fiber orientations as described by \\cite{menzel2021}: The in-plane fiber orientations were computed from peak pairs with a distance of $(180 \\pm 35)^{\\circ}$.\nThe previous angular SLI measurements shown in this paper were evaluated with SLIX as described in \\cite{menzel2021}, using no smoothing and the centroid of the peak tips in the line profiles to improve upon the angular resolution of $15^{\\circ}$.\n\nThe \\textit{fiber orientation map} is a simple way to visualize and interpret SLI measurements (see Fig.\\ \\ref{fig:visualization}): Each in-plane fiber orientation (direction angle) is mapped to a unique color based on the hue channel of the HSV color space (see color wheel at the top left). For each measured image pixel, up to four different fiber directions are stored in $2 \\times 2$ subpixels. Depending on the number of derived fiber orientations, the subpixels have one, two, three, or four different colors (see enlarged region on the right). \nImage pixels with no determined fiber direction are displayed in black.\nWith this approach, crossing regions are immediately visible in the fiber orientation map and no information is lost.\n\nAnother kind of visualization is the \\textit{fiber orientation distribution map} as shown in Figs.\\ \\ref{fig:chiasm}B and \\ref{fig:vervet}B,E. \nThis visualization sacrifices single-pixel accuracy in order to display the individual fiber orientations as unit vectors.\nFor this purpose, the determined fiber direction angles are converted to unit vectors for each image pixel and plotted as colored lines (cf.\\ Fig.\\ \\ref{fig:visualization} on the very right).\nTo make the fiber orientations visible for a large area, the vectors are displayed for less image pixels.\nHowever, without reducing the number of displayed vectors, the result will be somewhat equal to the fiber orientation map. \nOur approach is to increase the size of the vectors and keep the information density high at the same time. \nHence, instead of thinning out the vector image and showing the vectors e.\\,g.\\ for every $40^{\\text{th}}$ image pixel, all unit vectors in the region (e.\\,g.\\ the vectors of $40 \\times 40$ image pixels) are shown on top of each other. \nTo identify the dominating fiber orientation for each region, the vectors are assigned a low alpha value. \nThus, a single vector with a different orientation than the dominating orientation appears very faintly, but is still visible.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Fig3_Visualization.pdf}\n\\end{center}\n\\caption{Visualization of fiber orientations, shown exemplary for a region in Fig.\\ \\ref{fig:chiasm}A: The in-plane fiber orientations are displayed by different colors (color wheel at the top left). Every image pixel contains $2 \\times 2$ subpixels with up to four different colors, depending on the number of derived fiber orientations. Here, up to three different fiber orientations are shown (see enlarged region). Image pixels for which no fiber direction could be determined are displayed in black. Individual fiber orientations can be represented as colored lines (as shown on the right).} \n\\label{fig:visualization}\n\\end{figure}\n\n\n\\section{Results}\n\n\\subsection{Calibration Measurements}\n\\label{sec:comparison}\n\nTo study the radiation characteristics of the different LEDs and estimate their impact on the measured scattering patterns, calibration measurements were performed. For this purpose, a diffusor plate (with homogeneous scattering properties) was placed in the sample holder and illuminated by a square of $8 \\times 8$ LEDs. The square was moved along the center line of the LED display and an image was recorded for each of the 32 different positions ($x = -15.5, ..., 15.5$). The average light intensity of the inner $1000 \\times 1000$ image pixels was plotted against the angle of illumination: $\\theta = \\arctan\\left(\\frac{x\\,\\cdot\\,8\\,\\cdot\\,1.8\\,\\text{mm}}{13\\,\\text{cm}}\\right)$.\nFigure \\ref{fig:calibration} shows the corresponding curves for different illumination times (A) and different gain factors of the camera (B).\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Fig4_Calibration-curves.pdf}\n\\end{center}\n\\caption{Average transmitted light intensity of a diffusor plate (inner $1000 \\times 1000$ px) illuminated by $8 \\times 8$ LEDs from 32 different positions (along the center line of the LED display; $\\theta=0^{\\circ}$ corresponds to the middle position of the display, cf.\\ Fig.\\ \\ref{fig:setup}A). The different curves belong to measurements with (\\textbf{A}) different illumination times (100\\,ms, 500\\,ms, 1000\\,ms, 1500\\,ms, $\\dots$, 10\\,s) with gain 3, and (\\textbf{B}) different gain factors (1, 2, ..., 27) with 1\\,sec illumination.} \n\\label{fig:calibration}\n\\end{figure}\n\nWhen illuminating with larger angles, the transmitted light intensity decreases significantly, especially for large illumination times and gain factors. This can be explained by the fact that the distance between light source and sample increases with increasing illumination angle (the intensity decreases $\\propto 1\/r^2$ for spherical emitters) and the LEDs have a limited angle of radiation (view angle $\\lesssim 120^{\\circ}$) so that outer LEDs do not emit much light under large angles. An illumination time of more than one second is needed to achieve sufficient transmitted light intensities when illuminating from the outer border of the display.\nNote that the diffusor plate leads to more scattering and absorption than an object carrier with brain section. The required illumination time is therefore expected to be shorter (or comparable when using a smaller number of illuminated LEDs).\nThe illumination characteristics should be taken into account when interpreting the measured scattering patterns. For the resulting line profiles, which are computed by integrating from the center to the outer border of the scattering pattern, we do not expect qualitative changes (concerning e.\\,g.\\ the peak positions).\n\n\\subsection{Comparison to Previous SLI Measurements, Coherent Fourier Scatterometry, and Simulations}\n\\label{sec:comparison}\n\nTo validate the method of SLI scatterometry and enable a direct comparison to previous results from angular SLI measurements and coherent Fourier scatterometry, the same samples were used for the measurement: three crossing sections of human optic tracts (Fig.\\ \\ref{fig:chiasm}) and coronal vervet monkey brain sections (Fig.\\ \\ref{fig:vervet}). \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Fig5_Chiasm.pdf}\n\\end{center}\n\\caption{Three crossing sections of human optic tracts measured with angular\/scatterometry SLI. \\textbf{(A,B)} In-plane fiber orientations obtained from an angular SLI measurement with $15^{\\circ}$ azimuthal steps and px = 6.5\\,\\textmu m, performed 4 months after tissue embedding (as shown in \\cite{menzel2021} Fig.\\ 5, bottom). The fiber orientations are shown for each image pixel in different colors (A), and for $40 \\times 40$ image pixels together as colored lines (B). The three tissue layers are marked by arabic numbers. \\textbf{(C,D)} Scattering patterns and corresponding (smoothed) azimuthal line profiles obtained from an SLI scatterometry measurement of the same sample ($2\\times2$ illuminated LEDs, $50 \\times 50$ kernels, gain factor 10, illumination 5 sec, 15 months after tissue embedding), evaluated exemplary for image pixels located in one (i), two (ii), and three (iii) tissue layers, at the positions indicated in B. The concentric rings in the scattering patterns denote steps of $\\Delta\\theta = 10^{\\circ}$. The dashed vertical lines in D indicate the determined peak positions from which the nerve fiber orientations were computed (indicated by solid lines in C in the respective color). \\textbf{(E)} Scattering pattern and corresponding azimuthal line profile obtained from a coherent Fourier scatterometry measurement for the region marked by the dashed white circle in B (adapted from \\cite{menzel2020-BOEx}, Fig.\\ 5(iv); the measurement was performed with a laser with 1.12\\,mm diameter, a numerical aperture of 0.4, and 4 months after tissue embedding).}\n\\label{fig:chiasm}\n\\end{figure}\n\nFigure \\ref{fig:chiasm}C,D shows the SLI scattering patterns and corresponding line profiles for three selected image pixels in one, two, and three crossing tissue layers of the optic tracts. The scattering patterns show the scattering behavior of in-plane crossing nerve fibers as predicted by simulations \\cite{menzel2020} and observed in coherent Fourier scatterometry \\cite{menzel2020-BOEx}: In-plane nerve fibers generate scattering reflexes perpendicular to their orientation so that the orientations of crossing nerve fibers can be determined by the position of the peaks in the resulting line profiles. As expected, the scattering patterns show one, two, and three scattering reflexes which correspond to two, four, and six distinct peaks in the line profiles. From the positions of the peaks, the orientations of the nerve fibers were computed as described in Sec.\\ \\ref{sec:methods-visualization} and visualized as colored lines. The fiber orientations correspond very well to the fiber orientations computed from previous angular SLI measurements at the approximate same locations (see (i),(ii),(iii) in Fig.\\ \\ref{fig:chiasm}). \n\nA further comparison to a scattering pattern obtained from coherent Fourier scatterometry of the same location (triple tissue layers, dashed circle in B) shows that the resulting line profiles are very similar to each other (cf.\\ Fig.\\ \\ref{fig:chiasm}C(iii) and E). \nIt should be noted that SLI scatterometry shows the distribution of scattered light onto a plane (leading to much lower intensities at the borders of the scattering pattern), while coherent Fourier scatterometry shows the distribution of scattered light onto a hemisphere (projected onto the xy-plane). Also, the maximum angle of illumination is different (steps of $\\Delta\\theta=10^{\\circ}$ are marked by concentric rings in the scattering patterns, respectively). \nTaking into account that the measurements were performed at different times (A\/B: three months, C-E: 10-15 months after tissue embedding), the results correspond very well to each other, demonstrating that SLI scatterometry yields reliable scattering patterns.\n\nFigure \\ref{fig:vervet} shows the SLI scatterometry measurement of a coronal vervet brain section (C-E) in direct comparison to a previous angular SLI measurement of a neighboring section (A-B). \nThe crossing nerve fibers in the corona radiata (cr) are nicely visible, both in the fiber orientation distribution maps (B,E) and in the scattering pattern maps (D). The fiber orientations computed from the scattering patterns (E) correspond very well to the fiber orientations obtained from the angular SLI measurement (B).\n\nAngular SLI measurements have the drawback that pixels at the outer border of the image are illuminated under different polar angles $\\theta$ than pixels in the center of the image, which leads to asymmetries in the resulting line profiles so that peaks might not be detected and wrong\/perpendicular fiber orientations are computed. This phenomenon becomes visible when comparing left and right brain hemispheres which should show approximately mirror-inverted structures. In regions that are not located at the image border, the reconstructed fiber orientations in the corpus callosum (cc) follow the course of the bundle as expected (Fig.\\ \\ref{fig:vervet}B, left). At the right border of the image, some reconstructed fiber orientations run perpendicularly to the course of the bundle (Fig.\\ \\ref{fig:vervet}B, white arrow). \nIn SLI scatterometry, it is possible to determine the center of the scattering patterns (as described in Sec.\\ \\ref{sec:methods-scatteringpatterns}) and overcome this problem. The fiber orientation distribution maps of the corresponding regions (Fig.\\ \\ref{fig:vervet}E) show mirror-inverted structures and no artifacts at the border of the image.\n\nFinally, SLI scatterometry allows to study out-of-plane fiber structures.\nFigure \\ref{fig:vervet}C shows at the bottom the SLI scattering pattern obtained from an image pixel in the fornix (as indicated in A), where most fibers are running out of the coronal section plane. The top figure in C shows the simulated scattering pattern of a fiber bundle with an out-of-plane inclination angle of $50^{\\circ}$. While in-plane nerve fibers generate scattering reflexes perpendicular to their orientation, the simulation studies \\cite{menzel2020} predict that the light is scattered more and more in the direction of the fibers with increasing out-of-plane angle. As expected, the SLI scattering pattern shows a slightly curved reflex, demonstrating that SLI scatterometry does not only yield compatible results with previous measurements (angular SLI and coherent Fourier scatterometry), but also with simulation results and theoretical predictions.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Fig6_Vervet.pdf}\n\\end{center}\n\\caption{Coronal vervet monkey brain section measured with angular\/scatterometry SLI. \\textbf{(A,B)} In-plane fiber orientations obtained from an angular SLI measurement with $\\Delta\\phi=15^{\\circ}$-steps and px = 13.7\\,\\textmu m, performed one day after tissue embedding (section 512, as shown in \\cite{menzel2021} Fig.\\ 8c). The fiber orientations are shown in A for each image pixel in different colors; for the regions surrounded by the dashed rectangles, the fiber orientations for every $30 \\times 30$ image pixels are displayed on top of each other as colored lines in B. Relevant anatomical regions are labeled: corona radiata (cr), corpus callosum (cc), fornix (f). \\textbf{(C)} Scattering pattern of a region with out-of-plane nerve fibers in the fornix (bottom) obtained from the an SLI scatterometry measurement with $4 \\times 4$ illuminated LEDs, $40 \\times 40$ kernels, gain factor 10, illumination 10 sec, 10 months after tissue embedding, shown in direct comparison to a simulated scattering pattern (top) obtained from finite-difference time-domain simulations of an out-of-plane fiber bundle with $50^{\\circ}$ inclination (adapted from \\cite{menzel2020}, Fig. 7a). \\textbf{(D)} Scattering pattern maps of similar regions as in B, obtained from an SLI scatterometry measurement of section 493 measured with one illuminated LED, $50 \\times 50$ kernels, gain factor 27, illumination 10 sec, 15 months after tissue embedding (left), and $2 \\times 2$ LEDs, $80 \\times 80$ kernels, gain factor 10, illumination 5 sec, 12 months after tissue embedding (right), shown for every $150^{\\text{th}}$ image pixel (px = 3\\,\\textmu m). \\textbf{(E)} Fiber orientation distribution maps of the same regions: the fiber orientations were computed with SLIX from every $15^{\\text{th}}$ scattering pattern and displayed on top of each other as colored lines for every $10 \\times 10$ scattering patterns.}\n\\label{fig:vervet}\n\\end{figure}\n\n\\subsection{Evaluation of Human Brain Section}\n\\label{sec:human-section}\n\nFigure \\ref{fig:human} shows the results of an angular SLI measurement with azimuthal steps of $5^{\\circ}$ (B,C) and an SLI scatterometry measurement (D) for a coronal section of a left human brain hemisphere (A). \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{Fig7_Human.pdf}\n\\end{center}\n\\caption{Section of a left human hemisphere measured with angular\/scatterometry SLI. \\textbf{(A)} Transmittance image. \\textbf{(B)} In-plane fiber orientations obtained from an angular SLI measurement with $\\Delta\\phi=5^{\\circ}$ steps\/width, $\\theta = [42.5,47.5]^{\\circ}$, px = 3\\,\\textmu m, performed 8 months after tissue embedding with green light (gain factor 2, illumination 3 sec). Relevant anatomical structures are labeled in yellow (ilf: inferior longitudinal fascicle, OLV: occipital lateral ventricle, or:  optic radiation, ss:  sagittal stratum, tp: tapetum, V1: primary visual cortex, V2: prestriate visual cortex). \\textbf{(C)} Fiber orientation vectors shown for every $40^{\\text{th}}$ image pixel for the regions (1,2,3) marked in B. Vectors are shown if at least 8\\,\\% of the surrounding $40 \\times 40$ pixels have a defined fiber orientation. \\textbf{(D)} Scattering pattern maps obtained from SLI scatterometry measurements of the same sample ($4\\times4$ illuminated LEDs, gain factor 10, illumination 3 sec, 8 months after tissue embedding), shown for every $50^{\\text{th}}$ image pixel for the regions (4,5) marked in B. The white dashed lines indicate tissue borders for better reference.}\n\\label{fig:human}\n\\end{figure}\n\nThe fiber orientation maps in B show the \\textit{sagittal stratum (ss)} and surrounding white matter structures. The sagittal stratum contains highly parallel nerve fibers running mostly in rostro-caudal direction, perpendicular to the coronal section plane. Its environment, however, is characterized by fiber bundles, e.\\,g.\\ \\textit{inferior longitudinal fascicle (ilf)} and \\textit{tapetum (tp)}, entering or perforating the sagittal stratum from its lateral and medial interface.\nMultiple fiber directions in the white dashed rectangles are displayed as colored lines for every $40^{\\text{th}}$ image pixel in C. The fiber orientation vectors reveal astonishing details, even of very small fiber bundles: The fibers of the inferior longitudinal fascicle (cyan) in region (1), for instance, join fibers originating from the dorso-parietal cortex (red fibers) and proceed dorso-medially to the parasagittal cortex (yellow). However, another portion of the red fibers branches off, crossing the inferior longitudinal fascicle, and enters the sagittal stratum (magenta).\nRegion (2) reveals fiber bundles splitting off the tapetum on their ventral course to the lateral side (green, yellow), while other fibers coming from the lateral side and perforating the whole sagittal stratum turn ventrally, eventually entering the tapetum (red, magenta).\nThe vectors in region (3) retrace the typical fiber architecture of the \\textit{primary visual cortex (V1)} with the radial input (green, cyan) and the pronounced transversal fiber layer (blue, magenta) located in the center of the cortex (\\textit{Stria of Gennari}).\n \nThe scattering pattern maps in D show the transition zone between in-plane (tapetum) and out-of-plane fiber bundles (sagittal stratum) (4) as well as for different tissue types (5), as marked in B. The scattering patterns are shown for every $50^{\\text{th}}$ image pixel; the white dashed lines mark tissue borders for better reference. The transition becomes nicely visible in the different shapes of the scattering patterns: in-plane fibers show elongated scattering reflexes perpendicular to their orientation (lower right corner in 4, right stripe in 5), while out-of-plane fibers show broader, almost circular scattering reflexes (upper left corner in 4, middle stripe in 5). \n\nIn box (5), four areas can be distinguished by their scattering pattern size and anisotropy: Very weak scattering in the cortex (V2 in the lower right corner) does not allow for the detection of distinct scattering pattern anisotropy due to the low amount of myelinated fibers. In the white matter substrate, however, the radially oriented anisotropy increases considerably dominated by the tangential course of strongly myelinated terminals of the optic radiation. The next layer consists of terminals of the major forceps corporis callosi (medial layer), constituting the medial wall of the occipital horn of the lateral ventricle (lateral layer). The major forceps is characterized by a vertical scattering anisotropy. The lateral ventricle can be identified by a single strand of tiny isotropic scattering patterns. Eventually, the tapetum is again characterized by strong scattering with vertically oriented anistropy identifying transversal fibers (red area in the upper left area of 5).\n\nThe results demonstrate that SLI scatterometry is a powerful approach, revealing the intricate nerve fiber architecture of the human brain.\n\n\n\\section{Discussion}\n\nThe presented scatterometry measurements with Scattered Light Imaging (SLI) allow the simultaneous generation of complete scattering patterns for all image pixels in an investigated brain section. In this way, SLI scatterometry provides full structural information of complex nerve fiber structures, even in brain regions with densely packed fibers.\n\nCoherent Fourier scatterometry \\cite{menzel2020-BOEx} measures the complete scattering patterns with high detail, but requires mechanical rasterizing of the sample and the object-space resolution is limited to the minimum size of the laser beam ($> 100$\\,\\textmu m). With SLI scatterometry, a scattering pattern can be measured for every image pixel so that the object-space resolution is only limited by the available optics (here: px = 3\\,\\textmu m). The resolution of the scattering patterns, determining the accuracy with which the underlying nerve fiber structures can be reconstructed, is determined by the density of LEDs (here:1.8\\,mm pixel pitch) and the number of measurements (e.\\,g.\\ $40 \\times 40$ kernels). \nPrevious angular SLI measurements using 24 azimuthal illumination angles with $\\Delta\\phi = 15^{\\circ}$ steps and fixed polar angle \\cite{menzel2021} yield highly discretized line profiles and provide not enough details for a reliable analysis of the underlying fiber structures. \nWith SLI scatterometry, highly resolved line profiles (down to $\\Delta\\phi=1^{\\circ}$) can be generated making use of integration and bilinear interpolation (see Sec.\\ \\ref{sec:methods-scatteringpatterns}), which allow for a much more accurate determination of the in-plane nerve fiber orientations. In addition, the SLI scattering patterns enable more advanced studies of out-of-plane nerve fibers (cf.\\ Fig.\\ \\ref{fig:vervet}C). A significant improvement of SLI scatterometry over angular SLI measurements is that it allows to identify the center of the scattering pattern and prevent artifacts from asymmetric illumination of tissue voxels at the image borders (cf.\\ Fig.\\ \\ref{fig:vervet}B,E). \n\nWe could show that the SLI scattering patterns are compatible with previous results from coherent Fourier scatterometry and angular SLI measurements, and also agree with theoretical predictions from simulation studies (see Sec.\\ \\ref{sec:comparison}). \nThis demonstrates that SLI scatterometry can be used to measure scattering patterns and make valid predictions for the underlying nerve fiber structures.\n\nThe in-plane fiber directions can be reliably determined from the measured scattering patterns. Future studies should focus on how the out-of-plane fiber angles can be reliably extracted. A comparison of scattering patterns obtained from measurements with different wavelengths (red, green, blue) are also expected to yield additional information, revealing e.\\,g.\\ the size of scattering structures such as the fiber diameters.\n\nBy illuminating one LED at a time, the scattering patterns can be measured with high structural detail. However, with illumination times of at least one second (to provide sufficient brightness for illuminations under large angles) and $256 \\times 256$ LEDs, the measurement and processing of several thousand images is very time consuming. \nThe here presented SLI scatterometry measurements and high-resolution scattering patterns can be used as ground truth in order to develop more efficient measurement protocols that reduce the number of required measurements while maintaining accuracy and resolution of the resulting scattering patterns. Resampling the high-resolution SLI scattering patterns in different ways allows to study how much the number of measurements can be reduced without losing important details in the scattering patterns. Another idea is to use approaches from compressed sensing \\cite{duarte2011} and exploit the sparsity of the signal in order to significantly reduce the amount of measurements to recover the signal. In SLI scatterometry, every camera pixel can be understood as an independent single pixel camera \\cite{duarte2008} so that the same approaches from compressed sensing can be applied. As these approaches employ the illumination of many LEDs, this will also reduce the required illumination time.\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\n\\section*{Author Contributions}\n\nMM designed and supervised the research. MR conducted the calibration measurements. JR programmed the smoothing of the SLI profiles and visualized the fiber orientation and vector maps. MR and DG performed the SLI scatterometry measurements. DG assisted with anatomical labeling. MM prepared the results for the figures and wrote the first draft of the manuscript. JR and DG wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.\n\n\n\\section*{Funding}\n\nThis work was funded by the Helmholtz Association port-folio theme ``Supercomputing and Modeling for the Human Brain'', the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No.\\ 945539 (``Human Brain Project'' SGA3), and the National Institutes of Health under grant agreements No.\\ R01MH092311 and 5P40OD010965.\n\n\n\\section*{Acknowledgments}\nThe authors gratefully thank Roxana Kooijmans (Netherlands Institute for Neuroscience, Amsterdam) for providing the human brain tissue, Karl Zilles and Roger Woods (David Geffen School of Medicine at UCLA, USA) for collaboration in the vervet brain project, and the lab team of the INM-1 (Forschungszentrum J\\\"ulich GmbH, Germany) for preparing the brain sections.\n\n\n\\section*{Data Availability Statement}\nThe software \\textit{SLIX} \\cite{slix} that was used for the smoothing of the line profiles and for the visualization of the nerve fiber orientations is available on GitHub (\\url{https:\/\/github.com\/3d-pli\/slix}). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTheoretic evidence from lattice simulations \\cite{Re1} of QCD for\na phase transition from a hadron gas to a quark-gluon plasma at high\ntemperature has prompted experimental efforts to create this new phase\nof matter in the laboratory during the early stages of\nultrarelativistic heavy ion collisions \\cite{Re2}. However, because of\nthe estimated very short life time of the collision zone, the highly\nexcited quark-gluon system may spend a considerable fraction of its\nlife in a non-thermalized, pre-equilibrium state. The dynamical tool\nto treat dissipative processes in heavy ion collisions and the\napproach to local thermal equilibrium is in principle non-equilibrium\nquantum transport theory. Since QCD, the theory describing the\ninteractions between quarks and gluons, is a gauge theory, the kinetic\nequations should be gauge covariant. A relativistic and gauge\ncovariant kinetic theory for quarks and gluons has been derived\n\\cite{Re3}, both in a classical framework \\cite{Re4,Re5} and as a\nquantum kinetic theory \\cite{Re6,Re7} based on Wigner operators\ndefined in 8-dimensional phase space \\cite{Re8}. Some preliminary\napplications to the quark-gluon plasma, such as linear color response,\ncolor correlations and collective plasma oscillations \\cite{Re9,Re10}\nhave been discussed in this framework using a semiclassical expansion\nof the quantum transport theory. Apart from its relativistic and\ngauge covariance, an important aspect of the four-dimensional\napproach to transport theory is that the complex kinetic equation can\nbe split up into a constraint and a transport equation, where the\nformer is a quantum extension of the classical mass-shell condition,\nand the latter is a covariant generalization of the Vlasov-Boltzmann\nequation. The complementarity of these two ingredients is essential\nfor a physical understanding of quantum kinetic theory \\cite{henning}.\n\nAs a first step towards a full kinetic treatment of the quark-gluon\nplasma, a detailed study of relativistic transport theory\nfor electrodynamics was performed over the years (see \\cite{Re3,hakim}\nfor references). The absence of the complications arising from the\nnon-abelian character of QCD allows for a deeper insight into\nthe structure of the kinetic theory itself, and it is also easier to\nfind useful applications in this simpler case. Vasak, Gyulassy and\nElze \\cite{Re11} (to be quoted as VGE) discussed the relativistic\nquantum transport theory for spinor electrodynamics and gave a very\nlucid account of the covariant spinor decomposition for the Wigner\noperator of Dirac particles (see also \\cite{hakim}).\n\nIn classical transport theory, all the physical currents are connected\nwith the distribution function $f$. The quantum mechanical analogue of\n$f$ is the Wigner function, which is the ensemble expectation of the\nWigner operator. In the process of performing the ensemble average of\nthe kinetic equation for the Wigner operator, one encounters the\ntwo-body correlation function $\\langle F^{\\mu\\nu}\\hat W \\rangle$,\nwhere $F^{\\mu\\nu}$ is the gauge field strength. In the general case, the\ntwo-body correlation function again depends on higher order\ncorrelation functions; this generates the so called BBGKY hierarchy\n\\cite{Re8}. A popular method to obtain a closed kinetic equation for\nthe Wigner function $W = \\langle \\hat W \\rangle$ is to use the Hartree\napproximation where the gauge field is considered as a mean field\n$\\bar F^{\\mu\\nu}$, leading to the replacement $\\langle F^{\\mu\\nu} \\hat\nW \\rangle = \\bar F^{\\mu\\nu} \\langle \\hat W \\rangle$. In this\napproximation the BBGKY chain is truncated at the one body level, and\nthe kinetic equation for the Wigner function has the same form as that\nfor the Wigner operator. This version of the mean field approximation,\nwhich treats the particle fields quantum mechanically, but uses the\nclassical approximation for the gauge fields, is widely used in\nelectrodynamic transport theory \\cite{Re11,Re12,Re13}. It is\nappropriate for strong but slowly varying electromagnetic fields.  An\ninteresting example is the Schwinger process \\cite{Re14,Re15} for pair\nproduction in external electromagnetic fields. Since this\napproximation is sufficient for our purpose, we will also restrict\nourselves to the investigation of transport theory for particles\ninteracting with an external field.\n\nIn a recent paper Bialynicki-Birula, Gornicki and Rafelski \\cite{Re12}\n(to be quoted as BGR) proposed an equal-time transport equation for\nspinor electrodynamics. Unlike the covariant theory where the Wigner\noperator is defined as the four-dimensional Fourier transform of the\ngauge covariant density matrix $\\Phi_4(x,y)$ (see below),\n \\begin{equation}\n \\label{w4}\n   \\hat W_4(x,p) = \\int d^4y\\, e^{ip{\\cdot}y} \\, \\Phi_4(x,y) \\, ,\n \\end{equation}\nthey introduced an equal-time correlation function $\\Phi_3 (x,{\\bf y})\n= \\Phi({\\bf x},{\\bf y},t)$ and considered its spatial Fourier\ntransform which depends only on three momentum coordinates:\n \\begin{equation}\n \\label{w3}\n   \\hat W_3(x,{\\bf p}) = \\int d^3y\\,\n   e^{-i{\\bf p}{\\cdot}{\\bf y}} \\, \\Phi_3(x,{\\bf y}) \\, .\n \\end{equation}\n{}From this definition and the Dirac equation they obtained directly a\nkinetic equation for $\\hat W_3$. They showed that the Wigner function\n$\\langle \\hat W_3(x,{\\bf p}) \\rangle$ is a direct analogue of the\nclassical distribution function $f({\\bf x},{\\bf p},t)$ and that it\nprovides a systematic way of studying the phase-space dynamics of QED\nin the semiclassical limit; each spinor component of $\\langle \\hat W_3\n\\rangle$ corresponds to a definite physical distribution function. Of\ncourse, $\\hat W_3(x,{\\bf p})$ is not manifestly Lorentz covariant.\n\nRecently this equal-time method has been extended \\cite{Re13} to\nscalar electrodynamics in the Feshbach-Villars representation. The\nensuing applications of this so-called ``three-dimensional\" transport\ntheory to the problem of nonperturbative pair creation in both spinor\n\\cite{Re12} and scalar \\cite{Re16,Re17} QED indicated that some quantum\nproblems may be treated more easily in phase space in terms of the\nWigner operator than by conventional field theoretic methods. We will\nshow, however, that the published derivation of the equal-time\ntransport theory is incomplete. A complete treatment leads to\nadditional constraints on the Wigner function which for the specific\nsituations treated by previous authors actually simplifies its\nstructure; in general, however, it leads to complications whose\neffects cannot be neglected.\n\nWe show in this paper that the three-dimensional transport theory can\nalso be derived by a different method which does not suffer from this\nincompleteness. In particular, we discuss the relationship between the\ncovariant and three-dimensional approaches. From the comparison of\nEqs.(\\ref{w4}) and (\\ref{w3}) it is easily seen that there exists a\ngeneral connection between the four- and three-dimensional Wigner\noperators:\n \\begin{equation}\n \\label{w34}\n   \\hat W_3(x,{\\bf p}) = \\int dp_0\\ \\hat W_4(x,p)\\, .\n \\end{equation}\nThis relation shows that the three-dimensional transport theory can\nbe obtained by an energy average of its covariant version. The main\nformal difference between the energy averaging method used in this\npaper and the direct equal-time derivation of BGR is that our approach\nstarts with an explicitly relativistic formulation. This covariant\nformulation is shown to be complete, i.e. the equation of motion for\nthe Wigner operator is equivalent to the Dirac equation for the field\noperators. This completeness is preserved during the reduction from\nthe four- to the three-dimensional version via energy averaging. The\ncovariant formulation followed by energy averaging also allows us to\nobtain a three-dimensional transport equation for the scalar Wigner\noperator directly from the Klein-Gordon equation, rather than via the\nFeshbach-Villars $2\\times 2$ matrix formulation.\n\nWe proceed as follows. In section 2, we discuss the transport\ntheory for spinless charged particles. We first set up the covariant\nconstraint and transport equations, which are the basis of the\nreduction to the three-dimensional version. Next we consider the\nsemiclassical expansion in $\\hbar$ and the classical limit. Then we\naverage the covariant equations over the energy to obtain the\nthree-dimensional equations. With these we investigate the pair\ncreation of charged scalar particles in a spatially homogeneous but\ntime-dependent electric field. This problem was studied before\nin the Feshbach-Villars matrix representation \\cite{Re17}, but our\nprocedure turns out to be much simpler because of the scalar nature of\nour Wigner function. In Section 3, we study the kinetic theory for\nDirac fermions. We derive a complete set of two selfadjoint covariant\nequations, one of which corresponds to the four-dimensional version of\nthe BGR equation. We give a full mapping of the spinor components from\nthe four- to the three-dimensional representation. The two selfadjoint\ncovariant transport equations can be combined into a single complex\nequation, the VGE equation, which is then subjected to the energy\naveraging procedure. We discuss the classical limit of the resulting\nthree-dimensional equations and compare it with the results\n\\cite{Re18} from the equal-time method. We then show how to get a\ncomplete set of three-dimensional kinetic equations in the general,\nfully quantum mechanical case; these contain as a subset the various\nspinor components of the BGR equation, but also a set of additional\nconstraints. We discuss these results and summarize them in our\nconclusions.\n\n \\section{Scalar electrodynamics}\n \\subsection{Relativistic covariant kinetic equations}\n\nThe abelian gauge theory of scalar particles with mass $m$ and charge\n$e$ is defined through the Lagrangian density\n \\begin{equation}\n \\label{Sl}\n   {\\cal L} = (\\partial_\\mu-ieA_u)\\phi^\\dagger\n              (\\partial^\\mu+ieA^\\mu)\\phi - m^2\\phi^\\dagger\\phi\n              - {\\textstyle{1\\over 4}} F_{\\mu\\nu} F^{\\mu\\nu} \\ .\n \\end{equation}\nIt leads to the Klein-Gordon equation for the scalar field operator\n$\\phi$,\n \\begin{equation}\n \\label{Klein}\n   \\left[ (\\partial_\\mu + ieA_\\mu(x)) (\\partial^\\mu+ieA^\\mu(x))\n   + m^2 \\right] \\, \\phi(x) = 0 \\ ,\n \\end{equation}\nthe adjoint equation for $\\phi^\\dagger$, and to Maxwell's equations\nfor the field strength tensor $F^{\\mu\\nu}$,\n \\begin{equation}\n \\label{Ma}\n   \\partial_\\mu F^{\\mu\\nu}(x) = j^\\nu (x)  \\ ,\n   \\qquad\\qquad\n   \\partial_\\mu \\tilde F^{\\mu\\nu}(x) = 0 \\ ,\n \\end{equation}\nwhere $\\tilde F^{\\mu\\nu}(x) = {1\\over 2} \\epsilon^{\\mu\\nu\\rho\\sigma}\nF_{\\rho\\sigma}$ is the dual field tensor and $j^\\nu(x)$ is the charge\ncurrent of the scalar particles.\n\nThe relativistically covariant Wigner operator is the Fourier\ntransform Eq.~(\\ref{w4}) of the gauge covariant scalar field\ncorrelation function\n \\begin{equation}\n \\label{Sd1}\n   \\Phi_4(x,y) = \\phi(x)e^{-y\\cdot{\\cal D}^\\dagger\/2}\\cdot\n                        e^{y\\cdot{\\cal D}\/2}\\phi^\\dagger(x) \\ .\n \\end{equation}\nThe covariant derivatives ${\\cal D}_\\mu(x)=\\partial_\\mu + ieA_\\mu(x)$\nand ${\\cal D}^\\dagger_\\mu(x) = \\partial_\\mu - ieA_\\mu(x)$ in this\nexpression ensure the gauge covariance (actually: gauge independence\nin the case of a $U(1)$ gauge theory) of the density matrix.\nFollowing \\cite{Re6} and replacing the covariant derivatives here by a\nline integral of the gauge field $A_\\mu(x)$ along a straight line\nbetween the space-time points $x-{y\\over 2}$ and $x+{y\\over 2}$, we\nrecover the more familiar form \\cite{Re14}\n \\begin{equation}\n \\label{Sd2}\n   \\Phi_4(x,y) = \\phi\\left(x+{\\textstyle{1\\over 2}} y \\right) \\,\n                 \\exp\\left[ ie\\int^{1\\over 2}_{-{1\\over 2}}ds\\,\n                             A(x+sy){\\cdot}y \\right]\n                 \\phi^\\dagger\\left(x-{\\textstyle{1\\over 2}} y\\right) \\, .\n \\end{equation}\nThe resulting scalar Wigner operator is selfadjoint:\n \\begin{equation}\n \\label{Sw1}\n   \\hat W^\\dagger_4(x,p) = \\hat W_4(x,p) \\ .\n \\end{equation}\n\nSubstituting Eq.~(\\ref{Sd1}) into Eq.~(\\ref{w4}) and integrating over\n$y$ we see that the Wigner operator is just the density of particles\nat space-time point $x$ with kinetic momentum $p$,\n \\begin{equation}\n \\label{Sw2}\n   \\hat W_4(x,p) = \\phi(x) \\, \\delta^4 (p-\\hat\\pi(x)) \\,\n                   \\phi^\\dagger(x) \\, ,\n \\end{equation}\nwhere $\\hat\\pi(x) = {i\\over 2}({\\cal D}_\\mu(x) - {\\cal D}_\\mu^\\dagger\n(x))$ is the kinetic momentum operator. Eq.~(\\ref{Sw2}) motivates a\nstatistical interpretation of the Wigner function as a generalized\nphase-space density and generates simple relations between the Wigner\noperator and all physical space-time density operators. For example,\nthe operators for the charge current density and the energy momentum\ntensor are given by\n \\begin{eqnarray}\n \\label{CE}\n   \\hat j_\\mu(x)\n   &=& ie \\Bigl( \\phi^\\dagger(x) \\partial_\\mu \\phi(x)\n                - (\\partial_\\mu \\phi^\\dagger(x)) \\phi(x) \\Bigr)\n   \\nonumber\\\\\n   &=& e \\int d^4p \\, p_\\mu \\, \\hat W_4(x,p) \\, ,\n   \\nonumber\\\\\n   \\hat T_{\\mu\\nu}(x)\n   &=& {\\textstyle{1\\over 2}} \\Big( (\\partial_\\mu\\phi^\\dagger(x))\n                        (\\partial_\\nu \\phi(x))\n                      - (\\partial_\\nu \\phi^\\dagger(x))\n                        (\\partial_\\mu\\phi(x))\\Big)\n   \\nonumber\\\\\n   &=& \\int d^4p\\, p_\\mu p_\\nu \\, \\hat W_4(x,p) \\, .\n \\end{eqnarray}\n\nThe equations of motion for the Wigner operator are a direct\nconsequence of the field equations (\\ref{Klein}). We calculate the\nsecond-order derivatives of the correlator $\\Phi_4(x,y)$ with respect\nto $x$ and $y$,\n \\begin{eqnarray}\n \\label{Sd3}\n   \\left( {\\textstyle{1\\over 2}} \\partial_\\mu^x +\\partial_\\mu^y \\right)\n   \\left( {\\textstyle{1\\over 2}} \\partial_x^\\mu +\\partial_y^\\mu \\right) \\!\\!\\!\\!\\!\\!\\!\\!\n   && \\Phi_4(x,y) = - m^2\\Phi_4(x,y)\n   \\nonumber\\\\\n   &&- 2ie\\int^{1\\over 2}_{-{1\\over 2}}ds\n       \\left({\\textstyle{1\\over 2}}+s\\right) y_\\nu F^{\\nu\\mu}(x+sy)\n       \\left({\\textstyle{1\\over 2}}\\partial_\\mu^x + \\partial_\\mu^y\\right)\n       \\, \\Phi_4(x,y)\n   \\nonumber\\\\\n   && -(ie)^2 \\left[\\int^{1\\over 2}_{-{1\\over 2}}ds\n              \\left( {\\textstyle{1\\over 2}} + s \\right)\n              y{\\cdot}F(x+sy)\\right]^2\n              \\! \\Phi_4(x,y)\n   \\nonumber\\\\\n   && + ie\\int ^{1\\over 2}_{-{1\\over 2}}ds\n        \\left({\\textstyle{1\\over 2}}+s\\right)^2\n        y{\\cdot}j(x+sy)\\, \\Phi_4(x,y) \\, ,\n \\end{eqnarray}\nwhere we have employed the Klein-Gordon and Maxwell equations in the\nfirst and last terms on the right-hand side. Eq.~(\\ref{Sd3}) can be\nrewritten in a compact way,\n \\begin{equation}\n \\label{Sd4}\n   \\left[ {\\textstyle{1\\over 4}} D^\\mu D_\\mu - \\Pi^\\mu\\Pi_\\mu + m^2\n         - i\\Pi^\\mu D_\\mu\\right]\\,\\Phi_4(x,y) = 0\\, ,\n \\end{equation}\nwhere we defined the two Lorentz covariant operators $D$ and $\\Pi$ by\n \\begin{eqnarray}\n \\label{DP1}\n   D^\\mu(x,y)\n   &=& \\partial^\\mu_x + ie\\int^{1\\over 2}_{-{1\\over 2}}ds\n       \\, y_\\nu F^{\\nu\\mu}(x+sy) \\, ,\n   \\nonumber\\\\\n   \\Pi^\\mu(x,y)\n   &=& i\\left(\\partial^\\mu_y + ie\\int^{1\\over 2}_{-{1\\over 2}}\n       ds\\, s\\, y_\\nu F^{\\nu\\mu}(x+sy)\\right) \\, .\n \\end{eqnarray}\n\nPerforming the Fourier transform with respect to $y$, we obtain the\nfollowing exact quadratic kinetic equation for the scalar Wigner\noperator:\n \\begin{equation}\n \\label{Sw3}\n   \\left[ {\\textstyle{1\\over 4}} \\hbar^2 D^\\mu D_\\mu - \\Pi^\\mu \\Pi_\\mu + m^2\n         -i\\hbar\\Pi^\\mu D_\\mu \\right]\\, \\hat W_4(x,p) = 0 \\, .\n \\end{equation}\nThe operators $D$ and $\\Pi$ in this equation are now defined in phase\nspace and can be obtained from Eqs.~(\\ref{DP1}) by the replacements\n$y^\\mu \\to -i\\partial_p^\\mu$ and $\\partial_y^\\mu \\to -ip^\\mu$:\n \\begin{eqnarray}\n \\label{DP2}\n    D_\\mu(x,p)\n    &=& \\partial_\\mu - e\\int ^{1\\over 2}_{-{1\\over 2}} ds\\,\n        F_{\\mu\\nu} (x-i\\hbar s\\partial_p)\\, \\partial_p^\\nu \\, ,\n    \\nonumber \\\\\n    \\Pi_\\mu(x,p)\n    &=& p_\\mu - ie\\hbar \\int ^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n        F_{\\mu\\nu}(x-i\\hbar s\\partial_p) \\, \\partial_p^\\nu \\, .\n \\end{eqnarray}\nIn Eqs.~(\\ref{Sw3}) and (\\ref{DP2}) we reinstated the\n$\\hbar$-dependence explicitly in order to be able to discuss the\nsemiclassical expansion in the next subsection. (The speed\nof light $c$ is still omitted.) Obviously, the operators $D_\\mu$ and\n$\\Pi_\\mu$ are gauge covariant extensions of the partial derivative\n$\\partial_\\mu$ and the momentum $p_\\mu$, respectively. They both are\nself-adjoint: namely, $D_\\mu^\\dagger(x,p)=D_\\mu(x,p)$ and\n$\\Pi_\\mu^\\dagger(x,p) = \\Pi_\\mu(x,p)$.\n\nSince the scalar Wigner operator is self-adjoint, too, the real and\nimaginary parts of the complex equation (\\ref{Sw3}) have to vanish\nseparately:\n \\begin{eqnarray}\n \\label{Sc}\n   &&\\left({\\textstyle{1\\over 4}} \\hbar^2 D^\\mu D_\\mu - \\Pi^\\mu\\Pi_\\mu + m^2\\right)\n     \\hat W_4(x,p) = 0 \\,  ,\n \\\\\n \\label{St}\n   &&\\hbar\\Pi^\\mu D_\\mu \\hat W_4(x,p) = 0 \\, .\n \\end{eqnarray}\nThe first equation is usually called constraint equation since\nit is obviously a generalization of the classical mass-shell condition\n$p^2-m^2=0$. The second equation has the typical form of a transport\nequation and is the generalization of the classical Vlasov equation\nfor charged particles with abelian interactions $p^\\mu (\\partial_\\mu -\ne F_{\\mu\\nu}(x) \\partial_p^\\nu) f(x,p) = 0$, where $f(x,p)$ is the\nclassical distribution function. These two equations together give a\ncomplete description for the Wigner operator; they are equivalent\nto the original field equations of motion.\n\n \\subsection{Semiclassical expansion}\n\nIn order to better understand the structure of the constraint and\ntransport equations, we consider their semiclassical expansion in\n$\\hbar$ and their classical limit. The calculation of quantum\ncorrections to this limit can then be performed in a systematic way.\n\nAs stated in the Introduction, we will in this paper consider the\nelectromagnetic field as a classical (mean) field. In this\napproximation the equation of motion for the Wigner {\\em operator} and\nits ensemble expectation value, the Wigner {\\em function}, become\nformally identical. Therefore we will now work with the equations of\nmotion for the Wigner {\\em function} by leaving off the hats over the\ncorresponding operators.\n\nThe field strength $F^{\\mu\\nu}$ in Eqs.~(\\ref{DP2}), which has to be\nevaluated at the shifted argument $x-i\\hbar s\\partial_p$, is defined in\nterms of its Taylor expansion around $x$ and can be expressed in terms\nof the so-called ``triangle operator\" $\\triangle =\n\\partial_p{\\cdot}\\partial_x$ as\n \\begin{equation}\n \\label{Fs}\n   F_{\\mu\\nu}(x-is\\hbar\\partial_p) =\n      e^{is\\hbar \\triangle } F_{\\mu\\nu}(x)\\, .\n \\end{equation}\nThe $s$-integration can then be done, and expanding the result in\npowers of $\\hbar$ we obtain\n \\begin{eqnarray}\n \\label{DP3}\n   D_\\mu(x,p)\n   &=& \\partial_\\mu - e{\\sin(\\hbar \\triangle \/2)\n                        \\over \\hbar \\triangle\/2}\n       F_{\\mu\\nu}(x)\\, \\partial_p^\\nu\n \\nonumber\\\\\n   &=& \\Big(\\partial_\\mu -eF_{\\mu\\nu}(x)\\,\\partial_p^\\nu \\Big)\n      +{e\\over 24} \\hbar^2\\triangle^2F_{\\mu\\nu}(x)\\,\\partial_p^\\nu\n      +\\cdots \\, ,\n \\nonumber\\\\\n   \\Pi_\\mu(x,p)\n   &=& p_\\mu +{e\\hbar\\over 2}\\left({\\cos(\\hbar\\triangle\/2)\n                                    \\over \\hbar \\triangle\/2}\n                 -{\\sin(\\hbar\\triangle\/2)\n                   \\over (\\hbar\\triangle\/2)^2}\\right)\n       F_{\\mu\\nu}(x)\\, \\partial_p^\\nu\n \\nonumber\\\\\n   &=& p_\\mu - {e\\over 12}\\hbar^2\\triangle F_{\\mu\\nu}(x)\\,\\partial_p^\\nu\n                 +\\cdots \\, ,\n \\end{eqnarray}\nwhere the dots indicate corrections from higher orders of $\\hbar$ or,\nequivalently, of the derivative operator $\\triangle$.\n\nExpanding the Wigner function similarly in powers of $\\hbar$,\n \\begin{equation}\n \\label{Sw4}\n   W_4(x,p) = W_4^{(0)}(x,p) + \\hbar W_4^{(1)}(x,p) +\n                               \\hbar^2 W_4^{(2)}(x,p) + \\cdots \\, ,\n \\end{equation}\nand inserting these expressions into the constraint and transport\nequations (\\ref{Sc}) and (\\ref{St}), we obtain their semiclassical\nexpansion. In the zeroth order, there is no information from the\ntransport equation, and the constraint equation reduces to the\nclassical mass-shell condition,\n \\begin{equation}\n \\label{Ms1}\n   (p^2-m^2)\\, W_4^{(0)}(x,p) = 0 \\, .\n \\end{equation}\nThis equation has two elementary solutions corresponding to positive\nand negative energies, and we can write\n \\begin{equation}\n \\label{Sw5}\n    W_4^{(0)}(x,p) = W_4^{+(0)}(x, {\\bf p})\\,\\delta (p_0-E_p)\n       + W_4^{-(0)}(x, {\\bf p})\\,\\delta (p_0+E_p) \\, ,\n \\end{equation}\nwhere $E_p = + \\sqrt{{\\bf p}^2+m^2}$ is the classical on-shell energy.\n\nTo next order in $\\hbar$, the transport equation (\\ref{St}) begins to\ncontribute. It yields the Vlasov equation\n \\begin{equation}\n \\label{Ve1}\n   p^\\mu\\Big(\\partial_\\mu-eF_{\\mu\\nu}(x)\\partial_p^\\nu\\Big)\n   W_4^{(0)}(x,p) = 0  \\, ,\n \\end{equation}\nwhich must be evaluated with the ansatz (\\ref{Sw5}). The constraint\nequation yields at order $\\hbar$\n \\begin{equation}\n \\label{Ms1a}\n   (p^2-m^2)\\, W_4^{(1)}(x,p) = 0 \\, ,\n \\end{equation}\nwith the solution\n \\begin{equation}\n \\label{Sw6}\n    W_4^{(1)}(x,p) = W_4^{+(1)}(x, {\\bf p})\\,\\delta (p_0-E_p)\n       + W_4^{-(1)}(x, {\\bf p})\\,\\delta (p_0+E_p) \\, .\n \\end{equation}\n\nAt order $\\hbar^2$, Eq.~(\\ref{Sc}) becomes\n \\begin{equation}\n \\label{Sw7}\n  \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n  (p^2-m^2)\\, W_4^{(2)}(x,p) =\n  \\left[{\\textstyle{1\\over 4}}\\Big(\\partial_x -eF(x){\\cdot}\\partial_p\\Big)^2\n       +{\\textstyle{e\\over 12}}\n         \\Big(p^\\mu\\triangle F_{\\mu\\nu}\\,\\partial_p^\\nu\n       +j(x){\\cdot}\\partial_p^\\nu\\Big)\\right]\n    W_4^{(0)}(x,p) \\, ,\n \\end{equation}\nwhere $j_\\nu(x) = e \\int d^4p\\, p_\\nu\\, W(x,p)$ is the average charge\ncurrent of the scalar particles in the ensemble which generates the\nmean electromagnetic field via Maxwell's equation (\\ref{Ma}).\nEq.~(\\ref{St}) yields at order $\\hbar^2$\n \\begin{equation}\n \\label{Ve1a}\n    p^\\mu\\Big(\\partial_\\mu-eF_{\\mu\\nu}(x)\\partial_p^\\nu\\Big)\n      W_4^{(1)}(x,p) = 0  \\,  .\n \\end{equation}\nThe right-hand side of (\\ref{Sw7}) represents off-shell corrections to\nthe mass-shell condition due to quantum corrections. Eq.~(\\ref{Ve1a})\nis the Vlasov equation for the first order Wigner operator. Quantum\ncorrections to the Vlasov equation for $W_4^{(2)}$ will arise only at\norder $\\hbar^3$. The fact that quantum corrections affect the Wigner\nfunction only at second order in $\\hbar$ is a specific feature of\nscalar particles. In spinor QED quantum effects arise in first order\nof $\\hbar$ due to spin interactions.\n\nWe now carry out the $p_0$-average of the Vlasov equations\n(\\ref{Ve1},\\ref{Ve1a}). This is facilitated by their simple form\n(\\ref{Sw5},\\ref{Sw6}) resulting from the mass-shell conditions\n(\\ref{Ms1},\\ref{Ms1a}). For the corresponding three-dimensional\npositive and negative energy Wigner functions $W_3^\\pm$ (see\nEq.~(\\ref{w34})) we obtain\n \\begin{equation}\n \\label{Ve2}\n   \\partial_t W_3^{\\pm(i)}(x,{\\bf p})\n   \\pm ({\\bf v}{\\cdot}{\\bf \\nabla}) W_3^{\\pm(i)}(x,{\\bf p})\n   + e \\Big({\\bf E}(x) \\pm {\\bf v}\\times\n             {\\bf B}(x)\\Big){\\cdot}{\\bf \\nabla}_p\n   W_3^{\\pm(i)}(x,{\\bf p}) = 0 \\, ,\n \\end{equation}\nwhere ${\\bf v}={\\bf p}\/E_p$, $i=0,1$, and ${\\bf E}$ and ${\\bf B}$ are\nthe electric and magnetic field components of\n$F^{\\mu\\nu}$. We now introduce classical particle and\nantiparticle distribution functions, $f$ and $\\bar f$, via\n \\begin{eqnarray}\n \\label{Ve3}\n  f^{(i)}(x,{\\bf p}) &=& W_3^{+(i)}(x,{\\bf p}) \\, ,\n  \\nonumber\\\\\n  \\bar f^{(i)}(x,{\\bf p}) &=& W_3^{-(i)}(x,{-\\bf p}) \\, .\n \\end{eqnarray}\nThey satisfy the well-known three-dimensional Vlasov transport\nequations for scalar particles and antiparticles moving in an external\nelectromagnetic field:\n \\begin{eqnarray}\n \\label{Ve4}\n   \\partial_t f^{(i)}(x,{\\bf p})\n   + ({\\bf v}{\\cdot}{\\bf \\nabla})f^{(i)}(x,{\\bf p})\n   + e\\Big({\\bf E}(x)+{\\bf v}\\times {\\bf B}(x)\n           \\Big){\\cdot}{\\bf \\nabla}_p f^{(i)}(x,{\\bf p}) = 0\\, ,\n \\nonumber \\\\\n   \\partial_t \\bar f^{(i)}(x,{\\bf p})\n   + ({\\bf v}{\\cdot}{\\bf \\nabla})\\bar f^{(i)}(x,{\\bf p})\n   - e\\Big({\\bf E}(x)+{\\bf v}\\times {\\bf B}(x)\n           \\Big){\\cdot}{\\bf \\nabla}_p \\bar f^{(i)}(x,{\\bf p}) = 0\n   \\, .\n \\end{eqnarray}\n\n \\subsection{Pair production in transport theory}\n\nIn this subsection, we give an exact non-perturbative solution of the\nscalar transport theory in a spatially constant external electric\nfield. The solution describes pair production due to vacuum\nexcitation. In electrodynamics, it was first studied many years ago by\nSchwinger \\cite{Re14}, who connected the probability of pair creation\nwith the imaginary part of the effective action in QED. Recently it\nwas reinvestigated as an application of the equal-time transport\ntheory. For scalar electrodynamics in the Feshbach-Villars\nrepresentation, Best and Eisenberg \\cite{Re17} obtained from the\nkinetic theory the same result as Popov did previously in \\cite{Re19}\nwith field operator techniques. Here we consider the pair creation of\nscalar particles directly in the Klein-Gordon representation of the\nkinetic theory, to show how the energy averaging method works\nin a non-perturbative case and to discuss its difference from the\nFeshbach-Villars based equal-time method.\n\n \\subsubsection{Initial value problem}\n\nThe determination of any solution of the differential equations\n(\\ref{Sc}) and (\\ref{St}) needs initial conditions. For the pair\ncreation problem we should search for a free vacuum solution as the\ninitial condition. For ${\\bf E}={\\bf B}=0$, the energy average of the\nfull constraint and transport equations result in the following\nthree-dimensional expressions:\n \\begin{eqnarray}\n \\label{Ec}\n    \\Bigl(\\partial_t^2 -{\\bf \\nabla}^2 +4E_p^2\\Bigr) W_3(x,{\\bf p})\n     &=& 4\\, \\varepsilon(x,{\\bf p}) \\, ,\n \\nonumber\\\\\n    \\partial_t \\, \\rho(x,{\\bf p}) - {\\bf \\nabla}{\\cdot}{\\bf j}(x,{\\bf p})\n    &=& 0 \\, .\n \\end{eqnarray}\nHere we defined the phase-space densities of electric charge\n$\\rho(x,{\\bf p})$, electric current ${\\bf j}(x,{\\bf p})$, and energy\n$\\varepsilon(x,{\\bf p})$ by\n \\begin{eqnarray}\n \\label{Ecc}\n    \\rho(x,{\\bf p})\n    &=& \\int dp_0\\, j_0(x,p) = e\\int dp_0\\, p_0 \\, W_4(x,p) \\, ,\n \\nonumber\\\\\n    {\\bf j}(x,{\\bf p})\n    &=& \\int dp_0 \\, {\\bf j}(x,p) = e\\, {\\bf p} \\, W_3(x,{\\bf p}) \\, ,\n \\nonumber\\\\\n    \\varepsilon (x,{\\bf p})\n    &=& \\int dp_0\\, T_{00}(x,p) = \\int dp_0 \\, p_0^2 \\, W_4(x,p) \\, .\n \\end{eqnarray}\n\nThe simplest homogeneous solution of Eq.~(\\ref{Ec}) is\n \\begin{eqnarray}\n \\label{Va}\n    W_3(x,{\\bf p})  &=& {1\\over E_p} \\, ,\n \\nonumber \\\\\n    \\rho(x,{\\bf p}) &=& 0 \\, ,\n \\nonumber \\\\\n    {\\bf j}(x,{\\bf p}) &=& e{\\bf v}=e{{\\bf p}\\over E_p}\\, ,\n \\nonumber\\\\\n    \\varepsilon(x,{\\bf p}) &=& E_p \\, .\n \\end{eqnarray}\n\nSince purely magnetic fields do not produce pairs, we can restrict our\nattention for the Schwinger pair creation mechanism to electric\nfields. We assume for simplicity a spatially homogeneous but\ntime-dependent electric field. The spatial homogeneity of the external\nfields and the initial condition (\\ref{Va}) allow to reduce the\nconstraint and transport equations to ordinary differential equations\nin time. The three-dimensional kinetic equations are obtained through\nthe energy average as\n \\begin{eqnarray}\n \\label{P1}\n    \\Bigl(D_t^2 + 4E_p^2\n          + {e\\over 3}\\,(\\partial_t {\\bf E}){\\cdot}{\\bf \\nabla}_p\n    \\Bigr)\\, W_3(t,{\\bf p})\n    &=& 4\\, \\varepsilon (t,{\\bf p}) \\, ,\n \\nonumber \\\\\n    D_t \\rho(t,{\\bf p}) &=& 0 \\, ,\n\\end{eqnarray}\nwhere\n \\begin{equation}\n \\label{P2}\n   D_t = \\partial_t+e{\\bf E}\\cdot{\\bf \\nabla}_p \\, .\n \\end{equation}\n\nTo obtain a closed equation of motion for the Wigner function\n$W_3(t,{\\bf p})$, we eliminate the energy density $\\varepsilon$ from the\nfirst equation of (\\ref{P1}). To this end we multiply the transport\nequation (\\ref{St}) by $p_0$ from the left and then integrate it with\nrespect to $p_0$:\n \\begin{equation}\n \\label{P3}\n   D_t \\varepsilon (t,{\\bf p}) =\n   \\left({e\\over 12}(\\partial_t {\\bf E}){\\cdot}{\\bf \\nabla}_p D_t\n       + {e\\over 12}(\\partial^2_t{\\bf E}){\\cdot}{\\bf \\nabla}_p\n       + e {\\bf p}{\\cdot}{\\bf E}\\right)\n   W_3(t,{\\bf p}) \\, .\n \\end{equation}\nWe then apply the operator $D_t$ on the first equation (\\ref{P1}) and\ncombine it with Eq.~(\\ref{P3}) to eliminate $\\varepsilon$. We obtain\n \\begin{equation}\n \\label{P4}\n   \\Bigl( D_t^3 + 4E_p^2D_t + 4e{\\bf E}{\\cdot}{\\bf p} \\Bigr)\n   W_3(t,{\\bf p})=0 \\ ,\n \\end{equation}\nwhich, together with the initial condition\n \\begin{equation}\n \\label{P4a}\n   W_3(t=-\\infty)={1\\over E_p} \\, ,\n \\end{equation}\ncan be solved as an initial value problem for the Wigner function.\nWe assume that the external electric field is switched on\nadiabatically in the far past and switched off adiabatically in the\nfar future, ${\\bf E}(t=-\\infty)={\\bf E}(t=\\infty)=0$.\n\nThe momentum derivative hidden in the operator $D_t$ complicates the\nsolution of Eq.~(\\ref{P4}). We follow \\cite{Re12} and use the\nwell-known method of characteristics in order to separate the momentum\nderivative and obtain an ordinary differential equation in time. One\nintroduces a test Wigner function \\cite{Re12} $\\omega_3$ through\n \\begin{equation}\n \\label{P5}\n    W_3(t,{\\bf p}) = \\int d^3{\\bf p}_0\\, \\omega_3(t,{\\bf p}_0) \\,\n    \\delta^{(3)}\\bigl({\\bf p}-{\\bf p}(t,{\\bf p}_0)\\bigr) \\, ,\n \\end{equation}\nwhere the function ${\\bf p}(t,{\\bf p}_0)$ is a solution of the\nclassical equation of motion for a particle with initial momentum\n${\\bf p}_0$ in an external electric field:\n \\begin{equation}\n \\label{P6}\n   {d{\\bf p}(t,{\\bf p}_0)\\over dt}=e{\\bf E}(t) \\, .\n\\end{equation}\nIts explicit form reads\n \\begin{equation}\n \\label{P7}\n   {\\bf p}(t,{\\bf p}_0)={\\bf p}_0+e\\int_{-\\infty}^t dt'\\, {\\bf E}(t') \\, .\n \\end{equation}\nSubstituting (\\ref{P5}) into (\\ref{P4}), the momentum derivative is\nabsorbed into the classical motion, and the partial differential equation\nis converted into\n \\begin{equation}\n \\label{P8}\n   \\int d^3{\\bf p}_0\\, \\left[\\Bigl(\\partial^3_t + 4E_p^2\\partial_t\n      + 4e{\\bf E}{\\cdot}{\\bf p}\\Bigr) \\omega_3 (t,{\\bf p}_0) \\right]\\,\n      \\delta^{(3)}\\Bigl({\\bf p}-{\\bf p}(t,{\\bf p}_0)\\Bigr)=0 \\, .\n \\end{equation}\nThis is solved if $\\omega_3(t,{\\bf p}_0)$ satisfies the ordinary\ndifferential equation\n \\begin{equation}\n \\label{P9}\n   \\Bigl(\\partial^3_t + 4E_p^2(t,{\\bf p}_0)\\,\\partial_t\n                     + 4e{\\bf E}{\\cdot}{\\bf p}(t,{\\bf p}_0)\\Bigr)\n   \\omega_3(t,{\\bf p}_0)=0\n \\end{equation}\nwith the initial condition\n \\begin{equation}\n \\label{P9a}\n   \\omega_3(t=-\\infty,{\\bf p}_0)={1\\over E_p} \\, ,\n \\end{equation}\nwhere ${\\bf p}(t,{\\bf p}_0)$ satisfies Eq.~(\\ref{P7}) and $E_p(t,{\\bf\np}_0)$ is the correspondimg on-shell energy.\n\nPlease note the time dependence of the momentum ${\\bf p}$ and the\nparticle energy $E_p$ arising from the classical equation of motion:\nwhen inserting the solution of (\\ref{P9}) into (\\ref{P5}) in order to\nconstruct the Wigner function $W_3$ from the test function $\\omega_3$,\nall the classical time dependence of $W_3$ resides in these functions\n${\\bf p}(t,{\\bf p}_0)$, $E_p(t,{\\bf p}_0)$, while the additional time\ndependence from quantum effects is described by the differential\nequation (\\ref{P9}) for $\\omega_3$.\n\nFor later it will be useful to similarly introduce a test charge\ndensity $\\varrho (t,{\\bf p}_0)$, a test charge current ${\\bf \\cal\nJ}(t,{\\bf p}_0)$, and a test energy density ${\\bf \\cal E}(t,{\\bf\np}_0)$; for example,\n \\begin{equation}\n \\label{P90}\n   \\rho(t,{\\bf p})=\\int d^3{\\bf p}_0\\, \\varrho (t,{\\bf p}_0)\\,\n   \\delta^{(3)}\\Bigl({\\bf p}-{\\bf p}(t,{\\bf p}_0)\\Bigr) \\, .\n \\end{equation}\nThey satisfy for arbitrary initial momentum ${\\bf p}_0$ the following\nequations:\n \\begin{eqnarray}\n \\label{P10}\n   \\partial_t\\, \\varrho(t,{\\bf p}_0)\n   &=& 0 \\, ,\n \\nonumber \\\\\n   \\partial_t\\, {\\bf \\cal E}(t,{\\bf p}_0)\n   &=& {\\bf E}\\cdot {\\bf \\cal J}(t,{\\bf p}_0) \\, ,\n\\end{eqnarray}\nwhich follow from Eqs.(\\ref{P1}) and (\\ref{P3}), respectively. Their\nphysical meaning is evident: The first equation shows that the net\ncharge density always vanishes due to the homogeneous initial\nconditions (\\ref{Va}). The second one is just Poynting's theorem of\nenergy-momentum conservation.\n\n \\subsubsection{Pair density}\n\nAs pointed out in \\cite{Re19}, the problem of pair creation in a\nhomogeneous, but time-dependent electric field can be mapped onto the\nquantum mechanics problem of an oscillator with variable frequency.\nThis will be exploited in the following treatment. We introduce an\nauxiliary function $\\zeta(t)$ via\n \\begin{equation}\n \\label{O1}\n   \\omega_3(t) = |\\zeta(t)|^2 \\, ,\n \\end{equation}\nin terms of which the pair production problem (\\ref{P9},\\ref{P9a})) is\nreduced to the solution of a quantum oscillator problem:\n \\begin{equation}\n \\label{O2}\n   \\partial^2_t \\zeta + E_p^2(t) \\, \\zeta=0\\, ,\\qquad\n   |\\zeta(t=-\\infty)|= 1\/\\sqrt{E_p} \\, .\n \\end{equation}\nSince we are interested in the total pair production yield at time\n$t\\rightarrow \\infty$, it is sufficient to study the asymptotic\nsolutions of this equation. Due to its similarity with the\ntime-dependent barrier-potential problem \\cite{Re20} in\nnon-relativistic quantum mechanics, the asymptotic solutions can be\neasily written down in WKB approximation:\n \\begin{eqnarray}\n \\label{O4}\n    \\zeta(t\\to -\\infty)\n    &=& e^{-iE_p^{-\\infty}t} \/ \\sqrt{E_p^{-\\infty}} \\, ,\n \\nonumber \\\\\n    \\zeta(t\\to + \\infty)\n    &=& C_1e^{-iE_p^\\infty t}+C_2e^{iE_p^\\infty t} \\, .\n\\end{eqnarray}\nHere we have already taken into account the initial condition for\n$\\zeta$, and $E_p^{-\\infty}$ and $E_p^\\infty$ are the asymptotic\nparticle energies $E_p^{-\\infty}=\\sqrt{m^2+{\\bf p}_0^2}\\,$ and\n$E_p^\\infty = \\sqrt{m^2+({\\bf p}(t=\\infty,{\\bf p}_0))^2}$. The\ncondition for applicability of the WKB approximation \\cite{Re21},\n \\begin{equation}\n \\label{O3}\n   \\partial_t E_p = {e{\\bf p}{\\cdot}{\\bf E}\\over E_p} \\ll E_p^2 \\, ,\n \\end{equation}\nis satisfied in the limit $t\\to \\pm \\infty$ since the electric field\nvanishes in this limit.\n\n{}From these asymptotic solutions we see that the test Wigner function\nin the limit $t\\to +\\infty$ contains both oscillating and\nnon-oscillating parts:\n \\begin{equation}\n \\label{O5}\n    \\omega_3(t\\to \\infty) = \\vert \\zeta(t\\to \\infty) \\vert^2\n    = (|C_1|^2+|C_2|^2)+(C_1C_2^*e^{-2iE_p^\\infty t} + {\\rm c.c.}) \\, .\n \\end{equation}\nThe created pairs are separated and accelerated by the electric field\nin opposite directions, thereby generating a current. Writing this\ncurrent as ${\\bf \\cal J}(t) = e{\\bf v(t)} n(t)$, where $n(t)$ is the\ntotal particle density (positive plus negative particles), and\ncomparing with its expression in terms of the test Wigner function,\nnamely, ${\\bf \\cal J}(t) = e{\\bf p}(t)\\, \\omega_3(t)$, we find\n \\begin{equation}\n \\label{P11}\n    n(t) = E_p(t)\\, \\omega_3(t) \\, .\n \\end{equation}\nTaking the time average of Eq.~(\\ref{O5}) which removes the rapidly\noscillating parts, we thus find for the asymptotic value of the total\nparticle density\n \\begin{equation}\n \\label{P12}\n    n(t=\\infty)=E_p^\\infty \\Bigl( |C_1|^2 + |C_2|^2 \\Bigr) \\, .\n \\end{equation}\nDue to the conservation law\n \\begin{equation}\n \\label{P13}\n   \\partial_t\\Big[(\\partial_t \\zeta) \\zeta^* -\n                  (\\partial_t \\zeta^*) \\zeta \\Big] = 0 \\, ,\n \\end{equation}\nwhich is easily derived from (\\ref{O2}), the expression in the square\nbracket is the same at $t=+\\infty$ and $t=-\\infty$ and can thus be\nevaluated with the initial conditions (\\ref{O4}). We obtain\n \\begin{equation}\n \\label{P14}\n   |C_1|^2 - |C_2|^2 = {1\\over E_p^\\infty} \\, .\n\\end{equation}\nThus the oscillator is fully characterized by the ratio\n \\begin{equation}\n \\label{P15}\n    r = {|C_2|^2 \\over |C_1|^2} \\, ,\n \\end{equation}\nwhich can be interpreted (and calculated) as the transmission\ncoefficient of the non-relativistic barrier-potential problem\n\\cite{Re20}. In terms of this ratio, the asymptotic particle density\nis given as\n \\begin{equation}\n \\label{P16}\n   n(t=\\infty) = {1+r\\over 1-r} = 1+2{r\\over 1-r} \\, .\n \\end{equation}\nThe first term on the right-hand side is a vacuum contribution and\nstems from the Dirac sea of charged particles; it can not be measured\nand must be removed by renormalization \\cite{Re17}. The second term\narises from the pair creation. The factor of $2$ reminds us that both\nparticles and antiparticles contribute to the pair current. The\nasymptotic pair density in phase-space is thus finally obtained as\n \\begin{equation}\n \\label{P17}\n   n_{pair}(t=\\infty) = {r \\over 1-r} \\, ,\n \\end{equation}\nwhich is in full agreement with the results derived both from field\ntheory \\cite{Re19} and from the transport theory in Feshbach-Villars\nrepresentation \\cite{Re17}.\n\nEven though the final results of the two approaches to scalar\ntransport theory, the equal-time and the energy averaging methods, are\nthe same, we would like to point out one essential difference between\nthese two procedures: The equal-time formulation has to rely on the\nFeshbach-Villars representation, since it requires field equations\nwhich contain only first-order time derivatives. As a result, the\nWigner operator in the equal-time approach is a $2\\times 2$ matrix\nand first must be decomposed into its ``spinor'' components, in an\nanalogous way as required for spinor QED which will be discussed in\nthe following section. This matrix structure leads to a set of coupled\nkinetic equations for the ``spinor'' components of $W_3$. The\nenergy averaging method works directly with the covariant Klein-Gordon\nequation. This is a scalar equation, and no such complication arises.\n\n \\section{Spinor electrodynamics}\n \\subsection{Covariant version of the BGR equation}\n\nWe now turn to the investigation of the transport theory for\nspin-${1\\over 2}$ particles interacting with an\nelectromagnetic field. We start from the lagrangian density\n \\begin{equation}\n \\label{D1}\n   {\\cal L} = \\bar\\psi\\Big(i\\gamma^\\mu(\\partial_\\mu+ieA_\\mu)-m\\Big)\\psi\n            - {\\textstyle{1\\over 4}} F^{\\mu\\nu}F_{\\mu\\nu} \\, ,\n \\end{equation}\nwhich gives rise to the Dirac equations for the complex fields $\\psi$\nand $\\bar\\psi$,\n \\begin{eqnarray}\n \\label{D2}\n   i\\gamma^\\mu(\\partial_\\mu+ieA_\\mu)\\psi\n   &=& m\\psi \\, ,\n \\nonumber\\\\\n   i(\\partial_\\mu-ieA_\\mu)\\bar\\psi\\gamma^\\mu\n   &=& - m\\bar\\psi \\, .\n \\end{eqnarray}\nThe spinor Wigner operator is the four-dimensional Fourier transform\nof the gauge invariant density matrix\n \\begin{equation}\n \\label{D3}\n   \\Phi_4(x,y)= \\psi\\left(x+{\\textstyle{1\\over 2}} y\\right)\n      \\exp\\left[ ie\\int^{1\\over 2}_{-{1\\over 2}}ds\\,\n                 A(x+sy){\\cdot}y \\right]\n      \\bar\\psi\\left(x-{\\textstyle{1\\over 2}} y\\right) \\, .\n\\end{equation}\nUnlike scalar electrodynamics, $\\Phi_4$ is now a $4\\times 4$ matrix in\nspin space, and the Wigner operator $\\hat W$ is no longer\nself-adjoint. It behaves under hermitian conjugation like an\nordinary $\\gamma$-matrix,\n \\begin{equation}\n \\label{D4}\n   \\hat W_4^\\dagger(x,p)=\\gamma_0\\hat W_4(x,p)\\gamma_0 \\, .\n \\end{equation}\n\nThe evolution equation of the covariant Wigner operator again follows\nfrom the field equations of motion. Calculating the first-order\nderivatives of the density matrix and using the Dirac equations in a\nsimilar way as in the scalar case, we find a pair of equations for\n$\\Phi_4$ in terms of the operators $D$ and $\\Pi$,\n \\begin{eqnarray}\n \\label{D5}\n    \\left( {\\textstyle{1\\over 2}} D_\\mu(x,y) - i\\Pi_\\mu(x,y) \\right)\n         \\gamma^0 \\gamma^\\mu \\Phi_4(x,y)\n    &=& - i m \\gamma^0 \\Phi_4(x,y) \\, ,\n \\nonumber\\\\\n    \\left( {\\textstyle{1\\over 2}} D_\\mu(x,y) + i\\Pi_\\mu(x,y) \\right)\n         \\Phi_4(x,y) \\gamma^\\mu \\gamma^0\n    &=& i m \\Phi_4(x,y) \\gamma^0 \\, .\n \\end{eqnarray}\nThe $\\gamma_0$-matrices have been included in order to facilitate\ncomparison with the three-dimensional equal-time approach of\nBGR \\cite{Re12}.\n\nMultiplying (\\ref{D5}) by another $\\gamma_0$-matrix from the right and\ntaking the Wigner transform, we derive the following kinetic equations\nfor the Wigner operator:\n \\begin{eqnarray}\n \\label{D6}\n    &&\\left( {\\textstyle{1\\over 2}} \\hbar D_\\mu(x,p) - i\\Pi_\\mu(x,p)\\right)\n            \\gamma^0 \\gamma^\\mu \\hat W_4(x,p) \\gamma^0\n        = - i m \\gamma^0 \\hat W_4(x,p)\\gamma^0 \\, ,\n \\\\\n \\label{D61}\n   &&\\left( {\\textstyle{1\\over 2}} \\hbar D_\\mu(x,p) + i\\Pi_\\mu(x,p)\\right)\n           \\hat W_4(x,p) \\gamma^\\mu\n    = i m \\hat W_4(x,p) \\, .\n \\end{eqnarray}\nWith an eye on the semiclassical expansion below, we have again\ndisplayed the $\\hbar$-depen\\-dence explicitly. We note that these two\nequations of motion in phase space, like the two Dirac equations\n(\\ref{D2}) in coordinate space, are adjoints of each other. Therefore,\neither one of the Eqs.~(\\ref{D6},\\ref{D61}) provides a complete\ndescription of the Wigner operator. By adding and subtracting these\ntwo equations, respectively, we can get two different, now\nself-adjoint equations:\n \\begin{eqnarray}\n \\label{D7}\n    {\\textstyle{1\\over 2}} \\hbar D_\\mu \\{ \\gamma^0 \\gamma^\\mu,\\ \\hat W_4\\gamma^0\\}\n    - i \\Pi_\\mu [\\gamma^0\\gamma^\\mu, \\ \\hat W_4\\gamma^0]\n    &=& -i m [\\gamma^0, \\ \\hat W_4\\gamma^0] \\, ,\n \\\\\n \\label{D8}\n    {\\textstyle{1\\over 2}} \\hbar D_\\mu [\\gamma^0 \\gamma^\\mu, \\ \\hat W_4 \\gamma^0]\n    - i \\Pi_\\mu \\{\\gamma^0\\gamma^\\mu, \\ \\hat W_4\\gamma^0 \\}\n    &=& - i m \\{\\gamma, \\ \\hat W_4\\gamma^0 \\} \\, .\n \\end{eqnarray}\nObviously (\\ref{D7}) and (\\ref{D8}) are symmetric under an exchange of\ncommutators and anticommutators. Their self-adjoint properties are\neasily proven with the aid of Eq.~(\\ref{D4}). Since these two\nself-adjoint equations are equivalent to either of the equations\n(\\ref{D6},\\ref{D61}), we have now three choices for a full description\nof the Wigner operator: Eq.~(\\ref{D6}), Eq.(\\ref{D61}), or\nEq.(\\ref{D7}) together with Eq.(\\ref{D8}). Eqs.(\\ref{D7}) and\n(\\ref{D8}) do not separately provide a complete description.\n\nWe would like to note another important feature of Eq.(\\ref{D7}): The\ncommutator in the second term of the left-hand side leads to the\nautomatic disappearance of $p_0$ contained in the operator\n$\\Pi_0(x,p)$. Since the only remaining $p_0$-dependence is in $\\hat\nW_4$, this renders the energy average very simple. The result of an\nintegration over $p_0$ is just the BGR equation from the equal-time\nformulation,\n \\begin{equation}\n \\label{D9}\n    \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n    \\hbar D_t \\hat W_3(x,{\\bf p}) = - {\\textstyle{1\\over 2}} \\hbar\n    {\\bf D}{\\cdot}\\{ \\gamma^0 {\\mbox{\\boldmath$\\gamma$}}, \\ \\hat W_3(x,{\\bf p}) \\}\n    - i {\\bf \\Pi}{\\cdot}[ \\gamma^0 {\\mbox{\\boldmath$\\gamma$}}, \\ \\hat W_3(x,{\\bf p})]\n    - i m [\\gamma^0, \\ \\hat W_3(x,{\\bf p})] \\, ,\n \\end{equation}\nwith\n \\begin{eqnarray}\n \\label{D10}\n    D_t(x,{\\bf p})\n    &=& \\partial_t + e \\int^{1\\over 2}_{-{1\\over 2}} ds\\,\n        {\\bf E}({\\bf x}+is\\hbar{\\bf \\nabla}_p, t){\\cdot}{\\bf \\nabla}_p\n    \\, ,\n \\nonumber \\\\\n    {\\bf D}(x,{\\bf p})\n    &=& {\\bf \\nabla} + e \\int^{1\\over 2}_{-{1\\over 2}}ds\\,\n    {\\bf B}({\\bf x}+is\\hbar{\\bf \\nabla}_p, t) \\times {\\bf \\nabla}_p\n    \\, ,\n \\nonumber \\\\\n    {\\bf \\Pi}(x,{\\bf p})\n    &=& {\\bf p}-i e \\hbar \\int^{1\\over 2}_{-{1\\over 2}}ds\\, s\\,\n    {\\bf B}({\\bf x} + is\\hbar{\\bf \\nabla}_p, t) \\times {\\bf \\nabla}_p\n    \\, ,\n \\end{eqnarray}\nwhere we have employed the BGR definition of the equal-time\nWigner operator \\cite{Re12}:\n \\begin{eqnarray}\n \\label{D11}\n    && \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n       \\hat W_3(x,{\\bf p}) = \\int dp_0 \\, \\hat W_4(x,p)\\, \\gamma_0\n \\\\\n    && = \\int d^3{\\bf y}\\, e^{-i{\\bf p}\\cdot{\\bf y}}\\,\n        \\psi\\left({\\bf x}+{\\textstyle{1\\over 2}}{\\bf y},t\\right)\\,\n        \\exp\\left[ - i e \\int^{1\\over 2}_{-{1\\over 2}}ds\\,\n                   {\\bf A}({\\bf x}+s{\\bf y},t)\\cdot{\\bf p} \\right]\n        \\psi^\\dagger\\left({\\bf x}-{\\textstyle{1\\over 2}}{\\bf y},t\\right) \\, .\n \\nonumber\n \\end{eqnarray}\nClearly equation (\\ref{D7}) is the covariant version of the BGR\nequation. However, as we have stressed, (\\ref{D7}) is not complete.\nOnly together with the self-adjoint equation (\\ref{D8}) one obtains a\nfull description for the Wigner operator. Therefore the covariant\nversion of the BGR equation is only one  part of the full covariant\ntransport theory. The question thus arises whether on the\nthree-dimensional level the BGR equation can be complete or not. To\nanswer this question we will reduce Eq.~(\\ref{D8}) to the\nthree-dimensional level by performing an energy average, too, and\nstudy its implications.\n\n \\subsection{Energy average of full covariant theory}\n\nThe two self-adjoint equations (\\ref{D7}) and (\\ref{D8}) have a\ncomplicated structure due to the occurrence of commutators and\nanticommutators. But since they are equivalent to either equation\n(\\ref{D6}) or (\\ref{D61}), we can consider equation (\\ref{D6})\ninstead. To simplify the calculation further, we even remove the\n$\\gamma_0$-matrices and thus obtain the VGE \\cite{Re11} equation:\n \\begin{equation}\n \\label{V1}\n   \\left[ \\gamma^\\mu \\left(\\Pi_\\mu+{\\textstyle{1\\over 2}} i\\hbar D_\\mu \\right)\n          -m \\right] \\hat W_4(x,p) = 0 \\, .\n \\end{equation}\nAs in the scalar case, we will from now on consider the\nelectromagnetic field as classical and pass from the Wigner operator\nto the Wigner function by leaving off the hats.\n\nThe Wigner function in spinor electrodynamics is a complex $4\\times 4$\nmatrix. VGE discussed its spinor decomposition and derived a set of\ncoupled equations for the components of $W_4$. Because of their\ncharacteristic transformation properties under Lorentz\ntransformations, it is convenient to choose the 16 matrices\n$\\Gamma_i=\\{1,\\ i\\gamma_5, \\ \\gamma_\\mu, \\ \\gamma_\\mu\\gamma_5, \\\n{1\\over 2}\\sigma_{\\mu\\nu}\\}$ as the basis for an expansion of the\nWigner function in spin space:\n \\begin{equation}\n \\label{V2}\n    \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n    W_4(x,p) = {\\textstyle{1\\over 4}} \\left[ F(x,p) + i \\gamma_5 P(x,p)\n             + \\gamma_\\mu V^\\mu(x,p) + \\gamma_\\mu \\gamma_5 A^\\mu(x,p)\n             + {\\textstyle{1\\over 2}} \\sigma_{\\mu\\nu} S^{\\mu\\nu}(x,p)\\right] \\, .\n \\end{equation}\nAll the components $F, P, V_\\mu, A_\\mu,$ and $S_{\\mu\\nu}$ are\nreal functions since the basis elements $\\Gamma_i$ transform under\nhermitian conjugation like $W_4$ itself, $\\Gamma_i^\\dagger = \\gamma_0\n\\Gamma_i \\gamma_0$ (see Eq.~(\\ref{D4})). They can thus\nbe interpreted as physical phase-space densities. The expansion\n(\\ref{V2}) decomposes the VGE equation into 5 coupled Lorentz\ncovariant equations for the spinor components. Since these components\nare real and the operators $D$ and $\\Pi$ are self-adjoint, one can\nseparate the real and imaginary parts of these 5 complex equations to\nobtain 10 real equations:\n \\begin{eqnarray}\n \\label{V3}\n    &&\\Pi^\\mu V_\\mu = m F \\, ,\n \\nonumber \\\\\n    &&\\hbar D^\\mu A_\\mu = 2m P \\, ,\n \\nonumber \\\\\n    &&\\Pi_\\mu F - {\\textstyle{1\\over 2}} \\hbar D^\\nu S_{\\nu\\mu}=m V_\\mu \\, ,\n \\nonumber \\\\\n    &&-\\hbar D_\\mu P+\\epsilon_{\\mu\\nu\\sigma\\rho}\\Pi^\\nu S^{\\sigma\\rho}\n      = 2m A_\\mu \\, ,\n \\nonumber \\\\\n    &&{\\textstyle{1\\over 2}} \\hbar (D_\\mu V_\\nu-D_\\nu V_\\mu)\n      + \\epsilon_{\\mu\\nu\\sigma\\rho}\\Pi^\\sigma A^\\rho\n      = m S_{\\mu\\nu} \\, ,\n \\end{eqnarray}\nand\n \\begin{eqnarray}\n \\label{V3a}\n    &&\\hbar D^\\mu V_\\mu = 0 \\, ,\n \\nonumber \\\\\n    &&\\Pi^\\mu A_\\mu =0 \\, ,\n \\nonumber \\\\\n    &&{\\textstyle{1\\over 2}} \\hbar D_\\mu F = - \\Pi^\\nu S_{\\nu\\mu} \\, ,\n \\nonumber \\\\\n    &&\\Pi_\\mu  P= -{\\textstyle{1\\over 4}} \\hbar \\epsilon_{\\mu\\nu\\sigma\\rho} D^\\nu\n      S^{\\sigma\\rho} \\, ,\n \\nonumber \\\\\n    &&\\Pi_\\mu V_\\nu-\\Pi_\\nu V_\\mu = {\\textstyle{1\\over 2}} \\hbar\n      \\epsilon_{\\mu\\nu\\sigma\\rho} D^\\sigma A^\\rho  \\, .\n \\end{eqnarray}\n\nPeforming an energy average of the expansion of $W_4$, (\\ref{V2}), we\nobtain a full mapping from the four- to the three-dimensional\ncomponents introduced in \\cite{Re12},\n \\begin{eqnarray}\n \\label{V4}\n    W_3(x,{\\bf p}) =\n    && \\!\\!\\!\\!\\!\\!\n       {\\textstyle{1\\over 4}} \\Bigl[ f_0(x,{\\bf p}) + \\gamma_5 f_1(x,{\\bf p})\n                        -i \\gamma_0 \\gamma_5 f_2(x,{\\bf p})\n                        + \\gamma_0 f_3(x,{\\bf p})\n \\\\\n   && + \\gamma_5 {\\mbox{\\boldmath$\\gamma$}}{\\cdot}{\\bf g}_0(x,{\\bf p})\n      + \\gamma_0 {\\mbox{\\boldmath$\\gamma$}}{\\cdot}{\\bf g}_1(x,{\\bf p})\n      - i{\\mbox{\\boldmath$\\gamma$}}{\\cdot}{\\bf g}_2(x,{\\bf p})\n      - \\gamma_5 {\\mbox{\\boldmath$\\gamma$}}{\\cdot}{\\bf g}_3(x,{\\bf p})\\Bigr]\\, ,\n \\nonumber\n \\end{eqnarray}\nnamely,\n \\begin{eqnarray}\n \\label{V5}\n    && f_0(x,{\\bf p})=\\int dp_0 \\, V_0(x,p) \\, ,\n \\nonumber \\\\\n    && f_1(x,{\\bf p})=-\\int dp_0 \\, A_0(x,p) \\, ,\n \\nonumber \\\\\n    && f_2(x,{\\bf p})=\\int dp_0 \\, P(x,p) \\, ,\n \\nonumber \\\\\n    && f_3(x,{\\bf p})=\\int dp_0 \\, F(x,p) \\, ,\n \\nonumber \\\\\n    && {\\bf g}_0 (x,{\\bf p})=-\\int dp_0 \\, {\\bf A}(x,p) \\, ,\n \\nonumber \\\\\n    && {\\bf g}_1 (x,{\\bf p})=\\int dp_0 \\, {\\bf V}(x,p) \\, ,\n \\nonumber \\\\\n    && g_2^i (x,{\\bf p})=-\\int dp_0 \\, S^{0i}(x,p) \\, , \\quad\n       i=1,2,3\\, ,\n \\nonumber \\\\\n    && g_3^i (x,{\\bf p}) = {\\textstyle{1\\over 2}} \\epsilon^{ijk}\\int dp_0 \\,\n       S_{jk}(x,p) \\, , \\quad i=1,2,3 .\n \\end{eqnarray}\nAs shown in \\cite{Re12,Re18}, the physically interesting densities\nsuch as charge current, energy momentum tensor and angular momentum\ntensor, can also be expressed in terms of these spinor components. For\nexample, $f_0$ and ${\\bf g}_0$ are the charge density and the spin density,\nrespectively.\n\nThe VGE equations (\\ref{V3},\\ref{V3a}) can be divided into two groups.\nGroup I contains the operator $\\Pi_0(x,p)$ which involves $p_0$; group\nII contains no $p_0$-dependence except for the one in the Wigner\nfunction itself. For group I the energy average is straightforward.\nThe result is in complete agreement with the spinor decomposition of\nthe BGR equation \\cite{Re12}:\n \\begin{eqnarray}\n \\label{BGR}\n   &&\\hbar(D_t f_0+{\\bf D}{\\cdot}{\\bf g}_1)=0 \\, ,\n \\nonumber \\\\\n   &&\\hbar(D_t f_1+{\\bf D}{\\cdot}{\\bf g}_0)=-2m f_2  \\, ,\n \\nonumber \\\\\n   &&\\hbar D_t f_2+2{\\bf \\Pi}{\\cdot}{\\bf g}_3=2m f_1  \\, ,\n \\nonumber \\\\\n   &&\\hbar D_t  f_3-2{\\bf \\Pi}{\\cdot}{\\bf g}_2)=0 \\, ,\n \\nonumber \\\\\n   &&\\hbar(D_t {\\bf g}_0+{\\bf D} f_1)-2{\\bf \\Pi}\\times {\\bf g}_1=0 \\, ,\n \\nonumber \\\\\n   &&\\hbar(D_t {\\bf g}_1+{\\bf D} f_0)-2{\\bf \\Pi}\\times {\\bf g}_0\n     = -2m {\\bf g}_2 \\, ,\n \\nonumber \\\\\n   &&\\hbar(D_t {\\bf g}_2+{\\bf D}\\times {\\bf g}_3)+2{\\bf \\Pi} f_3\n     =2m {\\bf g}_1 \\, ,\n \\nonumber \\\\\n   &&\\hbar(D_t {\\bf g}_3-{\\bf D}\\times {\\bf g}_2)-2{\\bf \\Pi} f_2 =0 \\, .\n \\end{eqnarray}\n\nDue to the additional $p_0$-dependence from the operator $\\Pi_0$, the\nequations in group II can not be completely reduced to expressions\nfrom the set of three-dimensional components given in Eq.~(\\ref{V5}).\nThey contain additionally higher $p_0$-moments of the four-dimensional\ncomponents,\n \\begin{eqnarray}\n \\label{V6}\n    &&\\int dp_0\\, p_0\\, V_0 -{\\bf \\Pi}\\cdot {\\bf g}_1\n      + \\tilde\\Pi_0 f_0=m f_3 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, A_0 +{\\bf \\Pi}\\cdot {\\bf g}_0\n      - \\tilde\\Pi_0 f_1 = 0 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, P + {\\textstyle{1\\over 2}} \\hbar {\\bf D}\\cdot {\\bf g}_3\n      + \\tilde\\Pi_0 f_2 = 0 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, F - {\\textstyle{1\\over 2}} \\hbar {\\bf D}\\cdot {\\bf g}_2\n      + \\tilde\\Pi_0 f_3=m f_0 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, {\\bf A} + {\\textstyle{1\\over 2}} \\hbar {\\bf D}\\times {\\bf g}_1\n      + {\\bf \\Pi} f_1 - \\tilde\\Pi_0 {\\bf g}_0 = -m {\\bf g}_3 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, {\\bf V} - {\\textstyle{1\\over 2}} \\hbar {\\bf D}\\times {\\bf g}_0\n      - {\\bf \\Pi} f_0 + \\tilde\\Pi_0 {\\bf g}_1 = 0 \\, ,\n \\nonumber\\\\\n    &&\\int dp_0\\, p_0\\, S^{0i}{\\bf e}_i - {\\textstyle{1\\over 2}} \\hbar {\\bf D} f_3\n      + {\\bf \\Pi}\\times {\\bf g}_3 - \\tilde\\Pi_0 {\\bf g}_2 = 0 \\, ,\n \\nonumber \\\\\n    &&\\int dp_0\\, p_0\\, S_{jk} \\epsilon^{jki}{\\bf e}_i\n      - \\hbar{\\bf D} f_2 +2{\\bf \\Pi}\\times {\\bf g}_2\n      + 2 \\tilde\\Pi_0 {\\bf g}_3 =2m {\\bf g}_0 \\, ,\n \\end{eqnarray}\nwhere we introduced the three-dimensional operator\n \\begin{equation}\n \\label{V7}\n    \\tilde \\Pi_0 (x,{\\bf p}) = i e \\hbar\n    \\int^{1\\over 2}_{-{1\\over 2}}ds\\, s\\, {\\bf E}({\\bf x}\n         + i s \\hbar {\\bf \\nabla}_p,t){\\cdot}{\\bf \\nabla}_p \\, .\n \\end{equation}\nThis second set of coupled equations for the components of $W_3$ does\nnot include the operator $D_t$. It corresponds to the energy average\nof the spinor decomposition of the second self-adjoint equation\n(\\ref{D8}), where $D_t$ drops out due to the commutator in the first\nterm. These equations are therefore constraint equations rather than\nequations of motion. They arise in addition to the BGR equations.\n\n \\subsection{Classical limit}\n\nBefore discussing the non-perturbative consequences of Eqs.~(\\ref{V6})\nwe show how they constrain the transport equations in the classical\nlimit. As $\\hbar\\to 0$, the original VGE equation can be\nwritten in the following quadratic form:\n \\begin{equation}\n \\label{Cl1}\n   (p^2-m^2) \\, W_4(x,p) = 0 \\, .\n \\end{equation}\nThis shows that the classical Wigner operator and hence all its spinor\ncomponents are all on the mass shell, and thus they have positive and\nnegative energy solutions similar to Eq.(\\ref{Sw5}),\n \\begin{eqnarray}\n \\label{Cl10}\n    && f_i^{+(0)}(x,{\\bf p})\\,\\delta(p_0-E_p),\\quad\n       f_i^{-(0)}(x,{\\bf p})\\,\\delta(p_0+E_p)\\, ,\n \\nonumber\\\\\n    && {\\bf g}_i^{+(0)}(x,{\\bf p})\\,\\delta(p_0-E_p),\\quad\n       {\\bf g}_i^{-(0)}(x,{\\bf p})\\,\\delta(p_0+E_p)\\, ,\n \\nonumber\\\\\n    &&i=0,1,2,3\\, .\n \\end{eqnarray}\nWith this the remaining $p_0$-integrals in Eqs.~(\\ref{V6}) can be done,\nand everything can be fully expressed in terms of the\nthree-dimensional functions (\\ref{V5}). Eqs.~(\\ref{V6}) constribute\naltogether six independent constraints for the positive and negative\nenergy parts of the spinor components:\n \\begin{eqnarray}\n \\label{Cl2}\n   && f^{\\pm (0)}_1=\\pm {{\\bf p}{\\cdot}{\\bf g}^{\\pm (0)}_0 \\over E_p} \\, ,\n \\nonumber\\\\\n   && f^{\\pm (0)}_2=0 \\, ,\n \\nonumber\\\\\n   && f^{\\pm (0)}_3=\\pm {m\\over E_p}\\, f^{\\pm (0)}_0 \\, ,\n \\nonumber\\\\\n   && {\\bf g}^{\\pm (0)}_1=\\pm {{\\bf p}\\over E_p}\\, f^{\\pm (0)}_0 \\, ,\n \\nonumber\\\\\n   && {\\bf g}^{\\pm (0)}_2={{\\bf p}\\times {\\bf g}^{\\pm (0)}_0\\over m} \\, ,\n \\nonumber\\\\\n   && {\\bf g}^{\\pm (0)}_3=\\pm {E_p^2 {\\bf g}^{\\pm (0)}_0\n      -({\\bf p}{\\cdot}{\\bf g}^{\\pm (0)}_0){\\bf p}\\over m E_p} \\, .\n \\end{eqnarray}\nIn addition, the classical limit \\cite{Re18} of the BGR equations in\n(\\ref{BGR}) gives another four constraints. Three of them are,\nhowever, already included in (\\ref{Cl2}), namely, the equations for\n$f^{+(0)}_2, {\\bf g}^{+(0)}_1$ and ${\\bf g}^{+ (0)}_2$.  The last one,\n$f^{+(0)}_1 = {\\bf p}{\\cdot}{\\bf g}^{+(0)}_3\/m$, is not independent of\nthe above equations either, but can be obtained through a linear\ncombination of Eqs.~(\\ref{Cl2}). Therefore, Eqs.~(\\ref{Cl2}) provide a\ncomplete set of constraint equations for the three-dimensional spinor\ncomponents in the classical limit. There remain only two independent\ncomponents, for example, the charge density $f^{(0)}_0$ and the spin density\n${\\bf g}^{(0)}_0$. The classical limit of the BGR equations\nyields only the positive energy parts of the latter 4 constraints,\nand misses the first and the third additional equations from (\\ref{Cl2}).\n\nThe classical transport equations for $f^{(0)}_0$ and ${\\bf g}^{(0)}_0$\narise from the first order in $\\hbar$ of the kinetic equations. From\nthe BGR equations in (\\ref{BGR}), we have\n \\begin{eqnarray}\n \\label{Cl3}\n   && (\\partial_t + e {\\bf E}{\\cdot}{\\bf \\nabla}_p) f^{\\pm(0)}_0\n       + ({\\bf \\nabla} + e {\\bf B}\\times {\\bf \\nabla}_p)\n         \\cdot {\\bf g}^{\\pm(0)}_1 = 0 \\, ,\n \\nonumber\\\\\n   && (\\partial_t + e {\\bf E}{\\cdot}{\\bf \\nabla}_p) {\\bf g}^{\\pm(0)}_3\n       - ({\\bf \\nabla} + e {\\bf B}\\times {\\bf \\nabla}_p)\n         \\times {\\bf g}^{\\pm(0)}_2\n \\nonumber\\\\\n   && \\qquad\\qquad + {{\\bf p}\\over m}\n      \\Big((\\partial_t +e{\\bf E}{\\cdot}{\\bf \\nabla}_p) f^{\\pm(0)}_1\n          +({\\bf \\nabla}+e{\\bf B}\\times\n            {\\bf \\nabla}_p){\\cdot}{\\bf g}^{\\pm(0)}_0 \\Big) = 0 \\, .\n \\end{eqnarray}\nUsing the constraint equations, some straightforward manipulations\nlead to the following decoupled Vlasov-type\nequation for the charge density,\n \\begin{equation}\n \\label{Cl4}\n    \\partial_t f^{\\pm(0)}_0 \\pm {\\bf v}{\\cdot}{\\bf \\nabla} f^{\\pm(0)}_0\n    + e ({\\bf E} \\pm {\\bf v}\\times {\\bf B}){\\cdot}{\\bf \\nabla}_p\n         f^{\\pm(0)}_0\n    = 0 \\, ,\n \\end{equation}\n and the three-dimensional kinetic equation for the spin density,\n \\begin{eqnarray}\n \\label{Cl5}\n   &&\\partial_t {\\bf g}^{\\pm(0)}_0\n     \\pm ({\\bf v}{\\cdot}{\\bf \\nabla}) {\\bf g}^{\\pm(0)}_0\n     + e \\Big[({\\bf E}\\pm{\\bf v}\\times {\\bf B}){\\cdot}{\\bf \\nabla}_p \\Big]\n              {\\bf g}^{\\pm(0)}_0\n \\nonumber\\\\\n   &&\\qquad\\qquad  - {e\\over E_p^2} \\Big[\n               ({\\bf p}{\\cdot}{\\bf g}^{\\pm(0)}_0){\\bf E}\n                -({\\bf E}{\\cdot}{\\bf p}) {\\bf g}^{\\pm(0)}_0\n                \\Big]\n            \\pm {e\\over E_p}{\\bf B}\\times {\\bf g}^{\\pm(0)}_0 = 0 \\, .\n \\end{eqnarray}\n\nThrough the energy averaging method, it is easy to prove that the above\nequation is just the three-dimensional formulation of the covariant\ntransport equation for the classical axial vector $A_\\mu^{(0)}$,\n \\begin{equation}\n \\label{BMT1}\n    p^\\mu(\\partial_\\mu-eF_{\\mu\\nu}\\partial_p^\\nu)A_\\sigma^{(0)}\n    =eF_{\\sigma\\rho}A^{\\rho (0)} \\, ,\n \\end{equation}\nwhich can be derived from the first order in $\\hbar$ of the linear equations\n(\\ref{V3}) and (\\ref{V3a}). From the discussions in \\cite{Re11},\nthis transport\nequation for $A_\\mu^{(0)}$ is equivalent to the covariant\nBargmann-Michel-Telegdi (BMT) equation \\cite{Re5,Re11,BMT} for a spinning\nparticle in a constant external field,\n \\begin{equation}\n \\label{BMT2}\n    m{ds^\\mu\\over d\\tau} = eF^{\\mu\\nu}(\\tau)s_\\nu(\\tau) \\, ,\n \\end{equation}\nwhere $s_\\mu = A_\\mu^{(0)}\/(A^{(0)}{\\cdot} A^{(0)})^{1\/2}$ is the covariant\nspin\nphase-space density. Therefore, the kinetic equation (\\ref{Cl5}) for the\nspin density ${\\bf g}^{\\pm(0)}_0$ can be\nrecognized as the three-dimensional BMT equation. It describes the\nprocession of the spin-polarization in a homogeneous  external\nelectromagnetic field.\n\nEquations (\\ref{Cl4}) and (\\ref{Cl5})\ncan be further simplified by introducing classical\nparticle and antiparticle distribution functions in an analogous way\nto the case of scalar QED:\n \\begin{eqnarray}\n \\label {Cl6}\n  f_i(x,{\\bf p})\n    &=& f_i^{+(0)}(x,{\\bf p}) \\, ,\n  \\nonumber\\\\\n  {\\bf g}_i(x,{\\bf p})\n    &=& {\\bf g}_i^{+(0)}(x,{\\bf p}) \\, ,\n  \\nonumber\\\\\n  \\bar f_i(x,{\\bf p})\n    &=& f_i^{-(0)}(x,-{\\bf p}) \\, ,\n  \\nonumber\\\\\n  \\bar {\\bf g}_i(x,{\\bf p})\n    &=& {\\bf g}_i^{-(0)}(x,-{\\bf p}) \\, .\n \\end{eqnarray}\nThe resulting equations for the particle distributions read\n \\begin{equation}\n \\label{Cl7}\n    \\partial_t f_0 + {\\bf v}{\\cdot}{\\bf \\nabla} f_0\n    + e ({\\bf E}+{\\bf v}\\times {\\bf B}){\\cdot}{\\bf \\nabla}_p f_0\n    = 0 \\, ,\n \\end{equation}\n \\begin{equation}\n \\label {Cl8}\n    \\partial_t{\\bf g}_0 + ({\\bf v}{\\cdot}{\\bf \\nabla}){\\bf g}_0\n    + e \\Big[ ({\\bf E}+{\\bf v}\\times {\\bf B}){\\cdot}{\\bf \\nabla}_p\n        \\Big] {\\bf g}_0\n    - {e\\over E_p^2} \\Big[ ({\\bf p}{\\cdot}{\\bf g}_0){\\bf E}\n                          -({\\bf E}{\\cdot}{\\bf p}){\\bf g}_0\n                     \\Big]\n    + {e\\over E_p} {\\bf B}\\times {\\bf g}_0 = 0 \\, ;\n \\end{equation}\nthose for the antiparticle distributions $\\bar f_0$ and $\\bar {\\bf\ng}_0$ differ only by a minus sign in front of the electric charge $e$.\n\nThe other particle and antiparticle distribution functions can be\nderived from the constraints (\\ref{Cl2}):\n \\begin{eqnarray}\n \\label {Cl12}\n   &&     f_1 = {{\\bf p}{\\cdot}{\\bf g}_0\\over E_p}\\, , \\qquad\n     \\bar f_1 = {{\\bf p}{\\cdot}{\\bar{\\bf g}}_0 \\over E_p} \\, ,\n \\nonumber\\\\\n   &&     f_2 = \\bar f_2 = 0 \\, ,\n \\nonumber\\\\\n   &&     f_3 = {m\\over E_p}f_0 \\, , \\qquad\n     \\bar f_3 = -{m\\over E_p} \\bar f_0 \\, ,\n \\nonumber\\\\\n   &&   {\\bf g}_1 = {{\\bf p}\\over E_p}f_0\\, , \\qquad\n   \\bar {\\bf g}_1 = {{\\bf p}\\over E_p}\\bar f_0 \\, ,\n \\nonumber\\\\\n   &&   {\\bf g}_2 = {{\\bf p}\\times {\\bf g}_0\\over m}\\, , \\qquad\n   \\bar {\\bf g}_2 = - {{\\bf p}\\times \\bar {\\bf g}_0\\over m} \\, ,\n \\nonumber\\\\\n   &&   {\\bf g}_3 = {E_p^2{\\bf g}_0 - ({\\bf p}{\\cdot}{\\bf g}_0){\\bf p}\n                     \\over E_p m} \\, , \\qquad\n   \\bar {\\bf g}_3 = -{E_p^2 \\bar {\\bf g}_0\n                      - ({\\bf p}{\\cdot}{\\bar{\\bf g}}_0){\\bf p}\n                     \\over E_p m} \\, .\n \\end{eqnarray}\n\nThe discussion above demonstrates the incompleteness of the equal-time\nBGR formulation, (\\ref{D9}) and (\\ref{BGR}), in the classical limit.\nWithout the additional constraints (\\ref{V6}) one has \\cite{Re18} $4$\nindependent components, $f^{(0)}_0, f^{(0)}_3, {\\bf g}^{(0)}_0$ and\n${\\bf g}^{(0)}_3$, and one obtains two groups of coupled equations,\nwhere the first one couples $f^{(0)}_0$ to $f^{(0)}_3$ and the second\none couples ${\\bf g}^{(0)}_0$ to ${\\bf g}^{(0)}_3$. The\nadditional constraints arising from the complete covariant formulation\nallow to decouple $f^{(0)}_0$ from $f^{(0)}_3$ and ${\\bf g}^{(0)}_0$\nfrom ${\\bf g}^{(0)}_3$. Therefore, we need only one scalar and one\nvector component to completely describe the classical behavior of the\nWigner operator. In fact, already VGE \\cite{Re11} pointed out in their\ncovariant formulation that to any finite order in $\\hbar$ the\npseudoscalar $P$, vector $V^\\mu$ and antisymmetric tensor\n$S^{\\mu\\nu}$\ncomponents can be expressed in terms of the scalar $F$ and axial\nvector $A^\\mu$ components. Furthermore, the general relationship\n$\\Pi_\\mu A^\\mu = 0$ in (\\ref{V3}) shows that $A_0$ is not an\nindependent component either. Thus only 4 of the $16$ components of\nthe Wigner function are dynamically independent. Through our mapping\nbetween the four- and three-dimensional components, which results\nfrom taking the energy average, we are thus able to construct\ntransport equations with only $2$ independent densities,\nthe scalar\nmass density $f_3$ and the vector spin density ${\\bf g}_0$, to any\norder of the semiclassical expansion.\n\n \\subsection{General kinetic equations in three-dimensional form}\n\nIn this subsection, we go beyond the semiclassical expansion and study\nthe full quantum kinetic equations in three-dimensional form. This is\nimportant for the treatment of non-perturbative processes like pair\ncreation in strong electric fields. Beyond the classical limit, the\nclassical mass-shell condition is generally violated. The quantum\nWigner function is no longer a $\\delta$-function in $p_0$ located at\nthe mass-shell energy, and for the elimination of the higher\n$p_0$-moments of the covariant spinor components from Eqs.~(\\ref{V6})\nwe must follow a different path. Following the procedure from Section\n2.3.1, we multiply the equations in group I of (\\ref{V3},\\ref{V3a}) by\n$p_0$ from the left and then take the energy average of these\nequations. We find\n \\begin{eqnarray}\n \\label{G1}\n   &&D_t\\int dp_0\\, p_0\\, V_0 + {\\bf D}{\\cdot}\\int dp_0\\, p_0\\, {\\bf V}\n     + I f_0 + {\\bf J}{\\cdot}{\\bf g}_1 = 0 \\, ,\n \\nonumber \\\\\n   &&D_t\\int dp_0\\, p_0\\, A_0 + {\\bf D}{\\cdot}\\int dp_0\\, p_0\\, {\\bf A}\n     - I f_1 - {\\bf J}{\\cdot}{\\bf g}_0 = 2 m \\int dp_0\\, p_0\\, P \\, ,\n \\nonumber \\\\\n   &&D_t\\int dp_0\\, p_0\\, P + I f_2 + 2\\, {\\bf K}{\\cdot}{\\bf g}_3\n     = - {\\bf \\Pi}{\\cdot}\\int dp_0\\, p_0\\, \\epsilon^{ijk} S_{jk} {\\bf e}_i\n       - 2 m \\int dp_0\\, p_0\\, A_0 \\, ,\n \\nonumber \\\\\n   &&{\\textstyle{1\\over 2}} \\Big[ D_t \\int dp_0\\, p_0\\, F + I f_3\n      -2 \\, {\\bf K}{\\cdot}{\\bf g}_2 \\Big] =\n      - {\\bf \\Pi}{\\cdot}\\int dp_0\\, p_0\\, S^{0i}{\\bf e}_i \\, ,\n \\nonumber \\\\\n   &&D_t \\int dp_0\\, p_0\\, {\\bf A} + {\\bf D} \\int dp_0\\, p_0\\, A_0\n     - I {\\bf g}_0 - {\\bf J} f_1 +2 \\, {\\bf K}{\\times}{\\bf g}_1\n     = 2\\, {\\bf \\Pi}\\times \\int dp_0\\, p_0\\, {\\bf V} \\, ,\n \\nonumber \\\\\n   &&{\\textstyle{1\\over 2}} \\Big[ D_t \\int dp_0\\, p_0\\, {\\bf V}\n     + {\\bf D} \\int dp_0\\, p_0\\, V_0 + I {\\bf g}_1 + {\\bf J} f_0\n     - 2\\, {\\bf K}{\\times}{\\bf g}_0 \\Big] =\n \\nonumber\\\\\n   &&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n     - {\\bf \\Pi}{\\times}\\int dp_0\\, p_0\\, {\\bf A}\n     + m \\int dp_0\\, p_0\\, S^{0i}{\\bf e}_i \\, ,\n \\nonumber \\\\\n   &&D_t \\int dp_0\\, p_0\\, S^{0i}{\\bf e}_i\n     - {1\\over 2} {\\bf D}{\\times}\\int dp_0\\, p_0\\,\n        \\epsilon^{ijk} S_{jk}{\\bf e}_i\n     - I {\\bf g}_2 - {\\bf J}{\\times}{\\bf g}_3 - 2\\, {\\bf K} f_3 =\n \\nonumber\\\\\n   &&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n     2\\, {\\bf \\Pi}\\int dp_0\\, p_0\\, F\n     - 2 m \\int dp_0\\, p_0\\, {\\bf V} \\, ,\n \\nonumber\\\\\n   &&{\\textstyle{1\\over 2}} \\left[ {\\textstyle{1\\over 2}} D_t \\int dp_0\\, p_0\\,\n                        \\epsilon ^{ijk} S_{jk} {\\bf e}_i\n      + {\\bf D}{\\times}\\int dp_0\\, p_0\\, S^{0i}{\\bf e}_i\n      + I {\\bf g}_3 - {\\bf J}{\\times}{\\bf g}_2 - 2\\, {\\bf K} f_2\\right] =\n \\nonumber\\\\\n   &&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n     {\\bf \\Pi}\\int dp_0\\, p_0\\, P \\, ,\n \\end{eqnarray}\nwith\n \\begin{eqnarray}\n \\label{G2}\n   &&I(x,{\\bf p}) = i e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n     (\\partial_t{\\bf E})({\\bf x}+is{\\bf \\nabla}_p,t){\\cdot}{\\bf \\nabla}_p\n   \\, ,\n \\nonumber \\\\\n   &&{\\bf J}(x,{\\bf p}) = i e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n     (\\partial_t{\\bf B})({\\bf x}+is{\\bf \\nabla}_p,t)\n      \\times{\\bf \\nabla}_p - e \\int^{1\\over 2}_{-{1\\over 2}} ds\\,\n      {\\bf E}({\\bf x}+is{\\bf \\nabla}_p,t) \\, ,\n \\nonumber \\\\\n   &&{\\bf K}(x,{\\bf p}) = e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s^2\\,\n      (\\partial_t{\\bf B})({\\bf x}+is{\\bf \\nabla}_p,t)\n       \\times {\\bf \\nabla}_p + i e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n       {\\bf E}({\\bf x}+is{\\bf \\nabla}_p,t) \\, .\n \\end{eqnarray}\n\nWe now combine Eqs.~(\\ref{V6}) and (\\ref{G1}) and eliminate all higher\n$p_0$-moments. In order to arrive at a set of independent contraints\nwe also use the BGR equations and remove all information already\ncontained in them. After a straightforward but tedious calculation we\nfinally derive the following constraints:\n \\begin{eqnarray}\n \\label {G3}\n    &&L f_0+{\\bf M}\\cdot {\\bf g}_1=0 \\, ,\n \\nonumber \\\\\n    &&L f_1+{\\bf M}\\cdot {\\bf g}_0=0 \\, ,\n \\nonumber \\\\\n    &&L f_2+2{\\bf N}\\cdot {\\bf g}_3=0 \\, ,\n \\nonumber \\\\\n    &&L f_3-2{\\bf N}\\cdot {\\bf g}_2=0 \\, ,\n \\nonumber \\\\\n    &&L {\\bf g}_0 -{\\bf M} f_1 -2{\\bf N}\\times {\\bf g}_1 = 0 \\, ,\n \\nonumber \\\\\n    &&L {\\bf g}_1 +{\\bf M} f_0 +2{\\bf N}\\times {\\bf g}_0 =0 \\, ,\n \\nonumber \\\\\n    &&L {\\bf g}_2 +{\\bf M}\\times {\\bf g}_3 -2{\\bf N} f_3 = 0 \\, ,\n \\nonumber \\\\\n    &&L {\\bf g}_3 -{\\bf M}\\times {\\bf g}_2 +2{\\bf N} f_2 = 0 \\, ,\n \\end{eqnarray}\nwith\n \\begin{eqnarray}\n \\label{G4}\n    && \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n      L(x,{\\bf p}) = i e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n      \\Big({\\bf \\nabla}\\times {\\bf B}({\\bf x}+is{\\bf \\nabla}_p,t)\n      \\Big) \\cdot {\\bf \\nabla}_p \\, ,\n \\nonumber \\\\\n    && \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n      {\\bf M}(x,{\\bf p}) = i e \\int^{1\\over 2}_{-{1\\over 2}} ds\\, s\\,\n      {\\bf \\nabla} \\Big({\\bf E}({\\bf x}+is{\\bf \\nabla}_p,t){\\cdot}{\\bf\n          \\nabla}_p\\Big) + e \\int^{1\\over 2}_{-{1\\over 2}} ds\\,\n      \\Big({\\bf E}({\\bf x}+is{\\bf \\nabla}_p,t) - {\\bf E}(x)\\Big) \\, ,\n \\nonumber \\\\\n    && \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n       {\\bf N}(x,{\\bf p}) = {\\textstyle{1\\over 4}} e \\int^{1\\over 2}_{-{1\\over 2}}ds\\,\n       ({\\bf \\nabla}_p{\\cdot}{\\bf \\nabla})\\,\n       {\\bf E}({\\bf x}+is{\\bf \\nabla}_p,t) + {\\bf K}(x,{\\bf p}) \\, .\n \\end{eqnarray}\n\n{\\em The complete set of kinetic equations for the Wigner function\n$W_3(x,{\\bf p})$ (resp. its spinor components) is given by the BGR\nequations (\\ref{BGR}) and the constraints (\\ref{G3}).} These equations\ntogether form the complete transport theory in its three-dimensional\nformulation. They are fully equivalent to the covariant kinetic\napproach defined by either of the two equations (\\ref{D6}) or by the\npair of self-adjoint equations (\\ref{D7},\\ref{D8}). While the\nequations in (\\ref{G3}) do not involve time derivatives and thus do\nnot by themselves describe transport, they provide essential\nconstraints for the Wigner operator.\n\nThe constraints (\\ref{G3}) hold for arbitrary external fields. For\nhomogeneous fields, the spatial derivatives of the electric and\nmagnetic fields ${\\bf E}$ and ${\\bf B}$ in the operators $L, {\\bf M}$\nand ${\\bf N}$ vanish, and the constraints (\\ref{G3}) disappear\nidentically. This shows that the equal-time BGR theory is in fact\ncomplete for the case of a spatially constant external field, i.e. for\nthe case studied in \\cite{Re12}. The application of the BGR theory in\n\\cite{Re12} to the problem of pair creation in spatially homogeneous\nbut time-dependent electric fields is therefore safe.\n\n \\section{Conclusions}\n\nRelativistic kinetic theory, formulated in terms of gauge covariant\nWigner operators, provides a useful method to study transport problems\nin quantum theory, especially the phase-space evolution of high\ntemperature and density plasmas. There exist two paths to formulate\nsuch a transport theory, starting from 4-dimensional or from\n3-dimensional momentum space, respectively. In this paper we have\nshown a way to connect these two approaches directly and arrived\nat the 3-dimensional formulation by taking an energy average of the\ncovariant 4-dimensional formulation.\n\nWe demonstrated the method explicitly for scalar and spinor QED with\nexternal electromagnetic fields. In scalar QED, our approach started\ndirectly from the Klein-Gordon equation and thus avoided the\ncomplications connected with the $2\\times 2$ matrix structure of the\nFeshbach-Villars representation \\cite{Re13}. Using the energy\naveraging method and taking the semiclassical limit, we directly\narrived at the classical Vlasov equation for on-shell particles. We\npointed out that there are no first-order quantum corrections due to\nthe absence of spin. This may partially explain the success of the\nclassical Vlasov approach as a good approximation for the description\nof many quantum systems involving scalar fields \\cite{Re22}.\n\nBeyond the semiclassical expansion, we could still perform the energy\naverage analytically for the case of spatially constant\nelectromagnetic fields, and we recovered the correct result for the\npair creation rate in a homogeneous electric field. Our calculation\nwas much easier than the one based on 3-dimensional transport theory\nin the Feshbach-Villars representation, due to the Lorentz scalar\nnature of the Wigner function employed by us.\n\nIn spinor electrodynamics, we concentrated on the question of\ncompleteness of the 3-dimensional (BGR) transport theory. Employing\nthe energy averaging method, we identified a covariant version of the\nequal-time BGR equation, but found that this covariant equation is not\nequivalent to the original Dirac equation. Integrating instead the\ncovariant VGE equation, which forms a complete description for the\nspinor Wigner function, over the energy $p_0$, we derived a set of\n3-dimensional (equal-time) transport and constraint equations\nwhich is complete and contains the BGR equations as a subset.\nThe additional equations can be understood as resulting from the\nenergy average of the 4-dimensional constraint or generalized\nmass-shell condition. They cannot be obtained by the direct equal-time\napproach of BGR. In the semiclassical limit they allow to reduce the\nnumber of independent spinor components of the Wigner function from 8\nin the BGR approach to 4, one scalar charge density and 3 vector\ncomponents of the spin density. The equations for the charge density and\nfor the spin density decouple, and each of them satisfies a\nVlasov-type transport equation in (6+1)-dimensional phase space. We\nalso derived the additional constraints explicitly for the full\nquantum case and showed that they vanish only for homogeneous external\nelectromagnetic fields.\n\nEither of the two formulations has its advantages and disadvantages.\nThe approach based on 4-dimensional momentum space (8-dimensional\nphase space) is manifestly Lorentz covariant, but setting it up as an\ninitial value problem poses certain difficulties \\cite{Re12}: since\nthe covariant Wigner function is a 4-dimensional Wigner transform of\nthe density matrix, its calculation at $t=-\\infty$ requires knowledge\nof the fields at all times. Since this knowledge does not a priori\nexist, the covariant transport equations can only be solved with\nphenomenologically motivated forms for the Wigner function in the far\npast. The formulation in 3-dimensional momentum space (i.e.\n(6+1)-dimensional phase space) does not have this problem: it requires\nthe density matrix only at equal times, and the Wigner function at\n$t=-\\infty$ can be directly calculated from the fields at $t=-\\infty$.\nSetting up the initial value problem is therefore straightforward in\nthis approach. For the calculation of the pair creation rate in\nhomogeneous electric fields this seems to be crucial: also our\ncalculation was based on the 3-dimensional version after taking the\nenergy average.\n\nIn both formulations the transport equations are supplemented by\nconstraint equations. In the covariant approach this is essentially\none $4\\times 4$ matrix equation, and it is easily interpreted as a\ngeneralized mass-shell condition. In the equal-time approach the\nconstraints are less transparent, and their derivation via the energy\naveraging procedure is actually rather tedious. For homogeneous\nexternal fields, however, they disappear to all orders in $\\hbar$, so\nfor this particular case the equal-time approach has a decisive\nadvantage over the covariant approach where this does not happen. In\nall other cases a large number of constraints, Eqs.~(\\ref{G3}), have to\nbe solved together with the BGR kinetic equations (\\ref{BGR}).\n\n\\vspace{0.25cm}\n\\noindent{\\bf Acknowledgments} \\\\\nP.Z. wishes to thank the Alexander von Humboldt Foundation\nfor a fellowship. U.H. would like to thank the participants of the ECT*\nWorkshop in Trento\n(Oct. 1994) on Parton Production in the Quark-Gluon Plasma, in particular\nJ. Eisenberg, Y. Kluger and E. Mottola, for stimulating discussions\nfollowing a presentation of part of this work. We are also grateful to S. Ochs\nfor helpful remarks.\nThis work was supported by DFG, BMFT, and GSI.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}