diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbmjn" "b/data_all_eng_slimpj/shuffled/split2/finalzzbmjn" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbmjn" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nTopological order\\cite{W8987,W9039,WN9077} is a new kind of order beyond the\nsymmetry breaking orders\\cite{L3726} in gapped quantum systems. Topological\norders are patterns of \\emph{long-range entanglement}\\cite{CGW1038} in\n\\emph{gapped quantum liquids} (GQL)\\cite{ZW1490}. Based on the unitary modular\ntensor category (UMTC) theory for non-abelian\nstatistics\\cite{MS8977,BakK01,K062}, in \\Ref{KW1458,W150605768}, it is proposed\nthat 2+1D bosonic topological orders are classified by $\\{\\text{UMTC}\\}\\times\n\\{\\text{iTO}_B\\}$, where $\\{\\text{UMTC}\\}$ is the set of UMTCs and\n$\\{\\text{iTO}_B\\}$ is the set of invertible topological orders\n(iTO)\\cite{KW1458,F1478} for 2+1D boson systems. In fact $\\{\\text{iTO}_B\\}=\\Z$\nwhich is generated by the $E_8$ bosonic quantum Hall (QH) state, and a table of\nUMTCs was obtained in \\Ref{RSW0777,W150605768}. Thus, we have a table (and a\nclassification) of 2+1D bosonic topological orders.\n\nIn a recent work\\cite{LW150704673}, we show that 2+1D fermionic topological\norders are classified by $\\{\\mce{\\sRp(Z_2^f)}\\}\\times \\{\\text{iTO}_F\\}$,\nwhere $\\{\\mce{\\sRp(Z_2^f)}\\}$ is the set of non-degenerate unitary\nbraided fusion categories (UBFC) over the symmetric fusion category (SFC)\n$\\sRp(Z_2^f)$ (see Definition \\ref{defMTC}). We also require\n$\\mce{\\sRp(Z_2^f)}$s to have modular extensions.\n$\\{\\text{iTO}_F\\}$ is the set of invertible topological orders for 2+1D fermion\nsystems. In fact $\\{\\text{iTO}_F\\}=\\Z$ which is generated by the $p+\\ii p$\nsuperconductor. In \\Ref{LW150704673} we computed the table for\n$\\mce{\\sRp(Z_2^f)}$s, and obtained a table (and a classification) of\n2+1D fermionic topological orders.\n\nIn \\Ref{LW150704673}, we also point out the importance of modular extensions.\nIf a $\\mce{\\sRp(Z_2^f)}$ does not have a modular extension, it means\nthat the fermion-number-parity symmetry is not on-site (\\ie\nanomalous\\cite{W1313}). On the other hand, if a $\\mce{\\sRp(Z_2^f)}$\ndoes have modular extensions, then the $\\mce{\\sRp(Z_2^f)}$ is\nrealizable by a lattice model of fermions. In this case, a given $\\mce{\\sRp(Z_2^f)}$ may have several\nmodular extensions. We found that different modular extensions of\n$\\mce{\\sRp(Z_2^f)}$ contain information of iTO$_F$s.\n\nOur result on fermionic topological orders can be easily generalized to\ndescribe bosonic\/fermionic topological orders with symmetry. This will be the\nmain topic of this paper. (Some of the results are announced in\n\\Ref{LW150704673}). In this paper, we will consider symmetric GQL phases for\n2+1D bosonic\/fermionic systems. The notion of GQL was defined in \\Ref{ZW1490}.\nThe symmetry group of GQL is $G$ (for bosonic systems) or $G^f$ (for fermionic\nsystems). If a symmetric GQL has long-range entanglement (as defined in\n\\Ref{CGW1038,ZW1490}), it corresponds to a symmetry enriched topological\n(SET) order\\cite{CGW1038}. If a symmetric GQL has short-range entanglement, it\ncorresponds to a symmetry protected trivial (SPT) order [which is also\nknown as symmetry protected topological (SPT)\norder]\\cite{GW0931,PBT1225,CLW1141,CGL1314,CGL1204}.\n\nIn this paper, we are going to show that, 2+1D symmetric GQLs are classified by\n$\\mce{\\cE}$ plus their modular extensions and chiral central charge. In other\nwords, GQLs are labeled by triples $(\\cC,\\cM,c)$, where $\\cC$ is a\n$\\mce{\\cE}$, $\\cM$ a modular extension of $\\cC$, and $c$ the chiral central\ncharge of the edge state. (To be more precise, a modular extension of $\\cC$,\n$\\cM$, is a UMTC with a fully faithful embedding $\\cC \\to \\cM$. In particular,\neven if the UMTC $\\cM$ is fixed, different embeddings correspond to\ndifferent modular extensions.) Here the SFC $\\cE$ is given by $\\cE=\\Rp(G)$ for\nbosonic cases, or $\\cE=\\sRp(G^f)$ for fermionic cases. In yet another way to\nphrase our result: we find that the structure $\\cE \\hookrightarrow \\cC\n\\hookrightarrow \\cM$ classifies\nthe 2+1D GQLs with symmetry $\\cE$, where $\\hookrightarrow$ represents the\nembeddings and\n$\\cen{\\cE}{\\cM}=\\cC$ (see Definition \\ref{cendef}).\n\nAs a special case of the above result, we find that bosonic 2+1D SPT phase with\nsymmetry $G$ are classified by the modular extensions of $\\Rp(G)$, while\nfermionic 2+1D SPT phase with symmetry $G^f$ are classified by the modular\nextensions of $\\sRp(G^f)$ that have central charge $c=0$.\n\nWe like to mention that \\Ref{BBC1440} has classified bosonic GQLs with symmetry\n$G$, using $G$-crossed UMTCs. This paper uses a different approach so that we\ncan classify both bosonic and fermionic GQLs with symmetry. We also like to\nmention that there is a mathematical companion \\Ref{LW160205936} of this paper, where\n one can find detailed proof and explanations for related mathematical results. \n\n\nThe paper is organized as the following. In Section \\ref{GQLsymm}, we review\nthe notion of topological order and introduce category theory as a theory of\nquasiparticle excitations in a GQL. We will introduce a categorical way to\nview the symmetry as well. In Section \\ref{inv}, we discuss invertible GQLs\nand their classification based on modular extensions. In Sections \\ref{clGQL} and\n\\ref{clGQL2}, we generalize the above results and propose a classification of\nall GQLs. Section \\ref{stack} investigates the stacking operation from\nphysical and mathematical points of view. Section \\ref{howto} describes how to\nnumerically calculate the modular extensions and Section \\ref{examples}\ndiscusses some simple examples. For people with physics background, one way to\nread this paper is to start with the Sections \\ref{GQLsymm} and \\ref{clGQL2},\nand then go to Section \\ref{examples} for the examples.\n\n\n\n\\section{Gapped quantum liquids, topological order and symmetry}\n\\label{GQLsymm}\n\n\\subsection{The finite on-site symmetry and symmetric fusion category}\n\nIn this paper, we consider physical systems with an on-site symmetry described\nby a finite group $G$. For fermionic systems, we further require that $G$\ncontains a central $Z_2$ fermion-number-parity subgroup. More precisely,\nfermionic symmetry group is a pair $(G,f)$, where $G$ is a finite group, $f\\neq\n1$ is an element of $G$ satisfying $f^2=1,fg=gf,\\forall g\\in G$. We denote the\npair $(G,f)$ as $G^f$.\n\nThere is another way to view the on-site symmetries, which is nicer because\nbosonic and fermionic symmetries can be formulated in the same manner.\nConsider a bosonic\/fermionic product state $|\\psi\\ket$ that does not break the\nsymmetry $G$: $U_g|\\psi\\ket=|\\psi\\ket,\\ g\\in G$. Then the new way to view the\nsymmetry is to use the properties of the excitations above the product state to\nencode the information of the symmetry $G$.\n\nThe product state contain only local excitations that can be created by acting\nlocal operators $O$ on the ground state $O|\\psi\\ket$. For any group action\n$U_g$, $U_g O|\\psi\\ket=U_g O U_g^\\dag U_g|\\psi\\ket=U_g O U_g^\\dag |\\psi\\ket$ is\nan excited state with the same energy as $O|\\psi\\ket$. Since we assume the\nsymmetry to be on-site, $U_g OU_g^\\dag$ is also a local operator. Therefore,\n$U_g OU_g^\\dag|\\psi\\ket$ and $O|\\psi\\ket$ correspond to the degenerate local\nexcitations. We see that local excitations ``locally'' carry group\nrepresentations. In other words, different types of local excitations are\nlabeled by irreducible representations of the symmetry group. \n\nBy looking at how the local excitations (more precisely, their group\nrepresentations) fuse and braid with each other, we arrive at the mathematical\nstructure called symmetric fusion categories (SFC). By definition a SFC is a\nbraided fusion category where all the objects (the excitations) have trivial\nmutal statistics (\\ie centralize each other, see next section). \nA SFC is automatically a unitary braided fusion category. \n\nIn fact, there are only two kinds of SFCs: one is representation category of\n$G$: $\\Rp(G)$, with the usual braiding (all representations are bosonic); the\nother is $\\sRp(G^f)$ where an irreducible representation is bosonic if $f$ is\nrepresented trivially ($+1$), and fermionic if $f$ is represented\nnon-trivially($-1$).\n\nIt turns out SFC (or the fusion and braiding properties of the local\nexcitations) fully characterize the symmetry group. Therefore, it is\nequivalent to say finite on-site symmetry is given by a SFC $\\cE$. By Tannaka\nduality $\\cE$ gives rises to a unique finite group $G$ and by checking the\nbraiding in $\\cE$ we know whether it is bosonic or fermionic. This is the new\nway, the categorical way, to view the symmetry. Such a categorical view of\nbosonic\/fermionic symmetry allows us to obtain a classification of symmetric\ntopological\/SPT orders.\n\n\\subsection{Categorical description of topological excitations with symmetry}\n\nIn symmetric GQLs with topological order (\\ie with long range entanglement),\nthere can be particle-like excitations with local energy density, but they\ncannot be created by local operators. They are known as (non-trivial)\ntopological excitations. Topological excitations do not necessarily carry group\nrepresentations. Nevertheless, we can still study how they fuse and braid with\neach other; so we have a unitary braided fusion category (UBFC) to describe the\nparticle-like excitations. To proceed, we need the following key definition on\n``centralizers.''\n\\begin{dfn}\n The objects $X,Y$ in a UBFC $\\cC$ are said to \\emph{centralize} (mutually\n local to) each other if \n \\begin{align}\n c_{Y,X}\\circ c_{X,Y}=\\mathrm{id}_{X\\otimes Y},\n \\end{align}\n where $c_{X,Y}: X\\otimes Y\\cong Y\\otimes X$ is the braiding in $\\cC$.\n\\end{dfn}\n\n Physically, we say that $X$ and $Y$ have trivial mutual statistics.\n\n\\begin{dfn}\n\\label{cendef}\n Given a subcategory $\\cD\\subset \\cC$, its \\emph{centralizer}\n $\\cen{\\cD}{\\cC}$ in $\\cC$\n is the full subcategory of objects in $\\cC$ that centralize all the objects in\n $\\cD$. \n\\end{dfn}\n\nWe may roughly view a category as a ``set'' of particle-like excitations.\nSo the centralizer $\\cen{\\cD}{\\cC}$ is the ``subset'' of particles in $\\cC$\nthat have trivial mutual statistics with all the particles in $\\cD$.\n\n\\begin{dfn}\n\\label{defMTC}\n A UBFC $\\cE$ is a \\emph{symmetric} fusion category if $\\cen{\\cE}{\\cE}=\\cE$.\nA UBFC $\\cC$ with a fully faithful embedding $\\cE\\inj \\cen{\\cC}{\\cC}$ is called\na UBFC over $\\cE$. Moreover, $\\cC$ is called a non-degenerate UBFC over $\\cE$, or\n$\\mce{\\cE}$, if $\\cen{\\cC}{\\cC}=\\cE$.\n\\end{dfn}\n\n\\begin{dfn} \\label{def:hom-bfce}\nTwo UBFCs over $\\cE$, $\\cC$ and $\\cC'$ are equivalent if there is a unitary braided\nequivalence $F:\\cC\\to\\cC'$ such that it preserves the embeddings, i.e.,\nthe following diagram commute.\n \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{Ceq}\n\\xymatrix{\n \\cE\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cC\\ar[d]^{F}\n \\\\\n \\cE\\ar@{^{ (}->}[r]&\\cC'\n}\n\\end{align}\nWe denote the category of unitary braided auto-equivalences of $\\cC$ by $\\mathcal{A}\\mathrm{ut}(\\cC)$ and its underlining group by $\\mathrm{Aut}(\\cC)$. \n\\end{dfn}\n\nWe recover the usual definition of\nUMTC when $\\cE$ is trivial, \\ie the category of Hilbert spaces, denoted by\n$\\mathrm{Vec}=\\Rp(\\{1\\})$. In this case the subscript is omitted.\n\nPhysically, a UBFC $\\cC$ is the collection of all bulk topological excitations\nplus their fusion and braiding data. Requiring $\\cC$ to be a \n$\\mce{\\cE}$ means: (1) the set of local excitations, $\\cE$ (which is the set of\nall the irreducible representations of the symmetry group), is included in\n$\\cC$ as a subcategory; (2) $\\cC$ is anomaly-free, \\ie all the topological\nexcitations (the ones not in $\\cE$) can be detected by mutual\nbraiding\\cite{KW1458}. In other words, every topological excitation must have\nnon-trivial mutual statistics with some excitations. Those excitations that\ncannot be detected by mutual braiding (i.e., $\\cen{\\cC}{\\cC}$) are exactly the\nlocal excitations in $\\cE$. Moreover, we want the symmetry to be on-site\n(gaugeable), which requires the existence of modular extensions (see Definition \\ref{mextdef}). Such an understanding leads to the following\nconjecture:\n\\begin{conj}\n Bulk topological excitations of topological orders with symmetry $\\cE$ are\nclassified by $\\mce{\\cE}$'s that have modular extensions.\n\\end{conj}\n\\noindent\n\nWe like to remark that $\\mce{\\cE}$'s fail to classify\ntopological orders. This is because two different topologically ordered phases\nmay have bulk topological excitations with the same non-abelian statistics (\\ie\ndescribed by the same $\\mce{\\cE}$). However, $\\mce{\\cE}$'s, with modular\nextensions, do classify topological orders up to invertible ones. See next\nsection for details. The relation between anomaly and modular extension will\nalso be discussed later.\n\n\n\\section{Invertible GQLs and modular extension}\n\\label{inv}\n\n\\subsection{Invertible GQLs}\n\nThere exist non-trivial topological ordered states that have only trivial\ntopological excitations in the bulk (but non-trivial edge states). They are\n``invertible'' under the stacking operation\\cite{KW1458,F1478}\n(see Section \\ref{stack} for details). More generally,\nwe define \n\\begin{dfn}\nA GQL is invertible if its bulk topological excitations are all trivial\n(\\ie can all be created by local operators).\n\\end{dfn}\nConsider some invertible GQLs with the same symmetry $\\cE$. The bulk\nexcitations of those invertible GQLs are the same which are described by the\nsame SFC $\\cE$. Now the question is: How to distinguish those invertible GQLs?\n\n\nFirst, we believe that invertible bosonic topological orders with no symmetry\nare generated by the $E_8$ QH state (with central charge $c=8$) via\ntime-reversal and stacking, and form a $\\Z$ group. Stacking with an $E_8$ QH state only\nchanges the central charge by $8$, and does not change the bulk excitations or\nthe symmetry. So the only data we need to know to determine the invertible\nbosonic topological order with no symmetry is the central charge $c$. The\nstory is parallel for invertible fermionic topological orders with no symmetry,\nwhich are believed to be generated by the $p+\\ii p$ superconductor state with\ncentral charge $c=1\/2$.\n\nSecond, invertible bosonic GQLs with symmetry are generated by bosonic SPT\nstates and invertible bosonic topological orders (\\ie $E_8$ states) via\nstacking. We know that the bosonic SPT states with symmetry $G$ are\nclassified by the 3-cocycles in $H^3[G,U(1)]$. Therefore, bosonic invertible\nGQLs with symmetry $G$ are classified by $H^3[G,U(1)]\\times \\Z$ (where $\\Z$\ncorresponds to layers of $E_8$ states).\n\nHowever, this result and this point of view is not natural to generalize to\nfermionic cases or non-invertible GQLs. Thus, we introduce an equivalent point\nof view, which can cover boson, fermion, and non-invertible GQLs in the same\nfashion.\n\n\\subsection{Modular extension}\n\nFirst, we introduce the notion of modular extension of a $\\mce{\\cE}$:\n\\begin{dfn}\n\\label{mextdef}\n Given a $\\mce{\\cE}$ $\\cC$, its \\emph{modular extension} is a UMTC\n $\\cM$, together with a fully faithful embedding $\\iota_\\cM:\n \\cC\\hookrightarrow\\cM$, such that $\\cen{\\cE}{\\cM}=\\cC$, equivalently\n $\\dim(\\cM)=\\dim(\\cC)\\dim(\\cE)$.\n\n Two modular extensions $\\cM$ and $\\cM'$ are equivalent if\n there is an equivalence between the UMTCs ${F:\\cM\\to\\cM'}$ that preserves the\n embeddings, i.e., the following diagram commute.\n \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{MEeq}\n\\xymatrix{\n \\cC\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cM\\ar[d]^{F}\\\\\n \\cC\\ar@{^{ (}->}[r]&\\cM'\n}\n\\end{align}\n We\n denote the set of equivalent classes of modular extensions of $\\cC$ by $\\mathcal{M}_{ext}(\\cC)$.\n\\end{dfn}\n\\begin{rmk}\n Since the total quantum dimension of modular extensions of a given $\\cC$ is\nfixed, there are only finitely many different modular extensions, due to\n\\Ref{BNRW13}. In principle we can always perform a finite search to exhaust all\nthe modular extensions.\n\\end{rmk}\n\nRemember that $\\cC$ describes the particle-like excitations in our topological\nstate. Some of those excitations are local that have trivial mutual statistics\nwith all other excitations. Those local excitation form $\\cE \\subset \\cC$.\nThe modular extension $\\cM$ of $\\cC$ is obtained as adding particles that have\nnon-trivial mutual statistics with the local excitations in $\\cE$, so that\nevery particle in $\\cM$ will always have non-trivial mutual statistics with\nsome particles in $\\cM$. Since the particles in $\\cE$ carry ``charges'' (\\ie\nthe irreducible representations of $G$), the added particles correspond to\n``flux'' (\\ie the symmetry twists of $G$). So the modular extension correspond\nto gauging\\cite{LG1209} the on-site symmetry $G$. Since we can use the gauged\nsymmetry to detect SPT orders\\cite{HW1339}, we like to propose the following\nconjecture\n\\begin{conj}\\label{invclassB}\n Invertible bosonic GQLs with symmetry $\\cE=\\Rp(G)$ are classified by\n$(\\cM,c)$ where $\\cM$ is a modular extension of $\\cE$ and $c=0$ mod 8.\n\\end{conj}\n\n\n\\subsection{Classify 2+1D bosonic SPT states}\n\nInvertible bosonic GQLs described by $(\\cM,c)$ include both bosonic SPT states\nand bosonic topological orders. Among those, $(\\cM,c=0)$ classify bosonic SPT\nstates. In other words:\n\\begin{cor}\n2+1D bosonic SPT states with symmetry $G$ are classified by the modular\nextensions of $\\Rp(G)$ (which always have $c=0$).\n\\end{cor} \n\nIn \\Ref{CLW1141,CGL1314,CGL1204}, it was shown that 2+1D bosonic SPT states are\nclassified by $H^3[G,U(1)]$. Such a result agrees with our conjecture, due to\nthe following theorem, which follows immediately from results in \\Ref{dgno2007}. \n\n\\begin{thm} \\label{thm:1to1-repG}\nThe modular extensions of $\\Rp(G)$ 1-to-1 correspond to 3-cocycles in $H^3[G,U(1)]$. The central charge of these modular extensions are $c=0$ mod 8.\n\\end{thm}\n\n\\begin{rmk}\nIn Sec.\\,\\ref{sec:mext-repG}, we give more detailed explanation of the 1-to-1 correspondence in Theorem\\,\\ref{thm:1to1-repG}. Moreover, we will prove a stronger result in Theorem\\,\\ref{thm:spt}. It turns out that the set $\\cM_{ext}(\\Rp(G))$ of modular extensions of $\\Rp(G)$ is naturally equipped with a physical stacking operation such that $\\cM_{ext}(\\Rp(G))$ forms an abelian group, which is isomorphic to the group $H^3[G,U(1)]$. \n\\end{rmk}\n\n\n\n\\begin{rmk}\n$c\/8$ determines the number of layers of the $E_8$ QH states, which is the\ntopological order part of invertible bosonic symmetric GQLs. In other words\n\\begin{align}\n&\\ \\ \\ \\\n\\{ \\text{invertible bosonic symmetric GQLs} \\} \n\\nonumber\\\\\n&=\n\\{ \\text{bosonic SPT states} \\}\\times \\{ \\text{layers of $E_8$ states} \\}.\n\\end{align}\n\\end{rmk}\n\n\n\n\n\\subsection{Classify 2+1D fermionic SPT states}\n\nThe above approach also apply to fermionic case. Note that, the invertible\nfermionic GQLs with symmetry $G^f$\nhave bulk excitations described by SFC $\\cE=\\sRp(G^f)$.\nSo we would like to conjecture that\n\\begin{conj}\\label{invclassF}\n Invertible fermionic GQLs with symmetry $G^f$ are classified by\n$(\\cM,c)$, where $\\cM$ is a modular extension of $\\cE=\\sRp(G^f)$,\nand $c$ is the central charge determining the layers of $\\nu=8$ IQH states.\n\\end{conj}\n\\begin{rmk}\nNote that, the central charge $c$ mod 8 is determined by $\\cM$, while\n$ (c - \\text{mod}(c,8))\/8$ determines the number of layers of the\n$\\nu=8$ IQH states.\n\\end{rmk}\n\\begin{rmk}\nInvertible fermionic symmetric GQLs include both fermion SPT states and\nfermionic topological orders. $(\\cM,c)$ with $c=0$ classify fermionic SPT\nstates. \n\\end{rmk} \nIn other words, \n\\begin{cor}\n2+1D fermionic SPT states with symmetry $G$ are classified by the $c=0$ modular extensions of\n$\\sRp(G^f)$. \n\\end{cor} \n\\begin{rmk}\nUnlike the bosonic case, in general\n\\begin{align}\n&\\ \\ \\ \\\n\\{ \\text{invertible fermionic symmetric GQLs} \\} \n\\\\\n& \\neq\n\\{ \\text{fermionic SPT states} \\}\\times \\{ \\text{layers of $p+\\ii p$ states} \\}.\n\\nonumber \n\\end{align}\n\\end{rmk}\n\n\nWhen there is no symmetry, the invertible fermionic GQLs become the invertible\nfermionic topological order, which have bulk excitations described by\n$\\cE=\\sRp(Z_2^f)$. $\\sRp(Z_2^f)$ has 16 modular extensions, with central\ncharges $c=n\/2, n=0,1,2,\\dots,15$. There is only one modular extension with\n$c=0$, which correspond to trivial product state. Thus there is no non-trivial\nfermionic SPT state when there is no symmetry, as expected.\n\nThe modular extensions with $c=n\/2$ correspond to invertible fermionic\ntopological order formed by $n$ layers of $p+\\ii p$ states.\nSince the modular extensions can only determine $c$ mod 8,\nin order for the above picture to be consistent, we need to show the following\n\\begin{thm}\\label{pE8}\nThe stacking of 16 layers $c=1\/2$ $p+\\ii p$ states is equivalent to a $\\nu=8$\nIQH state, which is in turn equivalent to a $E_8$ bosonic QH state stacked with\na trivial fermionic product state.\n\\end{thm}\n\\begin{proof}\nFirst, two layers of $p+\\ii p$ states is equal to one layer of $\\nu=1$ IQH\nstate. Thus, 16 layers $c=1\/2$ $p+\\ii p$ states is equivalent to a $\\nu=8$ IQH\nstate. To show $\\nu=8$ IQH state is equivalent to $E_8$ bosonic QH state\nstacked with a trivial fermionic product state, we note that the $\\nu=8$ IQH\nstate is described by $K$-matrix $K_{\\nu=8}=I_{8\\times 8}$ which is a 8-by-8\nidentity matrix. While the $E_8$ bosonic QH state stacked with a trivial\nfermionic product state is described by $K$-matrix $K_{E_8\\boxtimes\n\\cF_0}=K_{E_8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm $, where $K_{E_8}$ is the matrix\nthat describe the $E_8$ root lattice. We also know that two odd\\footnote{An odd\nmatrix is a symmetric integer matrix with at least one of its diagonal elements\nbeing odd.} $K$-matrices $K_1$ and $K_2$ describe the same fermionic topological\norder if after direct summing with proper number of $\\bpm 1 &0 \\\\ 0& -1 \\epm\n$'s:\n\\begin{align}\nK_1'&=K_1\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm \\oplus \\cdots\n\\nonumber\\\\\nK_2'&=K_2\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm \\oplus \\cdots,\n\\end{align}\n$K_1'$ and $K_2'$ become equivalent, \\ie\n\\begin{align}\n K_1' = U K_2' U^T,\\ \\ \\ \\ U \\in SL(N,\\Z).\n\\end{align}\nNotice that $K_{\\nu=8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm$ and $K_{E_8\\boxtimes\n\\cF_0}$ have the same determinant $-1$ and the same signature. Using the result\nthat odd matrices with $\\pm 1$ determinants are equivalent if they have\nthe same signature, we find that $K_{\\nu=8}\\oplus \\bpm 1 &0 \\\\ 0& -1 \\epm$\nand $K_{E_8\\boxtimes \\cF_0}$ are equivalent. Therefore $\\nu=8$ IQH state is\nequivalent to $E_8$ bosonic QH state stacked with a trivial fermionic product\nstate.\n\\end{proof}\n\n\n\\section{A full classification of 2+1D GQLs with symmetry}\n\\label{clGQL}\n\nWe have seen that all invertible GQLs with symmetry $G$ (or $G^f$) have the\nsame kind of bulk excitations, described by $\\Rp(G)$ (or $\\sRp(G^f)$). To\nclassify distinct invertible GQLs that shared the same kind of bulk\nexcitations, we need to compute the modular extensions of $\\Rp(G)$ (or\n$\\sRp(G^f)$). This result can be generalized to non-invertible topological\norders.\n\nIn general, the bulk excitations of a 2+1D bosonic\/fermionic SET are described\nby a $\\mce{\\cE}$ $\\cC$. However, there can be many distinct SET orders that\nhave the same kind of bulk excitations described by the same $\\cC$. To\nclassify distinct invertible SET orders that shared the same kind of bulk\nexcitations $\\cC$, we need to compute the modular extensions of $\\cC$. This\nleads to the following\n\\begin{conj}\n\\label{classSET}\n 2+1D GQLs with symmetry $\\cE$ (\\ie the 2+1D SET orders) are classified by $(\\cC,\\cM,c)$,\n where $\\cC$ is a $\\mce{\\cE}$ describing the bulk topological excitations,\n $\\cM$ is a modular extension of $\\cC$ describing the edge state up to\n $E_8$ states, and $c$ is the central charge determining the layers of $E_8$\n states.\n\\end{conj}\n\nLet $\\cM$ be a modular extension of a $\\mce{\\cE}$ $\\cC$. We note that\nall the simple objects (particles) in $\\cC$ are contained in $\\cM$ as\nsimple objects. Assume that the particle labels of $\\cM$ are\n$\\{i,j,\\dots, x, y,\\dots\\}$, where $i,j,\\cdots $ correspond to the particles in\n$\\cC$ and $x,y,\\cdots $ the additional particles (not in $\\cC$). Physically,\nthe additional particles $x,y,\\cdots $ correspond to the symmetry twists of the\non-site symmetry\\cite{W1447}. The modular extension $\\cM$ describes the\nfusion and the braiding of original particles $i,j,\\cdots $ with the symmetry\ntwists. In other words, the modular extension $\\cM$ is the resulting\ntopological order after we gauge the on-site symmetry\\cite{LG1209}.\n\nNow, it is clear that the existence of modular extension is closely related to\nthe on-site symmetry (\\ie anomaly-free symmetry) which is gaugable (\\ie allows\nsymmetry twists). For non-on-site symmetry (\\ie anomalous\nsymmetry\\cite{W1313}), the modular extension does not exist since the symmetry\nis not gaugable (\\ie does not allow symmetry twists). We also have\n\\begin{conj}\n\\label{classSETA}\n 2+1D GQLs with anomalous symmetry\\cite{W1313} $\\cE$ are\nclassified by $\\mce{\\cE}$'s that have no modular extensions.\n\\end{conj}\n\n\nIt is also important to clarify the equivalence relation between the triples\n$(\\cC,\\cM,c)$. Two triples $(\\cC,\\cM,c)$ and $(\\cC',\\cM',c')$ are equivalent\nif: (1) $c=c'$; (2) there exists braided equivalences $F_\\cC:\\cC\\to\\cC'$ and\n$F_\\cM:\\cM\\to\\cM'$ such that all the embeddings are preserved, i.e., the\nfollowing diagram commutes.\n \\newdir^{ (}{{}*!\/-5pt\/@^{(}}\n\\begin{align}\n\\label{TOeq}\n\\xymatrix{\n \\cE\\ar@{^{ (}->}[r]\\ar@{=}[d]&\\cC\\ar@{^{ (}->}[r]\\ar[d]^{F_\\cC}\n &\\cM\\ar[d]^{F_\\cM}\\\\\n \\cE\\ar@{^{ (}->}[r]&\\cC'\\ar@{^{ (}->}[r]&\\cM'\n}\n\\end{align}\nThe equivalence classes will be in one-to-one\ncorrespondence with GQLs (\\ie SET orders and SPT orders).\n\nNote that the group of the automorphisms of a $\\mce{\\cE}$ $\\cC$, denoted by\n$\\mathrm{Aut}(\\cC)$ (recall Definition\\,\\ref{def:hom-bfce}), naturally acts on the\nmodular extensions $\\mathcal{M}_{ext}(\\cC)$ by changing the embeddings, i.e. $F\\in\\mathrm{Aut}(\\cC)$ acts as follows: \n$$\n(\\cC\\hookrightarrow\\cM)\\mapsto (\\cC\\xrightarrow{F}\\cC\\hookrightarrow \\cM)\n$$ \nFor a fixed $\\cC$, the above equivalence relation amounts to say that GQLs with bulk excitations described by a fixed $\\cC$ are in one-to-one correspondence with the quotient\n$\\mathcal{M}_{ext}(\\cC)\/\\mathrm{Aut}(\\cC)$ plus a central charge $c$. When $\\cC=\\cE$, the GQLs\nwith bulk excitations described by $\\cE$ and central charge $c=0$ are SPT phases. In this case, the group $\\mathrm{Aut}(\\cE)$, where $\\cE$ is viewed as the trivial $\\mce{\\cE}$, is trivial. Thus, SPT phases are classified by the modular extensions of\n$\\cE$ with $c=0$.\n\n\n\\section{Another description of 2+1D GQLs with symmetry}\n\\label{clGQL2}\n\n\nAlthough the above result has a nice mathematical structure, it is hard to\nimplement numerically to produce a table of GQLs. To fix this problem, we\npropose a different description of 2+1D GQLs. The second description is\nmotivated by a conjecture that the fusion and the spins of the particles,\n$(\\cN^{IJ}_K,\\cS_I)$, completely characterize a UMTC. We conjecture that\n\\begin{conj}\n\\label{NsNsNs}\nThe data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$, up\nto some equivalence relations, gives a one-to-one classification of\n2+1D GQLs with symmetry $G$ (for boson) or $G^f$ (for fermion), with a\nrestriction that the symmetry group can be fully characterized by the fusion\nring of its irreducible representations. The data $( \\tilde N^{ab}_c,\\tilde\ns_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$ satisfies the conditions described in\nAppendix \\ref{cnds} (see \\Ref{W150605768} for UMTCs). \n\\end{conj}\n\nHere $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$\nis closely related to $(\\cE;\\cC;\\cM;c)$ discussed above. The data $(\\tilde\nN^{ab}_c,\\tilde s_a)$ describes the symmetry (\\ie the SFC $\\cE$):\n$a=1,\\cdots,\\tilde N$ label the irreducible representations and $\\tilde\nN^{ab}_c$ are the fusion coefficients of irreducible representations. $\\tilde\ns_a =0$ or $1\/2$ depending on if the fermion-number-parity transformation $f$\nis represented trivially or non-trivially in the representation $a$. The data\n$(N^{ij}_k,s_i)$ describes fusion and the spins of the bulk particles\n$i=1,\\cdots,N$ in the GQL. The data $(N^{ij}_k,s_i)$ contains $(\\tilde\nN^{ab}_c,\\tilde s_a)$ as a subset, where $a$ is identified with the first\n$\\tilde N$ particles of the GQL. The data $(\\cN^{IJ}_K,\\cS_I)$ describes\nfusion and the spins of a UMTC, and it includes $(N^{ij}_k,s_i)$ as a subset,\nwhere $i$ is identified with the first $N$ particles of the UMTC. Also among\nall the particles in UMTC, only the first $N$ (\\ie $I=1,\\cdots,N$) have trivial\nmutual statistics with first $\\tilde N$ particles (\\ie $I=1,\\cdots,\\tilde N$).\nLast, $c$ is the chiral central charge of the edge state.\n\nIf the data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ fully characterized\nthe $\\mce{\\cE}$, then the Conjecture \\ref{NsNsNs} would be equivalent to the\nConjecture \\ref{classSET}.\nHowever, for non-modular tensor category, $( \\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ fails to to fully characterize a\n$\\mce{\\cE}$. In other words, there are different $\\mce{\\cE}$'s that have the\nsame data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$. We need to include\nthe extra data, such as the $F$-tensor and the $R$-tensor, to fully\ncharacterize the $\\mce{\\cE}$.\n\nIn Appendix \\ref{SETtbl}, we list the data $( \\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$ that satisfy the conditions in Appendix \\ref{cnds} (without the\nmodular extension condition) in many tables. Those tables include all the\n$\\mce{\\cE}$'s (up to certain total quantum dimensions), but the tables are not\nperfect: (1) some entries in the tables may be fake and do not correspond to\nany $\\mce{\\cE}$ (for the conditions are only necessary); (2) some entries in\nthe tables may correspond to more then one $\\mce{\\cE}$ (since $( \\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ does not fully characterize a $\\mce{\\cE}$).\n\nWe then continue to compute $(\\cN^{IJ}_K,\\cS_I;c)$, the modular extensions of\n$( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$. We find that the modular\nextensions can fix the imperfectness mentioned above. First, we find that the\nfake entries do not have modular extensions, and are ruled out. Second, as we\nwill show in Section \\ref{stack}, all $\\mce{\\cE}$'s have the same numbers of\nmodular extensions (if they exist); therefore, the entry that corresponds to\nmore $\\mce{\\cE}$'s has more modular extensions. The modular extensions can tell\nus which entries correspond to multiple $\\mce{\\cE}$'s. This leads to the\nconjecture that the full data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i;\n\\cN^{IJ}_K,\\cS_I;c)$ gives rise to an one-to-one classification of 2+1D GQLs, and allows us to calculate the\ntables of 2+1D GQLs, which include 2+1D SET states and 2+1D SPT states. Those\nare given in Section \\ref{examples}.\n\nAs for the equivalence relation, we only need to consider\n$(\\cN^{IJ}_K,\\cS_I;c)$, since the data $(\\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$ is included in $(\\cN^{IJ}_K,\\cS_I;c)$. Two such data\n$(\\cN^{IJ}_K,\\cS_I;c)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I;\\bar c)$ are called\nequivalent if $c=\\bar c$, and $(\\cN^{IJ}_K,\\cS_I)$ and $(\\bar \\cN^{IJ}_K,\\bar\n\\cS_I)$ are related by two permutations of indices in the range $N_{\\cM} \\geq I\n> N$ and in the range $N\\geq I > \\tilde N$, where $N_{\\cM}$ is the range of\n$I$. Such an equivalence relation corresponds to the one in eqn. (\\ref{TOeq})\nand will be called the TO-equivalence relation. We use the TO-equivalence\nrelation to count the number of GQL phases (\\ie the number of SET orders and\nSPT orders).\n\nWe can also define another equivalence relation, called ME-equivalence\nrelation: we say $(\\cN^{IJ}_K,\\cS_I;c)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I;\\bar\nc)$ to be ME-equivalent if $c=\\bar c$ and they only differ by a permutation\nof indices in range $I > N$. The ME-equivalence relation is closely related to\nthe one defined in eqn. (\\ref{MEeq}). We use the ME-equivalence relation to\ncount the number of modular extensions of a \\emph{fixed} $\\cC$. \n\nLast, let us explain the restriction on the symmetry group. In the Conjecture\n\\ref{NsNsNs}, we try to use the fusion $\\tilde N^{ab}_c$ of the irreducible\nrepresentations to characterize the symmetry group. However, it is known that\ncertain different groups may have identical fusion ring for their irreducible\nrepresentations. So we need to restrict the symmetry group to be the group\nthat can be fully characterized by its fusion ring. Those groups include\nsimple groups and abelian groups\\cite{Yuan16}. If we do not impose such a\nrestriction, then the Conjecture \\ref{NsNsNs} give rise to GQLs with a given\nsymmetry fusion ring, instead of a given symmetry group.\n\n\n\\section{The stacking operation of GQLs}\n\\label{stack}\n\n\\subsection{Stacking operation}\n\nConsider two GQLs $\\cC_1$ and $\\cC_2$. If we stack them together (without\nintroducing interactions between them), we obtain another GQL, which is denoted\nby $\\cC_1\\boxtimes \\cC_2$. The stacking operation $\\boxtimes$ makes the set of\nGQLs into a monoid. $ \\boxtimes $ does not makes the set of GQLs into a group,\nbecause in general, a GQL $\\cC$ may not have an inverse under $\\boxtimes$. \\ie\nthere is no GQL $\\cD$ such that $\\cC\\boxtimes \\cD$ becomes a trivial product\nstate. This is because when a GQL have non-trivial topological excitations,\nstacking it with another GQL can never cancel out those topological\nexcitations.\n\nWhen we are considering GQLs with symmetry $\\cE$, the simple stacking\n$\\boxtimes$ will ``double'' the symmetry, leads to a GQL with symmetry\n$\\cE\\bt\\cE$ ($\\Rp(G\\times G)$ or $\\sRp(G^f\\times G^f)$). In general we allow\nlocal interactions between the two layers to break some symmetry such that the\nresulting system only has the original symmetry $\\cE$ (In terms of the symmetry group, keep only the\nsubgroup $G\\hookrightarrow G\\times G$ with the diagonal embedding $g\\mapsto\n(g,g)$). This leads to the stacking between GQLs with symmetry $\\cE$,\ndenoted by $\\bt_\\cE$. Similarly, $\\bt_\\cE$ makes GQLs with symmetry $\\cE$ a\nmonoid, but in general not all GQLs are invertible.\n\nHowever, if the bulk excitations of $\\cC$ are all local (\\ie all described by\nSFC $\\cE$), then $\\cC$ will have an inverse under the stacking operation\n$\\boxtimes_\\cE$, and this is why we call such GQL invertible. Those invertible\nGQLs include invertible topological orders and SPT states.\n \n\n\\subsection{The group structure of bosonic SPT states}\n\\label{grpSPT}\n\nWe have proposed that 2+1D SPT states are classified by $c=0$ modular\nextensions of the SFC $\\cE$ that describes the symmetry. Since SPT states are\ninvertible, they form a group under the stacking operation $ \\boxtimes_\\cE $.\nThis implies that the modular extensions of the SFC should also form a group\nunder the stacking operation. So checking if the modular extensions of the\nSFC have a group structure is a way to find support for our conjecture.\n\nHowever, in this section, we will first discuss such stacking\noperation and group structure from a physical point of view.\nWe will only consider bosonic SPT states.\n\nIt has been proposed that the bosonic SPT states are described by group\ncohomology $\\cH^{d+1}[G,U(1)]$\\cite{CLW1141,CGL1314,CGL1204}. However, it has\nnot been shown that those bosonic SPT states form a group under stacking\noperation. Here we will fill this gap. An ideal bosonic SPT state of symmetry\n$G$ in $d+1$D is described the following path integral\n\\begin{align}\n Z =\\sum_{\\{g_i\\}} \\prod_{\\{i,j,\\cdots \\}} \\nu_{d+1}(g_i,g_j,\\cdots )\n\\end{align}\nwhere $\\nu_{d+1}(g_i,g_j,\\cdots )$ is a function $G^{d+1} \\to U(1)$, which is a\ncocycle $\\nu_{d+1}\\in \\cH^{d+1}[G,U(1)]$. Here the space-time is a complex whose\nvertices are labeled by $i,j,\\cdots $, and $\\prod_{\\{i,j,\\cdots \\}}$ is the\nproduct over all the simplices of the space-time complex.\nAlso $\\sum_{\\{g_i\\}}$ is a sum over all $g_i$ on each vertex.\n\nNow consider the stacking of two SPT states described by cocycle\n$\\nu_{d+1}'$ and \n$\\nu_{d+1}''$: \n\\begin{align}\n Z =\\sum_{\\{g'_i,g''_i\\}} \\prod_{\\{i,j,\\cdots \\}} \n\\nu'_{d+1}(g'_i,g'_j,\\cdots )\n\\nu''_{d+1}(g''_i,g''_j,\\cdots ) .\n\\end{align}\nSuch a stacked state has a symmetry\n$G \\times G$ and is a $G \\times G$ SPT state.\n\nNow let us add a term to break the $G \\times G$-symmetry to $G$-symmetry\nand consider\n\\begin{align}\n\\label{ZU}\n Z =\\sum_{\\{g'_i,g''_i\\}} \\prod_{\\{i,j,\\cdots \\}} &\n\\nu'_{d+1}(g'_i,g'_j,\\cdots )\n\\nu''_{d+1}(g''_i,g''_j,\\cdots ) \\times \n\\nonumber\\\\\n \\prod_i & \\ee^{-U|g_i'-g_i''|^2}\n,\n\\end{align}\nwhere $|g'-g''|$ is an invariant distance between group elements. As we change\n$U=0$ to $U=+ \\infty $, the stacked system changes into the system for an ideal\nSPT state described by the cocycle\n$\\nu_{d+1}(g_i,g_j,\\cdots)=\\nu_{d+1}'(g_i,g_j,\\cdots )\n\\nu_{d+1}''(g_i,g_j,\\cdots )$. If such a deformation does not cause any phase\ntransition, then we can show that the stacking of a $\\nu_{d+1}'$-SPT state with\na $\\nu_{d+1}''$-SPT state give rise to a $\\nu_{d+1}=\\nu_{d+1}'\\nu_{d+1}''$-SPT\nstate. Thus, the key to show the stacking operation to give rise to the group\nstructure for the SPT states, is to show the theory \\eqn{ZU} has no phase\ntransition as we change $U=0$ to $U= +\\infty $.\n\nTo show there is no phase transition, we put the system on a closed space-time\nwith no boundary, say $S^{d+1}$. In this case, $\\prod_{\\{i,j,\\cdots \\}} \n\\nu'_{d+1}(g'_i,g'_j,\\cdots ) \\nu''_{d+1}(g''_i,g''_j,\\cdots )=1$, since\n$\\nu_{d+1}'$ and $\\nu_{d+1}''$ are cocycles.\nThus the path integral \\eq{ZU} is reduced to\n\\begin{align}\n Z =\\sum_{\\{g'_i,g''_i\\}} \\prod_i \\ee^{-U|g_i'-g_i''|^2} \n= \\Big(|G| \\sum_g \\ee^{-U|1-g|^2}\\Big)^{N_v} ,\n\\end{align}\nwhere $N_v$ is the number of vertices and $|G|$ the order of the symmetry\ngroup. We see that the free energy density\n\\begin{align}\n f = -\\lim_{N_v\\to\\infty}\\ln Z\/N_v\n\\end{align}\nis a smooth function of $U$ for $U\\in [0, \\infty )$. There is indeed no phase\ntransition.\n\nThe above result is highly non trivial from a categorical point of view.\nConsider two 2+1D bosonic SPT states described by two modular extensions $\\cM'$\nand $\\cM''$ of $\\Rp(G)$. The natural tensor product $\\cM' \\boxtimes \\cM''$ is\nnot a modular extension of $\\Rp(G)$, but a modular extension of $\\Rp(G)\n\\boxtimes \\Rp(G)=\\Rp(G \\times G)$. So, $\\cM' \\boxtimes \\cM''$ describes a $G\n\\times G$-SPT state. According to the above discussion, we need to break the\n$G \\times G$-symmetry down to the $G$-symmetry to obtain the $G$-SPT state.\nSuch a symmetry breaking process correspond to the so call ``anyon\ncondensation'' in category theory. We will discuss such anyon condensation\nlater. The stacking operation $\\bt_\\cE$, with such a symmetry breaking process\nincluded, is the correct stacking operation that maintains the symmetry $G$.\n\n\n\\subsection{Mathematical construction of the stacking operation}\n\n\nWe have conjectured that a 2+1D topological order with symmetry $\\cE$ is\nclassified by $(\\cC,\\cM_\\cC,c)$, where $\\cC$ is a $\\mce{\\cE}$ , $\\cM_\\cC$ is a\nmodular extension of $\\cC$, and $c$ is the central charge. If we have another\ntopological order of the same symmetry $\\cE$ described by $(\\cC',\\cM_{\\cC'},c')$,\nstacking $(\\cC,\\cM_\\cC,c)$ and $(\\cC',\\cM_{\\cC'},c')$ should give a third\ntopological order described by similar data $(\\cC'',\\cM_{\\cC''},c'')$:\n\\begin{align}\n (\\cC,\\cM_\\cC,c) \\boxtimes_\\cE (\\cC',\\cM_{\\cC'},c') =\n(\\cC'',\\cM_{\\cC''},c'')\n\\end{align}\n\nIn this section, we will show that such a stacking operation can be defined\nmathematically. This is an evidence supporting our Conjecture \\ref{classSET}.\nWe like to point out that a special case of the above result for\n$\\cC=\\cC'=\\cC''=\\cE=\\Rp(G)$ was discussed in section \\ref{grpSPT}.\n\nTo define $\\boxtimes_\\cE$ mathematically, first, we like to introduce\n\\begin{dfn}\\label{alg}\n A \\emph{condensable algebra} in a UBFC $\\cC$ is a\n triple $(A,m,\\eta)$, $A\\in\\cC$,\n $m:A\\ot A\\to A$, $\\eta:\\one\\to A$ satisfying\n \\begin{itemize}\n \\item Associative: $m(\\id_A\\ot m)=m(m\\ot \\id_A)$\n \\item Unit: $m(\\eta\\ot\\id_A)=m(\\id_A\\ot\\eta)=\\id_A$\n \\item Isometric: $m m^\\dag=\\id_A$\n \\item Connected: $\\Hom(\\one,A)=\\C$\n \\item Commutative: $m c_{A,A}=m$\n \\end{itemize}\n\\end{dfn}\nPhysically, such an condensable algebra $A$ is a composite self-bosonic anyon\nsatisfies additional conditions such that one can condense $A$ to obtain\nanother topological phase.\n\n\\begin{dfn}\n A (left) \\emph{module} over a condensable algebra $(A,m,\\eta)$ in $\\cC$ is a\n pair $(X,\\rho)$, $X\\in\\cC$, $\\rho:A\\ot X\\to X$ satisfying\n \\begin{gather}\n \\rho(\\id_A\\ot\\rho)=\\rho(m\\ot \\id_M),\\nonumber\\\\\n \\rho(\\eta\\ot\\id_M)=\\id_M.\n \\end{gather}\n It is further a \\emph{local} module if\n \\begin{align*}\n \\rho c_{M,A} c_{A,M}=\\rho.\n \\end{align*}\n\\end{dfn}\n We denote the category of left $A$ modules by $\\cC_A$.\n A left module $(X,\\rho)$ is turned into a right module via the braiding,\n $(X,\\rho c_{X,A})$ or $(X,\\rho c_{A,X}^{-1})$, and thus an $A$-$A$ bimodule.\n The relative tensor functor $\\ot_A$ of bimodules then turns $\\cC_A$ into a fusion category.\n (This is known as $\\alpha$-induction in subfactor context.)\n In general there can be two monoidal structures on $\\cC_A$, since there are\n two ways to turn a left module into a bimodule (usually we pick one for\n definiteness\n when considering $\\cC_A$ as a fusion category).\n The two monoidal structures coincide for the fusion subcategory $\\cC_A^0$ of\n local $A$ modules. Moreover, $\\cC_A^0$ inherited the braiding from $\\cC$ and\n is also a UBFC. The local modules are nothing but the anyons in the\n topological phases after condensing $A$.\n\\begin{lem}[DMNO\\cite{dmno}]\n \\[\\dim(\\cC_A)=\\frac{\\dim(\\cC)}{\\dim(A)}.\\]\n If $\\cC$ is a UMTC, then so is $\\cC_A^0$, and\n \\[\\dim(\\cC_A^0)=\\frac{\\dim(\\cC)}{\\dim(A)^2}.\\]\n\\end{lem}\n A non-commutative algebra $A$ is also of interest. We have the left center\n $A_l$ of $A$, the maximal subalgebra such that $m c_{A_l,A}=m$, and the right\n center $A_r$, the maximal subalgebra such that $m c_{A,A_r}=m$. $A_l$ and\n $A_r$ are commutative subalgebras, thus condensable.\n\\begin{thm}[FFRS\\cite{FFRS03}]\n There is a canonical equivalence between the categories of local modules\n over the left and right centers, $\\cC_{A_l}^0=\\cC_{A_r}^0$.\n\\end{thm}\n\\begin{dfn}\n The Drinfeld center $Z(\\cA)$ of a monoidal category $\\cA$ is a monoidal category with\n objects as pairs $(X\\in\\cA,b_{X,-})$, where $b_{X,-}: X\\ot -\\to -\\ot X$ are\n half-braidings that satisfy similar conditions as braidings. Morphisms and\n the tensor product are naturally defined.\n\\end{dfn}\n $Z(\\cA)$ is a braided monoidal category. There is a forgetful tensor functor\n $for_\\cA:Z(\\cA)\\to \\cA$, $(X,b_{X,-})\\mapsto X$ that forgets the half-braidings.\n\\begin{thm}[M{\\\"u}ger\\cite{Mue01}]\n $Z(\\cA)$ is a UMTC if $\\cA$ is a fusion category and\n $\\dim(Z(\\cA))=\\dim(\\cA)^2$.\n\\end{thm}\n\\begin{dfn}\n Let $\\cC$ be a braided fusion category and $\\cA$ a fusion category, a tensor\n functor $F:\\cC\\to \\cA$ is called a central functor if it factorizes through\n $Z(\\cA)$, i.e., there exists a braided tensor functor $F':\\cC\\to Z(\\cA)$ such\n that $F=F'for_\\cA$.\n\\end{dfn}\n\n\\begin{lem}\n [DMNO\\cite{dmno}]\n Let $F:\\cC\\to\\cA$ be a central functor, and $R:\\cA\\to\\cC$ the right adjoint\nfunctor of $F$.\nThen the object $A=R(\\one) \\in\\cC$ has a canonical structure of condensable algebra.\n$\\cC_A$ is monoidally equivalent to the image of\n$F$, i.e. the smallest fusion subcategory of $\\cA$ containing $F(\\cC)$.\n\\end{lem}\n\n\\begin{exa}\n If $\\cC$ is a UBFC, it is naturally embedded into\n $Z(\\cC)$, so is $\\overline\\cC$. Therefore, $\\cC\\bt\\overline\\cC\\hookrightarrow Z(\\cC)$.\n Compose this embedding with the forgetful functor $for_\\cC:Z(\\cC)\\to\\cC$ we\n get a central functor\n\\begin{align*}\n \\cC\\bt\\overline\\cC &\\to \\cC\\\\\n X\\bt Y&\\mapsto X\\ot Y.\n\\end{align*}\nLet $R$ be its right adjoint functor, we obtain a condensable algebra\n$L_\\cC:=R(\\one)\\cong \\oplus_i ( i\\bt \\bar i) \\in \\cC\\bt\\overline\\cC$ ($\\bar i$\ndenotes the dual object, or anti-particle of $i$) and $\\cC=(\n\\cC\\bt\\overline\\cC)_{L_\\cC}$, $\\dim(L_\\cC)=\\dim(\\cC)$.\nIn particular, for a symmetric category $\\cE$, $L_\\cE$ is a condensable algebra\nin $\\cE\\bt\\cE$, and $\\cE=(\\cE\\bt\\cE)_{L_\\cE}=(\\cE\\bt\\cE)_{L_\\cE}^0$ for $\\cE$\nis symmetric, all $L_\\cE$-modules are local.\nCondensing $L_\\cE$ is nothing but breaking the symmetry from $\\cE\\bt\\cE$ to\n$\\cE$.\n\\end{exa}\n\n\nNow, we are ready to define the stacking operation for $\\mce{\\cE}$'s as well\nas their modular extensions.\n\\begin{dfn}\\label{stacking}\n Let $\\cC,\\cD$ be $\\mce{\\cE}$'s, and $\\cM_\\cC,\\cM_\\cD$ their \n modular extensions. The stacking is defined by:\n \\begin{align*}\n \\cC\\bt_\\cE\\cD:=(\\cC\\bt\\cD)^0_{L_\\cE},\\quad \\cM_\\cC\\bt_\\cE \\cM_\\cD:=(\\cM_\\cC\\bt\n \\cM_\\cD)_{L_\\cE}^0\n \\end{align*}\n\\end{dfn}\nNote that in Ref.~\\onlinecite{DNO11}, the tensor product $\\bt_\\cE$ for\n$\\mce{\\cE}$'s is defined as $(\\cC\\bt\\cD)_{L_\\cE}$. For $\\mce{\\cE}$'s the two\ndefinitions coincide $(\\cC\\bt\\cD)^0_{L_\\cE}=(\\cC\\bt\\cD)_{L_\\cE}$, for $L_\\cE$ lies in\nthe centralizer of $\\cC\\bt\\cD$ which is $\\cE\\bt\\cE$. But for the modular extensions we have to take\nthe unusual definition above.\n\n\\begin{thm}\n $\\cC\\bt_\\cE\\cD$ is a $\\mce{\\cE}$, and\n $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a modular extension of $\\cC\\bt_\\cE\\cD$.\n\\end{thm}\n\\begin{proof}\n The embeddings\n$\\cE=(\\cE\\bt\\cE)_{L_\\cE}^0\\hookrightarrow (\\cC\\bt\\cD)^0_{L_\\cE}=\\cC\\bt_\\cE\\cD\n\\hookrightarrow (\\cM_\\cC\\bt\\cM_\\cD)^0_{L_\\cE}=\\cM_\\cC\\bt_\\cE\\cM_\\cD$\nare obvious.\nSo $\\cC\\bt_\\cE\\cD$ is a UBFC over $\\cE$. Also\n\\begin{align*}\n \\dim(\\cC\\bt_\\cE\\cD)=\\frac{\\dim(\\cC\\bt\\cD)}{\\dim(L_\\cE)}\n =\\frac{\\dim(\\cC)\\dim(\\cD)}{\\dim(\\cE)},\n\\end{align*}\nand $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a UMTC,\n\\begin{align*}\n \\dim(\\cM_\\cC\\bt_\\cE\\cM_\\cD)\n =\\frac{\\dim(\\cM_\\cC\\bt\\cM_\\cD)}{\\dim(L_\\cE)^2}=\\dim(\\cC)\\dim(\\cD).\n\\end{align*}\n Thus, $\\cM_\\cC\\bt_\\cE\\cM_\\cD$ is a\n modular extension of $\\cC\\bt_\\cE\\cD$.\n\\end{proof}\n\n Take $\\cD=\\cE$. Note that $\\cC\\bt_\\cE\\cE=\\cC$. Therefore, for any\n modular extension $\\cM_\\cE$ of $\\cE$, $\\cM_\\cC\\bt_\\cE\\cM_\\cE$ is still a\n modular extension of $\\cC$. In the following we want to show the inverse,\n that one can extract the ``difference'', a modular extension of $\\cE$,\n between two modular extensions of $\\cC$.\n\n\\begin{lem}\\label{Lag}\n We have $(\\cC\\bt\\overline\\cC)_{L_\\cC}^0=\\cen{\\cC}{\\cC}$.\n\\end{lem}\n\\begin{proof}\n $(\\cC\\bt\\overline\\cC)_{L_\\cC}$ is equivalent to $\\cC$ (as a fusion\ncategory). Moreover, for $X\\in\\cC$ the equivalence gives the free module $\nL_\\cC\\ot(X\\bt\n\\one )\\cong L_\\cC\\ot(\\one\\bt X)$. $L_\\cC\\ot(X\\bt\\one )$ is a local $L_\\cC$ module if and only\nif $X\\bt \\one$ centralize $L_\\cC$. This is the same as $X\\in\n\\cen{\\cC}{\\cC}$. Therefore we have $(\\cC\\bt \\overline\\cC)_{L_\\cC}^0=\\cen{\\cC}{\\cC}$.\n\\end{proof}\n\n\n\\begin{thm}\\label{main}\n let $\\cM$ and $\\cM'$ be two modular extensions of the $\\mce{\\cE}$ $\\cC$. There\n exists a unique $\\cK\\in\\mathcal{M}_{ext}(\\cE)$ such that $\\cK\\bt_\\cE\\cM=\\cM'$. Such $\\cK$\n is given by\n \\begin{align*}\n \\cK=(\\cM'\\bt \\overline\\cM)_{L_\\cC}^0.\n \\end{align*}\n\\end{thm}\n\\begin{proof}\n $\\cK$ is a modular extension of $\\cE$. This follows\n Lemma \\ref{Lag}, that $\\cE=\\cen{\\cC}{\\cC}=(\\cC\\bt\\overline \\cC)^0_{L_\\cC}$ is a full\n subcategory of $\\cK$. $\\cK$ is a UMTC by construction, and\n $\\dim(\\cK)=\\frac{\\dim(\\cM)\\dim(\\cM')}{\\dim(L_\\cC)^2}=\\dim(\\cE)^2$.\n\n To show that $\\cK=(\\cM'\\bt\\overline\\cM)_{L_\\cC}$ satisfies\n $\\cM'=\\cK\\bt_\\cE\\cM$, note that\n $\\cM'=\\cM'\\bt\\mathrm{Vec}=\\cM'\\bt(\\overline\\cM\\bt\\cM)_{L_{\\overline\\cM}}^0$. It suffies that\n \\begin{align*}\n (\\cM'\\bt\\overline\\cM\\bt\\cM)_{\\one\\bt\n L_{\\overline\\cM}}^0=[(\\cM'\\bt\\overline\\cM)_{L_\\cC}^0\\bt\n \\cM]_{L_\\cE}^0\\\\\n =(\\cM'\\bt\\overline\\cM\\bt\\cM)_{(L_\\cC\\bt\\one)\\ot(\\one\\bt\n L_\\cE)}^0.\n \\end{align*}\n This follows that $\\one\\bt L_{\\overline\\cM} $ and $(L_\\cC\\bt\\one)\\ot(\\one\\bt\n L_\\cE)$ are left and right centers of the algebra $(L_\\cC\\bt\\one)\\ot(\\one\\bt\n L_{\\overline\\cM})$.\n\n If $\\cM'=\\cK\\bt_\\cE\\cM=(\\cK\\bt\\cM)_{L_\\cE}^0$,\n then\n \\begin{align*}\n \\cK= (\\cK\\bt\\cM\\bt\\overline\\cM)_{\\one\\bt\n L_\\cM}^0=\n (\\cK\\bt\\cM\\bt\\overline\\cM)_{(L_\\cE\\bt\\one)\\ot(\\one\\bt \n L_\\cC)}^0\\\\\n =[(\\cK\\bt_\\cE\\cM)\\bt\\overline\\cM]_{L_\\cC}^0\n =(\\cM'\\bt\\overline\\cM)_{L_\\cC}^0.\n \\end{align*}\n It is similar here that $\\one\\bt L_{\\cM} $ and\n $(L_\\cE\\bt\\one)\\ot(\\one\\bt\n L_\\cC)$ are the left and right centers of the algebra\n $(L_\\cE\\bt\\one)\\ot(\\one\\bt\n L_{\\cM})$. This proves the uniqueness of $\\cK$.\n\n\\end{proof}\n\n\nLet us list several consequences of Theorem \\ref{main}.\n\\begin{thm}\\label{hegroup}\n $\\mathcal{M}_{ext}(\\cE)$ forms an finite abelian group.\n \\end{thm}\n \\begin{proof}\n Firstly, there exists at least one modular extension of a symmetric fusion\n category $\\cE$,\n the Drinfeld center $Z(\\cE)$. So the set $\\mathcal{M}_{ext}(\\cE)$ is not empty.\n The multiplication is given by the stacking $\\bt_\\cE$.\n It is easy to verify that the stacking $\\bt_\\cE$ for modular extensions\n is associative and commutative. To show that they form a group we only need\n to find out the identity and inverse.\n In this case $\\cK=(\\cM'\\bt \\overline \\cM)^0_{L_\\cE}=\\cM'\\bt_\\cE\\overline \\cM$,\n Theorem \\ref{main} becomes $\\cM'\\bt_\\cE\\overline\\cM\\bt_\\cE\\cM=\\cM'$, for any\n modular extensions $\\cM,\\cM'$ of $\\cE$.\n Thus, $\\overline{\\cM'}\\bt_\\cE\n \\cM'=\\overline{\\cM'}\\bt_\\cE\n \\cM'\\bt_\\cE\\overline\\cM\\bt_\\cE\\cM\n =\\overline\\cM\\bt_\\cE\\cM$, i.e. $\\overline\\cM\\bt_\\cE\\cM$, is the same category\n for any extension $\\cM$, which turns out to be $Z(\\cE)$. It is exactly the identity element. It is then\n obvious that the inverse of $\\cM$ is $\\overline\\cM$.\n The finiteness follows from \\Ref{BNRW13}.\n \\end{proof}\n\n \\begin{exa}\n For bosonic case we find that $\\mathcal{M}_{ext}(\\Rp(G))=H^3(G,U(1))$, which is\n discussed in more detail in the next subsection. For fermionic case a\n general group cohomological classification is still lacking. We know some\n simple ones such as $\\mathcal{M}_{ext}(\\sRp(\\Z_2^f))=\\Z_{16}$, which agrees with\n Kitaev's\n 16-fold way\\cite{K062}.\n \\end{exa}\n\n \\begin{thm}\\label{hetorsor}\n For a $\\mce{\\cE}$ $\\cC$, if the modular extensions exist, $\\mathcal{M}_{ext}(\\cC)$ form\n a $\\mathcal{M}_{ext}(\\cE)$-torsor. In particular, $|\\mathcal{M}_{ext}(\\cC)|=|\\mathcal{M}_{ext}(\\cE)|$.\n \\end{thm}\n \\begin{proof}\n The action is given by the stacking $\\bt_\\cE$.\n For any two extensions $\\cM,\\cM'$, there is a unique extension $\\cK$ of\n $\\cE$, such that $\\cM\\bt_\\cE\\cK=\\cM'$. To see $Z(\\cE)$ acts trivially, note\n that $\\cM'\\bt_\\cE Z(\\cE)=\\cM\\bt_\\cE \\cK\\bt_\\cE Z(\\cE)=\\cM\\bt_\\cE\\cK=\\cM'$\n holds for any $\\cM'$. Due to uniqueness we also know that only $\\cZ_\\cE$\n acts trivially. Thus, the action is free and transitive.\n \\end{proof}\n This means that for any modular extension of $\\cC$,\n stacking with a nontrivial modular extensions of $\\cE$, one always obtains\n a different modular extension of $\\cC$; on the other hand, starting with a\n particular modular extension of $\\cC$, all the other modular extensions can\n be generated by staking modular extensions of $\\cE$ (in other words, there\n is only on orbit). However, in general, there is no preferred choice of the\n starting modular extension, unless $\\cC$ is the form $\\cC_0\\bt \\cE$ where\n $\\cC_0$ is a UMTC. \n\n \n\\subsection{Modular extensions of $\\Rp(G)$} \\label{sec:mext-repG}\n\n\\begin{figure}[tb]\n$$\n\\setlength{\\unitlength}{.5pt}\n\\begin{picture}(100, 185)\n \\put(-80,){\\scalebox{1}{\\includegraphics{pic-half-braiding}}}\n \\put(0,0){\n \\put(-18,-40){\n \n \\put(20, 265) { $e\\in \\Rp(G) \\subset \\cM$}\n \\put(-30, 170) { $-\\otimes A$}\n \\put(75, 81) { $x$}\n \\put(75,30) { $\\gamma_2$}\n \\put(75,132) { $\\gamma_1$}\n \\put(100, 210) {$\\cM$}\n \\put(230, 81) {$\\cM_A$}\n \\put(160,45) {$\\mathrm{Vec}$}\n \\put(0, 60) {$F(e)$}\n }\\setlength{\\unitlength}{1pt}}\n \\end{picture}\n \n$$\n\\caption{Consider a physical situation in which the excitations in the $2+1$D\n bulk are given by a modular extension $\\cM$ of $\\Rp(G)$, and those on the\n gapped boundary by the UFC $\\cM_A$. Consider a simple particle $e\\in \\Rp(G)$\n in the bulk moving toward the boundary. The bulk-to-boundary map is given by\n the central functor $-\\otimes A: \\cM \\to \\cM_A$, which restricted to $\\Rp(G)$\n is nothing but the forgetful functor $F:\\Rp(G) \\to \\mathrm{Vec}$. Let $x$ be a\n simple excitation in $\\cM_A$ sitting next to $F(e)$. We move $F(e)$ along the\n semicircle $\\gamma_1$ (defined by the half-braiding), then move along the\n semicircle $\\gamma_2$ (defined by the symmetric braiding in the trivial phase\n $\\mathrm{Vec}$).}\n\\label{fig:G-grading}\n\\end{figure}\n\nWe set $\\cE=\\Rp(G)$ throughout this subsection. Let $(\\cM,\\iota_\\cM)$ be a modular\nextension of $\\Rp(G)$. $\\iota_\\cM$ is the embedding\n$\\iota_\\cM:\\cE\\hookrightarrow\\cM$ that we need to consider explicitly in this\nsubsection. The algebra $A=\\mathrm{Fun}(G)$ is a condensable algebra in $\\Rp(G)$ and\nalso a condensable algebra in $\\cM$. Moreover, $A$ is a Lagrangian algebra in\n$\\cM$ because $(\\dim A)^2 = |G|^2=(\\dim \\Rp(G))^2 = \\dim \\cM$. Therefore, $\\cM\\simeq Z(\\cM_A)$, where $\\cM_A$ is the category of right $A$-modules in $\\cM$. In other words, $\\cM$ describes the bulk excitations in a 2+1D topological phase with a gapped boundary (see Fig.\\,\\ref{fig:G-grading}). Moreover, the fusion category $\\cM_A$ is pointed and equipped with a canonical fully faithful $G$-grading\\cite{dgno2007}, which means that \n$$\n\\cM_A=\\oplus_{g\\in G} (\\cM_A)_g, \\quad (\\cM_A)_g\\simeq \\mathrm{Vec}, \\,\\, \\forall g\\in G, \n$$ \n$$\n\\quad \\mbox{and} \\quad \\otimes: (\\cM_A)_g \\boxtimes (\\cM_A)_h \\xrightarrow{\\simeq} (\\cM_A)_{gh}.\n$$\nLet us recall the construction of this $G$-grading. The physical meaning of\nacquiring a $G$-grading on $\\cM_A$ after condensing the algebra $A=\\mathrm{Fun}(G)$ in\n$\\cM$ is depicted in Figure\\,\\ref{fig:G-grading}. The process in\nFigure\\,\\ref{fig:G-grading} defines the isomorphism $F(e) \\otimes_A x\n \\xrightarrow{z_{e,x}} x \\otimes_A F(e) = F(e) \\otimes_A x$,\n which further gives a monoidal automorphism $\\phi(x)\\in \\mathrm{Aut}(F)=G$ of\nthe fiber functor $F: \\Rp(G) \\to \\mathrm{Vec}$. \n\nSince $\\phi$ is an isomorphism, the associator of the monoidal category $\\cM_A$ determines a unique\n$\\omega_{(\\cM,\\iota_M)}\\in H^3(G, U(1))$ such that $\\cM_A \\simeq \\mathrm{Vec}_G^\\omega$ as $G$-graded fusion categories. \n\n\n\\begin{thm} \\label{thm:spt}\nThe map $(\\cM, \\iota_\\cM) \\mapsto \\omega_{(\\cM, \\iota_\\cM)}$ defines a group isomorphism $\\cM_{ext}(\\Rp(G)) \\simeq H^3(G, U(1))$. In particular, we have \n$$ \n(Z(\\mathrm{Vec}_G^{\\omega_1}),\\iota_{\\omega_1}) \\boxtimes_\\cE\n(Z(\\mathrm{Vec}_G^{\\omega_2}),\\iota_{\\omega_2}) \\simeq (Z(\\mathrm{Vec}_G^{\\omega_1+\\omega_2}), \\iota_{\\omega_1+\\omega_2}).\n$$\n\\end{thm}\n\nFor the proof and more related details, see also \\Ref{LW160205936}.\n\n\n\\subsection{Relation to numerical calculations}\nIn Section \\ref{clGQL2} we proposed another way to characterise GQLs, using\nthe data $( \\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$\nwhich is more friendly in numerical calculations. We would like to investigate how\nto calculate the stacking operation in terms of these data.\n\nAssuming that $\\cC$ and $\\cC'$ can be characterized by data\n$(N^{ij}_k,s_i)$ and $( N^{\\prime ij}_k,s^\\prime_i)$. Let $(\nN^{\\cD, ij}_k, s^\\cD_i)$ be the data that characterizes the stacked $\\mce{\\cE}$ $\\cD\n=\\cC\\bt_\\cE \\cC'$.\n\nTo calculate $(N^{\\cD, ij}_k, s^\\cD_i)$, let us first construct\n\\begin{align}\n N^{ii',jj'}_{kk'} = N^{ij}_{k} N^{\\prime i'j'}_{k'} \n,\\ \\ \\ \\ \\\ns_{ii'}=s_i+s'_{i'}.\n\\end{align}\nNote that, the above data describes a $\\mce{\\cE\\boxtimes \\cE}$ $\\cD' =\\cC\\bt\n\\cC'$ (\\ie with centralizer $\\cE\\bt \\cE$), which is not what we want. We need\nreduce centralizer from $\\cE\\bt \\cE$ to $\\cE$. This is the $G\\times G$ to $G$\nprocess and $\\cC$-$\\cC'$ coupling, or condensing the $L_\\cE$\nalgebra, as discussed above\n\nTo do the $\\cE\\bt \\cE$ to $\\cE$ reduction (\\ie to obtain the real stacking\noperation $\\bt_\\cE$), we can introduce an equivalence relation. Noting that the\nexcitations in $\\cD'=\\cC\\bt \\cC'$ are labeled by $ii'=i\\bt i'$, the equivalence\nrelation is \n\\begin{align}\n ii'\\sim jj', \\quad{\\text{if }} ii'\\ot L_\\cE=jj'\\ot L_\\cE.\n\\end{align}\nwhere $L_\\cE=\\oplus_a a\\bar a, a\\in\\cE$. In the simple case of abelian groups,\nwhere all the $a$'s are abelian particles, the equivalence relation reduces to\n\\begin{align}\n (a\\ot i)i'\\sim i(a\\ot i'),\\ \\ \\ \\\n\\forall \\ \\ i\\in \\cC,\\ i'\\in \\cC',\\ a\\in \\cE.\n\\end{align}\nMathematically, this amounts to consider only the free local $L_\\cE$ modules.\nThe equivalent classes $[ii']$ are then some composite anyons in\n$\\cD=\\cC\\bt_\\cE\\cC'$\n\\begin{align}\n [ii']=k\\oplus l \\oplus \\cdots , \\quad\\text{ for some }k,l,\\dots\\in\\cD.\n\\end{align}\nIn other words, they form a fusion sub ring of $\\cD$.\nMoreover, the spin of $ii'$ is the same as the direct summands\n\\begin{align}\n s_{ii'}=s_k^\\cD=s_l^\\cD=\\cdots\n\\end{align}\nSince it is\nlimited to a subset of data of $\\mce{\\cE}$'s, we can only give these necessary\nconditions. However, as we already give a large list of GQLs in terms of these data,\nthey are usually enough to pick the resulting $\\cC\\bt_\\cE\\cC'$ from the list.\n\n\\section{How to calculate the modular extension of a $\\mce{\\cE}$}\n\\label{howto}\n\n\n\\subsection{A naive calculation}\n\nHow do we calculate the modular extension $\\cM$ of $\\mce{\\cE}$ $\\cC$ from\nthe data of $\\cC$? Actually, we do not know how to do that. So here, we will\nfollow a closely related Conjecture \\ref{NsNsNs}, and calculate instead\n$(\\cN^{IJ}_K,\\cS_I,c)$ (that fully characterize $\\cM$) from the data $(\n\\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$ (that partially characterize $\\cC$).\nIn this section, we will describe such a calculation.\n\nWe note that all the simple objects (particles) in $\\cC$ are contained in $\\cM$\nas simple objects, and $\\cM$ may contain some extra simple objects. Assume\nthat the particle labels of $\\cM$ are $\\{I,J,\\cdots \\}=\\{i,j,\\dots, x,\ny,\\dots\\}$, where we use $i,j,\\cdots $ to label the particles in $\\cC$ and\n$x,y,\\cdots $ to label the additional particles (not in $\\cC$). Also let us\nuse $a,b,\\cdots $ to label the simple objects in the centralizer of $\\cC$:\n$\\cE=\\cC_\\cC^\\text{cen}$. Let $\\cN^{IJ}_K$, $\\cS_{I}$ be the fusion\ncoefficients and the spins for $\\cM$, and $N^{ij}_k,\\ s_i$ be the fusion\ncoefficients and the spins for $\\cC$. The idea is to find as many conditions on\n$(\\cN^{IJ}_K, \\cS_{I})$ as possible, and use those conditions to solve for\n$(\\cN^{IJ}_K, \\cS_{I})$. Since the data $(\\cN^{IJ}_K, \\cS_{I})$ describe the\nUMTC $\\cM$, they should satisfy all the conditions discussed in\n\\Ref{W150605768}. On the other hand, as a modular extension of $\\cC$,\n$(\\cN^{IJ}_K, \\cS_{I})$ also satisfy some additional conditions. Here, we will\ndiscuss those additional conditions.\n\nFirst, the modular extension $\\cM$ has a fixed total quantum dimension.\n\\begin{align}\n\\label{dimMCE}\n \\dim(\\cM)=\\dim(\\cE)\\dim(\\cC).\n\\end{align}\nIn other words\n\\begin{align}\n \\sum_{I\\in \\cM} d_I^2 = \\sum_{a\\in \\cE} d_a^2 \\sum_{i\\in \\cC} d_i^2.\n\\end{align}\n\nPhysically, the modular extension $\\cM$ is obtained by ``gauging'' the\nsymmetry $\\cE$ in $\\cC$ (\\ie adding the symmetry twists of $\\cE$). So the\nadditional particles $x,y,\\cdots$ correspond to the symmetry twists. Fusing an\noriginal particle $i\\in \\cC$ to a symmetry twist $x\\notin \\cC$ still give us a\nsymmetry twist. Thus\n\\begin{align}\n \\cN^{ix}_j = \\cN^{xi}_j = \\cN^{ij}_x =0.\n\\end{align}\n\nTherefore, $\\cN_i$ for $i\\in \\cC$ is block diagonal: \n\\begin{align}\n\\cN_i= N_i \\oplus \\hat N_i, \n\\end{align}\nwhere $( N_i)_{jk}=\\cN^{ij}_{k}=N^{ij}_{k}$ and $(\\hat N_i)_{x \ny}=\\cN^{iy}_{x}$. \n\nIf we pick a charge conjugation for the additional particles $x\\mapsto\n\\bar x$, the conditions for fusion rules reduce to\n\\begin{align}\n \\cN^{i x}_{y}=\\cN^{x i}_{y}=\\cN^{\\bar{x} y}_{i}=\\cN^{i \\bar{y}}_{\\bar{x}},\n\\nonumber \\\\\n \\sum_{k\\in\\cC} N^{ij}_k \\cN^{kx}_{y}= \\sum_{z \\notin\\cC} \\cN^{i z}_{ x} \\cN^{j y}_{z}.\n \\label{extN}\n\\end{align}\nWith a choice of charge conjugation, it is enough to construct (or search for)\nthe matrices $\\hat N_i$ and $\\cN^{xy}_z$ to determine all the extended fusion\nrules $\\cN^{IJ}_K$. \n\nBesides the general condition \\eqref{extN}, there are also some simple\nconstraints on $\\hat N_i$ that may speed up the numerical search.\nFirstly, observe that \\eqref{extN} is the same as\n\\begin{align}\n\\hat N_i \\hat N_j = \\sum_{k\\in \\cC} N^{ij}_k \\hat N_k,\n\\end{align}\nwhere $i,j,k \\in \\cF$. This means that $\\hat N_i$ satisfy the same fusion\nalgebra as $ N_i$, and $N^{ij}_k=\\cN^{ij}_k$ is the structure constant;\ntherefore, the eigenvalues of $\\hat N_i$ must be a subset of the eigenvalues\nof $ N_i$. \n\nSecondly, since $\\sum_{y\\notin\\cC} \\cN^{i x}_{y}d_{y}= d_i d_{x}$, by\nPerron-Frobenius theorem, we know that $d_i$ is the largest eigenvalue of\n$\\hat N_i$, with eigenvector $v, v_{x}=d_{x}$. ($d_i$ is also the largest\nabsolute values of the eigenvalues of $\\hat N_i$.) Note that ${\\hat N_{\\bar\ni} \\hat N_{i}= \\hat N_i \\hat N_{\\bar i},}$ ${ \\hat N_{\\bar i}=\\hat\nN_i^\\dag}$. Thus, $d_i^2$ is the largest eigenvalue of the positive\nsemi-definite Hermitian matrix $\\hat N_{i}^\\dag\\hat N_{i}$. For any unit\nvector $v$ we have $v^\\dag \\hat N_i^\\dag \\hat N_i v\\leq d_i^2$, in\nparticular,\n\\begin{align}\n (\\hat N_i^\\dag \\hat N_i)_{xx}=\\sum_{y} (\\cN^{ix}_{y})^2\\leq\n d_i^2.\n \\label{extentry}\n\\end{align}\nThe above result is very helpful to reduce the scope of numerical search.\n\nOnce we find the fusion rules, $\\cN^{IJ}_K$, we can then use the rational\nconditions and other conditions to determine the spins $\\cS_I$ (for details, see\n\\Ref{W150605768}). The set of data $(\\cN^{IJ}_K,\\cS_I)$ that satisfy all the\nconditions give us the set of modular extensions.\n\n\nThe above proposed calculation for modular extensions is quite expensive. If\nthe quantum dimensions of the particles in $\\cC$ are all equal to 1: $d_i=1$,\nthen there is another much cheaper way to calculate the fusion coefficient\n$\\cN^{IJ}_K$ of the modular extension $\\cM$. Such an approach is explained in\nAppendix \\ref{FRgroup}. We will also use such an approach in our calculation.\n\nLast, we would like to mention that two sets of data $(\\cN^{IJ}_K,\\cS_I)$ and\n$(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ describe the same modular extension of\n$\\cC$, if they only differ by a permutation of indices $x \\in \\cM$ but\n$x \\notin \\cC$. So some times, two sets of data $(\\cN^{IJ}_K,\\cS_I)$ and\n$(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ can describe different modular extensions,\neven through they describe the same UMTC. (Two sets of data\n$(\\cN^{IJ}_K,\\cS_I)$ and $(\\bar \\cN^{IJ}_K,\\bar \\cS_I)$ describe the same\nUMTC, if they are only different by a permutation of indices $I \\in \\cM$.)\n\nWhy we use such a permutation in the calculation of modular extensions. (which\nis the ME-equivalence relation discussed before)? This is because when we\nconsidering modular extensions, the particle $x \\in \\cM$ but $x \\notin \\cC$\ncorrespond to symmetry twists. They are extrinsic excitations that do not\nappear in the finite energy spectrum of the Hamiltonian. While the particle\n$i\\in \\cC$ are intrinsic excitations that do appear in the finite energy\nspectrum of the Hamiltonian. So $x \\notin \\cC$ and $i\\in \\cC$ are physically\ndistinct and we do not allow permutations that mix them. Also we should not\npermute the particles $a\\in \\cE$, because they correspond to symmetries. We\nshould not mix, for example, the $Z_2$ symmetry of exchange layers and the\n$Z_2$ symmetry of 180$^\\circ$ spin rotation.\n\n\n\\subsection{The limitations of the naive calculation}\n\nSince a $\\mce{\\cE}$ $\\cC$ is not modular, the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$ may not fully characterize $\\cC$. To fully characterize $\\cC$, we need to use additional data, such\nas the $F$-tensor and the $R$-tensor\\cite{K062,W150605768}. \n\nIn this paper, we will not use those additional data. As a result, the data\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ may correspond to several different\n$\\mce{\\cE}$ $\\cC$'s. In other words, $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$ is a one-to-many labeling of $\\mce{\\cE}$'s.\n\nSo in our naive calculation, when we calculate the modular extensions of\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$, we may actually calculate the\nmodular extension of several different $\\cC$'s that are described by the same\ndata $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$. But for $\\mce{\\cE}$'s that\ncan be fully characterized by the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i)$, our calculation produce the modular extensions of a single\n$\\cC$. For example, the naive calculation can obtain the correct modular\nextensions of $\\cC=\\Rp(G)$ and $\\cC=\\sRp(G^f)$, when $G$ and $G^f$ are abelian\ngroups, or simple finite groups\\cite{Yuan16}.\n\nIf the $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ happen to describe two\ndifferent $\\mce{\\cE}$'s, we find that our naive calculation will produce the\nmodular extensions for both of $\\mce{\\cE}$'s (see Section \\ref{Z2N5}). So by\ncomputing the modular extensions of $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$,\nwe can tell if $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ corresponds to none,\none, two, \\etc $\\mce{\\cE}$'s. This leads to the Conjecture \\ref{NsNsNs} that\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i,\\cN^{IJ}_K,\\cS_I;c)$ can fully and\none-to-one classify GQLs in 2+1D.\n\n\\section{Examples of 2+1D SET orders and SPT orders}\n\\label{examples}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The bottom two rows correspond to the two modular extensions of\n$\\Rp(Z_2)$ (denoted by $N_c^{|\\Th|}=2^{\\zeta^1_2}_0$). Thus we have two\ndifferent trivial topological orders with $Z_2$ symmetry in 2+1D (\\ie two $Z_2$\nSPT states). \n} \n\\label{mextZ2} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$2^{\\zeta_{2}^{1}}_{ 0}$ & $2$ & $1, 1$ & $0, 0$ & $\\Rp(Z_2)$ \\\\\n\\hline\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, 0, \\frac{1}{2}$ & $Z_2$ gauge\\\\\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{4}, \\frac{3}{4}$ & double semion\\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{The two modular extensions of $N^{|\\Th|}_{c}=3^{\\zeta_{2}^{1}}_{ 2}$.\n$3^{\\zeta_{2}^{1}}_{ 2}$ has a centralizer $\\Rp(Z_2)$. Thus we have two\ntopological orders with $Z_2$ symmetry in 2+1D which has only one type of\nspin-$1\/3$ topological excitations. We use $N^{|\\Th|}_{c}$ to label\n$\\mce{\\cE}$'s, where $\\Theta ={D}^{-1}\\sum_{i}\\ee^{2\\pi\\ii s_i} d_i^2=\n|\\Th|\\ee^{2\\pi \\ii c\/8}$ and $D^2=\\sum_id_i^2$.\n} \n\\label{mextZ2a} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$3^{\\zeta_{2}^{1}}_{ 2}$ & $6$ & $1, 1, 2$ & $0, 0, \\frac{1}{3}$ & \n\\tiny $K=\\begin{pmatrix}\n 2 & -1 \\\\\n -1 & 2 \\\\\n\\end{pmatrix}\n$\n\\\\\n\\hline\n$5^{ B}_{ 2}$ & $12$ & $1, 1, 2,\\zeta_{4}^{1},\\zeta_{4}^{1}=\\sqrt{3}$ & $0, 0, \\frac{1}{3}, \\frac{1}{8}, \\frac{5}{8}$ & $(A_1,4)$ \\\\\n$5^{ B}_{ 2}$ & $12$ & $1, 1, 2,\\zeta_{4}^{1},\\zeta_{4}^{1}$ & $0, 0, \\frac{1}{3}, \\frac{3}{8}, \\frac{7}{8}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\nIn this section, we will discuss simple examples of $\\mce{\\cE}$ $\\cC$'s, and\ntheir modular extensions $\\cM$. The triple $(\\cC, \\cM,c)$ describe a\ntopologically ordered or SPT phase. A single $\\mce{\\cE}$ $\\cC$ only describes\nthe set of bulk topological excitations, which correspond to topologically\nordered states up to invertible ones.\n\nHowever, in this section we will not discuss examples of $\\mce{\\cE}$ $\\cC$.\nWhat we really do is to discuss examples of the solutions $(\\tilde\nN^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ (which are not really $\\mce{\\cE}$'s, but\nclosely related). We will also discuss the modular extensions\n$(\\cN^{IJ}_K,\\cS_I;c)$ of $(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$.\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ will correspond to $\\mce{\\cE}$\n$\\cC$ if it has modular extensions $(\\cN^{IJ}_K,\\cS_I;c)$. This allows us to\nclassify GQLs in terms of the data $(\\tilde N^{ab}_c,\\tilde\ns_a;N^{ij}_k,s_i,\\cN^{IJ}_K,\\cS_I;c)$.\n\n\\subsection{$Z_2$ bosonic SPT states}\n\nTables \\ref{SETZ2-34}, \\ref{SETZ2-5}, and \\ref{SETZ2-6} list the solutions\n$(\\tilde N^{ab}_c,\\tilde s_a;N^{ij}_k,s_i)$ when $(\\tilde N^{ab}_c,\\tilde\ns_a)$ describes a SFC $\\Rp(Z_2)$. The table contains all $\\mce{\\Rp(Z_2)}$'s\nbut may contain extra fake entries. Physically, they describe possible\nsets of bulk excitations for $Z_2$-SET orders of bosonic systems. The sets of\nbulk excitations are listed by their quantum dimensions $d_i$ and spins $s_i$.\n\nFor example, let us consider the entry $N_c^{|\\Th|}=2_0^{\\zeta_2^1}$ in Table\n\\ref{SETZ2-34}. Such an entry has a central charge $c=0$. Also $N=2$, hence\nthe $Z_2$-SET state has two types of bulk excitations both with $d_i=1$ and\n$s_i=0$. Both types of excitations are local excitations; one is the trivial type\nand the other carries an $Z_2$ charge.\n\nThe first question that we like to ask is that ``is such an entry a fake entry,\nor it corresponds to some $Z_2$-symmetric GQL's?'' If it corresponds to\nsome $Z_2$-symmetric GQL's, how many distinct $Z_2$-symmetric GQL\nphases that it corresponds to? In other word, how many distinct\n$Z_2$-symmetric GQL phases are there, that share the same set of bulk\ntopological excitations described by the entry $2_0^{\\zeta_2^1}$?\n\nBoth questions can be answered by computing the modular extensions of\n$2_0^{\\zeta_2^1}$ (which is also denoted as $\\Rp(Z_2)$). We find that the\nmodular extensions exist, and thus $\\Rp(Z_2)$ does correspond to some \n$Z_2$-symmetric GQL's. In fact, one of the $Z_2$-symmetric GQL's is the\ntrivial product state with $Z_2$ symmetry. Other $Z_2$-symmetric GQL's are\n$Z_2$ SPT states.\n\nAfter a numerical calculation, we find that there are only two different\nmodular extensions of $\\Rp(Z_2)$ (see Table \\ref{mextZ2}). Thus there are two\ndistinct $Z_2$-symmetric GQL phases whose bulk excitations are described by the\n$\\Rp(Z_2)$. The first one corresponds to the trivial product states whose\nmodular extension is the $Z_2$ gauge theory which has four types of particles\nwith $(d_i,s_i)=(1,0), (1,0),(1,0),(1,\\frac12)$. (Gauging the $Z_2$ symmetry\nof the trivial product state gives rise to a $Z_2$ gauge theory.) The second\none corresponds to the only non-trivial $Z_2$ bosonic SPT state in 2+1D, whose\nmodular extension is the double-semion theory which has four types of particles\nwith $(d_i,s_i)=(1,0), (1,0),(1,\\frac14),(1,-\\frac14)$. (Gauging the $Z_2$\nsymmetry of the $Z_2$-SPT state gives rise to a double-semion theory\n\\cite{LG1209}.) So the $Z_2$-SPT phases are classified by $\\Z_2$, reproducing\nthe group cohomology result\\cite{CLW1141,CGL1314,CGL1204}. In general, the\nmodular extensions of $\\Rp(G)$ correspond to the bosonic SPT states in 2+1D\nwith symmetry $G$.\n\n\\subsection{$Z_2$-SET orders for bosonic systems}\n\n\\begin{table}[t] \n\\caption{\nThe fusion rule of the $N_c^{|\\Th|}=3_2^{\\zeta_2^1}$ $Z_2$-SET order.\nThe particle $\\textbf{1}$\ncarries the $Z_2$-charge $0$, and the particle $s$ carries the $Z_2$-charge\n$1$. From the table, we see that $\\sigma\\otimes\\sigma=\\textbf{1} \\oplus s\n\\oplus \\sigma$.\n} \n\\label{SET32} \n\\centering\n\\begin{tabular}{ |c|ccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{3}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 2$\\\\\n\\hline\n $3^{\\zeta_{2}^{1}}_{ 2}$ & $\\textbf{1}$ & $s$ & $\\sigma$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ s$ & $ \\sigma$ \\\\\n$s$ & $ s$ & $ \\textbf{1}$ & $ \\sigma$ \\\\\n$\\sigma$ & $ \\sigma$ & $ \\sigma$ & $ \\textbf{1} \\oplus s \\oplus \\sigma$ \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\n\\begin{table}[t] \n\\caption{\nThe fusion rules of the two $N_c^{|\\Th|}=4_1^{\\zeta_2^1}$ $Z_2$ symmetry\nenriched topological orders with identical $d_i$ and $s_i$. \nWe see that one has a $Z_2\\times Z_2$ fusion rule and\nthe other has a $Z_4$ fusion rule.\n} \n\\label{SETZ2-45} \n\\centering\n\\begin{tabular}{ |c|cccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{4}$ & $ \\frac{1}{4}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$\\\\\n\\hline\n $4^{\\zeta_{2}^{1}}_{ 1}$ & $\\textbf{00}$ & $\\textbf{01}$ & $\\textbf{10}$ & $\\textbf{11}$ \\\\\n\\hline\n$\\textbf{00}$ & $ \\textbf{00}$ & $ \\textbf{01}$ & $ \\textbf{10}$ & $ \\textbf{11}$ \\\\\n$\\textbf{01}$ & $ \\textbf{01}$ & $ \\textbf{00}$ & $ \\textbf{11}$ & $ \\textbf{10}$ \\\\\n$\\textbf{10}$ & $ \\textbf{10}$ & $ \\textbf{11}$ & $ \\textbf{00}$ & $ \\textbf{01}$ \\\\\n$\\textbf{11}$ & $ \\textbf{11}$ & $ \\textbf{10}$ & $ \\textbf{01}$ & $ \\textbf{00}$ \\\\\n\\hline\n\\end{tabular}\n~~~~~~\n\\begin{tabular}{ |c|cccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{4}$ & $ \\frac{1}{4}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$\\\\\n\\hline\n $4^{\\zeta_{2}^{1}}_{ 1}$ & $\\textbf{0}$ & $\\textbf{2}$ & $\\textbf{1}$ & $\\textbf{3}$ \\\\\n\\hline\n$\\textbf{0}$ & $ \\textbf{0}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{3}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{0}$ & $ \\textbf{3}$ & $ \\textbf{1}$ \\\\\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{0}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{1}$ & $ \\textbf{0}$ & $ \\textbf{2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table} \n\nThe entry $N_c^{|\\Th|}=3_2^{\\zeta_2^1}$ in Table \\ref{SETZ2-34} corresponds to\nmore non-trivial $\\mce{\\Rp(Z_2)}$. It describes the bulk excitations of\n$Z_2$-SET orders which has only one type of non-trivial topological\nexcitation(with quantum dimension $d=2$ and spin $s=1\/3$, see Table\n\\ref{SET32}). The other two types of excitations are local excitations with\n$Z_2$-charge $0$ and $1$. We find that $3_2^{\\zeta_2^1}$ has modular\nextensions and hence is not a fake entry.\n\nTo see how many SET orders that have such set of bulk excitations, we need to\ncompute how many modular extensions are there for $3_2^{\\zeta_2^1}$. We find\nthat $3_2^{\\zeta_2^1}$ has two modular extensions (see Table \\ref{mextZ2a}).\nThus there are two $Z_2$-SET orders with the above mentioned bulk excitations.\nIt is not an accident that the number of $Z_2$-SET orders with the same set of\nbulk excitations is the same as the number of $Z_2$ SPT states. This is\nbecause the different $Z_2$-SET orders with a fixed set of bulk excitations are\ngenerated by stacking with $Z_2$ SPT states.\n\nWe would like to point out that for any $G$-SET state, if we break the\nsymmetry, the $G$-SET state will reduce to a topologically ordered state\ndescribed by a UMTC. In fact, the different $G$-SET states described by the\nsame $\\mce{\\cE}$ (\\ie with the same set of bulk excitations) will reduce to the\nsame topologically ordered state (\\ie the same UMTC). In Appendix \\ref{SB}, we\ndiscussed such a symmetry breaking process and how to compute UMTC from\n$\\mce{\\cE}$. We found that the two $Z_2$-SET orders from $3_2^{\\zeta_2^1}$\nreduce to an abelian topological order described by a $K$-matrix $\\bpm 2& -1\\\\\n-1& 2 \\epm$. This is indicated by SB:$K=\\bpm 2& -1\\\\ -1& 2 \\epm$ in the\ncomment column of Table \\ref{SETZ2-34}. In other place, we use SB:$N^B_c$ or\nSB:$N^F_c({a \\atop b})$ to indicate the reduced topological order after the\nsymmetry breaking (for bosonic or fermionic cases). (The topological orders\ndescribed by $N^B_c$ or $N^F_c({a \\atop b})$ are given by the tables in\n\\Ref{W150605768} or \\Ref{LW150704673}.)\n\nAs we have mentioned, there are two $Z_2$-SET orders with the same bulk\nexcitations. But how to realize those $Z_2$-SET orders? We find that one of\nthe $Z_2$-SET orders is the double layer FQH state with $K$-matrix $\\bpm 2 &\n-1\\\\ -1 & 2\\\\ \\epm$ (same as the reduced topological order after symmetry\nbreaking), where the $Z_2$ symmetry is the layer-exchange symmetry. The\nquasiparticles are labeled by the $l$-vectors $l=\\bpm l_1\\\\l_2\\epm$. The two\nnon-trivial quasiparticles are given by \n\\begin{align} \nl\n&= \\bpm 1 \\\\ 0\\epm, \\ \\ \\bpm 0 \\\\ 1\\epm, \\ \\ \n\\end{align} \nwhose spins are all equal to $\\frac13$.\n\nSince the layer-exchange $Z_2$ symmetry exchanges $l_1$ and $l_2$, we see that\nthe two excitations $ \\bpm 1 \\\\ 0\\epm, \\ \\ \\bpm 0 \\\\ 1\\epm$ always have the\nsame energy. Despite the $Z_2$ symmetry has no 2-dim irreducible\nrepresentations, the above spin-1\/3 topological excitations has an exact\ntwo-fold degeneracy due to the $Z_2$ layer-exchange symmetry. This effect is\nan interplay between the long-range entanglement and the symmetry:\n\\emph{degeneracy in excitations may not always arise from high dimensional\nirreducible representations of the symmetry.}\n\nSuch two degenerate excitations are viewed as one type of topological\nexcitations with quantum dimension $d=2$ (for the two-fold degeneracy) and spin\n$s=\\frac13$ (see Table \\ref{SETZ2-34}). The $Z_2$ symmetry twist in such a\ndouble-layer state carry a non-abelian statistics with quantum dimension\n$d=\\sqrt{3}$. In fact, there are two such $Z_2$ symmetry twists whose spin\ndiffer by 1\/2.\n\nThe other $Z_2$-SET order can be viewed as the above\ndouble layer FQH state $K=\\bpm 2 & -1\\\\ -1 & 2\\\\ \\epm$ stacked with a $Z_2$ SPT\nstate.\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The four modular extensions of $N^{|\\Th|}_{c}=5^{\\zeta_{2}^{1}}_{ 0}$ with $Z_2\\times Z_2$ fusion.\n$5^{\\zeta_{2}^{1}}_{ 0}$ has a centralizer $\\Rp(Z_2)$. The first pair and the\nsecond pair turns out to be equivalent.\n} \n\\label{mextZ2b} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$5^{\\zeta_{2}^{1}}_{ 0}$ & $8$ & $1\\times 4, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0$ & \\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{15}{16}, \\frac{1}{16}, \\frac{7}{16},\n\\frac{9}{16}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{3}{16}, \\frac{13}{16}, \\frac{11}{16},\n\\frac{5}{16}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{-3\/2}$\\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{1}{16}, \\frac{15}{16}, \\frac{9}{16},\n\\frac{7}{16}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{-1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, 0, \\frac{13}{16}, \\frac{3}{16}, \\frac{5}{16},\n\\frac{11}{16}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n\\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{ The four modular extensions of $N^{|\\Th|}_{c}=5^{\\zeta_{2}^{1}}_{ 1}$ with $Z_2\\times Z_2$ fusion.\n$5^{\\zeta_{2}^{1}}_{ 1}$ has a centralizer $\\Rp(Z_2)$.\n} \n\\label{mextZ2c} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$5^{\\zeta_{2}^{1}}_{ 1}$ & $8$ & $1\\times 4, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}$ & \\\\\n\\hline\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{13}{16}, \\frac{13}{16}, \\frac{5}{16}, \\frac{5}{16}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n\\hline\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{15}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ &\\tiny $1\\times 4, 2,\\zeta_{2}^{1}\\times 4$ &\\tiny $0, 0,\n\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{3}{16}, \\frac{15}{16},\n\\frac{11}{16}, \\frac{7}{16}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{-1\/2}$\\\\\n\\hline \n\\end{tabular} \n\\end{table}\n\n\n\\subsection{Two other $Z_2$-SET orders for bosonic\nsystems}\n\nThe fourth and fifth entries in Table \\ref{SETZ2-34} describe the bulk\nexcitations of two other $Z_2$-SET orders. Those bulk excitations have\nidentical $s_i$ and $d_i$, but they have different fusion rules $N^{ij}_k$ (see\nTable \\ref{SETZ2-45}). \n\nBoth entries have two modular extensions, and correspond to two SET orders.\nAmong the two SET orders for the $Z_2\\times Z_2$ fusion rule, one of them is\nobtained by stacking a $Z_2$ \\emph{neutral} $\\nu=1\/2$ Laughlin state with a\ntrivial $Z_2$ product state. The other is obtained by stacking a $Z_2$ neutral\n$\\nu=1\/2$ Laughlin state with a non-trivial $Z_2$ SPT state. \n\nThe entry with $Z_4$ fusion rule also correspond to two SET orders. They are\nobtained by stacking a $Z_2$ \\emph{charged} $\\nu=1\/2$ Laughlin state with a\ntrivial or a non-trivial $Z_2$ SPT state. Here, \\emph{charged} means that the\nparticles forming the $\\nu=1\/2$ Laughlin state carry $Z_2$-charge 1. In this\ncase, the anyon in the $\\nu=1\/2$ Laughlin state carries a fractional\n$Z_2$-charge $1\/2$. So the fusion of two such anyons give us a $Z_2$-charge 1\nexcitation instead of a trivial neutral excitation. This leads to the $Z_4$\nfusion rule.\n\n\n\\subsection{The rank $N=5$ $Z_2$-SET orders for bosonic systems}\n\\label{Z2N5}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe first and the third entries in Table \\ref{mextZ2b} have different\nfusion rules, despite they have the same $(d_i,s_i)$.\n} \n\\label{frZ2_5_1} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ 0$ & $ \\frac{1}{16}$ & $ \\frac{7}{16}$ & $ \\frac{9}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 0}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ 0$ & $ \\frac{1}{16}$ & $ \\frac{7}{16}$ & $ \\frac{9}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 0}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ \\\\\n\\hline\n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe third and the fourth entries in Table \\ref{mextZ2c} have different\nfusion rules, despite they have the same $(d_i,s_i)$.\n} \n\\label{frZ2_5_3} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{3}{16}$ & $ \\frac{7}{16}$ & $ \\frac{11}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{3}{16}$ & $ \\frac{7}{16}$ & $ \\frac{11}{16}$ & $ \\frac{15}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe fusion rules of the first and the second entries in Table \\ref{mextZ2c}.\n} \n\\label{frZ2_5_3a} \n\\centering\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{1}{16}$ & $ \\frac{1}{16}$ & $ \\frac{9}{16}$ & $ \\frac{9}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ \\\\\n\\hline\n\\end{tabular}\n\\\\[3mm]\n\\begin{tabular}{ |c|ccccccccc|}\n \\hline \n $s_i$ & $0$ & $ 0$ & $ \\frac{1}{2}$ & $ \\frac{1}{2}$ & $ \\frac{1}{8}$ & $ \\frac{5}{16}$ & $ \\frac{5}{16}$ & $ \\frac{13}{16}$ & $ \\frac{13}{16}$\\\\\n $d_i$ & $1$ & $ 1$ & $ 1$ & $ 1$ & $ 2$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$ & $\\zeta_{2}^{1}$\\\\\n\\hline\n $9^{ 1}_{ 1}$ & $\\textbf{1}$ & $\\textbf{2}$ & $\\textbf{3}$ & $\\textbf{4}$ & $\\textbf{5}$ & $\\textbf{6}$ & $\\textbf{7}$ & $\\textbf{8}$ & $\\textbf{9}$ \\\\\n\\hline\n$\\textbf{1}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{7}$ & $ \\textbf{8}$ & $ \\textbf{9}$ \\\\\n$\\textbf{2}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{9}$ & $ \\textbf{6}$ & $ \\textbf{7}$ \\\\\n$\\textbf{3}$ & $ \\textbf{3}$ & $ \\textbf{4}$ & $ \\textbf{1}$ & $ \\textbf{2}$ & $ \\textbf{5}$ & $ \\textbf{8}$ & $ \\textbf{7}$ & $ \\textbf{6}$ & $ \\textbf{9}$ \\\\\n$\\textbf{4}$ & $ \\textbf{4}$ & $ \\textbf{3}$ & $ \\textbf{2}$ & $ \\textbf{1}$ & $ \\textbf{5}$ & $ \\textbf{6}$ & $ \\textbf{9}$ & $ \\textbf{8}$ & $ \\textbf{7}$ \\\\\n$\\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{2} \\oplus \\textbf{3} \\oplus \\textbf{4}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ \\\\\n$\\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ \\\\\n$\\textbf{7}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ \\\\\n$\\textbf{8}$ & $ \\textbf{8}$ & $ \\textbf{6}$ & $ \\textbf{6}$ & $ \\textbf{8}$ & $ \\textbf{7} \\oplus \\textbf{9}$ & $ \\textbf{2} \\oplus \\textbf{3}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{4}$ & $ \\textbf{5}$ \\\\\n$\\textbf{9}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{9}$ & $ \\textbf{7}$ & $ \\textbf{6} \\oplus \\textbf{8}$ & $ \\textbf{5}$ & $ \\textbf{2} \\oplus \\textbf{4}$ & $ \\textbf{5}$ & $ \\textbf{1} \\oplus \\textbf{3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\n\nThe first and the second entries in Table \\ref{SETZ2-5} describe two $N=5$\n$\\mce{\\Rp(Z_2)}$'s. They describe two different sets of bulk excitations for\n$Z_2$-SET orders. Those bulk excitations have identical $s_i$ and $d_i$,\nbut they have different fusion rules $N^{ij}_k$: the 4 $d=1$ particles have a\n$Z_2\\times Z_2$ fusion rule for the first entry, and they have a $Z_4$ fusion\nrule for the second entry (as indicated by F:$Z_2\\times Z_2$ or F:$Z_4$ in the\ncomment column of Table \\ref{SETZ2-5}).\n\n\\subsubsection{The first entry in Table \\ref{SETZ2-5}}\n\nLet us compute the modular extensions of the first entry (\\ie\n$5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion). Since the total quantum\ndimension of the modular extensions is $D^2=16$, the modular extensions must\nhave rank $N=13$ or less (since quantum dimension $d \\geq 1$).\n\n\nNow we would like to show $N=13$ is not possible. If a modular extension has\n$N=13$, then it must have 12 particles (labeled by $a=1,\\cdots,12$) with\nquantum dimension $d_a=1$, and one particle (labeled by $x$) with quantum\ndimension $d_x=2$, so that $12\\times 1^2+2^2=D^2=16$. In this case,\nwe must have the fusion rule\n\\begin{align}\n a\\otimes x=x,\\ \\ \\ \\ x\\otimes x= 1\\oplus 2\\oplus 3\\oplus 4.\n\\end{align}\nwhere $x\\otimes x$ is determined by the fusion rule of the $\\mce{\\Rp(Z_2)}$.\nThe above determines the fusion matrix $N_x$ defined as $(N_x)_{ij} \\equiv\nN^{xi}_{j}$. The largest eigenvalue of $N_x$ should be $2$, the quantum\ndimension of $x$. Indeed, we find that the largest eigenvalue of $N_x$ is $2$.\nBut we also require that $N_x$ can be diagonalized by a unitary matrix (which\nhappens to be the $S$-matrix). $N_x$ fails such a test. So $N$ cannot be 13.\n\n$N$ also cannot be 12. If $N=12$, then the modular extension will have 10\nparticles (labeled by $a=1,\\cdots,10$) with quantum dimension $d_a=1$, one\nparticle (labeled by $x$) with quantum dimension $d_x=2$, and one particle\n(labeled by $y$) with quantum dimension $d_y=\\sqrt 2$. The fusion of 10\n$d_a=1$ particles is described by an abelian group $Z_{10}$ or $Z_2\\times Z_5$.\nNone of them contain $Z_2\\times Z_2$ as subgroup. Thus $N=12$ is incompatible\nwith the $Z_2\\times Z_2$ fusion of the first four $d_a=1$ particles.\n\nWe searched the modular extensions with $N$ up to 11. We find four $N=9$\nmodular extensions (see Table \\ref{mextZ2b}), and thus the first entry\ncorresponds to valid $Z_2$-SET states. \n\n\nIn fact one of the $Z_2$-SET states is the $Z_2$ gauge theory with a $Z_2$\nglobal symmetry, where the $Z_2$ symmetry action exchange the $Z_2$-charge $e$\nand the $Z_2$-vortex $m$. The degenerate $e$ and $m$ give rise to the\n$(d,s)=(2,0)$ particle (the fifth particle in the table). The bound state of\n$e$ and $m$ is a fermion $f$. It may carry the $Z_2$-charge 0 or 1, which\ncorrespond to the third and the fourth particle with $(d,s)=(1,1\/2)$ in the\ntable. \n\nHowever, from the discussion in the last few sections, we know that a\n$\\mce{\\Rp(Z_2)}$ always has 2 modular extensions, corresponding to the 2\nbosonic $Z_2$-SPT states in 2+1D. This seems contradictory with the above result\nthat the $Z_2$-SET state, $5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion, has\nfour different modular extensions. \n\nIn fact, there is no contradiction. Here, we only use $(N^{ij}_k,s_i)$ to\nlabel different entries. However, a $\\mce{\\cE}$ is fully characterized by\n$(N^{ij}_k,s_i)$ plus the $F$-tensors and the $R$-tensors. \nTo see this point, we note that the Ising-like UMTC $N^B_c=3^B_{m\/2}$,\n$m=1,3,\\cdots,15$ (with central charge $c=m\/2$) has three particles: $1$, $f$\nwith $(d_f,s_f)=(1,1\/2)$, and $ \\sigma $ with $(d_\\sigma ,s_\\sigma\n)=(\\sqrt{2},m\/16)$. Its $R$-tensor is given by\\cite{K062}\n\\begin{align}\n R^{ff}_1 &=-1, &\n R^{ \\sigma f }_\\sigma &= \n R^{ f \\sigma }_\\sigma = -\\ii^m, \n\\\\\n R^{ \\sigma \\sigma }_1 &= (-1)^{\\frac{m^2-1}{8}} \\ee^{-\\ii \\frac{ \\pi }{8} m}, &\n R^{ \\sigma \\sigma }_f &= (-1)^{\\frac{m^2-1}{8}} \\ee^{\\ii \\frac{3 \\pi }{8} m},\n\\nonumber \n\\end{align}\nand some components of the $F$-tensor are given by\n\\begin{align}\n F^{f \\sigma \\sigma; \\sigma }_{ f;1 }=\n F^{ \\sigma \\sigma f; \\sigma }_{ f;1 }=1.\n\\end{align}\nThe values of $R^{ \\sigma f }_\\sigma$ and $R^{ f \\sigma }_\\sigma$ are not gauge\ninvariant. But if we fix the values of the $F$-tensor to be the ones given\nabove, this will fix the gauge, and we can treat $R^{ \\sigma f }_\\sigma$ and\n$R^{ f \\sigma }_\\sigma$ as if they are gauge invariant quantities.\n\nIf we stack $N^B_c=3^B_{m\/2}$ and $N^B_c=3^B_{m'\/2}$ together, the induced\nUMTC $ 3^B_{m\/2}\\boxtimes 3^B_{m'\/2}$ contains particles\n$\\textbf{1}=(1,1)$, $\\textbf{2}=(f,f')$, $\\textbf{3}=(f,1)$,\n$\\textbf{4}=(1,f')$, $\\textbf{5}=( \\sigma , \\sigma' )$. Those 5 particles are\nclosed under the fusion, and correspond to the 5 particles in $\\mce{\\Rp(Z_2)}$\n$5^{\\zeta_2^1}_{m+m'}$. We note that some components of the $R$-tensor of $\n3^B_{m\/2}\\boxtimes 3^B_{m'\/2}$ are given by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =-\\ii^m,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =-\\ii^{m'}.\n\\end{align}\n\nTaking $(m,m')=(-1,1)$ and $(1,-1)$, it is clear the $ 3^B_{-1\/2}\\boxtimes\n3^B_{ 1\/2}$ and $ 3^B_{ 1\/2}\\boxtimes 3^B_{-1\/2}$ give rise to two different\n$R$-tensors that have identical $(N^{ij}_k,s_i)$. So the first entry in\nTable \\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{0}$ with $Z_2\\times Z_2$ fusion) split\ninto two different entries if we include the $R$-tensors. Each give rise to two\nmodular extensions, and this is why we got four modular extensions. In\nTable \\ref{mextZ2b}, the first two modular extensions have the same\n$(N^{ij}_k,s_i)$, $F$-tensor and $R$-tensors when restricted to the first 5\nparticles. \nThe second pair of modular extensions also have the same $(N^{ij}_k,s_i)$,\n$F$-tensor and $R$-tensor when restricted to the first 5 particles, but their\n$R$-tensor is different from that of the first pair.\nHowever, note that under the exchange of the two fermions, the $R$-tensor of\nthe first pair becomes that of the second pair.\n\n\nWe like to stress that Table \\ref{mextZ2b} is obtained using the\nME-equivalence relation, \\ie the different entries are different under the\nME-equivalence relation (see Section \\ref{clGQL2}). We see that for each fixed\n$\\mce{\\Rp(Z_2)}$ (\\ie for each fixed set of $(N^{ij}_k,s_i)$, $F$-tensor and\n$R$-tensor), there are two modular extensions, which agrees with our general result\nfor modular extensions. However, if we ignore $F$-tensor and $R$-tensor, then\nfor each fixed set of $(N^{ij}_k,s_i)$, we get four modular extensions. This\nis because $(N^{ij}_k,s_i)$ is only a partial description of a\n$\\mce{\\Rp(Z_2)}$, and\nas discussed above, in this case there are two ways to assign $F$-tensor and $R$-tensor to\nthem.\nThis is why each fixed $(N^{ij}_k,s_i)$ has four modular extensions, while\neach fixed $(N^{ij}_k,s_i,F,R)$ has only two modular extensions.\n\nOn the other hand, under the TO-equivalence relation (see\nSection \\ref{clGQL2}),\nthe two ways to assign $F$-tensor and $R$-tensor are actually equivalent\n(related by exchanging the two fermions), and the first entry in\nTable \\ref{SETZ2-5} corresponds to only one $\\mce{\\Rp(Z_2)}$. Thus,\nthe first entry is equivalent to the third entry, and\nthe second entry is equivalent to the fourth entry in Table \\ref{mextZ2b}.\nSo the four entries of Table \\ref{mextZ2b} in fact represent only two distinct\n$Z_2$-SET orders.\n\nOne of the two $Z_2$-SET orders have been studied extensively. It corresponds\nto $Z_2$ gauge theory with a $\\Z_2$ global symmetry that exchanges the\n$Z_2$-gauge-charge $e$ and the $Z_2$-gauge-vortex $m$\\cite{W0303,KLW0834}. \n\n\\subsubsection{The second entry in Table \\ref{SETZ2-5}}\n\nNext, we compute the modular extensions of the second entry in Table\n\\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{0}$ with\n$Z_4$ fusion). Again, we can use the same argument to show that modular\nextensions of rank 12 and above do not exist. We searched the modular\nextensions with $N$ up to 11, and find that there is no modular extensions.\nSo the second entry is not realizable and does not correspond to any valid\nbosonic $Z_2$-SET in 2+1D. This is indicated by NR in the comment column of\nTable \\ref{SETZ2-5}.\n\nNaively, the (none existing) state from the second entry is very similar to\nthat from the first entry. It is also a $Z_2$ gauge theory with a $Z_2$ global\nsymmetry that exchange $e$ and $m$. However, for the second entry, the $f$\nparticles (the third and the fourth particles) are assigned fraction\n$Z_2$-charge of $\\pm 1\/2$. This leads to the $Z_4$ fusion rule. Our result\nimplies that such an assignment is not realizable (or is illegal).\nIt turns out that all the $5^{\\zeta_2^1}_{c}$'s with $Z_4$ fusion do not have\nmodular extensions. They are not realizable, and do not correspond to any 2+1D\nbosonic $Z_2$-SET orders.\n\n\\subsubsection{The third entry in Table \\ref{SETZ2-5}}\n\nThird, let us compute the modular extensions of the third entry in Table\n\\ref{SETZ2-5} (\\ie $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion). We find\nthat the entry has four modular extensions. In fact, the entry corresponds to\ntwo different $\\mce{\\Rp(Z_2)}$s, each with two modular extensions, as implied\nby the two $Z_2$-SPT states. The two $\\mce{\\Rp(Z_2)}$s have identical\n$(N^{ij}_k,s_i,c)$, but different $F$-tensors and $R$-tensors. Sometimes two\ndifferent $\\mce{\\cE}$'s (with different $F$-tensors and the $R$-tensors) can\nhave the same $(N^{ij}_k,s_i)$'s. The third, seventh,\\dots, entries of Table\n\\ref{SETZ2-5} provide such examples. We like to stress that this is different\nfrom the first entry in Table \\ref{SETZ2-5} which corresponds to one\n$\\mce{\\Rp(Z_2)}$.\n\nTo see those different $F$-tensors and $R$-tensors, we note that one of the two\n$5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion has modular extensions given by\n$3^B_{1\/2}\\boxtimes 3^B_{1\/2}$ and $3^B_{-3\/2}\\boxtimes 3^B_{5\/2}$. We find the\n$R$-tensor for this first $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion is\ngiven by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =-\\ii,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =-\\ii.\n\\end{align}\nThe second $5^{\\zeta_2^1}_{1}$ with $Z_2\\times Z_2$ fusion has modular\nextensions given by $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes\n3^B_{-1\/2}$. We find the $R$-tensor for the second $5^{\\zeta_2^1}_{1}$ with\n$Z_2\\times Z_2$ fusion is given by\n\\begin{align}\n R^{(f,1),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (f,1)}_{( \\sigma , \\sigma')} =\\ii,\n\\nonumber\\\\\n R^{(1,f'),( \\sigma , \\sigma')}_{( \\sigma , \\sigma')}\n&= R^{( \\sigma , \\sigma'), (1,f')}_{( \\sigma , \\sigma')} =\\ii.\n\\end{align}\nWe see that the two $5^{\\zeta_2^1}_{1}$'s with $Z_2\\times Z_2$ fusion are\nreally different $\\mce{\\Rp(Z_2)}$. Each $5^{\\zeta_2^1}_{1}$ has two modular\nextensions, and that is why we have four entries in Table \\ref{mextZ2c}.\n\nAgain, Table \\ref{mextZ2c} is obtained using the ME-equivalence relation,\nand is not a table of GQLs. Under the TO-equivalence relation, the third\nentry is equivalent to the fourth entry of Table \\ref{mextZ2c}. So the four\nentries in Table \\ref{mextZ2c} actually describe \\emph{three} different\n$Z_2$-SET orders. This has a very interesting consequence: \\emph{The $Z_2$-SET\nstate described by the third (or fourth) entry in \\ref{mextZ2c}, after stacked\nwith an $Z_2$-SPT state, still remains in the same phase.} This is an example\nof the following general statement made previously: \\emph{The GQLs with bulk\nexcitations described by $\\cC$ are in one-to-one correspondence with the\nquotient $\\mathcal{M}_{ext}(\\cC)\/\\mathrm{Aut}(\\cC)$ plus a central charge $c$.} \nIn such an example $\\mathrm{Aut}(\\cC)$ is non-trivial.\n\nIt is worth noting here that for the second $5^{\\zeta_2^1}_{1}$, two modular\nextensions $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes 3^B_{-1\/2}$\nare actually equivalent UMTCs. This is an example that different embedings\nleads to different modular extensions. For $3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ the\nfirst fermion in $5^{\\zeta_2^1}_{1}$ is embedded into $3^B_{-1\/2}$ and the\nsecond fermion is embedded into $3^B_{3\/2}$, while for $3^B_{3\/2}\\boxtimes\n3^B_{-1\/2}$ the first fermion is embedded into $3^B_{3\/2}$ and the second\nfermion is embedded into $3^B_{-1\/2}$. The equivalence between\n$3^B_{-1\/2}\\boxtimes 3^B_{3\/2}$ and $3^B_{3\/2}\\boxtimes 3^B_{-1\/2}$ that\nexchanges both fermions and symmetry twists fails to relate the two\nembeddings, as they differ by a non-trivial automorphism of $5^{\\zeta_2^1}_{1}$ that exchanges only the two fermions. This is an\nexample that the $\\mathrm{Aut}(\\cC)$ action permutes the modular extensions, as discussed in Section \\ref{clGQL}.\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe three modular extensions of $\\Rp(Z_3)$.\n} \n\\label{mextZ3} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$3^{\\zeta_{4}^{1}}_{ 0}$ & $3$ & $1, 1, 1$ & $0, 0, 0$ & $\\Rp(Z_3)$ \\\\\n\\hline\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, 0, 0, \\frac{1}{3}, \\frac{1}{3}, \\frac{2}{3}, \\frac{2}{3}$ & $Z_3$ gauge\\\\\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, \\frac{1}{9}, \\frac{1}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{7}{9}, \\frac{7}{9}$ & \\\\\n$9^{ B}_{ 0}$ & $9$ & $1\\times 9$ & $0, 0, 0, \\frac{2}{9}, \\frac{2}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{8}{9}, \\frac{8}{9}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe six modular extensions of $\\Rp(S_3)$.\n} \n\\label{mextS3} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$3^{\\sqrt{6}}_{ 0}$ & $6$ & $1, 1, 2$ & $0, 0, 0$ & $\\Rp(S_3)$ \\\\\n\\hline\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, 0, \\frac{1}{3}, \\frac{2}{3}, 0, \\frac{1}{2}$ & $S_3$ gauge\\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, 0, \\frac{1}{3}, \\frac{2}{3}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{1}{9}, \\frac{4}{9}, \\frac{7}{9}, 0, \\frac{1}{2}$ & $(B_4,2)$ \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{1}{9}, \\frac{4}{9}, \\frac{7}{9}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{2}{9}, \\frac{5}{9}, \\frac{8}{9}, 0, \\frac{1}{2}$ & $(B_4,-2)$ \\\\\n$8^{ B}_{ 0}$ & $36$ & $1, 1, 2, 2, 2, 2, 3, 3$ & $0, 0, 0, \\frac{2}{9}, \\frac{5}{9}, \\frac{8}{9}, \\frac{1}{4}, \\frac{3}{4}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\\subsection{$Z_3$, $Z_5$, and $S_3$ SPT orders for bosonic systems}\n\nWe also find that $\\Rp(Z_3)$ has 3 modular extensions (see Table \\ref{mextZ3}),\n$\\Rp(Z_5)$ has 5 modular extensions (see Table \\ref{mextZ5}), and $\\Rp(S_3)$\nhas 6 modular extensions (see Table \\ref{mextS3}). They correspond to the 3\n$Z_3$-SPT states, the 5 $Z_5$-SPT states and the 6 $S_3$-SPT states\nrespectively. These results agree with those from group cohomology\ntheory\\cite{CGL1314}.\n\nWe note that for $\\Rp(Z_2)$, $\\Rp(Z_3)$, and $\\Rp(S_3)$, their modular\nextensions all correspond to distinct UMTCs. However, for $\\Rp(Z_5)$, its 5\nmodular extensions only correspond to 3 distinct UMTCs. $\\Rp(Z_5)$ has 5\nmodular extensions because $\\Rp(Z_5)$ can be embedded into the same UMTC in\ndifferent ways. The different embeddings correspond to different modular\nextensions.\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table}[t] \n\\caption{\nThe 16 modular extensions of $\\sRp(Z_2^f)$. \n} \n\\label{mextZ2f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$2^{ 0}_{0}$ & $2$ & $1, 1$ & $0, \\frac{1}{2}$ & $\\sRp(Z_2^f)$ \\\\\n\\hline\n$4^{ B}_{ 0}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, 0, 0$ & $Z_2$ gauge\\\\\n\\hline\n$4^{ B}_{ 1}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{ 2}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{ 3}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{ 4}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{-3}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & F:$Z_4$ \\\\\n$4^{ B}_{-2}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & F:$Z_2\\times Z_2$ \\\\\n$4^{ B}_{-1}$ & $4$ & $1, 1, 1, 1$ & $0, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & F:$Z_4$ \\\\\n$3^{ B}_{ 1\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{1}{16}$ & $p+\\ii p$ SC\\\\\n$3^{ B}_{ 3\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{3}{16}$ & \\\\\n$3^{ B}_{ 5\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{5}{16}$ & \\\\\n$3^{ B}_{ 7\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{7}{16}$ & \\\\\n$3^{ B}_{-7\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{9}{16}$ & \\\\\n$3^{ B}_{-5\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{11}{16}$ & \\\\\n$3^{ B}_{-3\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{13}{16}$ & \\\\\n$3^{ B}_{-1\/2}$ & $4$ & $1, 1,\\zeta_{2}^{1}$ & $0, \\frac{1}{2}, \\frac{15}{16}$ & \\\\\n \\hline \n\\end{tabular} \n\\end{table}\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe five modular extensions of $\\Rp(Z_5)$.\n} \n\\label{mextZ5} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n \\hline \n$5^{\\sqrt{5}}_{ 0}$ & $5$ & $1\\times 5$ & $0, 0, 0, 0, 0$ & \\\\\n\\hline\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{1}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{2}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{3}{5}, \\frac{4}{5}, \\frac{4}{5}, \\frac{4}{5}, \\frac{4}{5}$ & $5^{ B}_{ 0}\\boxtimes 5^{ B}_{ 0}$\\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{1}{25}, \\frac{1}{25}, \\frac{4}{25}, \\frac{4}{25}, \\frac{6}{25}, \\frac{6}{25}, \\frac{9}{25}, \\frac{9}{25}, \\frac{11}{25}, \\frac{11}{25}, \\frac{14}{25}, \\frac{14}{25}, \\frac{16}{25}, \\frac{16}{25}, \\frac{19}{25}, \\frac{19}{25}, \\frac{21}{25}, \\frac{21}{25}, \\frac{24}{25}, \\frac{24}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{1}{25}, \\frac{1}{25}, \\frac{4}{25}, \\frac{4}{25}, \\frac{6}{25}, \\frac{6}{25}, \\frac{9}{25}, \\frac{9}{25}, \\frac{11}{25}, \\frac{11}{25}, \\frac{14}{25}, \\frac{14}{25}, \\frac{16}{25}, \\frac{16}{25}, \\frac{19}{25}, \\frac{19}{25}, \\frac{21}{25}, \\frac{21}{25}, \\frac{24}{25}, \\frac{24}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{2}{25}, \\frac{2}{25}, \\frac{3}{25}, \\frac{3}{25}, \\frac{7}{25}, \\frac{7}{25}, \\frac{8}{25}, \\frac{8}{25}, \\frac{12}{25}, \\frac{12}{25}, \\frac{13}{25}, \\frac{13}{25}, \\frac{17}{25}, \\frac{17}{25}, \\frac{18}{25}, \\frac{18}{25}, \\frac{22}{25}, \\frac{22}{25}, \\frac{23}{25}, \\frac{23}{25}$ & \\\\\n$25^{ B}_{ 0}$ & $25$ & $1\\times 25$ & $0, 0, 0, 0, 0, \\frac{2}{25}, \\frac{2}{25}, \\frac{3}{25}, \\frac{3}{25}, \\frac{7}{25}, \\frac{7}{25}, \\frac{8}{25}, \\frac{8}{25}, \\frac{12}{25}, \\frac{12}{25}, \\frac{13}{25}, \\frac{13}{25}, \\frac{17}{25}, \\frac{17}{25}, \\frac{18}{25}, \\frac{18}{25}, \\frac{22}{25}, \\frac{22}{25}, \\frac{23}{25}, \\frac{23}{25}$ & \\\\\n\\hline\n\\end{tabular} \n\\end{table*}\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 8 modular extensions of $\\sRp(Z_4^f)$.\n} \n\\label{mextZ4f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & \n$\\sRp(Z_4^f)$ \\\\\n\\hline\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & \\\\\n\\hline \n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{32}, \\frac{1}{32}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{9}{32}, \\frac{9}{32}, \\frac{17}{32}, \\frac{17}{32}, \\frac{25}{32}, \\frac{25}{32}$ & \\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{32}, \\frac{3}{32}, \\frac{11}{32}, \\frac{11}{32}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{19}{32}, \\frac{19}{32}, \\frac{27}{32}, \\frac{27}{32}$ & \\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{32}, \\frac{5}{32}, \\frac{13}{32}, \\frac{13}{32}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{21}{32}, \\frac{21}{32}, \\frac{29}{32}, \\frac{29}{32}$ & \\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{15}{16}, \\frac{15}{16}$ & $8^{ B}_{-1}\\boxtimes 2^{ B}_{-1}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{32}, \\frac{7}{32}, \\frac{15}{32}, \\frac{15}{32}, \\frac{23}{32}, \\frac{23}{32}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{31}{32}, \\frac{31}{32}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe two $c=0$ modular extensions of $\\sRp(Z_8^f)$ imply that\nthe $Z_8^f$ fermionic SPT phases are described by $\\Z_2$.\nAll other modular extensions only appear for integer $c$ and are all abelian (two modular extensions\nfor each integer $c$).\n} \n\\label{mextZ8f} \n\\centering\n\\begin{tabular}{ |c|c|l|p{5.3in}|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$8^{ 0}_{0}$ & $8$ & $1\\times 8$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}$ & \\\\\n\\hline\n$64^{ B}_{ 0}$ & $64$ & $1\\times 64$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}$, $\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$& \\\\\n$64^{ B}_{ 0}$ & $64$ & $1\\times 64$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}$, $\\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\n\n\\subsection{Invertible fermionic topological orders}\n\nWe find that $\\sRp(Z_2^f)$ has 16 modular extensions (see Table \\ref{mextZ2f})\nwhich correspond to invertible fermionic topological orders in 2+1D. One might\nthought that the invertible fermionic topological orders are classified by\n$\\Z_{16}$. But in fact, the invertible fermionic topological orders are\nclassified by $\\Z$, obtained by stacking the $c=1\/2$ $p+\\ii p$ states. The\ndiscrepancy is due to the fact that the modular extensions cannot see the $c=8$\n$E_8$ states. The 16 modular extensions exactly correspond to the invertible\nfermionic topological orders modulo the $E_8$ states. \n\nWe also find that the modular extensions with $c=$ even have a\n$Z_2\\times Z_2$ fusion rule, while the modular extensions with $c=$ odd have a\n$Z_4$ fusion rule (indicated by F:$Z_2\\times Z_2$ or F:$Z_4$ in the comment\ncolumn of Table).\n\nThe $Z_2^f$-SPT states for fermions is given by the modular extensions with\nzero central charge. We see that there is only one modular extension with\ncentral charge $c=0$. Thus there is no non-trivial 2+1D fermionic SPT states\nwith $Z_2^f$ symmetry. In general, the modular extensions of $\\sRp(G^f)$ with\nzero central charge correspond to the fermionic SPT states in 2+1D with\nsymmetry $G^f$.\n\n\n\nTo calculate the $Z_2\\times Z_2^f$ SPT orders for fermionic systems, we first\ncompute the modular extensions for $\\sRp(Z_2\\times Z_2^f)$. We note that\n$\\sRp(Z_2\\times Z_2^f)=\\sRp(Z_2^f\\times \\tilde Z_2^f)$. Thus, the modular\nextensions for $\\sRp(Z_2\\times Z_2^f)$ is the modular extensions of\n$\\sRp(Z_2^f\\times \\tilde Z_2^f)$. Some of the modular extensions\nof $\\sRp(Z_2^f\\times \\tilde Z_2^f)$ are given by the modular extensions of\n$\\sRp(Z_2^f)$ stacked (under $\\boxtimes$) with the modular extensions of\n$\\sRp(\\tilde Z_2^f)$. Some of the modular extensions of $\\sRp(Z_2\\times\nZ_2^f)$ are given by the modular extensions for $\\Rp(Z_2)$ stacked (under\n$\\boxtimes$) with the modular extensions of $\\sRp(Z_2^f)$.\n\nThe above mathematical statements correspond to the following physical picture:\nSome fermionic GQLs with $Z_2\\times Z_2^f$ symmetry can be viewed as bosonic\nGQLs with $Z_2$ symmetry stacked with fermionic GQLs with $Z_2^f$ symmetry.\nAlso some fermionic GQLs with $Z_2^f\\times \\tilde Z_2^f$ symmetry can be viewed\nas fermionic GQLs with $Z_2^f$ symmetry stacked with fermionic GQLs with\n$\\tilde Z_2^f$ symmetry.\n\nUsing \\eqn{Hgcnd}, we find that the modular extensions for $Z_2\\times Z_2^f$\nsymmetry must have ranks $7, 9, 10, 12, 16$. By direct search for those ranks,\nwe find that the modular extensions of $\\sRp(Z_2\\times Z_2^f)$ are given by\nTables \\ref{mextZ2Z2f}, \\ref{mextZ2Z2f12}, \\ref{mextZ2Z2f12b} and\n\\ref{mextZ2Z2f16}. The $N=9$ modular extensions of $\\sRp(Z_2\\times Z_2^f)$ in\nTable \\ref{mextZ2Z2f} are given by the stacking of the $N=3$ modular extensions\nof $\\sRp(Z_2^f)$ and the $N=3$ modular extensions of $\\sRp(\\tilde Z_2^f)$. The\n$N=16$ modular extensions of $\\sRp(Z_2\\times Z_2^f)$ in Table \\ref{mextZ2Z2f16}\nare given by the stacking of the $N=4$ modular extensions of $\\sRp(Z_2^f)$ and\nthe $N=4$ modular extensions of $\\sRp(\\tilde Z_2^f)$. There are also 64 $N=12$\nmodular extensions of $\\sRp(Z_2\\times Z_2^f)$ given by the stacking of the\n$N=4$ ($N=3$) modular extensions of $\\sRp(Z_2^f)$ and the $N=3$ ($N=4$) modular\nextensions of $\\sRp(\\tilde Z_2^f)$.\n\n\nMany of the modular extensions have non-trivial topological orders since the\ncentral charge $c$ is non-zero. There are eight modular extensions for each central\ncharge $c=0,1\/2,1,3\/2,\\dots,15\/2$, and in total $8\\times 16=128$ modular\nextensions. Those eight with $c=0$ correspond\nto the $Z_2\\times Z_2^f$ fermionic SPT states. Those are all the $Z_2\\times\nZ_2^f$ fermionic SPT states\\cite{GL1369}.\n\n\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N = 9$.\n} \n\\label{mextZ2Z2f} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, 0$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, 0$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, 0$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, 0$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n\\hline \n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{1}{8}$ & $3^{ B}_{ 1\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{1}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{1}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{1}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{1}{4}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{1}{4}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{1}{4}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{1}{4}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{3}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{3}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{3}{8}$ & $3^{ B}_{ 3\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{3}{8}$ & $3^{ B}_{-1\/2}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{1}{2}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{7}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{1}{2}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{1}{2}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{5}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{1}{2}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{5}{8}$ & $3^{ B}_{-7\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{5}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{7}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{5}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-3}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{5}{8}$ & $3^{ B}_{ 5\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{3}{4}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{3}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{3}{4}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{3}{4}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-2}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{7}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{3}{4}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{7}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{5}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{7}{8}$ & $3^{ B}_{-3\/2}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{7}{8}$ & $3^{ B}_{-5\/2}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$9^{ B}_{-1}$ & $16$ & $1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}, 2$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{7}{8}$ & $3^{ B}_{ 7\/2}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\n\n\\subsection{$Z_{2n}^f$ SPT orders for fermionic systems}\n\nWe also find the modular extensions for $\\sRp(Z_4^f)$, $\\sRp(Z_6^f)$, and\n$\\sRp(Z_8^f)$ (see Tables \\ref{mextZ4f}, \\ref{mextZ6f}, and \\ref{mextZ8f}).\nAgain, many of them has non-trivial topological orders since the central charge\n$c$ is non-zero. \n\nFor $Z_4^f$ group, only one of them have $c=0$. So there is no non-trivial\n$Z_4^f$ fermionic SPT states. For $Z_6^f$ group, only three of them have\n$c=0$. So, the $Z_6^f$ fermionic SPT states are described by $\\Z_3$. For\n$Z_8^f$ group, only two of them have $c=0$. So, the $Z_8^f$ fermionic SPT\nstates are described by $\\Z_2$. Those results are consistent with the results\nin \\Ref{KTT1429,CJWang}. However, the calculation present here is more\ncomplete.\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe first 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =12$.\n} \n\\label{mextZ2Z2f12} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{9}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{1}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $6^{ B}_{-1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{1}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $6^{ B}_{-1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{1}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{11}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{9}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{9}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $6^{ B}_{ 1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{3}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $6^{ B}_{ 1\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{3}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{3}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{13}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{11}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{11}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}$ & $6^{ B}_{ 3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{9}{16}$ & $6^{ B}_{ 3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{5}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{5}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{15}{16}$ & $4^{ B}_{ 0}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{13}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{13}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $6^{ B}_{ 5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{11}{16}$ & $6^{ B}_{ 5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{9}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{ 7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{9}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\section{Summary}\n\\label{sum}\n\nGQLs contain both topologically ordered states and SPT states. In this paper,\nwe present a theory that classify GQLs in 2+1D for bosonic\/fermionic systems\nwith symmetry.\n\nWe propose that the possible non-abelian statistics (or sets of bulk\nquasiparticles excitations) in 2+1D GQLs are classified by \n$\\mce{\\cE}$, where $\\cE=\\Rp(G)$ or $\\sRp(G^f)$ describing the symmetry in\nbosonic or fermionic systems. However, $\\mce{\\cE}$'s fail to\nclassify GQLs, since different GQL phases can have identical non-abelian\nstatistics, which correspond to identical $\\mce{\\cE}$. \n\nTo fix this problem, we introduce the notion of modular extensions for a\n$\\mce{\\cE}$. We propose to use the triple $(\\cC,\\cM,c)$ to\nclassify 2+1D GQLs with symmetry $G$ (for boson) or $G^f$ (for fermion). Here\n$\\cC$ is a $\\mce{\\cE}$ with $\\cE=\\Rp(G)$ or $\\sRp(G^f)$, $\\cM$ is a modular\nextension of $\\cC$ and $c$ is the chiral central charge of the edge state. We\nshow that the modular extensions of a $\\mce{\\cE}$ has a one-to-one\ncorrespondence with the modular extensions of $\\cE$. So the number of the\nmodular extensions is solely determined by the symmetry $\\cE$. Also, the $c=0$\nmodular extensions of a $\\cE$ ($\\cE=\\Rp(G)$ or $\\sRp(G^f)$) classify the 2+1D\nSPT states for bosons or fermions with symmetry $G$ or $G^f$.\n\nAlthough the above result has a nice mathematical structure, it is hard to\nimplement numerically to produce a table of GQLs. To fix this problem, we\npropose a different description of 2+1D GQLs. We propose to use the data $(\n\\tilde N^{ab}_c,\\tilde s_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$, up to some\npermutations of the indices, to describe 2+1D GQLs with symmetry $G$\n(for boson) or $G^f$ (for fermion), with a restriction that the symmetry group\n$G$ can be fully characterized by the fusion ring of its irreducible\nrepresentations (for example, for simple groups or abelian groups). Here the\ndata $(\\tilde N^{ab}_c,\\tilde s_a)$ describe the symmetry and the data\n$(N^{ij}_k,s_i)$ describes fusion and the spins of the bulk particles in the\nGQL. The modular extensions are obtained by ``gauging'' the symmetry $G$ or\n$G^f$. The data $(\\cN^{IJ}_K,\\cS_I)$ describes fusion and the spins of the\nbulk particles in the ``gauged'' theory. Last, $c$ is the chiral central\ncharge of the edge state.\n\nIn this paper (see Appendix \\ref{cnds}) and in \\Ref{W150605768}, we list the\nnecessary and the sufficient conditions on the data $(\\tilde N^{ab}_c,\\tilde\ns_a; N^{ij}_k,s_i; \\cN^{IJ}_K,\\cS_I;c)$, which allow us to obtain a\nlist of GQLs. However, in this paper, we did not give the list\nof GQLs directly. We first give a list of $(\\tilde N^{ab}_c,\\tilde s_a;\nN^{ij}_k,s_i)$, which is an imperfect list of $\\mce{\\cE}$'s. We then compute\nthe modular extensions $(\\cN^{IJ}_K,\\cS_I;c)$ for each entry $(\\tilde\nN^{ab}_c,\\tilde s_a; N^{ij}_k,s_i)$, which allows us to obtain a perfect list\nof GQLs (for certain symmetry groups). As a special case, we calculated the\nbosonic\/fermionic SPT states for some groups in 2+1D.\n\nIn \\Ref{LW160205936}, we will give a more mathematical description of our\ntheory. Certainly we hope to generalize the above framework to higher\ndimensions. We also hope to develop more efficient numerical codes to obtain\nbigger tables of GQLs.\n\n\\bigskip\n\n\\noindent {\\bf Acknowledgement}: \nWe like to thank Pavel Etingof, Dmitri Nikshych, Chenjie Wang, and Zhenghan\nWang for many helpful discussions. This research is supported by NSF Grant No.\nDMR-1506475, and NSFC 11274192. It is also supported by the John Templeton\nFoundation No. 39901. Research at Perimeter Institute is supported by the\nGovernment of Canada through Industry Canada and by the Province of Ontario\nthrough the Ministry of Research. LK is supported by the Center of\nMathematical Sciences and Applications at Harvard University.\n\n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nThe second 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =12$.\n} \n\\label{mextZ2Z2f12b} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{15}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{15}{16}$ & $4^{ B}_{ 1}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}$ & $6^{ B}_{ 7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{13}{16}$ & $6^{ B}_{ 7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-7\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{3}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{11}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{11}{16}, \\frac{11}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{15}{16}$ & $6^{ B}_{-7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{15}{16}$ & $6^{ B}_{-7\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{13}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-5\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{5}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{13}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{16}, \\frac{11}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 3\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $6^{ B}_{-5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{1}{16}, \\frac{9}{16}, \\frac{13}{16}, \\frac{13}{16}$ & $6^{ B}_{-5\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-3\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{15}{16}$ & $4^{ B}_{ 3}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{ 4}\\boxtimes 3^{ B}_{ 7\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{16}, \\frac{13}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-3}\\boxtimes 3^{ B}_{ 5\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $6^{ B}_{-3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{16}, \\frac{11}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $6^{ B}_{-3\/2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n$12^{ B}_{-1\/2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1,\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1},\\zeta_{2}^{1}$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{1}{16}, \\frac{9}{16}, \\frac{15}{16}, \\frac{15}{16}$ & $4^{ B}_{-1}\\boxtimes 3^{ B}_{ 1\/2}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the 32 modular extensions of $\\sRp(Z_2\\times Z_2^f)$ with $N =16$.\n} \n\\label{mextZ2Z2f16} \n\\centering\n\\begin{tabular}{ |c|c|l|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ & comment \\\\\n\\hline \n$4^{ 0}_{0}$ & $4$ & $1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}$ & $\\sRp(Z_2\\times Z_2^f)$ \\\\\n\\hline\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & $4^{ B}_{ 0}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 0}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-1}\\boxtimes 2^{ B}_{ 1}$\\\\\n\\hline\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 0}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 0}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 1}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{ 1}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 3}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, 0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}$ & $4^{ B}_{ 4}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{ 4}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{ 3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 4}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-3}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{4}, \\frac{3}{8}, \\frac{3}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{ 4}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $8^{ B}_{-3}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}$ & $4^{ B}_{-3}\\boxtimes 4^{ B}_{ 1}$\\\\\n$16^{ B}_{-2}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{ 3}\\boxtimes 4^{ B}_{ 3}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, 0, 0, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $4^{ B}_{-1}\\boxtimes 4^{ B}_{ 0}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{-2}\\boxtimes 2^{ B}_{ 1}$\\\\\n$16^{ B}_{-1}$ & $16$ & $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ & $0, 0, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{8}, \\frac{5}{8}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ & $8^{ B}_{-2}\\boxtimes 2^{ B}_{ 1}$\\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\n\\def1.25} \\setlength\\tabcolsep{3pt{1.25} \\setlength\\tabcolsep{3pt}\n\\begin{table*}[t] \n\\caption{\nAll the modular extensions of $\\sRp(Z_6^f)=\\sRp(Z_3\\times Z_2^f)$.\n} \n\\label{mextZ6f} \n\\centering\n\\begin{tabular}{ |c|c|l|l| } \n\\hline \n$N^{|\\Th|}_{c}$ & $D^2$ & $d_1,d_2,\\cdots$ & $s_1,s_2,\\cdots$ \\\\\n\\hline \n$6^{ 0}_{0}$ & $6$ & $1, 1, 1, 1, 1, 1$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}$ \\hfill $\\sRp(Z_6^f)$\\\\\n\\hline\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}$ \\\\\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{ 0}$ & $36$ & $1\\times 36$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, 0, 0, 0, 0, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n\\hline\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{11}{24}, \\frac{11}{24}, \\frac{11}{24}, \\frac{11}{24}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{19}{24}, \\frac{19}{24}, \\frac{19}{24}, \\frac{19}{24}, \\frac{5}{6}, \\frac{5}{6}$ \\\\\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{72}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{2}{9}, \\frac{2}{9}, \\frac{25}{72}, \\frac{25}{72}, \\frac{25}{72}, \\frac{25}{72}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{49}{72}, \\frac{49}{72}, \\frac{49}{72}, \\frac{49}{72}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{ 1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{1}{8}, \\frac{17}{72}, \\frac{17}{72}, \\frac{17}{72}, \\frac{17}{72}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{41}{72}, \\frac{41}{72}, \\frac{41}{72}, \\frac{41}{72}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{65}{72}, \\frac{65}{72}, \\frac{65}{72}, \\frac{65}{72}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{7}{12}, \\frac{7}{12}, \\frac{7}{12}, \\frac{7}{12}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{11}{12}, \\frac{11}{12}, \\frac{11}{12}, \\frac{11}{12}$ \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{36}, \\frac{1}{9}, \\frac{1}{9}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{5}{18}, \\frac{5}{18}, \\frac{13}{36}, \\frac{13}{36}, \\frac{13}{36}, \\frac{13}{36}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{25}{36}, \\frac{25}{36}, \\frac{25}{36}, \\frac{25}{36}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{ 2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{5}{36}, \\frac{5}{36}, \\frac{5}{36}, \\frac{5}{36}, \\frac{2}{9}, \\frac{2}{9}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{7}{18}, \\frac{7}{18}, \\frac{17}{36}, \\frac{17}{36}, \\frac{17}{36}, \\frac{17}{36}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{29}{36}, \\frac{29}{36}, \\frac{29}{36}, \\frac{29}{36}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{24}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{17}{24}, \\frac{17}{24}, \\frac{17}{24}, \\frac{17}{24}, \\frac{5}{6}, \\frac{5}{6}$ \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{19}{72}, \\frac{19}{72}, \\frac{19}{72}, \\frac{19}{72}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{43}{72}, \\frac{43}{72}, \\frac{43}{72}, \\frac{43}{72}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{67}{72}, \\frac{67}{72}, \\frac{67}{72}, \\frac{67}{72}$ \\\\\n$36^{ B}_{ 3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{11}{72}, \\frac{11}{72}, \\frac{11}{72}, \\frac{11}{72}, \\frac{5}{18}, \\frac{5}{18}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{3}{8}, \\frac{4}{9}, \\frac{4}{9}, \\frac{35}{72}, \\frac{35}{72}, \\frac{35}{72}, \\frac{35}{72}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{59}{72}, \\frac{59}{72}, \\frac{59}{72}, \\frac{59}{72}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{6}$ \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{7}{18}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{ 4}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{7}{24}, \\frac{7}{24}, \\frac{7}{24}, \\frac{7}{24}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{23}{24}, \\frac{23}{24}, \\frac{23}{24}, \\frac{23}{24}$ \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{13}{72}, \\frac{13}{72}, \\frac{13}{72}, \\frac{13}{72}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{37}{72}, \\frac{37}{72}, \\frac{37}{72}, \\frac{37}{72}, \\frac{5}{9}, \\frac{5}{9}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{13}{18}, \\frac{13}{18}, \\frac{61}{72}, \\frac{61}{72}, \\frac{61}{72}, \\frac{61}{72}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{-3}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{5}{72}, \\frac{5}{72}, \\frac{5}{72}, \\frac{5}{72}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{29}{72}, \\frac{29}{72}, \\frac{29}{72}, \\frac{29}{72}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{5}{8}, \\frac{53}{72}, \\frac{53}{72}, \\frac{53}{72}, \\frac{53}{72}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{12}, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{5}{12}, \\frac{5}{12}, \\frac{5}{12}, \\frac{5}{12}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{5}{6}, \\frac{5}{6}$ \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{11}{36}, \\frac{11}{36}, \\frac{11}{36}, \\frac{11}{36}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{23}{36}, \\frac{23}{36}, \\frac{23}{36}, \\frac{23}{36}, \\frac{13}{18}, \\frac{13}{18}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{8}{9}, \\frac{8}{9}, \\frac{35}{36}, \\frac{35}{36}, \\frac{35}{36}, \\frac{35}{36}$ \\\\\n$36^{ B}_{-2}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{7}{36}, \\frac{7}{36}, \\frac{7}{36}, \\frac{7}{36}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{19}{36}, \\frac{19}{36}, \\frac{19}{36}, \\frac{19}{36}, \\frac{11}{18}, \\frac{11}{18}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{3}{4}, \\frac{7}{9}, \\frac{7}{9}, \\frac{31}{36}, \\frac{31}{36}, \\frac{31}{36}, \\frac{31}{36}, \\frac{17}{18}, \\frac{17}{18}$ \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{5}{24}, \\frac{5}{24}, \\frac{5}{24}, \\frac{5}{24}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{13}{24}, \\frac{13}{24}, \\frac{13}{24}, \\frac{13}{24}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}$ \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{7}{72}, \\frac{7}{72}, \\frac{7}{72}, \\frac{7}{72}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{31}{72}, \\frac{31}{72}, \\frac{31}{72}, \\frac{31}{72}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{55}{72}, \\frac{55}{72}, \\frac{55}{72}, \\frac{55}{72}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{8}{9}, \\frac{8}{9}$ \\\\\n$36^{ B}_{-1}$ & $36$ & $1\\times 36$ &\\tiny $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{23}{72}, \\frac{23}{72}, \\frac{23}{72}, \\frac{23}{72}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{47}{72}, \\frac{47}{72}, \\frac{47}{72}, \\frac{47}{72}, \\frac{7}{9}, \\frac{7}{9}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{7}{8}, \\frac{17}{18}, \\frac{17}{18}, \\frac{71}{72}, \\frac{71}{72}, \\frac{71}{72}, \\frac{71}{72}$ \\\\\n\\hline\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{19}{48}, \\frac{19}{48}, \\frac{35}{48}, \\frac{35}{48}$ \\\\\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{41}{144}, \\frac{41}{144}, \\frac{89}{144}, \\frac{89}{144}, \\frac{137}{144}, \\frac{137}{144}$ \\\\\n$27^{ B}_{ 1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{1}{16}, \\frac{1}{16}, \\frac{1}{16}, \\frac{25}{144}, \\frac{25}{144}, \\frac{73}{144}, \\frac{73}{144}, \\frac{121}{144}, \\frac{121}{144}$ \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{25}{48}, \\frac{25}{48}, \\frac{41}{48}, \\frac{41}{48}$ \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{11}{144}, \\frac{11}{144}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{59}{144}, \\frac{59}{144}, \\frac{107}{144}, \\frac{107}{144}$ \\\\\n$27^{ B}_{ 3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{3}{16}, \\frac{3}{16}, \\frac{3}{16}, \\frac{43}{144}, \\frac{43}{144}, \\frac{91}{144}, \\frac{91}{144}, \\frac{139}{144}, \\frac{139}{144}$ \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{31}{48}, \\frac{31}{48}, \\frac{47}{48}, \\frac{47}{48}$ \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{29}{144}, \\frac{29}{144}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{77}{144}, \\frac{77}{144}, \\frac{125}{144}, \\frac{125}{144}$ \\\\\n$27^{ B}_{ 5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{13}{144}, \\frac{13}{144}, \\frac{5}{16}, \\frac{5}{16}, \\frac{5}{16}, \\frac{61}{144}, \\frac{61}{144}, \\frac{109}{144}, \\frac{109}{144}$ \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{5}{48}, \\frac{5}{48}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{37}{48}, \\frac{37}{48}$ \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{47}{144}, \\frac{47}{144}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{95}{144}, \\frac{95}{144}, \\frac{143}{144}, \\frac{143}{144}$ \\\\\n$27^{ B}_{ 7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{31}{144}, \\frac{31}{144}, \\frac{7}{16}, \\frac{7}{16}, \\frac{7}{16}, \\frac{79}{144}, \\frac{79}{144}, \\frac{127}{144}, \\frac{127}{144}$ \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{11}{48}, \\frac{11}{48}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{43}{48}, \\frac{43}{48}$ \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{17}{144}, \\frac{17}{144}, \\frac{65}{144}, \\frac{65}{144}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{113}{144}, \\frac{113}{144}$ \\\\\n$27^{ B}_{-7\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{1}{144}, \\frac{1}{144}, \\frac{49}{144}, \\frac{49}{144}, \\frac{9}{16}, \\frac{9}{16}, \\frac{9}{16}, \\frac{97}{144}, \\frac{97}{144}$ \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{1}{48}, \\frac{1}{48}, \\frac{17}{48}, \\frac{17}{48}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}$ \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{35}{144}, \\frac{35}{144}, \\frac{83}{144}, \\frac{83}{144}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{131}{144}, \\frac{131}{144}$ \\\\\n$27^{ B}_{-5\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{19}{144}, \\frac{19}{144}, \\frac{67}{144}, \\frac{67}{144}, \\frac{11}{16}, \\frac{11}{16}, \\frac{11}{16}, \\frac{115}{144}, \\frac{115}{144}$ \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{7}{48}, \\frac{7}{48}, \\frac{23}{48}, \\frac{23}{48}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{5}{144}, \\frac{5}{144}, \\frac{53}{144}, \\frac{53}{144}, \\frac{101}{144}, \\frac{101}{144}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}$ \\\\\n$27^{ B}_{-3\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{37}{144}, \\frac{37}{144}, \\frac{85}{144}, \\frac{85}{144}, \\frac{13}{16}, \\frac{13}{16}, \\frac{13}{16}, \\frac{133}{144}, \\frac{133}{144}$ \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, 0, \\frac{1}{6}, \\frac{1}{6}, \\frac{1}{3}, \\frac{1}{3}, \\frac{1}{2}, \\frac{1}{2}, \\frac{2}{3}, \\frac{2}{3}, \\frac{5}{6}, \\frac{5}{6}, \\frac{13}{48}, \\frac{13}{48}, \\frac{29}{48}, \\frac{29}{48}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{18}, \\frac{1}{18}, \\frac{2}{9}, \\frac{2}{9}, \\frac{7}{18}, \\frac{7}{18}, \\frac{5}{9}, \\frac{5}{9}, \\frac{13}{18}, \\frac{13}{18}, \\frac{8}{9}, \\frac{8}{9}, \\frac{23}{144}, \\frac{23}{144}, \\frac{71}{144}, \\frac{71}{144}, \\frac{119}{144}, \\frac{119}{144}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ \\\\\n$27^{ B}_{-1\/2}$ & $36$ &\\tiny $1\\times 18 , \\zeta_2^1\\times 9$ & $0, \\frac{1}{2}, 0, \\frac{1}{2}, 0, \\frac{1}{2}, \\frac{1}{9}, \\frac{1}{9}, \\frac{5}{18}, \\frac{5}{18}, \\frac{4}{9}, \\frac{4}{9}, \\frac{11}{18}, \\frac{11}{18}, \\frac{7}{9}, \\frac{7}{9}, \\frac{17}{18}, \\frac{17}{18}, \\frac{7}{144}, \\frac{7}{144}, \\frac{55}{144}, \\frac{55}{144}, \\frac{103}{144}, \\frac{103}{144}, \\frac{15}{16}, \\frac{15}{16}, \\frac{15}{16}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*} \n\n\\vfill\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\section{Introduction}\n\\input{introduction.tex}\n\n\\section{Preliminaries} \\label{sec:preliminary}\n\\input{preliminary.tex}\n\n\n\n\\section{Problem Formulation} \\label{sec:problem}\n\\input{problem.tex}\n\n\n\n\n\\section{Formula Synthesis Method}\\label{sec:formula_synthesis}\n\\input{synthesis.tex}\n\n\n\n\\section{Case Study}\\label{sec:case_studies}\n\\input{case.tex}\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\\input{conclusion.tex}\n\n\n\n\n\\subsection{Signals}\nWe define an n-dimensional discrete signal $\\mathbf{x}$ as a mapping from time domain $\\mathbb{N}^+$ to the real numbers $\\mathbb{R}^n$. A finite signal of length $K+1$ is shown as a sequence $\\mathbf{x} = x_0, x_1, ...,x_K$. We use $x_t^i$ to denote the projection of the state on the $i$th dimension at time $t$.\n\n\nA dataset of labeled signals is defined as \n\n\\begin{align}\\label{eq:dataset}\n \\mathcal{D} = \\{ (\\mathbf{x},\\mathbf{l}) \\mid \\mathbf{x} = & x_0, x_1, \\ldots,x_K, \\\\\n \\mathbf{l} = & l_0, l_1, \\ldots, l_K, \\text{ and } \\nonumber \\\\% K \\in \\mathbb{N} \\nonumber \\\\ \n x_t \\in &\\mathbb{R}^n, l_t \\in \\{0,1\\}, t=0,\\ldots,K\\}, \\nonumber\n\\end{align}\nwhere $l_t = 1$, means that at time $t$, an event of interest is occurred on signal $\\mathbf{x}$.\n\n\n\\subsection{Past Time Signal Temporal Logic}\nA Past Time Signal Temporal Logic (ptSTL) formula is defined with grammar:\n\\begin{align}\n\\phi = \\mathbf{T} | x^i\\sim c | \\neg\\phi | \\phi_1 \\wedge \\phi_2 | \\phi_1 \\vee \\phi_2 | \\phi_1 \\mathbf{S}_{[a,b]} \\phi_2 | \\mathbf{P}_{[a,b]} \\phi | \\mathbf{A}_{[a,b]} \\phi \\label{eq:ptstl} \n\\end{align}\nwhere $x^i$ is a signal variable, $\\sim \\in \\{>,<\\}$, and $c$ is a constant, $\\mathbf{T}$ is the Boolean constant $true$, $\\neg, \\wedge$ and $\\vee$ are the standard Boolean operators, $\\mathbf{S}_{[a,b]}$ $(since)$, $\\mathbf{P}_{[a,b]}$ $(previously)$, and $\\mathbf{A}_{[a,b]}$ $(always)$ are the temporal operators with time interval $[a,b]$. The semantics of a ptSTL formula is defined over a signal for a given time point. \n\nInformally, for signal $\\mathbf{x}$, at time $t$, formula $\\mathbf{P}_{[a,b]} \\phi $ is satisfied if $\\phi$ holds at some time in $[t-b,t-a]$, formula $\\mathbf{A}_{[a,b]} \\phi$ is satisfied if $\\phi$ holds everywhere in $[t-b,t-a]$, and $\\phi_1 \\mathbf{S}_{[a,b]} \\phi_2$ is satisfied if $\\phi_2$ holds at some time $t' \\in [t-b,t-a]$ and $\\phi_1$ holds since then. $(\\mathbf{x},t) \\models \\phi $ denotes that signal $\\mathbf{x}$ satisfies formula $\\phi$ at time $t$. Formally, the semantics are given as follows:\n\\begin{align}\n& (\\mathbf{x}, t) \\models \\mathbf{T} & & \\nonumber \\\\\n& (\\mathbf{x}, t) \\models x^i \\sim c &\\text{ iff } & x^i_t \\sim c , \\sim \\in \\{>,<\\}\\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\wedge \\phi_2 & \\text{ iff } & (\\mathbf{x}, t) \\models \\phi_1 \\text{ and } (\\mathbf{x}, t) \\models \\phi_2 \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\vee \\phi_2 & \\text{ iff } & (\\mathbf{x}, t) \\models \\phi_1 \\text{ or } (\\mathbf{x}, t) \\models \\phi_2 \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\mathbf{P}_{[a,b]} \\phi & \\text{ iff } & \\exists t' \\in I(t,[a,b]), (\\mathbf{x}, t') \\models \\phi \\label{eq:semantics} \\\\\n& (\\mathbf{x}, t) \\models \\mathbf{A}_{[a,b]} \\phi & \\text{ iff } & \\forall t' \\in I(t,[a,b]), (\\mathbf{x}, t') \\models \\phi \\nonumber \\\\\n& (\\mathbf{x}, t) \\models \\phi_1 \\mathbf{S}_{[a,b]} \\phi_2 & \\text{ iff } & \\exists t' \\in I(t,[a,b]), (\\mathbf{x}, t') \\models \\phi_2, \\nonumber \\\\\n& & & \\forall t'' \\in [t', t] (\\mathbf{x}, t'') \\models \\phi_1 \\nonumber, \n\\end{align}\n\\[ \\text{ where } I(t,[a,b]) = [t-b, t-a] \\cap [0,t] \\]\n\n\n\nNote that the previously ($\\mathbf{P}$) and always ($\\mathbf{A}$) operators are the special cases of the since operator, $\\mathbf{P}_{[a,b]} \\phi := \\mathbf{T}\\ \\mathbf{S}_{[a,b]} \\phi$ and $\\mathbf{A}_{[a,b]} \\phi := \\neg \\mathbf{P}_{[a,b]} \\neg \\phi$. We include them as they are used in the proposed methods.\n\n\n$Parametric$ $Past$ $Time$ $Signal$ $Temporal$ $Logic$ is an extension of ptSTL~\\citep{Asarin:2011}. In a parametric ptSTL formula, instead of numerical values in time interval bounds and predicates, parameters can be used. A parametric formula can be converted to a ptSTL formula by assigning a value to each parameter.\nAs an example consider the parametric formula $\\phi = \\mathbf{P}_{[p_1,p_2]} x < p_3$ with parameters $p_1, p_2$ and $p_3$. ptSTL formula $\\phi (v) = \\mathbf{P}_{[3,5]} x < 10.2$ is obtained with valuation $v = [3,5,10.2]$.\n \n\\subsection{Monotonicity of Parametric Signal Temporal Logic}\n\n\n\nMonotonicity properties for parametric STL is introduced by~\\cite{miningjournal}. A parametric STL formula $\\phi$ with parameters $[p_1, \\ldots, p_m]$ is $monotonically$ $increasing$ with parameter $p_i$ if (\\ref{eq:mon_inc}) holds along any signal $\\mathbf{x}$. Similarly, it is $monotonically$ $decreasing$ with parameter $p_i$ if (\\ref{eq:mon_dec}) holds.\n\\begin{align}\n&\\text{for all } v,v' \\text{ with } v(p_i) < v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad \\quad \\quad (\\mathbf{x},t) \\models \\phi(v) \\implies (\\mathbf{x},t) \\models \\phi(v') \\label{eq:mon_inc} \\\\\n&\\text{for all } v,v' \\text{ with } v(p_i) > v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad \\quad \\quad (\\mathbf{x},t) \\models \\phi(v) \\implies (\\mathbf{x},t) \\models \\phi(v') \\label{eq:mon_dec}\n\\end{align}\n\nEssentially, $\\phi$ is monotonically increasing with $p_i$ if the valuation can not change from satisfying to violating when only the value of the parameter $p_i$ is increased. \n\n \nOur aim in this work is to generate a ptSTL formula that represents the labels in a dataset~\\eqref{eq:dataset}. For this purpose, we generate label $\\mathbf{l}^\\phi= l^\\phi_0, l^\\phi_1, \\ldots, l^\\phi_N$ from a given signal $\\mathbf{x} = x_0, \\ldots, x_K$ using a given ptSTL formula $\\phi$ as follows:\n\\begin{align}\\label{eq:label_set}\n\tl^\\phi_t = &\\begin{cases} 1 \\text{ if } (\\mathbf{x}, t) \\models \\phi \\\\\n\t 0 \\text{ otherwise}\n\t\\end{cases}\n\\end{align}\n\nWe define number of positive labels $P^{\\#}(\\phi, \\mathbf{x} )$ (\\ref{eq:positive}) and number of negative labels $N^{\\#}(\\phi, \\mathbf{x} )$ (\\ref{eq:negative}) where $\\mathbf{l}^\\phi$ is generated by evaluating formula $\\phi$ along signal $\\mathbf{x}$ as defined in~\\eqref{eq:label_set}:\n\\begin{align}\\label{eq:positive}\n P^{\\#}(\\phi, \\mathbf{x}) = \\sum_{i=0}^{K} l_i^\\phi\n\n \\end{align}\n\\begin{align}\\label{eq:negative}\n N^{\\#}(\\phi, \\mathbf{x}) = \\sum_{i=0}^{K} \\neg l_i^\\phi\n \\end{align}\n \n \n\n \n\n\nAlso note that\n\\begin{equation}\\label{eq:totalformula}\n P^{\\#}(\\phi, \\mathbf{x}) + N^{\\#}(\\phi, \\mathbf{x}) = K + 1\n\\end{equation}\n\n\n\nWe derive monotonicity properties of $P^{\\#}(\\phi, \\cdot)$ for a parametric ptSTL formula $\\phi$ with respect to the monotonicity of $\\phi$. $P^{\\#}(\\phi,\\cdot)$ is monotonically increasing with $p_i$ if and only if the satisfaction value of $\\phi$ is monotonically increasing with $p_i$, i.e.,if~\\eqref{eq:mon_inc} holds along any signal $\\mathbf{x}$, then~\\eqref{eq:monoton_inc} holds:\n\\begin{align}\n&\\text{for all } v,v' \\text{ with } v(p_i) < v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad \\quad \\quad P^{\\#}(\\phi(v),\\mathbf{x}) \\leq P^{\\#}(\\phi(v'),\\mathbf{x}) \\label{eq:monoton_inc}\n\\end{align}\nSimilarly, if $\\phi$ is monotonically decreasing with $p_i$, then $P^{\\#}(\\phi, \\cdot)$ is also monotonically decreasing with $p_i$. Specifically, for any signal $\\mathbf{x}$,~\\eqref{eq:monoton_dec} holds when \\eqref{eq:mon_dec} holds:\n\\begin{align}\n&\\text{for all } v,v' \\text{ with } v(p_i) > v'(p_i), v(p_j) = v(p_j) \\text{ for each } i\\neq j, \\nonumber \\\\\n& \\quad \\quad \\quad P^{\\#}(\\phi(v),\\mathbf{x}) \\leq P^{\\#}(\\phi(v'),\\mathbf{x}) \\label{eq:monoton_dec}\n\\end{align}\n\n\nNote that, by~\\eqref{eq:totalformula}, $P^{\\#}(\\phi, \\cdot)$ and $N^{\\#}(\\phi, \\cdot)$ have the opposite monotonicity property, e.g., if $P^{\\#}(\\phi, \\cdot)$ is monotonically increasing with $p_i$ than $N^{\\#}(\\phi, \\cdot)$ is monotonically decreasing with $p_i$.\n\nIn our work, a parameter appears only once in a parametric ptSTL formula. Therefore, the considered formulas are monotonic in each parameter, i.e., either monotonically increasing or monotonically decreasing.\n\n\n\\subsection{Monotonicity for Temporal Parameters}\nThe number of positive labels $P^{\\#}(\\phi, \\mathcal{D})$ of a ptSTL formula $\\phi$ over a dataset $\\mathcal{D}$ is simply defined as the total number of positive labels, and derived from~\\eqref{eq:positive}. $N^{\\#}(\\phi, \\mathcal{D})$ is defined similarly.\n\\[ P^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x}, \\mathbf{l}) \\in \\mathcal{D}}^{K} P^{\\#}(\\phi, \\mathbf{x}) \\quad N^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x}, \\mathbf{l}) \\in \\mathcal{D}}^{K} N^{\\#}(\\phi, \\mathbf{x})\\]\n\nAs either a positive ($1$) or a negative ($0$) label is assigned to each data point, the equality $|\\mathcal{D}| \\times (K+1) = P^{\\#}(\\phi, \\mathcal{D}) + N^{\\#}(\\phi, \\mathcal{D})$ trivially holds. We define the number of correctly identified positive instances (\\textit{true positives}) with respect to the labels generated by the formula $\\phi$ using~\\eqref{eq:label_set} and the dataset labels as:\n\\begin{align}\\label{eq:tp}\n\n TP^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x},\\mathbf{l}) \\in \\mathcal{D}} \\sum_{i=0}^{K} l_i \\wedge l^\\phi _i\n\n \\end{align}\n \nSimilarly, the total number of incorrect positive results, i.e., the data points that have label $0$ in the given dataset and label $1$ according to the ptSTL formula $\\phi$ ($l_i^\\phi=1$) is defined as:\n\\begin{align}\\label{eq:fp}\n\n FP^{\\#}(\\phi, \\mathcal{D}) = \\sum_{(\\mathbf{x},\\mathbf{l})\\in \\mathcal{D}} \\sum_{i=1}^{K} \\neg l _i \\wedge l^\\phi_i \n \\end{align}\n \nThe derivations of $TP^{\\#}(\\cdot, \\cdot)$ and $FP^{\\#}(\\cdot, \\cdot)$ preserves monotonicity properties~\\eqref{eq:monoton_inc} and~\\eqref{eq:monoton_dec}. Therefore, if a parametric ptSTL formula $\\phi$ is increasing (or decreasing) with a parameter $p$, then both $TP^{\\#}(\\cdot, \\cdot)$ and $FP^{\\#}(\\cdot, \\cdot)$ are increasing (or decreasing) with $p$. \n\nWe use $\\mathcal{M}(p,\\phi)$ to denote the monotonicity property of parameter $p$ in $\\phi$ for the number of positives ($TP^{\\#}(\\cdot, \\cdot)$, $FP^{\\#}(\\cdot, \\cdot)$ or $P^{\\#}(\\cdot,\\cdot)$ ):\n\\begin{align}\\label{eq:monotonicity}\n\t\\mathcal{M}(p,\\phi) = &\\begin{cases}\n\t \\mathbf{I} \\text{ if } p \\text{ is monotonically increasing in } \\phi \\\\\n\t \\mathbf{D} \\text{ if } p \\text{ is monotonically decreasing in } \\phi\n\t\\end{cases}\n\\end{align}\n\nMonotonicity property, $\\mathcal{M}(\\cdot,\\cdot) $, for each parameter in a basic formula is given in Table~\\ref{tb:monotonicity_table}. \n \n\\begin{table}[h]\n\\begin{center}\n\\caption{\\textsc{Monotonicity Table}}\\label{tb:monotonicity_table}\n\\begin{tabular}{cc|cc}\n$\\phi$ & $\\mathcal{M}(p,\\phi)$ & $\\phi$ & $\\mathcal{M}(p,\\phi)$ \\\\ \\hline\\hline\n$x > p$ & $\\mathbf{D}$ & $x < p$ & $\\mathbf{I}$\\\\ \\hline\n$\\mathbf{A}_{[c, p]} \\varphi$ & $\\mathbf{D}$ & $\\mathbf{A}_{[p, c]} \\varphi$ & $\\mathbf{I}$\\\\ \\hline\n$\\mathbf{P}_{[c, p]}\\varphi$ & $\\mathbf{I}$ & $\\mathbf{P}_{[ p, c]}\\varphi$ & $\\mathbf{D}$ \\\\ \\hline\n$\\varphi_1 \\mathbf{S}_{[c, p]}\\varphi_2$ & $\\mathbf{I}$ & $\\varphi_1 \\mathbf{S}_{[p,c]}\\varphi_2$ & $\\mathbf{D}$ \\\\ \\hline\n\\end{tabular}\n\\end {center}\n\\end{table}\n\nNote that the preceding derivations are based on the number of positive labels. The number of correctly identified negative labels, $TN^{\\#}(\\phi, \\mathcal{D})$, and the number of incorrectly identified negative labels $FN^{\\#}(\\phi, \\mathcal{D})$ are defined similarly. These show the opposite monotonicity property, i.e., if $TP^{\\#}(\\phi, \\mathcal{D})$ is monotonically increasing in parameter $p$, then $TN^{\\#}(\\phi, \\mathcal{D})$ is monotonically decreasing in $p$. \nFurthermore, the negation operator ($\\neg$) inverts the monotonicity property. For example, while $\\mathcal{M}(p,\\mathbf{P}_{[a,b]} x < p)$ is $\\mathbf{I}$, $\\mathcal{M}(p,\\neg \\mathbf{P}_{[a,b]} x < p)$ is $\\mathbf{D}$. A parameter's monotonicity is determined by checking the syntax tree of the formula: each negation that appears from the root node to the parameter inverts the monotonicity of the parameter shown in Table~\\ref{tb:monotonicity_table}.\n\n\n\\begin{example}\n\\label{ex:pptstl_formula}\n\nConsider parametric ptSTL formula\n\\begin{equation}\\label{eq:exformula}\n\\phi = ( \\mathbf{P}_{[p_1, p_2]} ( qGust < p_3 ) ) \\wedge ( wGust < p_4 )\n \\end{equation} \nMonotonicity properties of $p_1, p_2, p_3$ and $p_4$ are $\n\\mathcal{M}(p_1,\\phi) = \\mathbf{D}$, $\n\\mathcal{M}(p_2,\\phi) = \\mathbf{I}$, $\n\\mathcal{M}(p_3,\\phi) = \\mathbf{I}$, $\n\\mathcal{M}(p_4,\\phi) = \\mathbf{I}.$\n\\end{example}\n\n\n\n\n\\subsection{Parameter Optimization Using Monotonicity}\\label{sec:parametersynthesis}\n\nWe now present an efficient method based on monotonicity to find parameters of a parametric ptSTL formula $\\phi$ from a given dataset $\\mathcal{D}$~\\eqref{eq:dataset} such that the number of correctly identified positives of the resulting formula is maximized while the number false positives is below a given threshold: \n\n\\begin{problem}\\label{prob:optimization} \nGiven a labeled dataset $\\mathcal{D}$~\\eqref{eq:dataset}, a parametric ptSTL formula $\\phi$ with $n$ parameters $p_1, p_2, \\ldots, p_n$, lower and upper bounds $l_i, u_i$ for each parameter $p_i$, an error bound $B \\in \\mathbb{N}$, find the valuation $v$ within the given limits that maximizes $TP^{\\#}(\\phi(v), \\mathcal{D})$ while guaranteeing that $ FP^{\\#}(\\phi(v), \\mathcal{D}) \\leq B$.\n\n\\end{problem}\n\nTo solve this problem, we first present an algorithm for parametric ptSTL formulas with two parameters, and then discuss how this approach is adapted for parametric ptSTL formulas with more than two parameters.\n\nWe present a diagonal search method to solve Prob.~\\ref{prob:optimization} in an efficient way when $n$ is 2, which adapts search problem of the product of an m element chain and an n element chain~\\citep{linial1985searching} for ptSTL parameter optimization. The diagonal search algorithm starts with a valuation $v$ with $v(p_1)$ is the bound on $p_1$ that maximizes $TP^{\\#}(\\phi, \\mathcal{D})$ (i.e. either $l_1$ or $u_1$) and $v(p_2)$ is the bound on $p_2$ that minimizes $TP^{\\#}(\\phi, \\mathcal{D})$. Given step sizes $\\delta_1$ and $\\delta_2$ for both parameters, the algorithm iteratively changes the value of a parameter according to the following rule: change $v(p_1)$ by $\\delta_1$ in the direction decreasing $P^{\\#}(\\phi, \\mathcal{D})$ if the error constraint does not hold at $v$, otherwise change $v(p_2)$ by $\\delta_2$ in the direction increasing $P^{\\#}(\\phi, \\mathcal{D})$. Thus, the algorithm moves along a diagonal of the product of the discretized parameter domains with the objective of satisfying the error bound or improving the optimization criteria. This diagonal method is summarized in Alg.~\\ref{alg:diagonal}\n\n\n\n\\begin{algorithm}\n\\caption{$DiagonalSearch(\\phi,B,\\mathcal{D}, l_1, u_1, \\delta_1, l_2, u_2, \\delta_2)$}\\label{alg:diagonal}\n\\begin{flushleft}\n\\begin{algorithmic}[1]\n\\Require{$\\phi$: A parametric ptSTL formula with parameters $p_1$ and $p_2$, $B$ : bound on $FP^{\\#}(\\phi(v), \\mathcal{D})$, $\\mathcal{D}$: dataset as in~\\eqref{eq:dataset}, $l_i, u_i, \\delta_i$: lower bound, upper bound and step size for parameter $p_i$, $i \\in \\{1,2\\}$}\n\\Ensure{$v_{best} = \\arg\\max_{ v } \\{TP^{\\#}(\\phi(v),\\mathcal{D}) \\mid FP^{\\#}(\\phi(v),\\mathcal{D}) < B\\}$}\n\\If {$\\mathcal{M}(p_1,\\phi) == \\mathbf{I}$} \\label{line:startinitialize}\n\t\\State $v(p_1) = u_1, \\bar \\delta_1 = -\\delta_1$\n\\Else\n\\State $v(p_1) = l_1, \\bar \\delta_1 = \\delta_1$\n\\EndIf\n\\If {$\\mathcal{M}(p_2,\\phi) == \\mathbf{I}$}\n\t\\State $v(p_2) = l_1, \\bar \\delta_2 = \\delta_2$\n\\Else\n\\State $v(p_2) = u_1, \\bar \\delta_2 = -\\delta_2$\n\\EndIf \\label{line:endinitialize}\n\\State $v_{best} = []$, $TP_{best} = 0$\n\\While{$l_1 \\leq v(p_1) \\leq u_1 \\wedge l_2 \\leq v(p_2) \\leq u_2$}\\label{line:loopstart}\n \\If{$B < FP^{\\#}(\\phi(v),\\mathcal{D})$} \\label{line:errorconstraint}\n \\State $v(p_1) = v(p_1) + \\bar \\delta_1$\n \\Else\n \n \\If{$TP^{\\#}(\\phi(v),\\mathcal{D}) \\geq TP_{best}$}\\label{line:bestknown}\n \\State $TP_{best} = TP^{\\#}(\\phi(v),\\mathcal{D})$, $v_{best} = v$\n \\EndIf\n \\State $v(p_2) = v(p_2) + \\bar \\delta_2$\n \\EndIf\n\\EndWhile\\label{line:loopend}\n\\State \\Return $v_{best}$\n\\end{algorithmic}\n\\end{flushleft}\n\\end{algorithm}\n\nIn lines~\\ref{line:startinitialize}-\\ref{line:endinitialize} of Alg.~\\ref{alg:diagonal}, the initial value and the update direction is defined for each parameter with respect to its monotonicity property. At each iteration of the main loop (lines~\\ref{line:loopstart}-\\ref{line:loopend}), exactly one parameter value is updated. If the error constraint (line~\\ref{line:errorconstraint}) is violated, the parameter that initialized to maximize $TP^{\\#}(\\phi,\\mathcal{D})$, $p_1$, is changed by $\\bar \\delta_1$ to reduce $FP^{\\#}(\\phi(v),\\mathcal{D})$. Otherwise, the current parameter assignment is a candidate solution, and it is checked against the best known solution (line~\\ref{line:bestknown}). Then, the parameter that initialized to minimize $FP^{\\#}(\\phi,\\mathcal{D})$, $p_2$, is changed by $\\bar \\delta_2$ to increase $TP^{\\#}(\\phi(v),\\mathcal{D})$. The iterations end when a parameter is out of the given bounds. Consequently, $O(m_1 + m_2)$ formula evaluations are performed over the given dataset, where $m_1 = \\frac{u_1 - l_1}{\\delta_1}, m_2 = \\frac{u_2 - l_2}{\\delta_2}$.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\resizebox{8.4 cm}{4.2 cm}{%\n\\begin{tikzpicture}[scale=.6]\n\\begin{scope}\n\n\\fill[black!30!red,opacity=0.6] (3,0) rectangle (5,1);\n\\fill[black!30!red,opacity=0.6] (5,0) rectangle (7,5);\n\\fill[black!30!red,opacity=0.6] (6,5) rectangle (7,6);\n\n\\fill[black!30!green,opacity=0.6] (1,0) rectangle (3,6);\n\\fill[black!30!green,opacity=0.6] (3,1) rectangle (5,6);\n\\fill[black!30!green,opacity=0.6] (5,5) rectangle (6,6);\n\\draw (1, 0) grid (7, 6);\n\n\\draw[very thick, scale=1] (1, 0) grid (3, 0);\n\\draw[very thick, scale=1] (3, 0) grid (3, 1);\n\\draw[very thick, scale=1] (3, 1) grid (5, 1);\n\\draw[very thick, scale=1] (5, 1) grid (5, 5);\n\\draw[very thick, scale=1] (5, 5) grid (6, 5);\n\\draw[very thick, scale=1] (6, 5) grid (6, 6);\n\\draw[very thick, scale=1] (6, 6) grid (7, 6);\n\\tikzset{anchor=west}\n\n\\setcounter{row7}{1}\n\\setrowseven {}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:l_1$\\end{turn}}{\\begin{turn}{90} \\tiny $\\: \\: \\:\\: \\: l_1 + \\delta_1$\\end{turn}}{\\tiny ..}{\\tiny .}{\\begin{turn}{90}\\tiny $ \\: \\: \\:\\: \\:u_1 - \\delta_1$\\end{turn}}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:u_1$\\end{turn}}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 $\\end{turn}}{1}{2}{2}{2}{3}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 1\\delta_2$\\end{turn}}{1}{2}{2}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 2\\delta_2$\\end{turn}}{1}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 3\\delta_2$\\end{turn}}{2}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 4\\delta_2$\\end{turn}}{2}{2}{3}{3}{4}{5}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 5\\delta_2$\\end{turn}}{3}{3}{4}{4}{4}{5}\n\n\\node[anchor=center] at (4.0, -0.5) {$FP^{\\#}(\\phi(v), \\mathcal{D})$};\n\\end{scope}\n\n\\begin{scope}[xshift=7.5cm]\n\n\\fill[black!30!red,opacity=0.6] (3,0) rectangle (5,1);\n\\fill[black!30!red,opacity=0.6] (5,0) rectangle (7,5);\n\\fill[black!30!red,opacity=0.6] (6,5) rectangle (7,6);\n\n\\fill[black!30!green,opacity=0.6] (1,0) rectangle (3,6);\n\\fill[black!30!green,opacity=0.6] (3,1) rectangle (5,6);\n\\fill[black!30!green,opacity=0.6] (5,5) rectangle (6,6);\n\\draw (1, 0) grid (7, 6);\n\n\\draw[very thick, scale=1] (1, 0) grid (3, 0);\n\\draw[very thick, scale=1] (3, 0) grid (3, 1);\n\\draw[very thick, scale=1] (3, 1) grid (5, 1);\n\\draw[very thick, scale=1] (5, 1) grid (5, 5);\n\\draw[very thick, scale=1] (5, 5) grid (6, 5);\n\\draw[very thick, scale=1] (6, 5) grid (6, 6);\n\\draw[very thick, scale=1] (6, 6) grid (7, 6);\n\n\\draw[line width=2pt,<-] (5.5,5.5)--(6.5,5.5) node[right]{};\n\\draw[line width=2pt,<-] (5.5,4.5)--(5.5,5.5) node[right]{};\n\\draw[line width=2pt,<-] (4.5,4.5)--(5.5,4.5) node[right]{};\n\\draw[line width=2pt,<-] (4.5,0.5)--(4.5,4.5) node[right]{};\n\\draw[line width=2pt,<-] (2.5,0.5)--(4.5,0.5) node[right]{};\n\\draw[line width=2pt,<-] (2.5,0)--(2.5,0.5) node[right]{};\n\\tikzset{anchor=west}\n\n\\setcounter{row7}{1}\n\n\\setrowseven {}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:l_1$\\end{turn}}{\\begin{turn}{90} \\tiny $\\: \\: \\:\\: \\: l_1 + \\delta_1$\\end{turn}}{\\tiny ..}{\\tiny .}{\\begin{turn}{90}\\tiny $ \\: \\: \\:\\: \\:u_1 - \\delta_1$\\end{turn}}{\\begin{turn}{90}\\tiny $\\: \\: \\:\\: \\:u_1$\\end{turn}}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2$\\end{turn}}{10}{11}{14} {15}{17} {18}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + \\delta_2$\\end{turn}}{10}{12}{15} {16}{17} {19}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 2\\delta_2$\\end{turn}}{11}{12}{15} {17}{18} {21}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 3\\delta_2$\\end{turn}}{13}{13}{17} {19}{21} {22}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 4\\delta_2$\\end{turn}}{14}{14}{18} {20}{22} {24}\n\\setrowseven {\\begin{turn}{-45}\\tiny $l_2 + 5\\delta_2$\\end{turn}}{15}{17}{22} {24}{28} {30}\n\n\n\\node[anchor=center] at (4.0, -0.5) {$TP^{\\#}(\\phi(v), \\mathcal{D})$};\n\\end{scope}\n\n\n\\end{tikzpicture}\n}\n\\end{center}\n\\caption{An example run of Alg.~\\ref{alg:diagonal}.} \\label{fig:grid}\n\\end{figure}\n\nAn example run of Alg.~\\ref{alg:diagonal} is shown in Fig.~\\ref{fig:grid} for illustration, where $B$ is $3$, both $\\mathcal{M}(p_1,\\phi)$ and $\\mathcal{M}(p_2,\\phi)$ are $\\mathbf{I}$. In Fig. \\ref{fig:grid}, each cell contains $FP^{\\#}(\\phi(v), \\mathcal{D})$ and $TP^{\\#}(\\phi(v), \\mathcal{D})$ on the left and right arrays, respectively. Cells with $ FP^{\\#}(\\phi(v), {\\mathcal{D}}) > B$ are marked with red (infeasible parameters) and the rest of the cells are marked with green. The algorithm takes a step to the left when it encounters a red cell and takes a step to the down when it encounters a green cell. The path the algorithm follows is shown on the grid. The algorithm returns $[u_1 - 2\\delta_1, l_2 + 4\\delta_2]$. Note that the algorithm evaluates the optimal in each row it finds the overall optimal for the given step sizes.\n\n\nWe now describe the proposed method to solve Prob.~\\ref{prob:optimization}. Let $\\phi$ be the given parametric ptSTL formula with n parameters $p_1, \\ldots, p_n$. If $n=1$, the optimal value is found with a binary search. If $n=2$, the optimal value is found with $DiagonalSearch$ method described in Alg.~\\ref{alg:diagonal}. Finally, if $n > 2$, $DiagonalSearch$ is run for $p_1$ and $p_2$ for all possible combinations of the last $n-2$ parameters, and the optimal parameters are returned. The whole process is referred as $ParameterSynthesis(\\phi, B, \\mathcal{D})$.\n\n\n\\begin{example}\n\\label{ex:optimization}\nConsider the parametric ptSTL formula~\\eqref{eq:exformula} from Ex. \\ref{ex:pptstl_formula} with parameter ranges: $l_1, l_2 = 0$, $u_1,u_2 = 30$, $l_3=-0.4$, $u_3= 0.3$, $l_4=-240$, $u_4=210$. \n$2,0.05,0.05$ and $30$ are set as the step sizes $\\delta_1\\delta_2,\\delta_3$ and $\\delta_4$, respectively. Since $n>2$, two of the parameters, namely $p_2, p_3$, are selected for $DiagonalSearch$ based on the size of the parameter domains. \nFor the remaining parameters $p_1$ and $p_4$, $\\phi^{i,j}$ is created as follows:\n\\begin{align}\n\\phi^{i,j} =& \\phi(v) \\text{ where } v = [i, p_2, p_3 , j ], \\\\ \\nonumber\n & i \\in \\{0,2, \\ldots 30\\}, j \\in \\{-240, -210, \\ldots 210\\\n\\end{align}\nAlg. \\ref{alg:diagonal} is run for each $\\phi^{i,j}$ with $B=5$ over the dataset $\\mathcal{D}$ defined in Ex.~\\ref{ex:formulation}. The maximum $TP^{\\#}(\\phi^{i,j}(v),\\mathcal{D})$ is attained when $i=4$ and $j=-20$ with $v_{best} = [10, 0]$, which corresponds to ptSTL formula \n$\\phi(v) = (\\mathbf{P}_{[4, 10]} qGust < 0 ) \\wedge ( wGust < -120 ))$ with $TP^{\\#}(\\phi,\\mathcal{D}) = 235$ and $FP^{\\#}(\\phi(v),\\mathcal{D}) = 4$. This formula explains approximately $50\\%$ of the large deviations from the normal behavior (label 1). Thus a possible approach would be adjusting internal commands generated from $pilot$ with respect to the wind behavior characterized by $\\phi(v)$.\nNote that while grid search requires 61440 valuations~\\citep{codit2018}, $ParameterSynthesis(\\phi, B, \\mathcal{D})$ can find the optimal solution with $4721$ valuations.\n \n\n\\end{example}\n\n\\subsection{Formula Synthesis}\nIn this section, we present the solution to the main problem (Prob.~\\ref{prob:main}) considered in this paper: find a ptSTL formula that represents labeled events in a dataset.\nIn general, an unexpected behavior\/fault can occur due to a number of different reasons. To utilize this property, we iteratively construct a formula for the given dataset $\\mathcal{D}$ as a disjunction of ptSTL formulas each representing a different reason. The goal in each iteration is to find a formula for a subset of the labeled instances, while limiting incorrectly labeled instances (FP) as this type of error propagates with disjunction operator. To find such a formula, we define the set of all parametric formulas as in~\\citep{codit2018}, and perform parameter optimization on each of them using $ParameterSynthesis$ method described in Sec.~\\ref{sec:parametersynthesis}. \n\nGiven the set of system variables, $\\{x^1, \\ldots,x^n\\}$, and a bound on the number of operators $N$, the set of all parametric ptSTL formulas with up to $N$ operators $\\mathcal{F}^{\\leq N}$ is recursively defined as:\n\\begin{align}\\label{eq:formula_space}\n\\mathcal{F}^{0} &= \\{ x^i \\sim p_i \\mid \\sim \\in \\{<,>\\}, i = 1,\\ldots,n\\} \\cup \\{\\mathbf{T}\\} \\\\\n\\mathcal{F}^{N} &= \\{\\neg \\phi , \\mathbf{P}_{[a,b]} \\phi, \\mathbf{A}_{[a,b]} \\phi \\mid \\phi \\in \\mathcal{F}^{N-1} \\} \\cup \\nonumber \\\\ \n&\\bigcup\\limits_{i=1}^{n-1} \\{ \\phi_{1} \\wedge \\phi_{2} , \\phi_{1} \\vee \\phi_{2} , \\phi_{1}\\mathbf{S}_{[a,b]}\\phi_{2} \\mid \\nonumber \\\\\n& \\quad\\quad\\quad\\quad \\phi_{1} \\in \\mathcal{F}^{i}, \\phi_{2} \\in \\mathcal{F}^{N-i-1}\\} \\nonumber \\\\\n\\mathcal{F}^{\\leq N} &= \\cup_{i=0}^{N} \\mathcal{F}^{i}\\nonumber\n\\end{align}\n\n\n\n\\begin{algorithm}\n\\caption{$FormulaSynthesis(\\mathcal{F}, B, \\mathcal{D}, p)$}\\label{alg:synthesize} \n\n\\begin{algorithmic}[1]\n\\Require{$\\mathcal{F}$: a set of parametric ptSTL formulas, $B$: bound on the number of false positives, $\\mathcal{D}$: a dataset as in~\\eqref{eq:dataset}, $p$: upper bound on the number of formulas concatenated with disjunction.} \n\n \\State $\\mathcal{F}^v= \\{ \\phi(v) = ParameterSynthesis( \\phi, B, \\mathcal{D}) \\mid \\phi \\in \\mathcal{F}\\}$ \\label{line:parametersyn}\n \\State $i=0, TP_{prev} = 0, TP=1$, $\\Phi = false$\n \\While{$TP > TP_{prev}$\\text{ and } $i < p$}\\label{line:loopstart2}\n \\State $\\phi(v)^* = \\arg \\max_{\\phi(v) \\in \\mathcal{F}^v} TP^{\\#}(\\Phi \\vee \\phi(v), \\mathcal{D})$\n \\State $\\Phi = \\Phi \\vee \\phi(v)^*$\n \\State $i=i+1$\n\n \\State $TP_{prev} = TP$, $TP = TP^{\\#}(\\Phi, \\mathcal{D})$\n \\EndWhile\\label{line:loopend2}\n\n \\State \\Return $\\Phi$\n\\end{algorithmic}\n\\end{algorithm}\n\nThe proposed formula synthesis approach is summarized in Alg.~\\ref{alg:synthesize}. The method takes a set of parametric formulas $\\mathcal{F}$, a bound on the number of false positives $B$, a labeled dataset $\\mathcal{D}$ and a bound $p$ on the number of ptSTL formulas, and generates a ptSTL formula $\\phi^\\star$ in the form of~\\eqref{eq:endformula} with at most $p$ sub-formulas, such that $FP^{\\#}(\\phi^\\star, \\mathcal{D}) < B p$ and $TP^{\\#}(\\phi^\\star, \\mathcal{D})$ is optimized. The set of parametric formulas can be defined as in~\\eqref{eq:formula_space}, or alternatively, an expert of the considered system can write a set of parametric formulas. In the algorithm, first, parameters are optimized for each parametric ptSTL formula $\\phi \\in \\mathcal{F}$ (line~\\ref{line:parametersyn}). Then, starting from $\\Phi = false$, iteratively, the formula $\\phi(v)^\\star$ maximizing the valuation of the combined formula $\\Phi \\vee \\phi(v)^\\star$ is selected from the set of ptSTL formulas $\\mathcal{F}^v$ until the sub-formula limit $p$ is reached, or concatenating new formulas does not improve the result (lines~\\ref{line:loopstart2}-\\ref{line:loopend2}). Note that at each iteration a formula $\\phi(v)^\\star$ is added to $\\Phi$ with disjunction.\n\nIn Alg.~\\ref{alg:synthesize}, $ParameterSynthesis( \\phi, B, \\mathcal{D}) $ is run only once for each parametric ptSTL formula. At every iteration of the algorithm, $TP^{\\#}(\\Phi \\vee \\phi(v), \\mathcal{D})$ is computed for each $\\phi(v) \\in \\mathcal{F}^v$ to select the formula $\\phi(v)^\\star$ that generate the highest increment in $TP$. Note that the resulting formula $\\phi^\\star$ might not be the optimal formula due to the iterative synthesis approach. Essentially, the fitness of the formula is upper bounded by the formula that would be obtained by performing parameter optimization on parametric formulas in the form of~\\eqref{eq:endformula} with $N \\times p$ parameters (as in~\\citep{codit2018}). \nHowever, due to the complexity of the parameter synthesis algorithm, this computation is not feasible for large formulas.\n\n\n\\begin{example}\n\\label{ex:synthesis}\nThe set of all parametric ptSTL formulas $\\mathcal{F}^{\\leq 2}$ with at most $2$ parameters over the system variables $\\{alpha, pilot, wGust, qGust\\}$ is generated according to~\\eqref{eq:formula_space}. The parameter domains are defined as:\n$p_a, p_b \\in \\{2i \\mid i=0, \\ldots,15 \\}$ for $\\mathbf{A}_{[p_a, p_b]}$,$\\mathbf{P}_{[p_a, p_b]}$, \\\\\n$p_{alpha} \\in \\{-0.5 + 0.05i \\mid i=0, \\ldots,20 \\}$, \\\\%$\\mathcal{V} = \\{x_0, x_1, x_2\\}$ \n$p_{pilot} \\in \\{-0.5 +05i \\mid i=0, \\ldots,20 \\}$, \\\\%$\\mathcal{V} = \\{x_0, x_1, x_2\\}$ \n$p_{wGust} \\in \\{-240 +30i \\mid i=0, \\ldots,15 \\}$, \\\\\n$p_{qGust} \\in \\{-0.4 - 0.05i \\mid i=0, \\ldots,14 \\}$. \n \nWe run Alg.~\\ref{alg:synthesize} with the parametric formula set $\\mathcal{F}^{\\leq 2}$, the dataset from Ex.~\\ref{ex:formulation}, bound $B=5$ and subformula limit $p=4$. The resulting formula is:\n\n\\begin{align}\n& \\phi = \\phi_1 \\vee \\phi_2 \\vee \\phi_3 \\vee \\phi_4 \\\\ \\nonumber\n& \\phi_1 = ( \\mathbf{P} _{[4, 10]} ( qGust < 0 ) ) \\wedge ( wGust < -120 ) \\\\ \\nonumber \n& \\phi_2 = ( wGust > 120 ) \\wedge ( \\mathbf{A} _{[14, 14]} ( pilot > -40 ) ) \\\\ \\nonumber \n& \\phi_3 =\\mathbf{P} _{[2, 2]} ( ( alpha < 30 ) \\wedge ( wGust < -120 ) )\\\\ \\nonumber\n& \\phi_4 =( \\mathbf{A} _{[4, 16]} ( qGust > 10 ) ) \\wedge ( pilot < -40 ) \\nonumber \n \\end{align}\n Each sub-formula $\\phi_1, \\phi_2 $, $\\phi_3$ and $ \\phi_4$ explains a condition that led to a disturbance in the pitch angle of the aircraft. \n The first formula $\\phi_1$ shows that a disturbance occurs when $wGust$ is less than $-120$ and $qGust$ was lower than $0$ for some time within the last 10 time steps to last 4 time steps. Formulas $\\phi_2 $, $\\phi_3$ and $ \\phi_4$ show that a disturbance occurs 2) when $wGust$ is greater than $120$ and $pilot$ was greater then $-40$ $14$ time steps ago, or 3) if $alpha$ was less than $30$ and $wGust$ was less than -120 two steps ago, or 4) if $qGust$ was higher than 10 for each step between last 4 and 16 steps and $pilot$ is less than -40 in the current step.\n\n $477$ out of $3000$ data points in $\\mathcal{D}$ are labeled with $1$.\n $TP^{\\#}(\\phi,\\mathcal{D}) $ and $FP^{\\#}(\\phi,\\mathcal{D})$ valuations are 419 and 18 respectively. Total mismatch count of 3000 points is computed as 76 which leads to an accuracy of 97.46\\%.\n\n\nThis result is found in 3350 seconds on a PowerEdge T430 machine with Intel Xeon E5-2650 12C\/24T processor. It is important to note that $\\phi$ includes $11$ operators and $15$ parameters and it is defined over $4$ system variables. This example shows that the proposed method can generate complex formulas from labeled datasets in an efficient way, since, due to the computational complexity, existing formula synthesis algorithms are validated on simpler formulas.\n\n\\end{example}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sect_introduction}\n\nIsotope abundance ratios provide a powerful tool for tracing stellar nucleosynthesis, evaluating the composition of stellar ejecta, and constraining the chemical evolution of the Milky Way \\citep{1994ARA&A..32..191W}. In particular, the $^{12}$C\/$^{13}$C ratio is one of the most useful tracers of the relative degree of primary to secondary processing. $^{12}$C is a predominantly primary nucleus formed by He burning in massive stars on short timescales \\citep[e.g.,][]{1995ApJS...98..617T}. $^{13}$C is produced on a longer timescale via CNO processing of $^{12}$C seeds from earlier stellar generations during the red giant phase in low- and intermediate-mass stars or novae \\citep[e.g.,][]{1994LNP...439...72H,1994ARA&A..32..153M,1994ARA&A..32..191W}. $^{12}$C\/$^{13}$C ratios are expected to be low in the central molecular zone (CMZ) and high in the Galactic outskirts, because the Galaxy formed from the inside out (e.g., \\citealt{2001ApJ...554.1044C}; \\citealt{2012A&A...540A..56P}).\n\nObservations indeed indicate a gradient of $^{12}$C\/$^{13}$C ratios across the Galaxy. \\citet{1976A&A....51..303W} and \\citet{1979MNRAS.188..445W} measured the $J_{Ka,Kc}=1_{1,0}-1_{1,1}$ lines of H$_2^{12}$CO and H$_2^{13}$CO near 5 GHz toward 11 and 24 Galactic continuum sources, respectively. While ignoring effects of photon trapping, the results suggested that the $^{12}$C\/$^{13}$C ratios may vary with galactocentric distance ($R_{\\rm GC}$). With the additional measurement of the $J_{Ka,Kc}=2_{1,1}-2_{1,2}$ line of H$_2$CO at 14.5 GHz, \\citet{1980A&A....82...41H, 1982A&A...109..344H, 1983A&A...127..388H, 1985A&A...143..148H} also reported a gradient after correcting for effects of optical depth and photon trapping. \\citet{1990ApJ...357..477L} used the optically thin lines of C$^{18}$O and $^{13}$C$^{18}$O to trace the carbon isotope ratios. They also found a systematic $^{12}$C\/$^{13}$C gradient across the Galaxy, ranging from about 20--25 near the Galactic center, to 30--50 in the inner Galactic disk, to $\\sim$70 in the local interstellar medium (ISM). \\citet{1996A&AS..119..439W}, complementing these investigations by also including the far outer Galaxy, encountered ratios in excess of 100 and demonstrated that the gradient found in the inner Galaxy continues farther out. \\citet{2005ApJ...634.1126M} obtained $^{12}$C\/$^{13}$C = 6.01$R_{\\rm GC}$ + 12.28 based on the CN measurements of \\citet{2002ApJ...578..211S}. Here and elsewhere, $R_{\\rm GC}$ denotes the galactocentric distance in units of kiloparsecs (kpc). By combining previously obtained H$_2$CO and C$^{18}$O results with these CN data, \\citet{2005ApJ...634.1126M} obtained $^{12}$C\/$^{13}$C = 6.21$R_{\\rm GC}$ + 18.71.\n\nMore recently, \\citet{2017ApJ...845..158H} reported observations of a variety of molecules (e.g., H$_2$CS, CH$_3$CCH, NH$_2$CHO, CH$_2$CHCN, and CH$_3$CH$_2$CN) and their $^{13}$C-substituted species toward Sgr B2(N). These authors obtained an average $^{12}$C\/$^{13}$C value of 24 $\\pm$ 7 in the Galactic center region, which is close to results using $^{12}$CH\/$^{13}$CH (15.8 $\\pm$ 2.4, Sgr B2(M)) by \\citet{2020A&A...640A.125J} and the particularly solid $^{12}$C$^{34}$S\/$^{13}$C$^{34}$S ratio (22.1$^{+3.3}_{-2.4}$, $+$50 km s$^{-1}$ Cloud) from \\citet{2020A&A...642A.222H} who use a variety of CS isotopologs and rotational transitions. \\citet{2019ApJ...877..154Y} proposed a linear fit of $^{12}$C\/$^{13}$C = (5.08 $\\pm$ 1.10)$R_{\\rm GC}$ + (11.86 $\\pm$ 6.60) based on a large survey of H$_2$CO. The latter includes data from the center to the outskirts of the Milky Way well beyond the Perseus Arm. However, data from the CMZ are not similar to those thoroughly traced by \\citet{2020A&A...642A.222H}, not many sources from the innermost Galactic disk could be included in this survey, and also the number of sources beyond the Perseus arm was small, meaning that there is still space for improvement.\n\nWhile the carbon isotope ratio has drawn much attention in the past, it is not the only isotope ratio that can be studied at radio wavelengths and that has a significant impact on our understanding of the chemical evolution of the Galaxy. The isotope ratios of sulfur are providing complementary information on stellar nucleosynthesis that is not traced by the carbon isotope ratio. Sulfur is special in that it provides a total of four stable isotopes, $^{32}$S, $^{34}$S, $^{33}$S, and $^{36}$S. In the Solar System, abundance ratios are 95.02 : 4.21 : 0.75 : 0.021, respectively \\citep{1989GeCoA..53..197A}. $^{32}$S and $^{34}$S are synthesized during stages of hydrostatic oxygen-burning preceding a type II supernova event or during stages of explosive oxygen-burning in a supernova of type Ia; $^{33}$S is synthesized in explosive oxygen- and neon-burning, which is also related to massive stars; and $^{36}$S may be an s-process nucleus. The comprehensive calculations of \\citet{1995ApJS..101..181W} indicate that $^{32}$S and $^{33}$S are primary (in the sense that the stellar yields do not strongly depend on the initial metallicity of the stellar model), while $^{34}$S is not a clean primary isotope; its yield decreases with decreasing metallicity. According to \\citet{1985ApJ...295..604T} and \\citet{1989A&A...210...93L}, $^{36}$S is produced as a purely secondary isotope in massive stars, with a possible (also secondary) contribution from asymptotic giant branch (AGB) stars. Only a small fraction of $^{36}$S is destroyed during supernova explosions (Woosley, priv. comm.). Comparing ``primary'' and ``secondary'' nuclei, we might therefore expect the presence of weak $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S gradients and a stronger $^{32}$S\/$^{36}$S gradient as a function of galactocentric radius.\n\nThere is a strong and widespread molecular species that allows us to measure carbon and sulfur isotope ratios simultaneously, namely carbon monosulfide (CS). CS is unique in that it is a simple diatomic molecule exhibiting strong line emission and possessing eight stable isotopologs, which allows us to determine the above-mentioned carbon and sulfur isotope ratios. Six isotopologs have been detected so far in the ISM (e.g., \\citealt{1996A&A...305..960C}; \\citealt{1996A&A...313L...1M}; \\citealt{2020A&A...642A.222H}; \\citealt{2020ApJ...899..145Y}). \n\nMaking use of the CS species, \\citet{1996A&A...305..960C} and \\citet{1996A&A...313L...1M} obtained average abundance ratios of 24.4 $\\pm$ 5.0, 6.3 $\\pm$ 1.0, and 115 $\\pm$ 17 for $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, and $^{34}$S\/$^{36}$S for the ISM, respectively. The latter is approximately half the solar value, but similar to the value found in IRC+10216 \\citep{2004A&A...426..219M}. Recently, \\citet{2020A&A...642A.222H} published $^{32}$S\/$^{34}$S ratios of 16.3$^{+2.1}_{-1.7}$ and 17.9 $\\pm$ 5.0 for the $+$50 km s$^{-1}$ cloud and Sgr B2(N) near the Galactic center, respectively. These are only slightly lower than the value of 22 in the Solar System. There is an obvious and confirmed $^{32}$S\/$^{34}$S gradient \\citep{1996A&A...305..960C, 2020ApJ...899..145Y} from the inner Galaxy out to a galactocentric distance of 12.0 kpc. Nevertheless, there is a lack of data at small and large galactocentric distances.\n\nWe are performing systematic observational studies on isotope ratios in the Milky Way, including $^{12}$C\/$^{13}$C \\citep{2019ApJ...877..154Y}, $^{14}$N\/$^{15}$N \\citep{2021ApJS..257...39C},$^{18}$O\/$^{17}$O (\\citealt{2015ApJS..219...28Z}, \\citeyear{2020ApJS..249....6Z}, \\citeyear{2020IAUGA..30..278Z}; \\citealt{2016RAA....16...47L}), and $^{32}$S\/$^{34}$S \\citep{2020ApJ...899..145Y}. We have thus performed a more systematic study on CS and its isotopologs toward 110 high-mass star-forming regions (HMSFRs). $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios can be directly derived from integrated $^{12}$C$^{34}$S\/$^{13}$C$^{34}$S (hereafter C$^{34}$S\/$^{13}$C$^{34}$S) and $^{13}$C$^{32}$S\/$^{13}$C$^{34}$S (hereafter $^{13}$CS\/$^{13}$C$^{34}$S, see Section \\ref{ratios_13c34s}) intensities, respectively. Also, $^{34}$S\/$^{33}$S and $^{34}$S\/$^{36}$S values could be obtained with measurements of C$^{34}$S, $^{12}$C$^{33}$S (hereafter C$^{33}$S), and $^{12}$C$^{36}$S (hereafter C$^{36}$S). Furthermore, $^{32}$S\/$^{33}$S and $^{32}$S\/$^{36}$S ratios can then be derived with the resulting $^{34}$S\/$^{33}$S and $^{34}$S\/$^{36}$S values combined with the $^{32}$S\/$^{34}$S ratios (see Sections \\ref{section_34s33s} to \\ref{section_32s36s}). In Section \\ref{sou_selection}, we describe the source selection and observations for our large sample. Section \\ref{results} presents our results on $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S ratios. Section \\ref{discussion} discusses potential processes that could contaminate and affect the isotope ratios derived in the previous section and provides a detailed comparison with results from earlier studies. Our main results are summarized in Section \\ref{summary}. \n\n\n\n\\section{Source selection and observations}\n\\label{sou_selection}\n\n\\subsection{Sample selection and distance}\n\\label{section_distance}\n\nIn 2019, we selected 18 HMSFRs from the Galactic center region to the outer Galaxy beyond the Perseus arm. To enlarge this sample, we chose 92 sources from the Bar and Spiral Structure Legacy (BeSSeL) Survey\\footnote{http:\/\/bessel.vlbi-astrometry.org} in 2020. These 92 targets were recently released by the BeSSeL project \\citep{2019ApJ...885..131R} and not observed by \\citet{2020ApJ...899..145Y}. In total, 110 objects in the Galaxy are part of our survey. The coordinates of our sample sources are listed in Table \\ref{table_sources}. Determining trigonometric parallaxes is a very direct and accurate method to measure the distance of sources from the Sun \\citep{2009ApJ...700..137R, 2014ApJ...783..130R, 2019ApJ...885..131R}. Over the past decade, mainly thanks to the BeSSeL project, the trigonometric parallaxes of approximately 200 HMSFRs have been determined across the Milky Way through dedicated high-resolution observations of molecular maser lines. Therefore, this is a good opportunity to investigate carbon and sulfur isotope ratios with well-determined distances across the Galaxy. The galactocentric distance ($R_{\\rm GC}$) can be obtained with the heliocentric distance $d$ from the trigonometric parallax data base of the BeSSeL project using \\begin{equation}\nR_{\\rm GC} = \\sqrt{( R_0 cos(l) - d )^2 + R_0^2 sin^2(l)}\n,\\end{equation}\n\\citep{2009ApJ...699.1153R}. $R_0$ = 8.178 $\\pm$ 0.013$\\rm _{stat.}$ $\\pm$ 0.022$\\rm _{sys.}$ kpc \\citep{2019A&A...625L..10G} describes the distance from the Sun to the Galactic center, $l$ is the Galactic longitude of the source, and $d$ is the distance either directly derived from the trigonometric parallax data based on the BeSSeL project or a kinematic distance in cases where no such distance is yet available. Because the uncertainty in $R_0$ is very small, it will be neglected in the following analysis. For 12 of our targets without trigonometric parallax data, we estimated their kinematic distances from the Revised Kinematic Distance calculator\\footnote{http:\/\/bessel.vlbi-astrometry.org\/revised\\_kd\\_2014} \\citep{2014ApJ...783..130R}. The resulting distances indicate that 6 of these 12 sources are located in the CMZ, namely SgrC, the $+$20 km~s$^{-1}$ cloud, the $+$50 km~s$^{-1}$ cloud, G0.25, G1.28$+$0.07, and SgrD. Four targets belong to the inner Galactic disk, namely PointC1, CloudD, Clump2, and PointD1. Two sources, W89-380 and WB89-391, are in the outer regions beyond the Perseus arm. The heliocentric distances ($d$) and the galactocentric distances ($R_{\\rm GC}$) for our sample are listed in Columns 6 and 7 of Table~\\ref{table_sources}.\n\n\n\n\\subsection{Observations}\nWe observed the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward 110 HMSFRs with the IRAM 30 meter telescope\\footnote{IRAM is supported by INSU\/CNRS (France), MPG (Germany) and IGN (Spain).} in 2019 June, July, and October under project 045-19 (PI, Christian Henkel) as well as in 2020 August within project 022-20 (PI, Hongzhi Yu). The on$+$off source integration times for our sources range from 4.0 minutes to 10.8 hours. These values are given in Appendix~\\ref{appendix_table} (Table \\ref{fitting_all}). The EMIR receiver with two bands, E090 and E150, was used to cover a bandwidth of $\\sim$16 GHz (from 90.8 to 98.2 GHz and 138.4 to 146.0 GHz) simultaneously in dual polarisation. We used the wide-mode FTS backend with a resolution of 195 kHz, corresponding to $\\sim$0.6 km s$^{-1}$ and $\\sim$0.4 km s$^{-1}$ at 96 GHz and 145 GHz, respectively. The observations were performed in total-power position-switching mode and the off position was set at 30$\\arcmin$ in azimuth. Pointing was checked every 2 hours using nearby quasars. Focus calibrations were done at the beginning of the observations and during sunset and sunrise toward strong quasars. The main beam brightness temperature, $T_{\\rm MB}$, was obtained from the antenna temperature $T_{\\rm A}^*$ via the relation $T_{\\rm MB}$ = $T_{\\rm A}^*\\times F_{\\rm eff}$\/$B_{\\rm eff}$ ($F_{\\rm eff}$: forward hemisphere efficiency; $B_{\\rm eff}$: main beam efficiency) with corresponding telescope efficiencies\\footnote{https:\/\/publicwiki.iram.es\/Iram30mEfficiencies}: $F_{\\rm eff}$\/$B_{\\rm eff}$ are 0.95\/0.81 and 0.93\/0.73 in the frequency ranges of 90.8-98.2 GHz and 138.4-146.0 GHz, respectively. The system temperatures were 100-160 K and 170-300 K on a $T_{\\rm A}^*$ scale for the E090 and E150 band observations. The half power beam width (HPBW) for each transition was calculated as HPBW($\\arcsec$)=2460\/$\\nu$(GHz). Rest frequencies, excitations of the upper levels above the ground state, Einstein coefficients for spontaneous emission, and respective beam sizes are listed in Table~\\ref{table_linelist}.\n\n\n\n\n\\begin{table*}[h]\n\\caption{Observed spectral line parameters$^a$.}\n\\centering\n\\begin{tabular}{cccccc}\n\\hline\\hline\nIsotopolog & Transition & $\\nu_0$\\tablefootmark{b} & $E_{\\rm up}$\\tablefootmark{c} & $A_{\\rm u,l}$\\tablefootmark{d} & HPBW\\tablefootmark{e} \\\\\n & \\ & (MHz) & (K) & (s$^{-1}$) & ($\\arcsec$) \\\\\n\\hline\n\\label{table_linelist}\nCS & 2-1 & 97980.953 &7.1 & 1.68 $\\times$ 10$^{-5}$ & 25.1 \\\\\nC$^{33}$S & 2-1 & 97172.064 &7.0 & 1.64 $\\times$ 10$^{-5}$ & 25.3 \\\\\nC$^{34}$S & 2-1 & 96412.95 &6.9 & 1.60 $\\times$ 10$^{-5}$ & 25.5 \\\\\nC$^{36}$S & 2-1 & 95016.722 &6.8 & 1.53 $\\times$ 10$^{-5}$ & 25.9 \\\\\n$^{13}$CS & 2-1 & 92494.308 &6.7 & 1.41 $\\times$ 10$^{-5}$ & 26.6 \\\\\n$^{13}$C$^{33}$S & 2-1 & 91685.241 &6.6 & 1.38 $\\times$ 10$^{-5}$ & 26.8 \\\\\n$^{13}$C$^{34}$S & 2-1 & 90926.026 &6.5 & 1.34 $\\times$ 10$^{-5}$ & 27.1 \\\\\nC$^{33}$S & 3-2 & 145755.732 &14.0 & 5.92 $\\times$ 10$^{-5}$ & 16.9 \\\\\nC$^{34}$S & 3-2 & 144617.101 &13.9 & 5.78 $\\times$ 10$^{-5}$ & 17.0 \\\\\nC$^{36}$S & 3-2 & 142522.785 &13.7 & 5.54 $\\times$ 10$^{-5}$ & 17.3 \\\\\n$^{13}$CS & 3-2 & 138739.335 &13.3 & 5.11 $\\times$ 10$^{-5}$ & 17.7 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\n\\tablefoottext{a}{From the Cologne Database for Molecular Spectroscopy \\citep[CDMS,][]{2005JMoSt.742..215M,2016JMoSp.327...95E}.}\n\\tablefoottext{b}{Rest frequency.}\n\\tablefoottext{c}{Upper energy level.}\n\\tablefoottext{d}{Einstein coefficient for spontaneous emission from upper $u$ to lower $l$ level.}\n\\tablefoottext{e}{Half power beam width.}}\n\\end{table*}\n\n\\subsection{Data reduction}\n\\label{datareduction}\n\nWe used the GILDAS\/CLASS\\footnote{https:\/\/www.iram.fr\/IRAMFR\/GILDAS\/} package to analyze the spectral line data. The spectra of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward one of our targets, DR21, are shown in Fig.~\\ref{fig_dr21}, after subtracting first-order polynomial baselines and applying Hanning smoothing. The spectra of all 110 targets, also after first-order polynomial-baseline removal and Hanning smoothing, are presented in Appendix~\\ref{appendix_spectra} (Fig.~\\ref{spectra_all}). \n\n\n\n\nAmong our sample of 110 targets, we detected the $J$ = 2-1 line of CS toward 106 sources, which yields a detection rate of 96\\%. The $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, C$^{33}$S, and $^{13}$C$^{34}$S were successfully detected in 90, 82, 46, and 17 of our sources with signal-to-noise (S\/N) ratios of greater than 3, respectively. The $J$ = 3-2 lines of C$^{34}$S, $^{13}$CS, and C$^{33}$S were detected in 87, 71, and 42 objects with S\/Ns of $\\ge$3.0. Line parameters from Gaussian fitting are listed in Appendix~\\ref{appendix_table} (Table \\ref{fitting_all}). Relevant for the evaluation of isotope ratios is the fact that for 17 sources with 19 velocity components, the S\/Ns of the $J$ = 2-1 transition of $^{13}$C$^{34}$S are greater than 3, which allows us to determine the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios directly with the $J$ = 2-1 lines of C$^{34}$S, $^{13}$CS, and $^{13}$C$^{34}$S. Toward 82 targets with 90 radial velocity components, the $J$ = 2-1 transitions of C$^{34}$S and $^{13}$CS were both detected with S\/Ns of $\\ge$3.0. The $J$ = 3-2 lines of C$^{34}$S and $^{13}$CS were both found in 71 objects with 73 radial velocity components and S\/Ns of $\\ge$3.0. Furthermore, the $J$ = 2-1 and $J$ = 3-2 transitions of C$^{34}$S and C$^{33}$S were detected with S\/Ns of $\\ge$3.0 toward 46 and 42 sources, respectively.\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.3\\textwidth]{spectra\/DR21.eps}\n\\caption{Line profiles of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward one typical target (DR21) of our large sample of 110 sources, after subtracting first-order polynomial baselines. The main beam brightness temperature scales are presented on the left hand side of the profiles. The spectra of all 110 objects in our sample are shown in Appendix~\\ref{appendix_spectra} (Fig.~\\ref{spectra_all}).}\n \\label{fig_dr21}\n\\end{figure}\n\nThe C$^{36}$S $J$ = 2-1 line was successfully detected with S\/Ns of $\\ge$3.0 toward three targets, namely W3OH, the $+$50 km~s$^{-1}$ cloud near the Galactic center, and DR21. As C$^{36}$S and $^{13}$C$^{33}$S are the least abundant among the CS isotopologs, tentative detection with S\/Ns of $\\sim$2.0 are also presented here but not included in further analyses. In another five objects, the C$^{36}$S $J$ = 2-1 line was tentatively detected. For the C$^{36}$S $J$ = 3-2 transitions, we report one detection with an S\/N larger than 3.0 toward Orion-KL and five tentative detections. The $J$ = 2-1 lines of $^{13}$C$^{33}$S were tentatively detected toward three sources, namely Orion-KL, W51-IRS2, and DR21.\nIntegration times and 1$\\sigma$ noise levels of the observed transitions are listed in Columns 3 and 4 of Table~\\ref{fitting_all} for each target.\n\n\n\n\n\\section{Results}\n\\label{results}\nIn the following, we first estimate the optical depths of the various lines to avoid problems with line saturation that might affect our results. We then present the carbon and sulfur isotope ratios derived from different detected CS isotopologs.\n\n\\subsection{Optical depth}\n\\label{section_opacities}\n\nThe main isotopolog, CS, is usually optically thick in massive star-forming regions (e.g. \\citealt{1980ApJ...235..437L}; \\citealt{2020ApJ...899..145Y}). Therefore, the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios cannot be determined from the line intensity ratios of $I$(CS)\/$I$($^{13}$CS) and $I$(CS)\/$I$(C$^{34}$S). However, assuming that the $J$ = 2-1 transitions of CS, C$^{34}$S, and $^{13}$CS share the same beam filling factor and excitation temperature, we can estimate the maximum optical depth of the $^{13}$CS $J$ = 2-1 line from:\n\\begin{equation}\n\\frac{T_{\\rm {mb}}(^{12}\\rm C\\rm S)}{T_{\\rm mb}(^{13}\\rm C\\rm S)} \\sim \\frac{1 - e^{-\\tau(^{13}\\rm C\\rm S)R_C}}{1 - e^{-\\tau(^{13}\\rm C\\rm S)}},\\,R_C = \\frac{^{12}\\rm C}{^{13}\\rm C},\n\\end{equation}\nwhere $T_{\\rm {mb}}$ is the peak main beam brightness temperature derived from the best Gaussian-fitting result and listed in Column 8 of Table~\\ref{fitting_all}. In this case, the $^{12}$C\/$^{13}$C ratios can be derived from the integrated line intensities of C$^{34}$S and $^{13}$C$^{34}$S with the assumption of $\\tau(\\rm C^{34}\\rm S) \\textless 1.0$, which then also implies $\\tau(^{13}\\rm C^{34}\\rm S) \\textless 1.0$ (see details in Section \\ref{ratios_13c34s}). Multiplying $\\tau(^{13}\\rm CS)$ by $R_C$ = $^{12}$C\/$^{13}$C, we can get the peak opacity $\\tau(\\rm CS)$ = $\\tau(^{13}{\\rm CS})R_C$. The maximum optical depth of C$^{34}$S can be obtained from:\n\\begin{equation}\n\\frac{T_{\\rm mb}(\\rm C\\rm ^{34}S)}{T_{\\rm mb}(^{13}\\rm C\\rm S)} = \\frac{1 - e^{-\\tau(\\rm C\\rm ^{34}S)}}{1 - e^{-\\tau(^{13}\\rm C\\rm S)}},\n\\end{equation}\nwhere $T_{\\rm {mb}}$ is the peak main beam brightness temperature derived from the best Gaussian-fitting result and listed in Column 8 of Table~\\ref{fitting_all}. As shown in Table \\ref{table_13c34sresults}, the peak optical depths of the $J$ = 2-1 lines of CS, C$^{34}$S, and $^{13}$CS for our 17 targets with detections of $^{13}$C$^{34}$S range from 1.29 to 8.79, 0.12 to 0.55, and 0.05 to 0.34, respectively. Therefore, C$^{34}$S and $^{13}$CS in these 17 objects are optically thin, even though they belong, on average, to the more opaque ones, being successfully detected\nin $^{13}$C$^{34}$S (see below). Nevertheless, the corrections for optical depth are applied to C$^{34}$S and $^{13}$CS with factors of $f_1$ and $f_2$, respectively.\n\\begin{equation}\nf_1=\\frac{\\tau(\\rm C\\rm ^{34}S)}{1-e^{-\\tau(\\rm C\\rm ^{34}S)}} {\\quad\\rm and}\n\\end{equation}\n\\begin{equation}\nf_2=\\frac{\\tau(^{13}\\rm C\\rm S)}{1-e^{-\\tau(^{13}\\rm C\\rm S)}}\n\\end{equation}\nare listed in Columns 7 and 8 of Table~\\ref{table_13c34sresults}, respectively.\n\nFor those 82 sources with detections of $J$ = 2-1 CS, C$^{34}$S, and $^{13}$CS, the optical depths were calculated based on the $^{12}$C\/$^{13}$C gradient that we derived from our C$^{34}$S and $^{13}$C$^{34}$S measurements (for details, see Section \\ref{ratios_13c34s}). In Table \\ref{table_doubleisotope}, the peak opacities of the $J$ = 2-1 lines of CS, C$^{34}$S, and $^{13}$CS for these 82 targets range from 0.34 to 14.48, 0.02 to 0.74, and 0.01 to 0.39, respectively. The CS $J$ = 2-1 lines are optically thick with $\\tau(\\rm CS) \\textgreater 1.0$ in most sources (89\\%) of our sample, while they tend to be optically thin in seven objects, namely Point C1 ($\\tau(\\rm CS) \\leq 0.59$), Sgr C ($\\tau(\\rm CS) \\leq 0.54$), Cloud D ($\\tau(\\rm CS) \\leq 0.64$), G1.28$+$0.07 ($\\tau(\\rm CS) \\leq 0.82$), Sgr D ($\\tau(\\rm CS) \\leq 0.45$), and Point D1 ($\\tau(\\rm CS) \\leq 0.34$). In contrast, the transitions from rare isotopologs, the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines in our sample, are all optically thin, as their maximum optical depths are less than 0.8 and 0.4, respectively. In the following, we are therefore motivated to consider all CS isotopologs as optically thin, except CS itself. This allows us to use ratios of integrated intensity of all the rare CS isotopologs ---but not CS itself--- to derive the carbon and sulfur isotope ratios we intend to study. Small corrections accounting for the optical depths are applied to the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines with factors of $f_1$ and $f_2$, respectively, and are listed in Columns 7 and 8 of Table~\\ref{table_doubleisotope}. The optical depths of the $J$ = 3-2 transitions cannot be estimated, as the CS $J$ = 3-2 line was not covered by our observations because of bandwidth limitations. However, the $^{32}$S\/$^{34}$S ratios for a given source obtained through the double isotope method from the $J$ = 2-1 and $J$ = 3-2 transitions are in good agreement, indicating that the C$^{34}$S and $^{13}$CS $J$ = 3-2 lines in our sample are also optically thin (see details in Section~\\ref{section_double_32s34s}). \n\nThe RADEX non Local Thermodynamic Equilibrium (LTE) model \\citep{2007A&A...468..627V} was used to calculate the variation of excitation temperature, $T_{ex}$, with optical depth. Frequencies, energy levels, and Einstein A coefficients for spontaneous emission were taken from the Cologne Database for Molecular Spectroscopy \\citep[CDMS;][]{2005JMoSt.742..215M,2016JMoSp.327...95E}. Recent collision rates for CS with para- and ortho-H$_2$ \\citep{2018MNRAS.478.1811D} were used. Figure~\\ref{fig_c34stex} shows the excitation temperatures and opacities of C$^{34}$S $J$ = 2-1 for a kinetic temperature of 30 K and a molecular hydrogen density of 10$^5$ cm$^{-3}$. Variations of $T_{ex}$ within about 2 K for our sample targets with optical depths of 0.02 $\\leq \\tau(\\rm C^{34}S) \\leq 0.74$ can barely affect our results.\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.5\\textwidth]{newFigures\/c34s21-tau-tex.png}\n\\caption{Excitation temperature, $T_{ex}$, as a function of optical depth for the $J$ = 2-1 transition of C$^{34}$S. The gray dashed lines indicate the range of opacities for our sample sources.}\n \\label{fig_c34stex}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[width=460pt]{newFigures\/1allratio12c_13c-lowerlimits.pdf}\n \\caption{$^{12}$C\/$^{13}$C isotope ratios from C$^{34}$S\/$^{13}$C$^{34}$S, CN\/$^{13}$CN, C$^{18}$O\/$^{13}$C$^{18}$O, H$_2$CO\/H$_2^{13}$CO, CH$^+$\/$^{13}$CH$^+$, and CH\/$^{13}$CH are plotted as functions of the distance from the Galactic center. The red symbol $\\odot$ indicates the $^{12}$C\/$^{13}$C isotope ratio of the Sun. The filled black circles are the results obtained from C$^{34}$S with corrections of opacity in the current work, and the resulting first-order polynomial fit is plotted as a solid line, with the gray-shaded area showing the 1$\\sigma$ interval of the fit. The open black circles are the 3$\\sigma$ lower limits obtained from nondetections of $^{13}$C$^{34}$S in the current work. The blue triangles, orange pentagons, yellow stars, green squares, and green diamonds are values determined from CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$ \\citep{2011ApJ...728...36R}, and CH \\citep{2020A&A...640A.125J}, respectively, using the most up-to-date distances. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n \\label{fig_gradient_12C13C}\n\\end{figure*}\n\n\n\\begin{table*}[h]\n\\caption{Isotope ratios derived with the $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, and $^{13}$C$^{34}$S.}\n\\centering\n\\small\n\\begin{tabular}{lcc|ccc|cc|cc}\n\\hline\\hline\nSource & $V_{\\rm LSR}$ & $R_{GC}$ & \\multicolumn{3}{c}{Optical depth} & \\multicolumn{2}{c}{Corrections for} & $^{12}$C\/$^{13}$C & $^{32}$S\/$^{34}$S \\\\\n & & & & & & \\multicolumn{2}{c}{optical depth} & & \\\\\n & (km s$^{-1}$) & (kpc) & CS & C$^{34}$S & $^{13}$CS & $f_1$ & $f_2$ & & \\\\\n\\hline\n\\label{table_13c34sresults}\nW3OH & -46.96 & 9.64 $\\pm$ 0.03 & 8.289 $\\pm$ 0.083 & 0.292 $\\pm$ 0.003 & 0.127 $\\pm$ 0.001 & 1.2 & 1.1 & 75.30 $\\pm$ 4.27 & 29.80 $\\pm$ 1.68 \\\\\nOrion-KL & 8.17 & 8.54 $\\pm$ 0.00 & 3.722 $\\pm$ 0.037 & 0.180 $\\pm$ 0.002 & 0.074 $\\pm$ 0.001 & 1.1 & 1.0 & 54.89 $\\pm$ 12.90 & 21.24 $\\pm$ 4.67 \\\\\nG359.61$-$00.24 & 19.33 & 5.51 $\\pm$ 0.15 & 3.531 $\\pm$ 0.035 & 0.204 $\\pm$ 0.002 & 0.089 $\\pm$ 0.001 & 1.1 & 1.0 & 43.85 $\\pm$ 11.95 & 17.31 $\\pm$ 4.79 \\\\\n$+$50~km~s$^{-1}$~cloud & 46.79 & 0.02 $\\pm$ 0.04 & 4.278 $\\pm$ 0.043 & 0.192 $\\pm$ 0.002 & 0.151 $\\pm$ 0.002 & 1.1 & 1.1 & 31.22 $\\pm$ 2.06 & 21.71 $\\pm$ 1.40 \\\\\nSgrB2 & 53.18 & 0.55 $\\pm$ 0.05 & 1.290 $\\pm$ 0.013 & 0.140 $\\pm$ 0.001 & 0.113 $\\pm$ 0.001 & 1.1 & 1.1 & 12.18 $\\pm$ 0.73 & 9.77 $\\pm$ 0.58 \\\\\nSgrB2 & 66.54 & 0.44 $\\pm$ 0.70 & 8.119 $\\pm$ 0.081 & 0.523 $\\pm$ 0.005 & 0.341 $\\pm$ 0.003 & 1.3 & 1.2 & 30.62 $\\pm$ 2.04 & 18.99 $\\pm$ 1.31 \\\\\nSgrB2 & 83.38 & 0.41 $\\pm$ 0.02 & 2.417 $\\pm$ 0.024 & 0.097 $\\pm$ 0.001 & 0.077 $\\pm$ 0.001 & 1.0 & 1.0 & 32.90 $\\pm$ 5.65 & 26.24 $\\pm$ 4.49 \\\\\nG006.79$-$00.25 & 20.87 & 4.75 $\\pm$ 0.25 & 8.789 $\\pm$ 0.088 & 0.484 $\\pm$ 0.005 & 0.242 $\\pm$ 0.002 & 1.3 & 1.1 & 45.82 $\\pm$ 8.39 & 19.46 $\\pm$ 3.63 \\\\\nG010.32$-$00.15 & 11.99 & 5.34 $\\pm$ 0.29 & 5.039 $\\pm$ 0.050 & 0.237 $\\pm$ 0.002 & 0.111 $\\pm$ 0.001 & 1.1 & 1.1 & 50.95 $\\pm$ 11.58 & 21.55 $\\pm$ 5.05 \\\\\nG019.36$-$00.03 & 26.40 & 5.58 $\\pm$ 0.49 & 6.789 $\\pm$ 0.068 & 0.332 $\\pm$ 0.003 & 0.141 $\\pm$ 0.001 & 1.2 & 1.1 & 56.41 $\\pm$ 14.37 & 21.19 $\\pm$ 5.53 \\\\\nG024.78$+$00.08 & 110.72 & 3.51 $\\pm$ 0.15 & 8.038 $\\pm$ 0.080 & 0.554 $\\pm$ 0.006 & 0.322 $\\pm$ 0.003 & 1.3 & 1.2 & 32.56 $\\pm$ 4.25 & 16.08 $\\pm$ 2.12 \\\\\nG028.39$+$00.08 & 78.01 & 4.83 $\\pm$ 0.17 & 8.705 $\\pm$ 0.087 & 0.443 $\\pm$ 0.004 & 0.291 $\\pm$ 0.003 & 1.2 & 1.2 & 37.04 $\\pm$ 10.05 & 18.74 $\\pm$ 5.12 \\\\\nG028.83$-$00.25 & 87.19 & 4.50 $\\pm$ 0.48 & 7.283 $\\pm$ 0.073 & 0.388 $\\pm$ 0.004 & 0.183 $\\pm$ 0.002 & 1.2 & 1.1 & 47.93 $\\pm$ 10.00 & 19.73 $\\pm$ 4.16 \\\\\nG030.70$-$00.06 & 89.94 & 4.20 $\\pm$ 0.10 & 7.088 $\\pm$ 0.071 & 0.448 $\\pm$ 0.004 & 0.263 $\\pm$ 0.003 & 1.2 & 1.1 & 33.39 $\\pm$ 2.77 & 18.07 $\\pm$ 1.52 \\\\\nG030.74$-$00.04 & 91.82 & 5.76 $\\pm$ 0.36 & 5.924 $\\pm$ 0.059 & 0.404 $\\pm$ 0.004 & 0.191 $\\pm$ 0.002 & 1.2 & 1.1 & 37.63 $\\pm$ 10.87 & 15.61 $\\pm$ 4.59 \\\\\nG030.81$-$00.05 & 98.89 & 5.73 $\\pm$ 0.24 & 4.691 $\\pm$ 0.047 & 0.306 $\\pm$ 0.003 & 0.187 $\\pm$ 0.002 & 1.2 & 1.1 & 29.07 $\\pm$ 2.99 & 14.97 $\\pm$ 1.44 \\\\\nW51-IRS2 & 61.07 & 6.22 $\\pm$ 0.06 & 1.883 $\\pm$ 0.019 & 0.119 $\\pm$ 0.001 & 0.052 $\\pm$ 0.001 & 1.1 & 1.0 & 38.26 $\\pm$ 4.35 & 16.70 $\\pm$ 1.92 \\\\\nDR21 & -2.49 & 8.10 $\\pm$ 0.00 & 7.079 $\\pm$ 0.071 & 0.270 $\\pm$ 0.003 & 0.106 $\\pm$ 0.001 & 1.1 & 1.1 & 76.22 $\\pm$ 7.57 & 27.48 $\\pm$ 2.84 \\\\\nNGC7538 & -57.12 & 9.47 $\\pm$ 0.07 & 3.433 $\\pm$ 0.034 & 0.131 $\\pm$ 0.001 & 0.050 $\\pm$ 0.001 & 1.1 & 1.0 & 73.19 $\\pm$ 12.54 & 26.72 $\\pm$ 4.57 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{ Velocities were obtained from measurements of C$^{34}$S, see Table \\ref{fitting_all} in Appendix \\ref{appendix_table}.}\n\\end{table*}\n\n\\subsection{$^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios derived directly from $^{13}$C$^{34}$S}\n\\label{ratios_13c34s}\n\n\\subsubsection{$^{12}$C\/$^{13}$C ratios}\n\\label{section_results_12c13c}\n\nThe $^{12}$C\/$^{13}$C ratios derived from the integrated intensity ratios of C$^{34}$S and $^{13}$C$^{34}$S with corrections of optical depth are listed in Table~\\ref{table_13c34sresults}. Figure~\\ref{fig_gradient_12C13C} shows our results as filled black circles. A gradient of $^{12}$C\/$^{13}$C is obtained with an unweighted least-squares fit:\n\\begin{equation}\n^{12}{\\rm C}\/^{13}{\\rm C} = (4.77 \\pm 0.81)R_{\\rm GC}+(20.76 \\pm 4.61).\n\\end{equation}\nThe correlation coefficient is 0.82. Around the CMZ toward the $+$50 km~s$^{-1}$ cloud and SgrB2, four velocity components of C$^{34}$S and $^{13}$C$^{34}$S were detected and then an average $^{12}$C\/$^{13}$C value of 27~$\\pm$~3 is derived. The uncertainties given here and below are standard deviations of the mean. Eleven objects within a range of 3.50 kpc < $R_{\\rm GC}$ < 6.50 kpc in the inner Galactic disk lead to an average $^{12}$C\/$^{13}$C value of 41~$\\pm$~9. In the Local arm near the Sun, the $^{13}$C$^{34}$S lines were detected toward two sources, Orion-KL and DR21. These provide an average $^{12}$C\/$^{13}$C value of 66~$\\pm$~10, which is lower than the Solar System ratio. The other two targets beyond the solar neighborhood belong to the Perseus arm and show a slightly higher value of 74~$\\pm$~8.\n\nFor sources with detections of C$^{34}$S and nondetections of $^{13}$C$^{34}$S, 3$\\sigma$ lower limits of the $^{12}$C\/$^{13}$C ratio have been derived and are shown as open black circles in Fig.~\\ref{fig_gradient_12C13C}. All these lower limits are below the $^{12}$C\/$^{13}$C gradient we describe above. \n\n\n\n\n\n\\subsubsection{$^{32}$S\/$^{34}$S ratios}\n\\label{section_ratios3234_13c34s}\n\nThe $^{32}$S\/$^{34}$S ratios directly derived from the integrated intensity ratios of $^{13}$CS\/$^{13}$C$^{34}$S from the $J$ = 2-1 lines with corrections of optical depth are listed in Table~\\ref{table_13c34sresults} and are plotted as a function of galactocentric distance in Fig.~\\ref{fig_gradient_32S34S}. With an unweighted least-squares fit, a gradient with a correlation coefficient of 0.47 can be obtained:\n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} =(0.73 \\pm 0.36)R_{\\rm GC}+(16.50 \\pm 2.07).\n\\end{equation}\nAn average $^{32}$S\/$^{34}$S ratio of 19~$\\pm$~2 is obtained in the CMZ, which is based on the measurements from two sources, namely the $+$50 km~s$^{-1}$ cloud next to the Galactic center and Sgr B2. In the inner Galactic disk at a range of 3.50 kpc < $R_{\\rm GC}$ < 6.50 kpc, $^{13}$C$^{34}$S was detected toward 11 objects, leading to an average $^{32}$S\/$^{34}$S value of 18~$\\pm$~4. For sources in the Local and the Perseus arm beyond the Sun, the $^{32}$S\/$^{34}$S ratios are 24~$\\pm$~4 and 28~$\\pm$~3, respectively. This reveals a gradient from the inner Galactic disk to the outer Galaxy, but none from the CMZ to the inner disk.\n\nFor sources with detections of $^{13}$CS and nondetections of $^{13}$C$^{34}$S, we determined 3$\\sigma$ lower limits to the $^{32}$S\/$^{34}$S ratio, which are shown as open black circles in Fig.~\\ref{fig_gradient_32S34S}. All these lower limits are below the $^{32}$S\/$^{34}$S gradient we describe above. \n\n\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[height=600pt]{newFigures\/1Allratio32s_34s_new-lower.pdf}\n \\caption{$^{32}$S\/$^{34}$S isotope ratios as functions of the distance from the Galactic center. The symbol $\\odot$ indicates the $^{32}$S\/$^{34}$S isotope ratio in the Solar System. In the upper panel, the $^{32}$S\/$^{34}$S ratios directly derived from $^{13}$CS\/$^{13}$C$^{34}$S in the $J$ = 2-1 transition and obtained from the double isotope method in the $J$ = 2-1 transition with corrections for optical depth are plotted as black and green dots, respectively. The $^{32}$S\/$^{34}$S ratios without corrections of opacity in the $J$ = 3-2 transition are plotted as light blue dots. The 3$\\sigma$ lower limits of $^{32}$S\/$^{34}$S ratios obtained from nondetections of $^{13}$C$^{34}$S in the current work are shown as open black circles. The $^{32}$S\/$^{34}$S ratios in \\citet{2020ApJ...899..145Y} derived from the double isotope method in the $J$ = 2-1 transitions are shown as blue dots in the lower panel. The $^{32}$S\/$^{34}$S values in the CMZ obtained from $^{13}$CS\/$^{13}$C$^{34}$S in \\citet{2020A&A...642A.222H} are plotted as red stars in both panels. The resulting first-order polynomial fits to $^{32}$S\/$^{34}$S ratios with the direct method and the double isotope method from the $J$ = 2-1 transition in this work are plotted as black and red solid lines in the two panels, respectively, with the gray and yellow shaded areas showing the 1~$\\sigma$ intervals of the fits. The magenta and cyan dashed-dotted lines show the $^{32}$S\/$^{34}$S gradients from \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n \\label{fig_gradient_32S34S}\n\\end{figure*}\n\n\n\n\\subsection{$^{32}$S\/$^{34}$S ratios obtained through the double isotope method}\n\\label{section_double_32s34s}\n\nThe $^{32}$S\/$^{34}$S values can also be derived from measurements of C$^{34}$S and $^{13}$CS using the carbon gradient obtained from our $^{13}$C$^{34}$S measurements above by applying the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{34}{\\rm S}} = R_{\\rm C} \\frac{I(^{13}{\\rm CS})}{I(\\rm C^{34}{\\rm S})},\n\\end{equation}\nwhere $R_{\\rm C}$ is the $^{12}$C\/$^{13}$C ratio derived from equation (6). The uncertainty on this latter is also included in our error budget. The $^{32}$S\/$^{34}$S ratios in the $J$ = 2-1 transitions were calculated with corrections of optical depth for 83 targets with 90 radial velocity components, in which the C$^{34}$S and $^{13}$CS $J$ = 2-1 lines were both detected, and are listed in Column 7 of Table \\ref{table_doubleisotope}. An unweighted least-squares fit to these values yields \n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} (2-1) = (0.75 \\pm 0.13)R_{\\rm GC}+(15.52 \\pm 0.78), \n\\end{equation}\nwith a correlation coefficient of 0.54. The C$^{34}$S and $^{13}$CS $J$ = 3-2 lines were both detected in 71 objects with 73 radial velocity components. The $^{32}$S\/$^{34}$S ratios derived with equation (8) from the $J$ = 3-2 transition are shown in Column 8 of Table \\ref{table_doubleisotope}. An unweighted least-squares fit to the $J$ = 3-2 transition yields \n\\begin{equation}\n^{32}{\\rm S}\/^{34}{\\rm S} (3-2) = (0.99 \\pm 0.14)R_{\\rm GC}+(16.05 \\pm 0.95), \n\\end{equation}\nwhich is within the errors, and is consistent with the trend obtained from the $J$ = 2-1 transition (equation (9)). However, we note that we do not have CS $J$ = 3-2 data, and therefore no opacity corrections could be applied to our C$^{34}$S $J$ = 3-2 spectra (see also Sect.~\\ref{section_34s33s}).\n\n\n\n\n\n\n\\subsection{$^{34}$S\/$^{33}$S ratios}\n\\label{section_34s33s}\n\nThe $^{34}$S\/$^{33}$S ratios can be determined directly from the intensity ratios of C$^{34}$S\/C$^{33}$S. The $^{34}$S\/$^{33}$S ratios from the $J$ = 2-1 lines were then corrected for optical depths derived in Section~\\ref{section_opacities}. However, both the $J$ = 2-1 and $J$ = 3-2 transitions of C$^{33}$S are split by hyperfine structure (HFS) interactions \\citep{1981CPL....81..256B}, which may affect the deduced values of $^{34}$S\/$^{33}$S. \n\nThe C$^{33}$S $J$ = 2-1 line consists of eight hyperfine components distributed over about 9.0 MHz \\citep{2005JMoSt.742..215M,2016JMoSp.327...95E}, which corresponds to a velocity range of about 28~km~s$^{-1}$. Following the method introduced in Appendix D in \\citet{2021A&A...646A.170G}, and assuming that the intrinsic width of each HFS line is 1~km~s$^{-1}$, the $J$ = 2-1 line profile can be reproduced by four components (see Fig.~\\ref{fig_c33s_hfs}, upper left panel). In this case, the main component ($I_{main}$) consists of four HFS lines ($F$=7\/2-5\/2, $F$=5\/2-3\/2, $F$=1\/2-1\/2, $F$=3\/2-5\/2), which account for 70\\% of the total intensity. Among the 46 sources with detections of the C$^{33}$S $J$ = 2-1 line, all of the four components were detected in 10 targets. Toward 16 objects, only the three components with the lowest velocities were detected, accounting for 98\\% of the total intensity. For the remaining 20 sources, only the main component was detected. Based on the above assumptions, 30\\% of the total intensity would be missed. The situation is different when the main component becomes broader. If the line width of the main component is larger than 10~km~s$^{-1}$, 87\\% of the total intensity is covered by the main spectral feature. When the line width of the main component is larger than 19.0~km~s$^{-1}$, then almost all HFS lines are included. In Fig.~\\ref{fig_model_hfs}, we show the dependence of the HFS factor for the $J$ = 2-1 line, $f_{\\rm 21HFS}$, on the line width of the main component. Depending on the specific condition of each target, we derived $f_{\\rm 21HFS}$ for each source. The values are listed in Table~\\ref{table_3433}. \n\nThe C$^{33}$S $J$ = 3-2 line consists of nine hyperfine components covering about 8.0 MHz \\citep{2005JMoSt.742..215M,2016JMoSp.327...95E}, corresponding to a velocity range of about 16~km~s$^{-1}$. Assuming that the intrinsic width of each HFS line is 1~km~s$^{-1}$, the $J$ = 3-2 line profile can be characterized by three components (see also Fig.~\\ref{fig_c33s_hfs}). All these three components are detected in only two sources, Orion-KL and DR21. The main component consists of four HFS lines ($F$=5\/2-3\/2, $F$=3\/2-1\/2, $F$=7\/2-5\/2, $F$=9\/2-7\/2), which account for 86\\% of the total intensity. When the line width becomes larger than 9.2~km~s$^{-1}$, almost all of the HFS lines overlap. The HFS factors ($f_{\\rm 32HFS}$) of the $J$ = 3-2 transition obtained individually for each source are listed in Table~\\ref{table_3433}.\n\nWe calculated the $^{34}$S\/$^{33}$S intensity ratios and present them in Table~\\ref{table_3433}. Applying corrections accounting for the effect of hyperfine splitting, the $^{34}$S\/$^{33}$S ratios are derived and are \nalso listed in Table~\\ref{table_3433}. The $^{34}$S\/$^{33}$S values obtained from the $J$ = 2-1 transitions are always higher than the ones derived from the $J$ = 3-2 lines toward the same source, with the exception of G097.53$+$03.18. This difference could be caused by the lack of corrections of optical depth in the $J$ = 3-2 transition. The average values of $^{34}$S\/$^{33}$S toward our sample are 4.35~$\\pm$~0.44 and 3.49~$\\pm$~0.26 in the $J$ = 2-1 and $J$ = 3-2 transitions, respectively. The $^{34}$S\/$^{33}$S ratios were found to be independent of galactocentric distance (Fig.~\\ref{fig_all33S}). \n\nAfter applying the opacity correction of the $J$ = 2-1 transition to the $J$ = 3-2 line in the same source, $^{34}$S\/$^{33}$S ratios in the $J$ = 3-2 transition are higher than the $^{34}$S\/$^{33}$S values in the $J$ = 2-1 transition in seven targets, suggesting that the C$^{34}$S $J$ = 3-2 lines in these seven sources are less opaque than the C$^{34}$S $J$ = 2-1 lines. The seven targets are the $+$20~km~s$^{-1}$~cloud, G023.43$-$00.18, G028.39$+$00.08, G028.83$-$00.25, G073.65$+$00.19, G097.53$+$03.18, and G109.87$+$02.11. A comparison of the corrected $^{34}$S\/$^{33}$S ratios in these two transitions is shown in Fig.~\\ref{fig_33S_compared}. On the other hand, toward the other 32 objects of the whole sample of 39 sources with detections in these two transitions, the $^{34}$S\/$^{33}$S ratios in the $J$ = 3-2 transition are still lower than the $^{34}$S\/$^{33}$S values in the $J$ = 2-1 transitions. This suggests that the C$^{34}$S $J$ = 3-2 lines may be more opaque than the C$^{34}$S $J$ = 2-1 lines. The ratios of $^{34}$S\/$^{33}$S values without corrections of opacity in the $J$ = 3-2 transition and the corrected $^{34}$S\/$^{33}$S values from the $J$ = 2-1 lines in 31 targets of these 32 sources are within the range of 1.02 to 1.71, which suggest that the optical depths of C$^{34}$S in the $J$ = 3-2 transition in these objects range from 0.05 to 1.20 based on equation (4). The maximum optical depth of the C$^{34}$S $J$ = 3-2 line toward the additional source, G024.85$+$00.08, which has not considered until now, is estimated to be 2.6.\n\n\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[width=500pt]{figure\/hfs_c33s.pdf}\n \\caption{Synthetic C$^{33}$S (2-1) and C$^{33}$S (3-2) spectra for two intrinsic line widths, 1.0~km~s$^{-1}$ (left panels) and 4.0~km~s$^{-1}$ (right panels).}\n \\label{fig_c33s_hfs}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[width=230pt]{figure\/plot_fHFS21.pdf}\n \\includegraphics[width=230pt]{figure\/plot_fHFS32.pdf}\n \\caption{Blue lines are curves showing the theoretical dependencies of the HFS factors, $f_{\\rm 21HFS}$ (left) and $f_{\\rm 32HFS}$ (right), on the line width of the C$^{33}$S main component for sources in which only the main component was detected. The red dotted vertical and horizontal lines indicate the values of the minimal line widths where the factors are reaching almost 1.0, 19.0~km~s$^{-1}$ for the $J$ = 2-1 transition ($f_{\\rm 21HFS} \\sim$ 0.99) and 9.2~km~s$^{-1}$ for the $J$ = 3-2 transition ($f_{\\rm 32HFS} \\sim$ 0.99). }\n \\label{fig_model_hfs}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{$^{32}$S\/$^{33}$S ratios}\n\\label{section_32s33s}\n\nThe $^{32}$S\/$^{33}$S values can also be derived from the $^{34}$S\/$^{33}$S ratios in Section~\\ref{section_34s33s} using the $^{32}$S\/$^{34}$S ratios ---which we directly obtained from $^{13}$CS\/$^{13}$C$^{34}$S and present in Section~\\ref{section_ratios3234_13c34s}--- by applying the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{33}{\\rm S}} = \\frac{^{34}\\rm S}{^{33}\\rm S} \\times \\frac{^{32}\\rm S}{^{34}\\rm S} .\n\\end{equation}\nFor sources where we did not detect $^{13}$C$^{34}$S, the $^{32}$S\/$^{34}$S ratios derived from the double isotope method in Section~\\ref{section_double_32s34s} are used. Their uncertainty is also included in our error budget. The resulting $^{32}$S\/$^{33}$S ratios are listed in Table~\\ref{table_3233}. As in the case of the $^{34}$S\/$^{33}$S ratios, the $^{32}$S\/$^{33}$S values obtained from the $J$ = 2-1 transitions with corrections for optical depth are slightly larger than the ones from the $J$ = 3-2 transitions without opacity corrections toward the same source. \n\nIn the CMZ, $^{32}$S\/$^{33}$S $J$ = 2-1 ratios toward four targets, namely the $+$20 km~s$^{-1}$ cloud, the $+$50 km~s$^{-1}$ cloud, Sgr B2, and Sgr D, lead to an average value of 70 $\\pm$ 16. In the inner Galaxy, in a galactocentric distance range of 2.0~kpc~$\\le R_{\\rm GC} \\le$~6.0~kpc, an average $^{32}$S\/$^{33}$S ratio of 82 $\\pm$ 19 was derived from values in 20 sources. Near the solar neighborhood, in a galactocentric distance range of 7.5~kpc~$\\le R_{\\rm GC} \\le$~8.5~kpc, $^{32}$S\/$^{33}$S ratios were obtained toward four objects, Orion-KL, G071.31$+$00.82, DR21, and G109.87$+$02.11, resulting in an average value of 88 $\\pm$ 21. For the outer Galaxy, beyond the local arm, we were able to deduce an average $^{32}$S\/$^{33}$S ratio of 105 $\\pm$ 19 from $^{32}$S\/$^{33}$S values in four sources. All these average values provide us with an indication of the existence of a $^{32}$S\/$^{33}$S gradient in the Galaxy. An unweighted least-squares fit to the $J$ = 2-1 transition data from 46 sources yields\n\\begin{equation}\n^{32}{\\rm S}\/^{33}{\\rm S} (2-1) = (2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57). \n\\end{equation}\nIn the $J$ = 3-2 transition with data from 42 targets, an unweighted least-squares fit can be obtained with\n\\begin{equation}\n^{32}{\\rm S}\/^{33}{\\rm S} (3-2) = (2.80 \\pm 0.59)R_{\\rm GC}+(59.30 \\pm 4.22). \n\\end{equation}\nThe $^{32}$S\/$^{33}$S gradients derived from these two transitions have similar slopes but obviously different intercepts. The lower intercept in the $J$ = 3-2 transition could be due to the fact that we could not correct the values for optical-depth effects. If this difference is caused only by the opacity, then an average optical depth in the C$^{34}$S $J$ = 3-2 transition of about 0.4 can be derived with equation (4). \n\n\\setcounter{table}{3}\n\n\\begin{table*}[h]\n\\centering\n\\caption{\\label{table_3233} $^{32}$S\/$^{33}$S isotope ratios}\n\\begin{tabular}{lccc}\n\\hline\\hline\nSource & $R_{GC}$ & \\multicolumn{2}{c}{$^{32}$S\/$^{33}$S} \\\\\n & (kpc) & $J$ = 2-1 & $J$ = 3-2 \\\\\n\\hline\nWB89-380 & 14.19 $\\pm$ 0.92 & 90.74 $\\pm$ 19.53 & 83.93 $\\pm$ 17.90 \\\\\nWB89-391 & 14.28 $\\pm$ 0.94 & 127.07 $\\pm$ 40.26 & 65.46 $\\pm$ 14.69 \\\\\nW3OH & 9.64 $\\pm$ 0.03 & 105.49 $\\pm$ 7.26 & 74.20 $\\pm$ 4.35 \\\\\nOrion-KL & 8.54 $\\pm$ 0.00 & 74.50 $\\pm$ 16.85 & 69.12 $\\pm$ 13.51 \\\\\n$+$20 km~s$^{-1}$ cloud & 0.03 $\\pm$ 0.03 & 82.63 $\\pm$ 18.20 & 78.98 $\\pm$ 17.86 \\\\\nG359.61$-$00.24 & 5.51 $\\pm$ 0.15 & 75.75 $\\pm$ 14.99 & 71.69 $\\pm$ 16.86 \\\\\n$+$50 km~s$^{-1}$ cloud & 0.02 $\\pm$ 0.04 & 76.49 $\\pm$ 16.96 & 66.68 $\\pm$ 14.92 \\\\\nG000.31$-$00.20 & 5.25 $\\pm$ 0.36 & 55.61 $\\pm$ 14.84 & $\\cdots$ \\\\\nSgrB2 & 0.44 $\\pm$ 0.70 & 42.98 $\\pm$ 9.62 & 35.66 $\\pm$ 7.84 \\\\\nSgrD & 0.45 $\\pm$ 0.07 & 77.48 $\\pm$ 20.88 & 50.61 $\\pm$ 13.66 \\\\\nG006.79$-$00.25 & 4.75 $\\pm$ 0.25 & 86.65 $\\pm$ 17.01 & 70.08 $\\pm$ 13.79 \\\\\nG007.47$+$00.05 & 12.35 $\\pm$ 2.49 & 107.98 $\\pm$ 30.91 & $\\cdots$ \\\\\nG010.32$-$00.15 & 5.34 $\\pm$ 0.29 & 88.27 $\\pm$ 17.82 & 79.56 $\\pm$ 15.99 \\\\\nG010.62$-$00.33 & 4.26 $\\pm$ 0.21 & 106.56 $\\pm$ 33.99 & $\\cdots$ \\\\\nG011.10$-$00.11 & 5.49 $\\pm$ 0.56 & 72.13 $\\pm$ 29.12 & $\\cdots$ \\\\\nG016.86$-$02.15 & 5.97 $\\pm$ 0.47 & 86.63 $\\pm$ 17.00 & 76.23 $\\pm$ 15.01 \\\\\nG017.02$-$02.40 & 6.40 $\\pm$ 0.36 & 100.03 $\\pm$ 20.54 & 86.36 $\\pm$ 17.47 \\\\\nG017.63$+$00.15 & 6.77 $\\pm$ 0.04 & 73.12 $\\pm$ 23.04 & $\\cdots$ \\\\\nG018.34$+$01.76 & 6.31 $\\pm$ 0.07 & 92.66 $\\pm$ 18.57 & 86.44 $\\pm$ 16.87 \\\\\nG019.36$-$00.03 & 5.58 $\\pm$ 0.49 & 94.55 $\\pm$ 18.78 & 80.16 $\\pm$ 15.89 \\\\\nG023.43$-$00.18 & 3.63 $\\pm$ 0.49 & 72.63 $\\pm$ 16.26 & 80.58 $\\pm$ 16.63 \\\\\nG024.78$+$00.08 & 3.51 $\\pm$ 0.15 & 70.88 $\\pm$ 14.22 & 55.64 $\\pm$ 11.12 \\\\\nG024.85$+$00.08 & 3.85 $\\pm$ 0.23 & 134.87 $\\pm$ 43.15 & 48.25 $\\pm$ 11.88 \\\\\nG028.30$-$00.38 & 4.71 $\\pm$ 0.26 & $\\cdots$ & 76.82 $\\pm$ 20.06 \\\\\nG028.39$+$00.08 & 4.83 $\\pm$ 0.17 & 85.30 $\\pm$ 17.17 & 76.11 $\\pm$ 15.02 \\\\\nG028.83$-$00.25 & 4.50 $\\pm$ 0.48 & 78.57 $\\pm$ 15.87 & 80.37 $\\pm$ 15.83 \\\\\nG030.70$-$00.06 & 4.20 $\\pm$ 0.10 & 76.54 $\\pm$ 15.18 & 62.68 $\\pm$ 12.57 \\\\\nG030.74$-$00.04 & 5.76 $\\pm$ 0.36 & 86.28 $\\pm$ 17.07 & 72.73 $\\pm$ 14.12 \\\\\nG030.78$+$00.20 & 4.19 $\\pm$ 0.05 & $\\cdots$ & 72.50 $\\pm$ 17.46 \\\\\nG030.81$-$00.05 & 5.73 $\\pm$ 0.24 & 88.21 $\\pm$ 17.23 & 72.72 $\\pm$ 14.52 \\\\\nG031.24$-$00.11 & 7.48 $\\pm$ 2.00 & $\\cdots$ & 90.57 $\\pm$ 21.01 \\\\\nG032.74$-$00.07 & 4.55 $\\pm$ 0.23 & 75.67 $\\pm$ 14.99 & 68.63 $\\pm$ 13.88 \\\\\nG032.79$+$00.19 & 5.26 $\\pm$ 1.57 & 79.87 $\\pm$ 16.07 & 76.72 $\\pm$ 14.93 \\\\\nG034.41$+$00.23 & 5.99 $\\pm$ 0.06 & 80.54 $\\pm$ 15.82 & 72.63 $\\pm$ 14.63 \\\\\nG034.79$-$01.38 & 6.21 $\\pm$ 0.09 & 103.60 $\\pm$ 22.03 & 82.58 $\\pm$ 16.21 \\\\\nG036.11$+$00.55 & 5.45 $\\pm$ 0.43 & 39.43 $\\pm$ 13.64 & $\\cdots$ \\\\\nG037.42$+$01.51 & 6.78 $\\pm$ 0.05 & 110.92 $\\pm$ 21.76 & 76.06 $\\pm$ 17.77 \\\\\nG040.28$-$00.21 & 6.02 $\\pm$ 0.10 & 110.40 $\\pm$ 31.44 & 71.86 $\\pm$ 17.32 \\\\\nG045.45$+$00.06 & 6.41 $\\pm$ 0.50 & 83.04 $\\pm$ 21.44 & 76.44 $\\pm$ 16.33 \\\\\nG048.99$-$00.29 & 6.18 $\\pm$ 0.02 & 107.90 $\\pm$ 27.60 & 87.21 $\\pm$ 20.50 \\\\\nW51-IRS2 & 6.22 $\\pm$ 0.06 & 74.69 $\\pm$ 14.46 & 66.43 $\\pm$ 12.72 \\\\\nG071.31$+$00.82 & 8.02 $\\pm$ 0.16 & 95.37 $\\pm$ 31.90 & 106.87 $\\pm$ 30.60 \\\\\nG073.65$+$00.19 & 13.54 $\\pm$ 2.90 & 102.16 $\\pm$ 32.61 & 128.66 $\\pm$ 31.17 \\\\\nG075.29$+$01.32 & 10.69 $\\pm$ 0.58 & 122.06 $\\pm$ 29.02 & 92.08 $\\pm$ 19.53 \\\\\nDR21 & 8.10 $\\pm$ 0.00 & 86.80 $\\pm$ 16.37 & 80.51 $\\pm$ 15.38 \\\\\nG090.92$+$01.48 & 10.13 $\\pm$ 0.63 & 87.88 $\\pm$ 19.68 & 95.73 $\\pm$ 20.14 \\\\\nG097.53$+$03.18 & 11.81 $\\pm$ 0.70 & 74.99 $\\pm$ 14.50 & 87.92 $\\pm$ 16.65 \\\\\nG109.87$+$02.11 & 8.49 $\\pm$ 0.01 & 94.41 $\\pm$ 18.40 & 98.52 $\\pm$ 23.17 \\\\\nNGC7538 & 9.47 $\\pm$ 0.07 & 104.53 $\\pm$ 19.54 & $\\cdots$ \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The $^{32}$S\/$^{33}$S isotope ratios from the $J$ = 2-1 transition are corrected for the line saturation effects, while the ones in the 3-2 line are not corrected because of the missing main isotopic species.}\n\\end{table*}\n\n\n\n\\subsection{$^{34}$S\/$^{36}$S ratios}\n\\label{section_34s36s}\n\nThe detection of C$^{36}$S in the $J$ = 2-1 and $J$ = 3-2 transitions allows us to also calculate $^{34}$S\/$^{36}$S ratios. Around the CMZ, the C$^{36}$S $J$ = 2-1 line in the $+$50 km~s$^{-1}$ cloud was detected, resulting in a $^{34}$S\/$^{36}$S ratio of 41~$\\pm$~4. In the local arm toward DR21 and Orion-KL \\citep{2007A&A...474..515M,2013ApJ...769...15X}, the C$^{36}$S $J$ = 2-1 and $J$ = 3-2 transitions were detected, respectively, leading to $^{34}$S\/$^{36}$S values of 117~$\\pm$~24 and 83~$\\pm$~7. This yields an average $^{34}$S\/$^{36}$S ratio of 100~$\\pm$~16 in the ISM near the Sun. In the Perseus arm beyond the Solar System \\citep{2006Sci...311...54X}, we detected the C$^{36}$S $J$ = 2-1 line toward W3OH and obtained a $^{34}$S\/$^{36}$S value of 140~$\\pm$~16. These results reveal the possibility of the existence of a $^{34}$S\/$^{36}$S gradient from the Galactic center region to the outer Galaxy. Five tentative detections in the C$^{36}$S $J$ = 2-1 line and also five tentative detections in the $J$ = 3-2 line provide additional $^{34}$S\/$^{36}$S ratios but with large uncertainties. All of the $^{34}$S\/$^{36}$S values are listed in Table~\\ref{table_all36s} and are plotted as a function of the distance to the Galactic center in Fig.~\\ref{fig_all36S}. \n\n\n\\subsection{$^{32}$S\/$^{36}$S ratios}\n\\label{section_32s36s}\n\nAs in the case of the $^{32}$S\/$^{33}$S ratios, the $^{32}$S\/$^{36}$S values could also be obtained from the resulting $^{34}$S\/$^{36}$S ratios in Section~\\ref{section_34s36s} and the $^{32}$S\/$^{34}$S ratios in Section~\\ref{section_ratios3234_13c34s} using the following equation:\n\\begin{equation}\n\\frac{^{32}{\\rm S}}{^{36}{\\rm S}} = \\frac{^{34}\\rm S}{^{36}\\rm S} \\times \\frac{^{32}\\rm S}{^{34}\\rm S}.\n\\end{equation}\nThe uncertainties of both isotope ratios in the product on the right-hand side of the equation are included in our error budget. The resulting $^{32}$S\/$^{36}$S ratios are listed in Table~\\ref{table_all36s}. In the CMZ, a $^{32}$S\/$^{36}$S ratio of 884~$\\pm$~104 is obtained toward the $+$50 km~s$^{-1}$ cloud. $^{32}$S\/$^{36}$S values of 1765~$\\pm$~414 and 3223~$\\pm$~742 are derived toward Orion-KL and DR21, leading to an average $^{32}$S\/$^{36}$S value of 2494 $\\pm$ 578 in the local regions near the Solar System. In the Perseus arm beyond the solar neighborhood toward W3OH, we obtain the highest $^{32}$S\/$^{36}$S value, 4181~$\\pm$~531. All these results indicate that there could be a positive $^{32}$S\/$^{36}$S gradient from the Galactic center region to the outer Galaxy. Figure~\\ref{fig_all36S} shows the $^{32}$S\/$^{36}$S ratios plotted as a function of the distance to the Galactic center.\n\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[height=620pt]{newFigures\/0Allratio33s-corrected.pdf}\n \\caption{ $^{34}$S\/$^{33}$S and $^{32}$S\/$^{33}$S isotope ratios (for the latter, see equations (12) and (13)) plotted as functions of the distance from the Galactic center. \\textbf{Top:} $^{34}$S\/$^{33}$S ratios derived from C$^{34}$S\/C$^{33}$S in the $J$ = 2-1 and $J$ = 3-2 transitions plotted as black and green dots, respectively. The red solid and the two dashed lines show the average value and its standard deviation, 4.35~$\\pm$~0.44, of $^{34}$S\/$^{33}$S with corrections of optical depth toward our sample in the $J$ = 2-1 transition. The yellow solid and the two dashed lines show the average value and its standard deviation, 3.49~$\\pm$~0.26, of $^{34}$S\/$^{33}$S toward our sample in the $J$ = 3-2 transition. The red symbol $\\odot$ indicates the $^{34}$S\/$^{33}$S isotope ratio in the Solar System. \\textbf{Bottom:} Black and green dots show the $^{32}$S\/$^{33}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions, respectively. The red symbol $\\odot$ indicates the $^{32}$S\/$^{33}$S value in the Solar System. The resulting first-order polynomial fits to the $^{32}$S\/$^{33}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions in this work are plotted as red and green solid lines, respectively, with the pink and yellow shaded area showing the 1~$\\sigma$ standard deviations. The red crosses are the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n \\label{fig_all33S}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[height=270pt]{newFigures\/0Allratio33sCompared2.pdf}\n \\caption{Comparison of $^{34}$S\/$^{33}$S ratios between the $J$ = 2-1 and 3-2 data. The $J$ = 2-1 ratios are opacity corrected, while the same correction factors were also applied to the $J$ = 3-2 data. The red solid line indicates that the $^{34}$S\/$^{33}$S ratios are equal in these two transitions.}\n \\label{fig_33S_compared}\n\\end{figure}\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[height=550pt]{newFigures\/0Allratio36s-corrected.pdf}\n \\caption{ $J$ = 2-1 (opacity corrected) and $J$ = 3-2 (no opacity corrections) $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios (see equations (18) and (19)) plotted as functions of the distance from the Galactic center. \\textbf{Top:} Filled black circles and filled black triangle present the $^{34}$S\/$^{36}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions derived from C$^{34}$S\/C$^{36}$S in this work with detections of C$^{36}$S, respectively. The open gray circles and open gray triangles present the $^{34}$S\/$^{36}$S ratios in the $J$ = 2-1 and $J$ = 3-2 transitions derived from C$^{34}$S\/C$^{36}$S in this work with tentative detections of C$^{36}$S, respectively. The blue diamonds show the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M}. The red symbol $\\odot$ indicates the $^{34}$S\/$^{36}$S isotope ratio in the Solar System. The resulting first-order polynomial fit to the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M} and this work, excluding the values from tentative detections, is plotted as a black solid line, with the green shaded area showing the 1~$\\sigma$ interval of the fit. \\textbf{Bottom:} $^{32}$S\/$^{36}$S ratios obtained from $^{34}$S\/$^{36}$S ratios combined with the $^{32}$S\/$^{34}$S ratios derived in this work. The filled black circles and filled black triangle present the values in the $J$ = 2-1 and $J$ = 3-2 transitions from this work with detections of C$^{36}$S, respectively. The open gray circles and open gray triangles present ratios in the $J$ = 2-1 and $J$ = 3-2 transitions from this work with tentative detections of C$^{36}$S, respectively. The $^{32}$S\/$^{36}$S ratios derived with $^{34}$S\/$^{36}$S values from \\citet{1996A&A...313L...1M} are plotted as blue diamonds. The red symbol $\\odot$ indicates the $^{32}$S\/$^{36}$S isotope ratio in the Solar System. The $^{32}$S\/$^{36}$S gradient, excluding tentative detections, is plotted as a black solid line, with the pink shaded area showing the 1~$\\sigma$ interval of the fit. The red crosses visualize results from the GCE model of \\citet[][see Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. }\n \\label{fig_all36S}\n\\end{figure*}\n\n\\begin{table*}[h]\n\\centering\n\\caption{\\label{table_all36s}Isotope ratios of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S}\n\\begin{tabular}{lc|cc|cc}\n\\hline\\hline\nSource & $R_{\\rm GC}$ & \\multicolumn{2}{c}{$^{34}$S\/$^{36}$S} & \\multicolumn{2}{c}{$^{32}$S\/$^{36}$S} \\\\\n & (kpc) & $J$ = 2-1 & $J$ = 3-2 & $J$ = 2-1 & $J$ = 3-2 \\\\\n\\hline\nW3OH & 9.64 $\\pm$ 0.03 & 140 $\\pm$ 16 & 174 $\\pm$ 29\\tablefootmark{*} & 4181 $\\pm$ 531 & 5195 $\\pm$ 919\\tablefootmark{*} \\\\\nOrion-KL & 8.54 $\\pm$ 0.00 & 92 $\\pm$ 53\\tablefootmark{*} & 83 $\\pm$ 7 & 1954 $\\pm$ 1207\\tablefootmark{*} & 1765 $\\pm$ 414 \\\\\n$+$20 km~s$^{-1}$ cloud & 0.03 $\\pm$ 0.03 & 69 $\\pm$ 11\\tablefootmark{*} & $\\cdots$ & 1015 $\\pm$ 278\\tablefootmark{*} & $\\cdots$ \\\\\n$+$50 km~s$^{-1}$ cloud & 0.02 $\\pm$ 0.04 & 41 $\\pm$ 4 & $\\cdots$ & 884 $\\pm$ 104 & $\\cdots$ \\\\\nG024.78$+$00.08 & 3.51 $\\pm$ 0.15 & 151 $\\pm$ 39\\tablefootmark{*} & 226 $\\pm$ 66\\tablefootmark{*} & 2424 $\\pm$ 699\\tablefootmark{*} & 3640 $\\pm$ 1164\\tablefootmark{*} \\\\\nG030.81$-$00.05 & 5.73 $\\pm$ 0.24 & 55 $\\pm$ 15\\tablefootmark{*} & $\\cdots$ & 829 $\\pm$ 233\\tablefootmark{*} & $\\cdots$ \\\\\nW51-IRS2 & 6.22 $\\pm$ 0.06 & 109 $\\pm$ 27\\tablefootmark{*} & 203 $\\pm$ 45\\tablefootmark{*} & 1814 $\\pm$ 506\\tablefootmark{*} & 3397 $\\pm$ 855\\tablefootmark{*} \\\\\nDR21 & 8.10 $\\pm$ 0.00 & 117 $\\pm$ 24 & 159 $\\pm$ 64\\tablefootmark{*} & 3223 $\\pm$ 742 & 4384 $\\pm$ 1829\\tablefootmark{*} \\\\\nNGC7538 & 9.47 $\\pm$ 0.07 & $\\cdots$ & 109 $\\pm$ 28\\tablefootmark{*} & $\\cdots$ & 2912 $\\pm$ 897\\tablefootmark{*} \\\\\n\\hline\n\\multicolumn{6}{c}{Below are the isotope ratios of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S from \\citet{1996A&A...313L...1M}} \\\\\n\\hline\nW3OH & 9.64 $\\pm$ 0.03 & $\\cdots$ & 181 $\\pm$ 49 & $\\cdots$ & 4118 $\\pm$ 1124 \\\\\nOrion-KL & 8.54 $\\pm$ 0.00 & $\\cdots$ & 104 $\\pm$ 7 & $\\cdots$ & 2281 $\\pm$ 174 \\\\\nIRAS15491\\tablefootmark{**} & 5.48 $\\pm$ 0.31 & $\\cdots$ & 108 $\\pm$ 15 & $\\cdots$ & 2120 $\\pm$ 307 \\\\\nIRAS15520\\tablefootmark{**} & 5.67 $\\pm$ 0.31 & 128 $\\pm$ 32 & 133 $\\pm$ 11 & 2531 $\\pm$ 641 & 2629 $\\pm$ 243 \\\\\nIRAS16172\\tablefootmark{**} & 5.03 $\\pm$ 0.29 & 121 $\\pm$ 18 & 119 $\\pm$ 14 & 2334 $\\pm$ 361 & 2296 $\\pm$ 287 \\\\\nNGC6334A & 6.87 $\\pm$ 0.12 & 108 $\\pm$ 15 & 154 $\\pm$ 20 & 2232 $\\pm$ 322 & 3183 $\\pm$ 431 \\\\\nNGC6334B & 6.87 $\\pm$ 0.12 & 163 $\\pm$ 46 & 156 $\\pm$ 26 & 3369 $\\pm$ 960 & 3225 $\\pm$ 552 \\\\\nW51(M) & 6.22 $\\pm$ 0.004 & 105 $\\pm$ 11 & 104 $\\pm$ 15 & 2119 $\\pm$ 237 & 2099 $\\pm$ 313 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{\\tablefoottext{*}{Values with large uncertainties are derived from the tentative detection of C$^{36}$S lines.} \\tablefoottext{**}{For these three sources without parallax data, their kinematic distances were estimated (for details, see Section~\\ref{section_distance}).} The $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios in this work from the $J$ = 2-1 transition are corrected for the optical depth effects, while the ones for the 3-2 line could not be corrected.}\n\\end{table*}\n\n\\section{Discussion}\n\\label{discussion}\n\nWith the measurements of optically thin lines of the rare CS isotopologs, C$^{34}$S, $^{13}$CS, C$^{33}$S, $^{13}$C$^{34}$S, and C$^{36}$S, we derived $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S isotope ratios. Combined with accurate galactocentric distances, we established a $^{32}$S\/$^{33}$S gradient for the first time and confirmed the existing gradients of $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S, as well as uniform $^{34}$S\/$^{33}$S ratios across the Milky Way, which are lower than previously reported. Furthermore, we may have detected $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S gradients for the first time. In Section~\\ref{section_comparisons_12c13c}, we compare the $^{12}$C\/$^{13}$C gradient obtained in this work with previous published ones derived from a variety of molecular species. A comparison between the $^{32}$S\/$^{34}$S gradients we obtained and previously published ones is presented in Section~\\ref{section_comparisons_32s34s}. The condition of LTE for C$^{33}$S with its HFS line ratios is discussed in Section~\\ref{section_hfs_c33s}. We then also compare our results on $^{34}$S\/$^{33}$S ratios with previously published values and discuss the $^{32}$S\/$^{33}$S gradient. In Section~\\ref{section_discussion_all36} we evaluate whether or not $^{34}$S\/$^{36}$S, $^{33}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S ratios may show gradients with galactocentric distance. Observational bias due to distance effects, beam size effects, and chemical fractionation are discussed. A comparison of several isotopes with respect to primary or secondary synthesis is provided in Section~\\ref{section_discussion_all}. Results from a Galactic chemical evolution (GCE) model, trying to simulate the observational data, are presented in Section~\\ref{section_discussion_model}.\n\n\n\n\n\n\\subsection{Comparisons of $^{12}$C\/$^{13}$C ratios determined with different species}\n\\label{section_comparisons_12c13c}\n\n$^{12}$C\/$^{13}$C ratios have been well studied in the CMZ where the value is about 20--25 \\citep[e.g.,][]{1983A&A...127..388H,1985A&A...149..195G,1990ApJ...357..477L,2005ApJ...634.1126M,2010A&A...523A..51R,2013A&A...559A..47B,2017ApJ...845..158H,2020A&A...642A.222H}, similar to the results that we obtained from C$^{34}$S in the current work. In the inner Galaxy, the $^{12}$C\/$^{13}$C ratios are higher than the values in the CMZ, which were $\\sim$50 as derived from H$_2^{12}$CO\/H$_2^{13}$CO in \\citet{1985A&A...143..148H} and 41~$\\pm$~9 from C$^{34}$S\/$^{13}$C$^{34}$S in this work. As can be inferred from Fig.~\\ref{fig_gradient_12C13C}, $^{12}$C\/$^{13}$C ratios at 3.0 kpc $\\le R_{\\rm GC} \\le$ 4.0 kpc might be as low as in the CMZ, but this is so far tentative and uncertain, as we only have one detection (G024.78$+$00.08) in this region and data from other groups referring to this small galactocentric interval are also relatively few. In the local regions near the Solar System, an average $^{12}$C\/$^{13}$C ratio of 66~$\\pm$~10 was obtained from C$^{34}$S\/$^{13}$C$^{34}$S in this work, which is consistent with $^{12}$C\/$^{13}$C values derived from other molecular species and their $^{13}$C isotopes, that are 75~$\\pm$~9 from C$^{18}$O \\citep{1998ApJ...494L.107K}, 60~$\\pm$~19 from CN \\citep{2005ApJ...634.1126M}, 74~$\\pm$~8 from CH$^+$ \\citep{2011ApJ...728...36R}, and 53~$\\pm$~16 from H$_2$CO \\citep{2019ApJ...877..154Y}. All these $^{12}$C\/$^{13}$C values for the solar neighborhood are well below the value for the Sun \\citep[89,][]{1989GeCoA..53..197A,2007ApJ...656L..33M}. This indicates that $^{13}$C has been enriched in the local ISM during the last 4.6 billion years following the formation of the Solar System. Beyond the Sun, at a galactocentric distance of about 10~kpc, our results from C$^{34}$S\/$^{13}$C$^{34}$S show a slightly higher value of 67~$\\pm$~8, which is similar to $^{12}$C\/$^{13}$C ratios from CN (66~$\\pm$~20), C$^{18}$O (69~$\\pm$~10), and H$_2$CO (64~$\\pm$~10). These values are still below the value for the Sun. In the far outer Galaxy at 13.8~kpc toward WB89~437, \\citet{1996A&AS..119..439W} found a 3$\\sigma$ lower limit of 201~$\\pm$~15 from C$^{18}$O\/$^{13}$C$^{18}$O. This suggests that the $^{12}$C\/$^{13}$C gradient extends well beyond the solar neighborhood to the outer Galaxy. Additional sources with large galactocentric distances have to be measured to further improve the statistical significance of this result.\n\nPreviously published $^{12}$C\/$^{13}$C ratios derived from CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$ \\citep{2011ApJ...728...36R}, and CH \\citep{2020A&A...640A.125J} are shown in Fig.~\\ref{fig_gradient_12C13C}, but with respect to the new distance values (see details in Section~\\ref{section_distance}). In Fig.~\\ref{distribution_12C13C}, the $^{12}$C\/$^{13}$C values from different molecular species are projected onto the Galactic plane. This also visualizes the gradient from the Galactic center to the Galactic outer regions beyond the Solar System. In Table~\\ref{table_all12C13Cgradients}, the fitting results for the old and new distances are presented. The comparison shows that the adoption of the new distances has indeed an effect on the fitting results, such as for the $^{12}$CN\/$^{13}$CN gradient. In \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}, the slope and intercept become (6.75~$\\pm$~1.44) and (5.77~$\\pm$~11.29) from (6.01~$\\pm$~1.19) and (12.28~$\\pm$~9.33), respectively. The fitting for H$_2^{12}$CO\/H$_2^{13}$CO from \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y} becomes (5.43~$\\pm$~1.04) and (13.87~$\\pm$~6.38) from (5.08~$\\pm$~1.10) and (11.86~$\\pm$~6.60), respectively. The Galactic $^{12}$C\/$^{13}$C gradients derived based on measurements of CN, C$^{18}$O, and H$_2$CO are in agreement with our results from C$^{34}$S and therefore indicate that chemical fractionation has little effect on the $^{12}$C\/$^{13}$C ratios. It is noteworthy that all these fits show a significant discrepancy with observations from the Galactic center. Indeed, they suggest values of 5--17 at $R_{\\rm GC}$=0, substantially below observed values of 20--25 (see also Tables~\\ref{table_all12C13Cgradients}, \\ref{table_allratios}). While the values in the CMZ are clearly lower than in the inner Galactic disk (with the potential exception at galactocentric distances of 2.0--4.0 kpc), they are larger than suggested by a linear fit encompassing the entire inner 12.0 kpc of the Galaxy.\n\n\n\n\\subsection{The $^{32}$S\/$^{34}$S gradient across the Milky Way}\n\\label{section_comparisons_32s34s}\n\nThe existence of a $^{32}$S\/$^{34}$S gradient was first proposed by \\citet{1996A&A...305..960C} based on observations of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines toward 20 mostly southern HMSFRs with galactocentric distances of between 3.0 and 9.0 kpc. Very recently, \\citet{2020ApJ...899..145Y} confirmed the existence of this $^{32}$S\/$^{34}$S gradient and enlarged the sample of measurements of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines to a total of 61 HMSFRs from the inner Galaxy out to a galactocentric distance of 12.0 kpc. In the CMZ, \\citet{2020A&A...642A.222H} found $^{32}$S\/$^{34}$S ratios of 16.3$^{+2.1}_{-1.7}$ and 17.9~$\\pm$~5.0 for the $+$50 km s$^{-1}$ cloud and Sgr B2(N), which is consistent with our values derived from $^{13}$C$^{34}$S and also with our results using the double isotope method. In the inner disk at 2.0~kpc $\\le R_{\\rm GC} \\le$ 6.0~kpc, a similar $^{32}$S\/$^{34}$S value of 18~$\\pm$~4 was derived based on our results in Sections~\\ref{section_ratios3234_13c34s} and \\ref{section_double_32s34s}. While $^{12}$C\/$^{13}$C ratios in the inner disk at $R_{\\rm GC} \\ge$ 4.0~kpc are clearly higher than in the CMZ (see details in Section~\\ref{section_comparisons_12c13c}), this is not the case for $^{32}$S\/$^{34}$S. On the contrary, $^{32}$S\/$^{34}$S ratios in the CMZ and inner disk are similar, as already suggested for the first time by \\citet{2020A&A...642A.222H}. In the local ISM, our results lead to an average $^{32}$S\/$^{34}$S ratio of 24~$\\pm$~4, which is close to the value in the Solar System \\citep[22.57,][]{1989GeCoA..53..197A}. That is also differing from the $^{12}$C\/$^{13}$C ratio and its clearly subsolar value in the local ISM. A more detailed discussion is given in Section~\\ref{section_discussion_all}. \n\n\nFor the first time, we established a $^{32}$S\/$^{34}$S gradient directly from measurements of $^{13}$CS and $^{13}$C$^{34}$S (see Section~\\ref{section_ratios3234_13c34s} for details). Similar $^{32}$S\/$^{34}$S gradients were also found in the $^{32}$S\/$^{34}$S values derived by the double isotope method with the $J$ = 2-1 and $J$ = 3-2 transitions (for details, see Section~\\ref{section_double_32s34s}). A gradient of $^{32}$S\/$^{34}$S = (0.75 $\\pm$ 0.13)$R_{\\rm GC}$+(15.52 $\\pm$ 0.78) was obtained based on a large dataset of 90 values from our detections of $^{13}$CS and C$^{34}$S $J$ = 2-1 lines with corrections of opacity, which is flatter than previous ones presented by \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}. Following \\citet{2020ApJ...899..145Y}, the gradient does not significantly change when the ratios in the CMZ or in the outer regions are excluded, indicating that the $^{32}$S\/$^{34}$S gradient is robust. \n\n\n\n\n\n\n\n\\begin{table*}[h]\n\\caption{Measurements of the $^{12}$C\/$^{13}$C gradient.}\n\\centering\n\\begin{tabular}{c|cc|cc}\n\\hline\\hline\n &\\multicolumn{2}{c}{Previous fitting results} & \\multicolumn{2}{c}{This work} \\\\\n & slope & intercept & slope & intercept \\\\\n\\hline\n\\label{table_all12C13Cgradients}\nCN\\tablefootmark{a} & 6.01 $\\pm$ 1.19 & 12.28 $\\pm$ 9.33 & 6.75 $\\pm$ 1.44 & 5.57 $\\pm$ 11.29 \\\\ \nC$^{18}$O\\tablefootmark{b} & 5.41 $\\pm$ 1.07 & 19.03 $\\pm$ 7.90 & 5.72 $\\pm$ 1.20 & 14.56 $\\pm$ 9.25 \\\\ \nH$_2$CO\\tablefootmark{c} & 5.08 $\\pm$ 1.10 & 11.86 $\\pm$ 6.60 & 5.43 $\\pm$ 1.04 & 13.87 $\\pm$ 6.38 \\\\ \nC$^{34}$S (this work) & $\\cdots$ & $\\cdots$ & 4.77 $\\pm$ 0.81 & 20.76 $\\pm$ 4.61 \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{Fitting results for the old (left) and new (right) distances, respectively. \n\\tablefoottext{a}{From \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}.}\n\\tablefoottext{b}{From \\citet{1990ApJ...357..477L}, \\citet{1996A&AS..119..439W}, and \\citet{1998ApJ...494L.107K}.}\n\\tablefoottext{c}{From \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y}.}}\n\\end{table*}\n\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[width=520pt]{newFigures\/MW_12C13C_corrected.png}\n \\caption{Distribution of $^{12}$C\/$^{13}$C ratios from 93 sources in the Milky Way. The background image is the structure of the Milky Way from the artist's impression [Credit:\nNASA\/JPL-Caltech\/ESO\/R. Hurt]. The $^{12}$C\/$^{13}$C isotope ratios with corrections for optical depth from C$^{34}$S\/$^{13}$C$^{34}$S in this work are plotted as circles. The triangles, pentagons, stars, squares, and diamonds indicate the $^{12}$C\/$^{13}$C ratios derived from CN\/$^{13}$CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O\/$^{13}$C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, H$_2$CO\/H$_2^{13}$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y}, CH$^+$\/$^{13}$CH$^+$ \\citep{2011ApJ...728...36R}, and CH\/$^{13}$CH \\citep{2020A&A...640A.125J}, respectively. The red symbol $\\odot$ indicates the position of the Sun. The color bar on the right-hand side indicates the range of the $^{12}$C\/$^{13}$C ratios. }\n \\label{distribution_12C13C}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n \\includegraphics[width=520pt]{newFigures\/MW_32S34S_corrected.png}\n \\caption{Distribution of $^{32}$S\/$^{34}$S ratios from 112 sources in the Milky Way. The background image is the structure of the Milky Way from the artist's impression [Credit:\nNASA\/JPL-Caltech\/ESO\/R. Hurt]. The $^{32}$S\/$^{34}$S isotope ratios with corrections of opacity derived in this work from $^{13}$CS\/$^{13}$C$^{34}$S and the double isotope method in the $J$ = 2-1 transition are plotted as circles and stars, respectively. The triangles indicate the results from the CMZ obtained by \\citet{2020A&A...642A.222H}. The red symbol $\\odot$ indicates the position of the Sun. The color bar on the right-hand side indicates the range of the $^{32}$S\/$^{34}$S ratios.}\n \\label{distribution_32S34S}\n\\end{figure*}\n\n\n\n\n\n\\subsection{C$^{33}$S}\n\\label{section_hfs_c33s}\n\n We detected at least three components of the C$^{33}$S $J$ = 2-1 line toward 26 sources. This will guide us to obtain information with respect to LTE or nonLTE conditions. As mentioned in Section~\\ref{section_34s33s}, the main component ($I_{main}$) consists of four HFS lines ($F$=7\/2-5\/2, $F$=5\/2-3\/2, $F$=1\/2-1\/2, $F$=3\/2-5\/2). Under LTE conditions in the optically thin case, the ratio between the other identified components (not belonging to the main one) and the main component can be obtained:\n\\begin{equation}\n\\begin{aligned}\nR_{211} &= \\frac{I(F=3\/2-1\/2 + F=5\/2-5\/2)}{I_{main}}\\\\ &= 0.25,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nR_{212} &= \\frac{I(F=3\/2-3\/2)}{I_{main}}\\\\ &= 0.15,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\nR_{213} &= \\frac{I(F=1\/2-3\/2)}{I_{main}}\\\\ &= 0.02.\n\\end{aligned}\n\\end{equation}\nA detailed examination of our 26 objects is presented in Table~\\ref{table_c33s_hfs}. Except for Orion-KL there is no evidence for nonLTE effects. All the remaining sources are found to be compatible with LTE conditions. The spectra of CS, C$^{34}$S, and $^{13}$CS toward Orion-KL show broad wings (see Fig.~\\ref{fig_orion}), which might lead to a highly complex C$^{33}$S $J$ = 2-1 line shape, which may be what is causing apparent LTE deviations in Orion-KL. \n\n\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=220pt]{figure\/Orion-KL-c33s.eps}\n \\caption{Line profiles of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, and $^{13}$CS toward Orion-KL.}\n \\label{fig_orion}\n\\end{figure}\n\n\nNo systematic dependence of the $^{34}$S\/$^{33}$S ratios on galactocentric distance was found in previous studies. The average $^{34}$S\/$^{33}$S values were 6.3~$\\pm$~1.0 and 5.9~$\\pm$~1.5 in \\citet{1996A&A...305..960C} and \\citet{2020ApJ...899..145Y}, respectively. A particularly low $^{34}$S\/$^{33}$S ratio of 4.3~$\\pm$~0.2 was found in the Galactic center region by \\citet{2020A&A...642A.222H}. These authors speculated about the possible presence of a gradient with low values at the center. However, in view of the correction for HFS in Section~\\ref{section_34s33s}, it appears that this is simply an effect of different HFS correction factors, with the one at the Galactic center with its wide spectral lines being 1.0, thus (exceptionally) not requiring a downward correction. \\citet{2020ApJ...899..145Y} considered the effect of HFS, while they overestimated the ratio of the main component to the total intensity of the C$^{33}$S $J$ = 2-1 line (for details, see their Section~4.2). This resulted in higher $^{34}$S\/$^{33}$S values. However, the $^{34}$S\/$^{33}$S ratios appear to be independent of galactocentric distance based on our results (see details in Section~\\ref{section_34s33s}). The average $^{34}$S\/$^{33}$S value with corrections of optical depth toward our sample in the $J$ = 2-1 transition is 4.35~$\\pm$~0.44, which is similar to the value in the Galactic center region derived by \\citet{2020A&A...642A.222H} and lower than the value of 5.61 in the Solar System \\citep{1989GeCoA..53..197A}. This indicates that no systematic variation exists in the $^{34}$S\/$^{33}$S ratios in our Galaxy, that $^{33}$S is (with respect to stellar nucleosynthesis) similar to $^{34}$S, and that the Solar System (see Fig.~\\ref{fig_all33S}) must be peculiar. An approximately constant $^{34}$S\/$^{33}$S ratio with opacity correction from the $J$ = 2-1 transition across the Galactic plane leads to a $^{32}$S\/$^{33}$S gradient in our Galaxy as we already mentioned in Section~\\ref{section_32s33s}: $^{32}{\\rm S}\/^{33}{\\rm S}$ = $(2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57)$, with a correlation coefficient of 0.46.\n\n\n\n\n\\begin{table*}[h]\n\\caption{Line intensity ratios of C$^{33}$S $J$ = 2-1 hyperfine components.}\n\\centering\n\\begin{tabular}{lcccc}\n\\hline\\hline\nSource & FWHM & $R_{211}$ & $R_{212}$ & $R_{213}$ \\\\\n & (km s$^{-1}$) & & & \\\\\n\\hline\n\\label{table_c33s_hfs}\nW3OH & 4.41 & 0.29 $\\pm$ 0.04 & 0.15 $\\pm$ 0.03 & 0.05 $\\pm$ 0.03 \\\\ \nOrion-KL & 4.25 & 0.60 $\\pm$ 0.02 & 0.30 $\\pm$ 0.01 & 0.44 $\\pm$ 0.02 \\\\ \nG359.61$-$00.24 & 3.17 & 0.40 $\\pm$ 0.05 & 0.62 $\\pm$ 0.05 & 0.14 $\\pm$ 0.04 \\\\ \nG006.79$-$00.25 & 2.73 & 0.28 $\\pm$ 0.02 & 0.17 $\\pm$ 0.02 & $\\cdots$ \\\\ \nG010.32$-$00.15 & 2.66 & 0.27 $\\pm$ 0.04 & 0.23 $\\pm$ 0.06 & $\\cdots$ \\\\ \nG016.86$-$02.15 & 3.32 & 0.30 $\\pm$ 0.05 & 0.18 $\\pm$ 0.03 & $\\cdots$ \\\\ \nG017.02$-$02.40 & 3.74 & 0.17 $\\pm$ 0.03 & 0.17 $\\pm$ 0.08 & $\\cdots$ \\\\ \nG018.34$+$01.76 & 2.30 & 0.24 $\\pm$ 0.07 & 0.13 $\\pm$ 0.06 & $\\cdots$ \\\\ \nG019.36$-$00.03 & 3.02 & 0.29 $\\pm$ 0.06 & 0.14 $\\pm$ 0.04 & $\\cdots$ \\\\ \nG023.43$-$00.18 & 3.81 & 0.40 $\\pm$ 0.14 & 1.15 $\\pm$ 0.34 & $\\cdots$ \\\\ \nG024.78$+$00.08 & 4.29 & 0.34 $\\pm$ 0.03 & 0.21 $\\pm$ 0.03 & 0.08 $\\pm$ 0.03 \\\\ \nG028.39$+$00.08 & 3.04 & 0.26 $\\pm$ 0.04 & 0.24 $\\pm$ 0.06 & $\\cdots$ \\\\ \nG028.83$-$00.25 & 2.27 & 0.25 $\\pm$ 0.04 & 0.18 $\\pm$ 0.03 & $\\cdots$ \\\\ \nG030.70$-$00.06 & 4.85 & 0.33 $\\pm$ 0.02 & 0.23 $\\pm$ 0.03 & 0.11 $\\pm$ 0.02 \\\\ \nG030.74$-$00.04 & 3.55 & 0.19 $\\pm$ 0.03 & 0.25 $\\pm$ 0.06 & $\\cdots$ \\\\ \nG030.81$-$00.05 & 6.15 & 0.27 $\\pm$ 0.03 & 0.16 $\\pm$ 0.02 & 0.14 $\\pm$ 0.02 \\\\ \nG032.74$-$00.07 & 4.54 & 0.36 $\\pm$ 0.04 & 0.16 $\\pm$ 0.03 & $\\cdots$ \\\\ \nG032.79$+$00.19 & 4.78 & 0.09 $\\pm$ 0.05 & 1.72 $\\pm$ 0.28 & $\\cdots$ \\\\ \nG034.41$+$00.23 & 4.56 & 0.32 $\\pm$ 0.04 & 0.28 $\\pm$ 0.04 & 0.12 $\\pm$ 0.03 \\\\ \nG034.79$-$01.38 & 2.40 & 0.44 $\\pm$ 0.15 & 0.14 $\\pm$ 0.06 & $\\cdots$ \\\\ \nG037.42$+$01.51 & 3.18 & 0.20 $\\pm$ 0.05 & 0.10 $\\pm$ 0.02 & $\\cdots$ \\\\ \nW51-IRS2 & 8.55 & 0.13 $\\pm$ 0.09 & 3.63 $\\pm$ 0.36 & 0.71 $\\pm$ 0.13 \\\\ \nDR21 & 2.71 & 0.28 $\\pm$ 0.01 & 0.20 $\\pm$ 0.01 & 0.14 $\\pm$ 0.01 \\\\ \nG097.53$+$03.18 & 5.22 & 0.30 $\\pm$ 0.05 & 0.13 $\\pm$ 0.04 & 0.14 $\\pm$ 0.04 \\\\ \nG109.87$+$02.11 & 3.19 & 0.23 $\\pm$ 0.06 & 0.38 $\\pm$ 0.07 & $\\cdots$ \\\\ \nNGC7538 & 3.71 & 0.22 $\\pm$ 0.01 & 0.21 $\\pm$ 0.01 & $\\cdots$ \\\\ \n\\hline\n\\end{tabular}\n\\tablefoot{ Full width at half maximum values of the main component were obtained from measurements of C$^{33}$S; see Table \\ref{fitting_all}. The errors provided here are 1$\\sigma$.\n}\n\\end{table*}\n\n\\subsection{C$^{36}$S}\n\\label{section_discussion_all36}\n\nAs mentioned in Section~\\ref{section_34s36s}, we find novel potential indications for a positive $^{34}$S\/$^{36}$S gradient with galactocentric radius. The $^{36}$S-bearing molecule C$^{36}$S was first detected by \\citet{1996A&A...313L...1M}. These authors observed the $J$ = 2-1 and 3-2 transitions toward eight Galactic molecular hot cores at galactocentric distances of between 5.0 kpc and 10.0 kpc. \\citet{1996A&A...313L...1M} reported an average $^{34}$S\/$^{36}$S ratio of 115~$\\pm$~17, which is smaller than the value in the Solar System \\citep[200.5,][]{1989GeCoA..53..197A}. This is consistent with this nucleus being of a purely secondary nature. Combining the ratios of \\citet{1996A&A...313L...1M} --- after applying new distances (see details in Table~\\ref{table_all36s})--- with our results in the $J$ = 2-1 transition, the following fit could be achieved:\n\\begin{equation}\n^{34}{\\rm S}\/^{36}{\\rm S} = (10.34 \\pm 2.74)R_{\\rm GC}+(57.45 \\pm 18.59), \n\\end{equation}\nwith a correlation coefficient of 0.71. As the $^{34}$S\/$^{33}$S ratios show a uniform distribution across our Galaxy (see details in Section~\\ref{section_hfs_c33s}), a $^{33}$S\/$^{36}$S gradient is also expected. We obtain (2.38~$\\pm$~0.67)$R_{\\rm GC}$+(13.21~$\\pm$~4.48). After applying our $^{32}$S\/$^{34}$S gradient to the $^{34}$S\/$^{36}$S ratios in \\citet{1996A&A...313L...1M} with equation (9), $^{32}$S\/$^{36}$S ratios were then derived and listed in Table~\\ref{table_all36s}. Combined with our results in the $J$ = 2-1 transition, a linear fit to the $^{32}$S\/$^{36}$S ratios is obtained:\n\\begin{equation}\n ^{32}{\\rm S}\/^{36}{\\rm S} = (314 \\pm 55)R_{\\rm GC}+(659 \\pm 374), \n \\end{equation}\nwith a correlation coefficient of 0.84. The $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S ratios are plotted as functions of galactocentric distances in Fig.~\\ref{fig_all36S}. Measurements of $^{34}$S\/$^{36}$S, $^{33}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S are still not numerous. More sources with detected C$^{36}$S lines would be highly\ndesirable, especially in the CMZ and the inner disk within $R_{\\rm GC}$~=~5.0~kpc.\n\n\n\\subsection{Observational bias due to distance effects}\n\\label{section_discussion_distance}\n\nWhile we have so far analyzed isotope ratios as a function of galactocentric distances, there might be a bias in the sense that the ratios could at least in part also depend on the distance from Earth, a bias caused by different linear resolutions. In Appendix~\\ref{appendix_spectra}, the $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{34}$S\/$^{33}$S, and $^{32}$S\/$^{33}$S, as well as the $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios are plotted as functions of the distance from the Sun and shown in Figs.~\\ref{fig_12C13C_2Sun} to \\ref{fig_all36S_2Sun}, respectively. No apparent gradients can be found, which indicates that any observational bias on account of distance-dependent effects is not significant for $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{32}$S\/$^{36}$S. This agrees with the findings of \\citet[][see their Section~4.5]{2020ApJ...899..145Y}.\n\n\n\\subsection{Beam size effects}\n\\label{section_beam}\n\nA good way to check whether the different beam sizes for different lines could affect our results is to compare the isotope ratios derived from different transitions at different frequencies observed with different beam sizes. As shown in Section~\\ref{section_double_32s34s}, $^{32}$S\/$^{34}$S ratios obtained from the double isotope method in the $J$ = 2-1 and 3-2 transitions are in good agreement, suggesting that the effect of beam size is negligible. Furthermore, \\citet{2020A&A...642A.222H} found an average $^{32}$S\/$^{34}$S ratio of 17.9~$\\pm$~5.0 in the envelope of Sgr B2(N) with the Atacama Large Millimetre\/submillimetre Array (ALMA) with 1$\\farcs$6 beam size, which is consistent with our results in the CMZ from the IRAM 30 meter telescope with beam sizes of about 27$\\arcsec$. \\citet{2020ApJ...899..145Y} derived similar $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S ratios from different telescopes ---that is, the IRAM 30 meter and the ARO 12 meter--- toward six HMSFRs and concluded that beam-size effects are insignificant (see details in their Section~4.1). All this suggests that beam-size effects could not obviously affect our results. \n\n\n\n\n\\subsection{Chemical fractionation}\n\\label{section_fractionation}\n\nIsotopic fractionation could possibly affect the isotope ratios derived from the molecules in the interstellar medium. \\citet{1976ApJ...205L.165W} firstly proposed that gas-phase CO should have a tendency to be enriched in $^{13}$CO because of the charge-exchange reaction of CO with $^{13}$C$^+$. Several theoretical studies also support this mechanism \\citep[e.g.,][]{1984ApJ...277..581L,2020MNRAS.497.4333V}, which was then extended to other carbon bearing species \\citep{2020MNRAS.498.4663L}. Formaldehyde forming in the gas phase is suggested to be depleted in the $^{13}$C bearing isotopolog \\citep[e.g.,][]{1984ApJ...277..581L}. However, if H$_2$CO originates from dust grain mantles, then the $^{13}$C bearing isotopolog might be enhanced relative to species like methanol and CO \\citep{2012LPI....43.1611W,2019ApJ...877..154Y}. Recently, \\citet{2020A&A...640A..51C} performed a new gas-grain chemical model and proposed that molecules formed starting from atomic carbon could also show $^{13}$C enhancements through the reaction $^{13}$C + C$_{3}$ $\\rightarrow$ $^{12}$C + $^{13}$CC$_{2}$. As already mentioned in Section~\\ref{section_comparisons_12c13c}, the Galactic $^{12}$C\/$^{13}$C gradient derived from C$^{34}$S in this work is in good agreement with previous results based on measurements of CN, C$^{18}$O, and H$_2$CO. Therefore, chemical fractionation cannot greatly affect the carbon isotope ratios. \n\nTo date, little is known about sulfur fractionation. \\citet{2019MNRAS.485.5777L} proposed a low $^{34}$S enrichment through the reaction of $^{34}$S$^+$ + CS $\\rightarrow$ S$^+$ + C$^{34}$S in dense clouds. A slight enrichment in $^{13}$C was predicted for CS with the $^{13}$C$^{+}$ + CS $\\rightarrow$ C$^+$ + $^{13}$CS reaction \\citep{2020MNRAS.498.4663L}. $^{32}$S\/$^{34}$S ratios derived directly from $^{13}$CS and $^{13}$C$^{34}$S and the double isotope method involving $^{12}$C\/$^{13}$C ratios (equation 8) turn out to agree very well, suggesting that sulfur fractionation is negligible as previously suggested by \\citet{2020A&A...642A.222H} and in this work. \n\n\\subsection{Interstellar C, N, O, and S isotope ratios}\n\\label{section_discussion_all}\n\nThe data collected so far allow us to evaluate the status of several isotopes with respect to primary or secondary synthesis in stellar objects. From the data presented here in Table~\\ref{table_allratios}, we choose the $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios because $^{12}$C is mostly primary \\citep{1995ApJS...98..617T} while $^{32}$S is definitely a primary nucleus \\citep{1995ApJS..101..181W}, against which the other isotopes can be evaluated. A question arises as to whether all these ratios, as well as those from nitrogen and oxygen, can be part of the same scheme.\n\n Comparing the CMZ with the ratios in the inner disk, the CMZ with the outer Galaxy, the inner disk with the ratios in the outer Galaxy, and the local ISM values with those of the Solar System, we obtain increases in values for $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios. All of these values are listed in Table~\\ref{table_allratios_com}. Percentages are clearly highest between $^{32}$S and $^{36}$S. These indicate that $^{36}$S is, as opposed to $^{32}$S, definitely secondary. Percentages between $^{12}$C and $^{13}$C are also high but not as extreme, presumably because $^{12}$C is also synthesized in longer lived stars of intermediate mass (e.g., \\citealt{2020ApJ...900..179K}). More difficult to interpret are the $^{32}$S\/$^{33}$S and $^{32}$S\/$^{34}$S ratios, where percentages are smaller, indicating that $^{33}$S and $^{34}$S are, as already mentioned in Section~\\ref{section_hfs_c33s}, neither fully primary nor secondary. However, percentages in the case of the $^{32}$S\/$^{33}$S ratio systematically surpass those of the $^{32}$S\/$^{34}$S ratio, suggesting a more secondary origin of $^{33}$S with respect to $^{34}$S, even though $^{34}$S\/$^{33}$S appears to be constant across the Galaxy. Finally, local interstellar $^{32}$S\/$^{34}$S and $^{32}$S\/$^{33}$S ratios behave strikingly differently with respect to solar values. While $^{32}$S\/$^{34}$S is (almost) solar, $^{32}$S\/$^{33}$S is far below the solar value. Peculiar Solar System abundance ratios may be the easiest way to explain this puzzling situation. Most likely there is an overabundance of $^{34}$S in the gas and dust that formed the Solar System.\n\n\nAnother clearly primary isotope is $^{16}$O, which allows us to look for $^{16}$O\/$^{18}$O and $^{16}$O\/$^{17}$O ratios \\citep{1993A&A...274..730H,1994LNP...439...72H,1999RPPh...62..143W,2008A&A...487..237W,2020ApJS..249....6Z}. The high percentages in the $^{16}$O\/$^{17}$O ratios show that $^{17}$O is more secondary than $^{18}$O, which is consistent with models of stellar nucleosynthesis, because $^{17}$O is a main product of the CNO cycle while $^{18}$O can also be synthesized by helium burning in massive stars.\n\n$^{14}$N\/$^{15}$N can also be measured; both nuclei can be synthesized in rotating massive stars and AGB stars as primary products \\citep[e.g.,][]{2002A&A...390..561M,2011MNRAS.414.3231K,2020ApJ...900..179K,2018ApJS..237...13L}. However, most of the $^{14}$N is produced through CNO cycling, and is therefore secondary \\citep[e.g.,][]{2011MNRAS.414.3231K,2014PASA...31...30K}. The production of $^{15}$N remains to be understood and may be related to novae \\citep[e.g.,][]{2020ApJ...900..179K,2022arXiv221004350R}. None of the stable nitrogen isotopes are purely primary. While $^{14}$N appears to be less secondary than $^{15}$N \\citep{1975A&A....43...71A,1994LNP...439...72H,1999RPPh...62..143W,2012ApJ...744..194A,2015ApJ...804L...3R,2018A&A...609A.129C,2021ApJS..257...39C}, in this case we do not have a clear calibration against an isotope that can be considered to be mainly primary. Remarkably, \\citet{2022arXiv220910620C} reported a rising $^{14}$N\/$^{15}$N gradient that peaks at $R_{\\rm GC}$ = 11 kpc and then decreases, and suggested that $^{15}$N could be mainly produced by novae on long timescales.\n\n\n\\begin{table*}[h]\n\\caption{Interstellar C, N, O, and S isotope ratios.}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n & Molecule & CMZ & Inner disk &Local ISM & Outer Galaxy & Solar System\\tablefootmark{*} \\\\\n\\hline\n\\label{table_allratios}\n$^{12}$C\/$^{13}$C & C$^{34}$S\\tablefootmark{a}& 27 $\\pm$ 3 & 41 $\\pm$ 9 & 66 $\\pm$ 10 & 74 $\\pm$ 8 & 89 \\\\ \n & CN\\tablefootmark{b} & $\\cdots$ & 44 $\\pm$ 12 & 41 $\\pm$ 11 & 66 $\\pm$ 19 & \\\\ \n & C$^{18}$O\\tablefootmark{c} & 24 $\\pm$ 1 & 41 $\\pm$ 2 & 60 $\\pm$ 5 & 70 $\\pm$ 10 & \\\\ \n & H$_2$CO\\tablefootmark{d} & $\\cdots$ & 40 $\\pm$ 7 & 50 $\\pm$ 13 & 64 $\\pm$ 10 & \\\\ \n & average & 25 $\\pm$ 2 & 42 $\\pm$ 9 & 54 $\\pm$ 10 & 69 $\\pm$ 12 & \\\\\n\\hline\n$^{14}$N\/$^{15}$N & CN\\tablefootmark{e} & $\\cdots$ & 269 $\\pm$ 59 & 314 $\\pm$ 104 & 289 $\\pm$ 85 & 270 \\\\ \n & HCN\\tablefootmark{f} & $\\cdots$ & 284 $\\pm$ 63 & 398 $\\pm$ 48 & 388 $\\pm$ 32 & \\\\ \n & HNC\\tablefootmark{f} & $\\cdots$ & 363 $\\pm$ 100 & 378 $\\pm$ 79 & 395 $\\pm$ 74 & \\\\ \n & N$_2$H$^{+}$\\tablefootmark{g} & $\\cdots$ & 900 $\\pm$ 250 & 496 $\\pm$ 65 & 581 $\\pm$ 140 & \\\\ \n & NH$_3$\\tablefootmark{h} & 40 $\\pm$ 13 & 175 $\\pm$ 46 & 297 $\\pm$ 99 & 96 $\\pm$ 44 & \\\\\n\\hline\n$^{16}$O\/$^{18}$O & H$_2$CO\\tablefootmark{i} & 263 $\\pm$ 45 & 327 $\\pm$ 32 & 560 $\\pm$ 25 & 625 $\\pm$ 144 & 490 \\\\\n$^{18}$O\/$^{17}$O & CO\\tablefootmark{j} & 3.4 $\\pm$ 0.1 & 3.6 $\\pm$ 0.2 & 3.9 $\\pm$ 0.4 & 4.8 $\\pm$ 0.6 & 5.5 \\\\\n$^{16}$O\/$^{17}$O\\tablefootmark{**} & & 894 $\\pm$ 155 & 1177 $\\pm$ 132 & 2184 $\\pm$ 244 & 3000 $\\pm$ 786 & 2625 \\\\\n\\hline\n$^{32}$S\/$^{34}$S\\tablefootmark{a} & & 19 $\\pm$ 2 & 18 $\\pm$ 4 & 24 $\\pm$ 4 & 28 $\\pm$ 3 & 23 \\\\\n$^{34}$S\/$^{33}$S\\tablefootmark{a} & & 4.2 $\\pm$ 0.2 & 4.3 $\\pm$ 0.4 & 4.2 $\\pm$ 0.5 & 4.1 $\\pm$ 0.3 & 5.6 \\\\\n$^{32}$S\/$^{33}$S\\tablefootmark{a} & & 70 $\\pm$ 16 & 82 $\\pm$ 19 & 88 $\\pm$ 21 & 105 $\\pm$ 19 & 127 \\\\\n$^{34}$S\/$^{36}$S\\tablefootmark{a} & & 41 $\\pm$ 4 & 122 $\\pm$ 18 & 111 $\\pm$ 16 & 161 $\\pm$ 32 & 200 \\\\\n$^{32}$S\/$^{36}$S\\tablefootmark{a} & & 884 $\\pm$ 104 & 2382 $\\pm$ 368 & 2752 $\\pm$ 458 & 4150 $\\pm$ 828 & 4525 \\\\\n\\hline\n\\end{tabular}\n\\tablefoot{The inner disk values refer to the mean values at galactocentric distances of 2.0 kpc~$\\le R_{\\rm GC} \\le$~6.0 kpc. The local ISM values refer to 7.5 kpc~$\\le R_{\\rm GC} \\le$~8.5 kpc. The outer Galaxy values point to 9.0 kpc~$\\le R_{\\rm GC} \\le$~11.0 kpc. \n\\tablefoottext{*}{From \\citet{1989GeCoA..53..197A}.} \n\\tablefoottext{a}{This work.}\n\\tablefoottext{b}{From \\citet{2002ApJ...578..211S} and \\citet{2005ApJ...634.1126M}.}\n\\tablefoottext{c}{From \\citet{1990ApJ...357..477L,1993ApJ...408..539L}, \\citet{1996A&AS..119..439W}, and \\citet{1998ApJ...494L.107K}.}\n\\tablefoottext{d}{From \\citet{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H} and \\citet{2019ApJ...877..154Y}.} \n\\tablefoottext{e}{From \\citet{2012ApJ...744..194A}, \\citet{2015ApJ...804L...3R}, and \\citet{2015ApJ...808L..46F}. } \n\\tablefoottext{f}{From \\citet{2012ApJ...744..194A} and \\citet{2018A&A...609A.129C}} \n\\tablefoottext{g}{From \\citet{2015ApJ...804L...3R}. } \n\\tablefoottext{h}{From \\citet{2021ApJS..257...39C}. } \n\\tablefoottext{i}{From \\citet{1981MNRAS.194P..37G} and \\citet{1994ARA&A..32..191W}. } \n\\tablefoottext{j}{From \\citet{2020ApJS..249....6Z}. } \n\\tablefoottext{**}{The $^{16}$O\/$^{17}$O ratios are derived with values of $^{16}$O\/$^{18}$O and $^{18}$O\/$^{17}$O.} }\n\\end{table*}\n\n\\begin{table*}[h]\n\\caption{Comparison of isotope ratios at different galactocentric distances. Given are percentage enhancements.}\n\\centering\n\\begin{tabular}{ccccc}\n\\hline\\hline\n & CMZ & CMZ & Inner disk & Local ISM \\\\\n & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ & $\\downarrow$ \\\\\n & Inner disk & Outer Galaxy & Outer Galaxy & Solar System \\\\\n\\hline\n\\label{table_allratios_com}\n$^{12}$C\/$^{13}$C & 68 $\\pm$ 33 & 176 $\\pm$ 54 & 64 $\\pm$ 21 & 65 $\\pm$ 31 \\\\\n\\hline\n$^{16}$O\/$^{18}$O & 24 $\\pm$ 9 & 138 $\\pm$ 61 & 91 $\\pm$ 43 & -12 $\\pm$ 5 \\\\\n$^{16}$O\/$^{17}$O & 32 $\\pm$ 8 & 236 $\\pm$ 111 & 155 $\\pm$ 73 & 20 $\\pm$ 13 \\\\\n\\hline\n$^{32}$S\/$^{34}$S & -5 $\\pm$ 11 & 47 $\\pm$ 10 & 56 $\\pm$ 18 & -4 $\\pm$ 17 \\\\\n$^{32}$S\/$^{33}$S & 17 $\\pm$ 8 & 50 $\\pm$ 16 & 28 $\\pm$ 6 & 44 $\\pm$ 34 \\\\\n$^{32}$S\/$^{36}$S & 169 $\\pm$ 50 & 369 $\\pm$ 125 & 74 $\\pm$ 31 & 64 $\\pm$ 27 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Galactic chemical environment}\n\\label{section_discussion_model}\n\n\n\\citet{2020ApJ...900..179K} established a Galactic chemical evolution (GCE) model based on the GCE model in \\citet{2011MNRAS.414.3231K} with updates with respect to new solar abundances and also accounting for failed supernovae, super-AGB stars, the s-process from AGB stars, and various r-process sites. Based on this GCE model, the predicted $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, and $^{32}$S\/$^{36}$S ratios at $R_{\\rm GC}$~=~2.0, 4.0, 6.0, 8.5, 12.0, and 17.0~kpc are obtained and plotted in Figs.~\\ref{fig_gradient_12C13C}, \\ref{fig_gradient_32S34S}, \\ref{fig_all33S}, and \\ref{fig_all36S}. The initial mass function and nucleosynthesis yields are the same for different galactic radii but star formation and inflow timescales ($\\tau_{\\rm s}$ and $\\tau_{\\rm i}$) depend on the Galactic radius (see \\citealt{2000ApJ...539...26K} for the definition of the timescales). Adopted values are $\\tau_{\\rm s}$~=~1.0, 2.0, 3.0, 4.6, 6.5, and 8.8 Gyr as well as $\\tau_{\\rm i}$~=~4.0, 5.0, 5.0, 5.0, 7.0, and 50.0 Gyr for $R_{\\rm GC}$~=~2.0, 4.0, 6.0, 8.5, 12.0, and 17.0~kpc, respectively. The predicted $^{12}$C\/$^{13}$C ratios are in good agreement with our results, while $^{32}$S\/$^{34}$S and $^{32}$S\/$^{36}$S ratios show significant deviations at larger galactocentric distances. $^{32}$S\/$^{33}$S ratios show an offset along the entire inner 12 kpc of the Milky Way. This indicates that current models of Galactic chemical evolution are still far from perfect. In this context, our data will serve as a useful guideline for further even more refined GCE models.\n\nVery recently, \\citet{2022arXiv220910620C} predicted $^{12}$C\/$^{13}$C gradients with four different models addressing nova systems (see details in their Table~2 and Sect.~4), following \\citet{2017MNRAS.470..401R,2019MNRAS.490.2838R,2021A&A...653A..72R}. The gradients from these four models are shown in Fig.~\\ref{fig_12C13C_GCEmodels}. The results from model 1 show a large deviation with respect to the observed values. The other three models could reproduce the ratios within the dispersion at galactocentric radii beyond the solar neighborhood, while the inner Galaxy is not as well reproduced.\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=280pt]{newFigures\/1allratio12c_13c-mosels.pdf}\n \\caption{$^{12}$C\/$^{13}$C isotope ratios from observations in this work and GCE models. The red symbol $\\odot$ indicates the $^{12}$C\/$^{13}$C isotope ratio of the Sun. The $^{12}$C\/$^{13}$C gradient obtained from C$^{34}$S in the current work is plotted as a black solid line, with the gray shaded area showing the 1$\\sigma$ interval of the fit. The red crosses visualize the results from the GCE model of \\citet[][see also Section~\\ref{section_discussion_model}]{2011MNRAS.414.3231K,2020ApJ...900..179K}. The dark green, light green, magenta, and pink lines refer to the predicted gradients from models in Table 2 from \\citet{2022arXiv220910620C}. }\n \\label{fig_12C13C_GCEmodels}\n\\end{figure}\n\n\\section{Summary}\n\\label{summary}\n\nWe used the IRAM 30 meter telescope to perform observations of the $J$ = 2-1 transitions of CS, C$^{33}$S, C$^{34}$S, C$^{36}$S, $^{13}$CS, $^{13}$C$^{33}$S, and $^{13}$C$^{34}$S as well as the $J$ = 3-2 transitions of C$^{33}$S, C$^{34}$S, C$^{36}$S, and $^{13}$CS toward a large sample of 110 HMSFRs. The CS $J$ = 2-1 line was detected toward 106 sources, with a detection rate of 96\\%. The $J$ = 2-1 transitions of C$^{34}$S, $^{13}$CS, C$^{33}$S, $^{13}$C$^{34}$S, and C$^{36}$S were successfully detected in 90, 82, 46, 17, and 3 of our sources, respectively. The $J$ = 3-2 lines of C$^{34}$S, $^{13}$CS, C$^{33}$S, and C$^{36}$S were detected in 87, 71, 42, and 1 object(s). All the detected rare CS isotopologs exhibit optically thin lines and allow us to measure the isotope ratios of $^{12}$C\/$^{13}$C, $^{32}$S\/$^{34}$S, $^{32}$S\/$^{33}$S, $^{32}$S\/$^{36}$S, $^{34}$S\/$^{33}$S, $^{34}$S\/$^{36}$S, and $^{33}$S\/$^{36}$S with only minor saturation corrections. Our main results are as follows:\n\\begin{itemize}\n \\item Based on the measurements of C$^{34}$S and $^{13}$C$^{34}$S $J$ = 2-1 transitions, we directly measured the $^{12}$C\/$^{13}$C ratios with corrections of opacity. With accurate distances obtained from parallax data \\citep{2009ApJ...700..137R, 2014ApJ...783..130R, 2019ApJ...885..131R}, we confirm the previously determined $^{12}$C\/$^{13}$C gradient. A least-squares fit to our data results in $^{12}\\rm C$\/$^{13}\\rm C$ = (4.77~$\\pm$~0.81)$R_{\\rm GC}$+(20.76~$\\pm$~4.61), with a correlation coefficient of 0.82. \n \\item The Galactic $^{12}$C\/$^{13}$C gradients derived based on measurements of CN \\citep{2002ApJ...578..211S,2005ApJ...634.1126M}, C$^{18}$O \\citep{1990ApJ...357..477L,1996A&AS..119..439W,1998ApJ...494L.107K}, and H$_2$CO \\citep{1980A&A....82...41H,1982A&A...109..344H,1983A&A...127..388H,1985A&A...143..148H,2019ApJ...877..154Y} are in agreement with our results from C$^{34}$S and emphasize that chemical fractionation has little effect on $^{12}$C\/$^{13}$C ratios.\n \\item While previously it had been assumed that a linear fit would provide a good simulation of carbon isotope ratios as a function of galactocentric distance, our analysis reveals that this does not hold for the Galactic center region. While $^{12}$C\/$^{13}$C ratios are lowest in this part of the Milky Way, they clearly surpass values expected from a linear fit to the Galactic disk sources. This indicates that there is no strict linear correlation of carbon isotope ratios across the Galaxy.\n \\item We confirm the previously determined $^{32}$S\/$^{34}$S gradients \\citep{1996A&A...305..960C,2020ApJ...899..145Y,2020A&A...642A.222H} with the direct method from $^{13}$CS and $^{13}$C$^{34}$S, as well as the double isotope method also using $^{12}$C\/$^{13}$C ratios in the $J$ = 2-1 and $J$ = 3-2 transitions. Opacity corrections could be applied to the $J$ = 2-1 transitions, but not to the $J$ = 3-2 lines that may show, on average, slightly higher opacities. A $^{32}$S\/$^{34}$S gradient of (0.75 $\\pm$ 0.13)$R_{\\rm GC}$+(15.52 $\\pm$ 0.78) was obtained based on a large dataset of 90 values from our double isotope method in the $J$ = 2-1 transition. The 19 sources permitting the direct determination of this ratio with $^{13}$CS\/$^{13}$C$^{34}$S yield $^{32}$S\/$^{34}$S=(0.73 $\\pm$ 0.36)$R_{\\rm GC}$+(16.50 $\\pm$ 2.07). \n \\item Differences between the behavior of the $^{12}$C\/$^{13}$C and $^{32}$S\/$^{34}$S ratios as a function of galactocentric distance are reported and should be used as input for further chemical models: (a) In the inner disk the $^{12}$C\/$^{13}$C ratios at $R_{\\rm GC} \\ge$ 4.0 kpc are clearly higher than the value in the CMZ, while the $^{32}$S\/$^{34}$S ratios in the CMZ and inner disk are similar, as already suggested for the first time by \\citet{2020A&A...642A.222H}. (b) In the local ISM, the $^{12}$C\/$^{13}$C ratio is well below the Solar System value but $^{32}$S\/$^{34}$S is still quite close to it. All of this indicates that, unlike $^{13}$C, $^{34}$S is not a clean secondary isotope.\n \\item There is no notable $^{34}$S\/$^{33}$S gradient across the Galaxy. Ratios are well below the values commonly reported in earlier publications. This is a consequence of accounting for the full hyperfine structure splitting of the C$^{33}$S lines. The average value of $^{34}$S\/$^{33}$S derived from the $J$ = 2-1 transition lines after corrections for opacity toward our sample is 4.35~$\\pm$~0.44.\n \\item While there is no $^{34}$S\/$^{33}$S gradient with galactocentric radius, interstellar\n $^{34}$S\/$^{33}$S values near the solar neighborhood are well below the Solar System ratio, most likely suggesting the Solar System ratio is peculiar, and perhaps also the $^{18}$O\/$^{17}$O ratio. A comparison of local interstellar and Solar System $^{32}$S\/$^{34}$S and $^{34}$S\/$^{33}$S ratios suggests that the Solar System may have been formed from gas and dust with a peculiarly high $^{34}$S abundance. The data also indicate that $^{33}$S is not a clean primary or secondary product of nucleosynthesis, similarly\nto $^{34}$S .\n \\item For the first time, we report a $^{32}$S\/$^{33}$S gradient in our Galaxy: $^{32}{\\rm S}\/^{33}{\\rm S}$ = $(2.64 \\pm 0.77)R_{\\rm GC}+(70.80 \\pm 5.57)$, with a correlation coefficient of 0.46. \n \\item We find first potential indications for a positive $^{34}$S\/$^{36}$S gradient with galactocentric radius. Combined $^{34}$S\/$^{36}$S ratios from \\citet{1996A&A...313L...1M} and our new data with corrections of opacity in the $J$ = 2-1 transition and applying new up-to-date distances yield a linear fit of $^{34}{\\rm S}\/^{36}{\\rm S}$ = $(10.34 \\pm 2.74)R_{\\rm GC}+(57.45 \\pm 18.59)$, with a correlation coefficient of 0.71. Considering the uniform $^{34}$S\/$^{33}$S ratios in our Galaxy, a $^{33}$S\/$^{36}$S gradient of (2.38~$\\pm$~0.67)$R_{\\rm GC}$+(13.21~$\\pm$~4.48) is also obtained. \n \\item For the first time, we report a tentative $^{32}$S\/$^{36}$S gradient with galactocentric radius: $^{32}{\\rm S}\/^{36}{\\rm S}$ = $(314 \\pm 55)R_{\\rm GC}+(659 \\pm 374)$, with a correlation coefficient of 0.84. Our measurements are consistent with $^{36}$S being a purely secondary nucleus. However, observations of $^{34}$S\/$^{36}$S and $^{32}$S\/$^{36}$S isotope ratios are still relatively few, especially in the CMZ and the inner disk within $R_{\\rm GC}$~=~5.0~kpc.\n \\item The predicted $^{12}$C\/$^{13}$C ratios from the latest Galactic chemical evolution models \\citep[e.g.,][]{2020ApJ...900..179K,2021A&A...653A..72R,2022arXiv220910620C} are in good agreement with our results, while $^{32}$S\/$^{34}$S and $^{32}$S\/$^{36}$S ratios show significant differences at larger galactocentric distances. $^{32}$S\/$^{33}$S ratios even show clear offsets along the entire inner 12 kpc of the Milky Way. Taken together, these findings provide useful guidelines for further refinements of models of the chemical evolution of the Galaxy.\n\\end{itemize}\n\n \n\\begin{acknowledgements}\nWe wish to thank the referee for useful comments. Y.T.Y. is a member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne. Y.T.Y. would like to thank the China Scholarship Council (CSC) and the Max-Planck-Institut f\\\"{u}r Radioastronomie (MPIfR) for the financial support. Y.T.Y. also thanks his fiancee, Siqi Guo, for her support during this pandemic period. C.K. acknowledges funding from the UK Science and Technology Facility Council through grant ST\/R000905\/1 and ST\/V000632\/1. We thank the IRAM staff for help provided during the observations.\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSimple organisms like fungi and slime moulds are able to display complex behaviours. This is surprising given that their network-like body plan lacks any central organizing centre. The slime mould \\emph{Physarum polycephalum}\\ has emerged as a model system to study the complex dynamics these organisms use to adapt to their environment. The organism has been shown to find the shortest path through a maze \\cite{Nakagaki:2000} and connect food sources in an efficient and at the same time robust network comparable to man-made transport networks \\cite{Tero:2010}. Furthermore, the slime mould distributes its body mass among several resources to obtain an optimal diet \\cite{Dussutour:2010} and is able to anticipate recurring stimuli \\cite{Saigusa:2008}.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{cutsite2.pdf}\n\\caption{Wound healing process in \\emph{P. polycephalum}\\ illustrated at four time points using bright field images. The cut occurred at \\SI{18}{\\minute} and the fan grown at cut site reached its maximal size at \\SI{60}{\\minute}. The network morphology was restored after \\SI{85}{\\minute}.}\n\\label{img:cutsite}\n\\end{figure*}\n\n\\emph{P. polycephalum}\\ is a true slime mould that forms a plasmodial network. Nuclei keep on dividing without forming cell walls, which results in a syncytial web-like network. The cytoplasm within this tubular network flows back and forth in a shuttle flow \\cite{Kamiya:1981}. These cytoplasmic flows are driven by cross-sectional contractions of the actin-myosin meshwork lining the gel-like tube walls \\cite{WohlfarthBottermann:1979}. Flows are organized across the entire network in a peristaltic wave of contractions that matches organism size \\cite{Alim:2013}. Flows generated in the organism are optimized for transport as contractions increase the effective dispersion of particles way beyond molecular diffusivity by a mechanism called Taylor dispersion \\cite{Marbach:2016}.\n\n\\emph{P. polycephalum}\\ adapts its network-like morphology to its environment by chemotaxis \\cite{Ueda:1976,DURHAM:1976,Chet:1977}. Here, stimulants are classified by being an attractant or a repellant depending on the organism's response to migrate toward or away from the stimulant. Stimulants have also been shown to affect cross-sectional contractions organism-wide by an increase in their frequency and amplitude for an attractant or a decrease for a repellant \\cite{Miyake:1994,Hejnowicz:1980}. A variety of chemical stimuli have been discussed for \\emph{P. polycephalum}, with glucose being a prominent attractant and salts like NaCl being effective repellants \\cite{Kincaid:1978,HIROSE:1982,MCCLORY:1985}. Temperature \\cite{Matsumoto:1988,Takamatsu:2004} and light \\cite{WohlfarthBottermann:1982,Nakagaki:1999} have also been found to act as stimulants that trigger organism-wide restructuring of the transport networks' morphology. In fact, the cytoplasmic flows themselves serve as the medium by which stimuli pervade the organism \\cite{Alim:2017}. \n\nA lot less is known about the impact of mechanical perturbations on the organism. In its natural habitat the slime mould suffers predation from grazing invertebrates causing severing that disrupts the transport network and its cytoplasmic flows. In experiments it has been found that quickly stretching a strand to 10-20\\% of its length while keeping it intact increases the amplitude of oscillations \\cite{Kamiya:1972}. Excising a single strand from a plasmodial network has been observed to lead to a roughly 20 minute cessation of contractions in the strand until recovery \\cite{Yoshimoto:1978}. This phenomenon was not observed for strands excised from the growing fan region of the slime mould resulting in speculations about the motive force being limited to the fan only. Yet, the cessation of contractions turned out to be hard to reproduce, see \\cite{Cieslawska:1984} and references therein. Among these discordant observations what remains established is local gelation of cytoplasmic flows upon touch without severing the organism \\cite{Achenbach:1981}. Despite the limited knowledge, wounding the organism by severing the network is part of daily laboratory routines and an eminent perturbation in natural habitat. \n\nHere we investigate \\emph{P. polycephalum} 's dynamics during wound healing following the quick and complete severing of a tube within the organism's network. We follow the process of wound healing across the individual's entire body, over the course of one hour after severing. The exemplary quantitative analysis of organism-wide contractions reveals a stepwise response spanning four different states. Briefly after severing, the contractions are often marked by an increase in amplitude and frequency, followed by a several minutes long cessation of contractions and stalling of cytoplasmic flows. This resting state is terminated by a sudden restart of vigorous contractions as the severed tube re-fuses. The vigorous state then transitions into a state of network-spanning contractions and continuous fan growth at the wounding site until the organism reverts back to pre-stimulus dynamics. Timing and significance of individual steps varies with the severity of cutting and cutting site location within the network. For example, stalling is found to be less pronounced when the network is cut in fan-like region. Overall, quick and complete severing triggers a response pattern with characteristics of the response to an attractive stimulus, including an increase in amplitude and frequency and net movement to stimulus site, see Fig.~\\ref{img:cutsite}. The reproducibility of stalling clarifies earlier contradictions and at the same time opens new avenues to investigate the biochemical dynamics behind the highly coordinated acto-myosin contractions underlying \\emph{P. polycephalum}'s arguably fascinating dynamics. \n\\section{Methods}\n\\subsection{Culturing and data acquisition}\nThe plasmodium is prepared from microplasmodia grown in liquid medium. The recipe for the medium is inspired by \\cite{Fessel:2012}, see Sec. S1. The advantage of this method over growing the plasmodium on oat flakes or bacteria is the ability to precisely control the nutritional state and amount of the organism. Also, plasmodia grown this way are free from oat flake residues or vacuoles containing food, which provides a cleaner sample for imaging. To prepare the plate for imaging, 0.2-0.5 mL of the microplasmodia grown in a shaking culture at $30^{\\circ}$C are transferred to an 1.5\\% agar plate and stored in a closed, but not sealed dish in the dark. After 12-24 hours, the microplasmodia fuse into a single plasmodium. The plasmodium is ready for imaging when there are no visible traces of liquid medium and the organism assumed its characteristic network shape, which usually occurs up to 36 hours after plating.\n\nThe imaging is performed with a Zeiss Axio Zoom V.16 microscope, equipped with a Zeiss PlanNeoFluar 1x\/0.25 objective and a Hamamatsu ORCA-Flash 4.0 digital camera. A green filter (550\/50nm) is placed over the transmission light source of the microscope to diminish \\emph{P. polycephalum}'s response to the light, and a humidity chamber prevents the sample from drying out. The acquisition of the images is done in Zeiss ZEN 2 (Blue Edition) software with bright-field setting. During the acquisition, the illumination of the sample is kept constant, and an image is taken every 3 seconds. The plasmodium is imaged for $\\sim$1 hour before the application of the mechanical stimulus to allow for the accommodation to the light \\cite{DURHAM:1976}. The stimulus is applied manually, using a microinjection needle with a blunt tip. The needle tip is held above the surface of the agar at a small angle and quickly dragged across the chosen plasmodial tube. The cut is severe and complete if the two parts of the tube separate completely. The plasmodium is then further imaged for more than 1 hour.\n\nUsing microplasmodia is so far the optimal way of obtaining non-severed networks, where the size and nutritional state are reproducible. However, there are challenges during the imaging that decrease the reproducibility of the experiment. In particular, plasmodia are highly motile and change their morphology accordingly. Furthermore, the organism tends to develop very large foraging fronts, which are not a suitable input for the presented comprehensive data analysis as they lack network characteristics. Lastly, the microscope light can act as stimulus \\cite{WohlfarthBottermann:1982,Nakagaki:1999,Tero:2010}, and even the green-filtered low-intensity illumination may cause the network to respond and change its behaviour to escape the imaging region. These challenges combined make the reproducibility and required stability of the network morphology over time challenging.\n\n\\subsection{Comprehensive network-based contraction analysis}\nTo quantify contraction dynamics we analyse bright field recordings in two different ways: for two morphologically static networks (see E2 and E3 in the experiment list) we perform an exhaustive network-based analysis as outlined in the following (see Fig.~\\ref{img:05_lineplots} and Fig. S4). For the additional 19 specimen which alter their network morphology dramatically over the course of the experiment, we analyse kymographs along static parts of the network as described in detail in Sec. S3 (see exemplary E1 and Mov. S5).\n\nImages recorded as a time series are processed as 8-bit uncompressed TIFs. At first every image is processed separately, then the results are stitched together, largely following Ref.~\\cite{Alim:2013}, and lastly the collective is analysed. On every single image, background is removed with the rolling-ball method. Then the image is used to create a mask, a binary image, with an intensity threshold that separates the network from the background. The mask is enhanced further, i.e.~only the biggest structure is considered, small holes are filled and single-pixel edges are smoothed. Subsequently, the resulting mask is used as a template for extracting the network's skeleton with a thinning method. In the skeletonized mask each pixel can be understood as a data point representing local intensity and diameter (see Fig.~\\ref{img:02_skel}). Local diameter is calculated as the largest fitting disk radius around the point within the mask. Within this disk the average intensity is computed and saved as intensity at the considered data point. Intensity and diameter anti-correlate due to the optical density of the slime mould and can therefore be used interchangeably considering Beer-Lambert law. Individual data points are attributed to a specific network branch of the network skeleton. To represent network topology, the network is broken down into vertices and edges where vertices describe pixel positions of branching points and edges represent two connected vertices. Each edge then acts as a parent for one specific branch. In this sense edges are abstracted simple connections and branches represent pixel-based resolution of a tube.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{02_skeleton_170220.pdf}\n\\caption{Scheme of intensity and diameter data extraction based on \\emph{P. polycephalum}\\ bright field images. The light grey area depicts the network mask based on the bright field images. Dark grey lines represent the network skeleton and the corresponding topology is shown in blue. Each pixel of the skeleton acts as a reference point for data derived during the analysis. The diameter is set as the distance from the reference point to the next non-mask pixel. The intensity is calculated by averaging individual pixel intensities over a corresponding disk (red).} \\label{img:02_skel}\n\\end{figure}\n\nAfter the network is extracted in space, the edges, vertices, diameters, and intensities are concatenated in time. To map intensity and diameter over time, a reference image is used, usually from an early time point. For every data point the shortest distance to any pixel in the reference image is calculated. This gives a quasi-static (x, y, t) $\\rightarrow$ (intensity, diameter) dataset, i.e.~the topology and vertex positions stay the same, but intensity and diameter can vary. This is justified as long as growth of the organism and vertex movement is minimal. The oscillatory behaviour of tubes in a certain time window can be described by four time dependent variables, namely amplitude $A$, frequency $f$ (or period $P$), phase $\\varphi$ and trend (base diameter) $d$. Each can be calculated from the time-evolution of the diameter or the intensity data, but if not stated otherwise the following results are only derived from intensity analysis. \n\nThe trend $d(t)$ is obtained with a moving-average filter with a kernel width of \\SI{200}{s} on each time trace (see Fig.~\\ref{img:03_linecomp}). The dataset is detrended with the calculated trend and smoothed with a Gaussian using a kernel width of \\SI{39}{s}. The kernel widths were chosen to extract the characteristic contraction pattern which usually has a frequency of \\SI{\\sim90}{\\second}. The values at every data point are stored as a complex valued time array, with the detrended and smoothed intensity representing the real part and the corresponding Hilbert transform representing the complex part, see S2 for more details. This time array, denoted analytic signal, serves as a basis to get instantaneous phase, frequency and amplitude by computing the angle or absolute value of the complex time series. Finally, the results are mapped back onto the network structure for each time point. In this fashion one can follow oscillatory behaviour resolved in time and space. Furthermore, the maps can be clustered in sub-networks and averaged separately to pinpoint local events in time. It should be mentioned that averaging of results for line plots, i.e.~Fig.~\\ref{img:05_lineplots}, is always done after the data-point based analysis took place. In this way for example, the apparent amplitude of the averaged intensity (Fig.~\\ref{img:05_lineplots}D) can be lower than the amplitude of each data point averaged (Fig.~\\ref{img:05_lineplots}B).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{03_linecomp_170306.pdf}\n\\caption{Derivation of oscillation specific parameters, i.e.~amplitude $A$(t), frequency $f$(t) and trend $d$(t), from single pixel time series. The trend is calculated using a moving average with a kernel width of \\SI{200}{s}. Intensity is filtered with a Gaussian of width \\SI{39}{s}. Amplitude and frequency are calculated from the absolute value and angle of the complex-valued analytic signal, respectively.} \\label{img:03_linecomp}\n\\end{figure}\n\\section{Results}\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=\\textwidth]{04_maps_170303.pdf}\n\\caption{Time evolution of an exemplary network and its spatially mapped oscillation parameters at \\SI{13}{\\minute}, \\SI{27}{\\minute}, \\SI{33}{\\minute} and \\SI{55}{\\minute}. The network was cut in the centre at \\SI{17.3}{\\minute} (\\emph{scissor icon}). Top row depicts the raw bright field data, middle row the local amplitude, and bottom row the local frequency. Amplitude and frequency decrease locally, first at the lower sub-network (\\emph{small dotted arc}) at \\SI{27}{\\minute}, subsequently at upper sub-network (\\emph{large dotted arc}) at \\SI{33}{\\minute}. At \\SI{38}{\\minute} cytoplasmic flows are re-established at the wounding site. Finally, amplitude and frequency values recover.} \\label{img:04_maps}\n\\end{figure*}\n\\subsection{Wounding induces fan growth at cut site}\nWe observe specimens before and after a quick and complete severing of a tube to follow the response of \\emph{P. polycephalum}\\ to wounding (see Fig.~\\ref{img:04_maps}A, Mov. S1 and Mov. S5). Bright field movies reveal that cutting of main tubes distal to fans triggers cessation of contractions followed by stalling of cytoplasmic flow (n=15 out of 21). After contractions resume the severed tube fuses back together (n=21 out of 21), i.e.~flow is re-established, and a fan starts to grow at the cut site. Furthermore, we observe accumulation of body mass close to the cut site which is most prominent in peripheral cuts (Fig. S2). However, the growth is transient and after a given time the initial morphology is restored and the organism returns to typical behaviour comparable to before wounding.\\\\\n\nIn consideration of previously mentioned technical limits, we selected one representative dataset with prominent discernible features for network-based analysis. The following findings are derived from this dataset and later compared with other experiments. The specific timing of events in the representative data set is as follows (see Fig.~\\ref{img:04_maps}). Two tubes are severed at \\SI{17.3}{\\minute} effectively dividing the network into two parts. In both sub-networks, the size-wise bigger and smaller part, flows stall transiently around \\SI{30}{\\minute}. At \\SI{38}{\\minute} a connecting tube is reinstated and starts to re-establish cytoplasmic flows across the cut site. Until about \\SI{63}{\\minute} a transient fan is created at the cut site. At \\SI{90}{\\minute} the initial morphology is restored and fans are grown elsewhere.\n\n\\subsection{Spatial mapping reveals localized stalling}\nWe perform network-based analysis on the wounded specimen to extract the interplay of contractions during the healing response. In particular, we map out the amplitude and frequency of contractions spatially (see Fig.~\\ref{img:04_maps}, Mov. S2 and Mov. S3). This allows us to exactly localize the onset of stalling as it goes hand in hand with low values of amplitude and frequency. Likewise, patterns in contraction dynamics in a region of interest are identified by spatially averaging amplitude and frequency in this region (see Fig.~\\ref{img:05_lineplots}).\n\nIn the representative dataset, wounding separates the network into two sub-networks. Spatial mapping reveals that oscillations cease on different time-scales in the two sub-networks. By identifying the two sub-networks as separate regions of interest, we quantify the patterns in contraction taking the spatial average of the respective contraction variables in each region. The small sub-network shows a drop in amplitude at \\SI{21.5}{\\minute} by \\SI{63}{\\percent} and only recovers eight and a half minutes later to comparable values. Here, the percentage is given as ratio of time averages before, during and after stalling. In detail, the averages of the first \\si{21.5} minutes, the \\si{9.5} minutes during stalling and \\si{15} minutes after stalling were considered. The bigger sub-network drops significantly later at \\SI{28}{\\minute} by \\SI{51}{\\percent} and recovers to \\SI{29}{\\percent} below the initial value nine minutes later. In the same time frames the frequency drops by \\SI{32}{\\percent} and \\SI{45}{\\percent} for the small and big sub-network, respectively. Yet, neither sub-network recovers its frequency fully right after the stalling phase. Only the small sub-network recovers 35 minutes later to initial frequencies whereas the bigger region levels off \\SI{35}{\\percent} below the initial value.\\\\\nFurthermore, the phase patterns over time (see Mov. S4) reveal changes in the travelling waves upon cutting. Initially (0 to \\SI{17.3}{\\minute}) one can observe peristaltic waves from the tail (right-hand side) to the front (left-hand side) which finally merge into concentric patterns in the fan regions. Then, at 18 to \\SI{30}{\\minute}, the small sub-network slows down noticeably (see change in frequency) and the big sub-network contracts with less apparent spatial correlation, i.e.~the peristaltic wave pattern is temporarily lost.\n\n\\subsection{Fan growth phase coincides with stable network-spanning contractions}\nAfter re-fusing of the two sub-networks, another distinct phase characterized by stable network-spanning contraction dynamics can be observed. In Fig.~\\ref{img:05_lineplots}D contractions appear uniform from \\SI{44}{\\minute} until \\SI{63}{\\minute}. During this phase, amplitude and frequency level off to a stable value with little fluctuations. The small sub-network shows a slight increase in frequency over this period and has more fluctuations in the average intensity data than the big sub-network. Note, that the time frame of these contractions coincide with fan formation at the cut site. Furthermore, the end of this phase also coincides with the largest fan in respect to area.\\\\\nNetwork-spanning contractions are further supported by the phase time series. When considering the phase development one can already observe a peristaltic wave travelling towards the cut site in the small sub-network as early as \\SI{30}{\\minute}. A spanning pattern in the large sub-network is reinstated around the \\SI{35}{\\minute} mark and a global pattern (small and large sub-network) appears roughly three minutes after re-fusing (\\SI{40}{\\minute}). Then a standing wave pattern appears between the central region including the cut site and the periphery. It is stable and network-spanning until \\SI{63}{\\minute}. Subsequently the phase pattern breaks into a peristaltic wave similar to pre-cut and propagates from the tail and the small sub-network into fan regions in the large sub-network.\n\n\\subsection{Stalling and fan growth periods are bridged by distinct transition periods}\nCloser analysis of contraction dynamics over time reveals that the time point of the cut, the stalling phase and the fan growth phase are transitioned by phases of high fluctuations. Particularly in the presented case, before stalling occurs, amplitude and frequency peak shortly in both sub-networks (see arrows in Fig.~\\ref{img:05_lineplots}). In the small sub-network this peak coincides with the cut, whereas another ten minutes pass for the big sub-network before the amplitude reaches its maximum. Surprisingly, here the frequency decline occurs three minutes before the amplitude drops. After stalling the amplitude increases sharply in both sub-networks, yet stays below previous values in the big sub-network. The small network undergoes a phase of roughly \\SI{13}{\\minute} where the amplitude oscillates vigorously. This also coincides with a second frequency drop even though there is no apparent drop in amplitude at this time point. After the fan growth phase, amplitude and frequency show slight gradients once more. Here behaviour becomes comparable to the pre-cut state as the slime mould develops a preferred growth direction in the periphery and continues foraging.\n\n\\begin{figure}[hpt]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{05_lineplots_170303.pdf}\n\\caption{Comparison of oscillation parameters in the big- and small-sub-network depicted in (A) which result from the cut. Grey area (cut site) is not considered in the analysis. Time series of amplitude (B), frequency (C) and intensity (D) are averaged in the respective domains and compared; top : big sub-network, bottom: small sub-network. In each of these plots the black dashed line indicates the moment of the cut. The first grey dashed line marks the time point of fusion and the second the moment of maximal fan size. Black bars underline periods of stalling in (B) and fan growth in (D). In (D) the solid line represents the Gaussian filtered intensity (kernel width = 39s) and markers show raw averaged data. Black arrows indicate respective extremal peaks in the transition periods. Four black dots on the time line correspond to the four time points chosen in Fig.\\ref{img:04_maps}.\n}\\label{img:05_lineplots}\n\\end{figure}\n\n\\subsection{Fan creation and stalling is reproducible for complete severing}\nFor comparison we analysed a second dataset with the same network-based method (see Fig. S4). The key features, i.e.~cut repair, stalling, a transition phase, stable network-spanning contractions and return to pre-cut behaviour are found likewise, but the succession and timing of the specific events vary. This dataset has a weaker fan growth at the cut site and the time point of maximal fan size follows immediately after fusion. Given the short period of fan growth global network-spanning contractions are not observed. However, standing phase wave patterns are visible in the larger sub-network before fusion. Lastly, the transition phase shows peaking amplitude and fluctuating frequencies and reverberates for more than 30 minutes. At \\SI{70}{\\minute} the network reinstates a peristaltic wave toward peripheral fan regions resuming pre-cut dynamics.\\\\\n\nIn further experiments analysed with a kymograph based approach, we confirmed stalling to be a common response after a cut (see Fig. E1, n=15 out of 21). However, the degree and duration of stalling is varying between experiments and is most reproducible for a severe cut close to the centre of the network.\\\\\nIn detail, we observe that both the degree and duration of stalling, depend on the network size and morphology, cut location, possibilities of re-routing the flow through neighbouring tubes and presence of large fans. Also, a network undergoing quick changes in morphology due to a presumed light shock is less likely to show stalling. Varying cut location shows that complete severing of a tube, with a diameter comparably large in size and few neighbouring tubes, results in strong stalling, see experiments E[2, 3, 5, 6, 8, 9, 12, 13, and 18]. The effect is even more pronounced in smaller networks and on tubes close to the centre of the network (E[2, 3, 5, 8, and 18]). Stalling is less pronounced, as measured by relative change in amplitude and frequency as well as visual inspection of bright field data, if severing was applied to fan-like regions or peripheral tubes (E[10, 11, 14, 15, 16, 17, 19, 20, and 21]). If a severed tube had alternative routes with a comparable flow direction, neighbouring tubes inflated shortly after the cut, indicating a re-routing of flow. Yet, in this case stalling severity ranged from non-existent (E19) to full-stop (E1). In all data sets fan growth is observed around the cut site, yet duration and fan sized varied greatly (see E2 and E9 as maximal and minimal examples). \\\\\nIn all 15 experiments that show stalling, the period lasted for a minimum of three minutes. The exact time point of stalling onset and its duration varied. Duration of transition periods also varied from complete omission up to \\num{22} minutes between cut and stalling. In 7 out of 15 experiments, a vigorous phase of increase in frequency or amplitude fluctuations could be observed in the transition phases.\n\n\\section{Discussion}\nWe investigate \\emph{P. polycephalum}'s response to wounding in the form of a quick and complete severing of tubes using bright field microscopy and quantitative analysis of contraction patterns. Mapping out the contractions amplitude and frequency in space and time allows us to uncover a multi-step pattern of wound healing in \\emph{P. polycephalum}. \n\nThe key of our network-based analysis is mapping contraction variables onto a few pixels serving as the skeletonized backbone of the complete network. This representation allows us to capture contraction dynamics across the entire network over the course of several hours with handleable amount of data. Furthermore, spatial mapping visualizes abstract variables in an approachable way which outlines region of interests or patterns in space. For example, in the representative data set the time-shift in the response pattern between the two sub-domains of the network would have been lost when averaging contraction dynamics across the entire network (see Fig. S1).\n\nAmong the multiple steps in the response to wounding the cessation of oscillations and stalling of the cytoplasmic streaming is most striking. The phenomenon of stalling of cytoplasmic flows has been observed previously ~\\cite{Kamiya:1972,Yoshimoto:1978}, but its reproducibility was deemed questionable \\cite{Cieslawska:1984}. Our work shows that cut location and severity are crucial parameters for inducing reproducible stalling. The stalling period is omitted when a tube is not completely severed, or cut in a way that allows the cut ends to rejoin quickly. In addition, the specific body plan affects the impact of a cutting stimulus. For example, severed fan-like regions show less pronounced stalling. However, we find reproducible strong stalling in networks where the affected tubes are crucial connections that cannot be re-routed easily - thereby clarifying previously discordant observations.\\\\\n\nStimuli are commonly classified into attractants or repellants. The response of \\emph{P. polycephalum}\\ to an attractive stimulus includes fan growth and mass transport towards the stimulus site, often accompanied with an increase of oscillation frequency and amplitude. When we apply a wounding stimulus resulting in complete cutting of a tube, we observe a multi-step response pattern where only two out of four steps show a noticeable increase in amplitude and frequency. Yet, wounding implies that the network architecture is perturbed. Taken into account that contraction frequency decreases as organism size decreases \\cite{Kuroda:2015} the impairment of network architecture itself might counteract any increase in frequency. Despite the weak indication from contraction frequency and amplitude, we always observe fan growth and movement of mass toward the cut site regardless of the tube hierarchy, plasmodium size or the severity of the cut. Fan growth is a lot bigger than initial spillage of cytoplasm due to cutting. Furthermore, we often identify a specific fan growth phase of network-spanning contractions well separated in time from the cutting event by the stalling phase. We therefore identify wounding as an attractive stimulus. The observation of network-spanning oscillations during fan outgrowth adds to our confidence about cutting being an attractive stimulus since the observed phase patterns resemble contraction patterns found in earlier work with attractive stimuli using glucose as a stimulant \\cite{Alim:2017}.\\\\\n\nEmploying spatial data analysis we uncovered that wounding triggers a choreography of multiple successive steps to heal the severed tube. The mere duration of the healing response now defines a suggested minimal wait time after trimming for \\emph{P. polycephalum}\\ experiments. The complexity of the response hints at an intricate signalling pattern underlying the coordination of contractions. It is likely that also the response to classical attractants and repellants, when scrutinized, reveal multiple steps. Unravelling the workings behind \\emph{P. polycephalum}'s ability to adapt, is arguably a fascinating albeit challenging question. Here, the reproducible cessation of contractions arising during this wound-healing response may open up new avenues to investigate the biochemical wiring underlying \\emph{P. polycephalum}'s complex behaviours. Furthermore, it is fascinating that the impact of wounding can be weakened by network architecture. This suggests that \\emph{P. polycephalum}'s body plan itself could be part of the organisms strategy to not only adapt to its environment, but also specifically prevent severe consequences of wounding. \n\n\\section*{Acknowledgements}\nWe thank Christian Westendorf for instructions on growing microplasmodia, as well as for invaluable discussions and advice. M.K. and F.B. acknowledge support by IMPRS for Physics of Biological and Complex Systems.\n\n\\section*{Bibliography}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAmong AGNs, blazars are the most luminous and violent objects in the\nuniverse and emit $\\gamma$-rays at energies higher than 100 MeV. They are\ndivided into two main subgroups: Highly\nvariable quasars, sometimes called optically violent variable (OVV) quasars,\nand BL Lacertae objects (BL Lac). Since one of the jets of blazars is\npointed toward the Earth, we see the jet emission strongly Doppler enhanced and\nhighly variable. According to the unification scheme \\citep{up95}, radio\ngalaxies are the mis-aligned parent population of blazars. Synchrotron peak\nfrequencies of BL Lac objects cover a large range from IR to X-rays, and based\non its location they are called low-frequency peaked BL Lac objects (LBL;\nsynchrotron peak in the IR), intermediate BL Lac objects (IBL; synchrotron\npeak in the optical\/UV), or high-frequency peaked BL Lac objects (HBL;\nsynchrotron peak in the X-rays).\n\nWhile there is little evidence for dense radiation environments\nin the nuclear regions of BL~Lac objects --- in particular, HBLs\n---, strong line emission in Flat Spectrum Radio Quasars (FSRQs)\nas well as the occasional detection of emission lines in the spectra of some\nBL~Lac objects \\citep[e.g.,][]{vermeulen95} indicates dense nuclear\nradiation fields in those objects. This is supported by spectral modeling\nof the SEDs of blazars using leptonic models which prefer scenarios\nbased on external radiation fields as sources for Compton scattering\nto produce the high-energy radiation in FSRQs, LBLs and also some\nIBLs \\citep[e.g.,][]{ghisellini98,madejski99,bb00,acciari08,abdo11}. If the\nVHE $\\gamma$-ray emission is indeed produced in the high-radiation-density\nenvironment of the broad line region (BLR) and\/or the dust torus of an\nAGN, it is expected to be strongly attenuated by $\\gamma\\gamma$ pair\nproduction \\citep[e.g.][]{pb97,donea03,reimer07,liu08,sb08}.\n \\cite{akc08} have suggested that such intrinsic $\\gamma\\gamma$ absorption may be\nresponsible for producing the unexpectedly hard intrinsic (i.e., after\ncorrection for $\\gamma\\gamma$ absorption by the extragalactic background\nlight) VHE $\\gamma$-ray spectra of some blazars at relatively high redshift.\nA similar effect has been invoked by \\cite{ps10} to explain the spectral\nbreaks in the {\\it Fermi} spectra of $\\gamma$-ray blazars.\nThis absorption process will lead to the development of Compton-supported\npair cascades in the circumnuclear environment \\citep[e.g.,][]{bk95,sb10,rb10,rb11}.\n\nIn \\cite{rb10,rb11}, we considered the full 3-dimensional development of\nCompton-supported VHE $\\gamma$-ray induced cascades in the external radiation\nfields in AGN environments.\n In those works, we have left the origin (leptonic SSC or IC, or hadronic) \nof the primary VHE $\\gamma$-ray emission deliberately unspecified in order to \ninvestigate the cascade development in as model-independent a way as possible.\nWe have shown that even very weak magnetic fields $(B\\lesssim\n\\mu$G) may be sufficient for efficient quasi-isotropization of the cascade emission.\nWe applied this idea to fit the {\\it Fermi}\n$\\gamma$-ray emission of the radio galaxies NGC~1275 and Cen~A. In\n\\cite{rb10,rb11}, parameters were chosen such that the synchrotron emission\nfrom the cascades was negligible.\n\nIn this paper, we present a generalization of the Monte-Carlo cascade\ncode developed in \\cite{rb11} to non-negligible magnetic fields and consider\nthe angle dependent synchrotron emission from the cascades. In section\n\\ref{setup}, we will outline the general model setup and assumptions\nand describe the modified Monte-Carlo code. Numerical results for generic\nparameters will be presented in section \\ref{parameterstudy}. We confirm\nthat for the objects Cen~A and NGC~1275 the synchrotron radiation from the\ncascades is negligible for those parameters used in \\cite{rb10,rb11}. In section\n\\ref{degeneracy}, we investigate the effect of the magnetic field and its\ndegeneracy. We show that by studying only the high energy emission from\nthe cascades, the magnetic field can not be determined, and additional\nconstraints are needed from the synchrotron emission component. This\nis illustrated for the case of NGC~1275. In section \\ref{3C279}, we will\ndemonstrate that for moderately strong magnetic fields the synchrotron\nemission from the cascades can produce a signature resembling the big blue\nbump (BBB) observed in several blazars and demonstrate \nthat this may make a non-negligible contribution to UV -- soft X-ray\nSED. We illustrate this for the case of 3C~279. We summarize in Section \n\\ref{summary}.\n\n\n\\section{\\label{setup}Model Setup and Code Description}\n\nThe general model setup used for this work is described in \\cite{rb10,rb11}.\nThe primary VHE $\\gamma$-ray emission is represented as a mono-directional\nbeam of $\\gamma$-rays propagating along the X axis, described by a power-law \nwith photon spectral index $\\alpha$ and a high-energy cut-off at $E_{\\gamma, max}$. \nWe assume that the primary $\\gamma$-rays interact via $\\gamma\\gamma$\nabsorption and pair production with an isotropic radiation field with \narbitrary spectrum within a fixed boundary, given by a radius $R_{\\rm ext}$.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f1.eps}\n\\caption{\\label{diskabs}$\\gamma\\gamma$ opacity due to accretion\ndisk photons as a function of height $z$ (in units of gravitational\nradii $r_g = GM\/c^2$) of the emission region above the black hole,\nfor three different VHE $\\gamma$-ray photon energies, and different\nblack-hole masses and disk luminosities. }\n\\end{figure}\n\nThe assumption of an isotropic external radiation field is appropriate\nfor line emission from the BLR, for distances from the central engine comparable\nto the size of the BLR ($\\sim 10^{17}$ -- $10^{18}$~cm), and for infrared emission\nfrom cold dust in the nuclear environment, on typical scales of $\\sim$~parsec.\nClose to the central black hole and accretion disk, direct emission from the\naccretion disk may dominate the radiation energy density. However, for moderate\ndistances from the disk, the primary $\\gamma$-ray beam will interact with the\naccretion disk emission under an unfavorable angle for $\\gamma\\gamma$ pair\nproduction. To illustrate this point, we plot in Figure \\ref{diskabs} the opacity\nto $\\gamma\\gamma$ absorption in the radiation field of an optically thick,\ngeometrically thin \\citep{ss73} accretion disk as a function of height\n$z$ of the emission region from the black hole. The figure shows that\ntypically, the accretion-disk $\\gamma\\gamma$ opacity drops below one\nat distances of a few 100 -- $10^3 \\, r_g$ from the black hole. This\nis of the order of the characteristic height of the emission region in\nleptonic models of blazar emission, and much smaller than the size of\nthe BLR or the dust torus. We therefore conclude that the neglect of\nthe accretion-disk emission in our simulations is a good approximation\nthroughout almost all of our simulation volume.\nFor the case of thermal blackbody radiation fields considered below, we\nchoose energy densities and blackbody temperatures characteristic of the\nobserved properties (temperatures and total luminosities) of thermal \ninfrared emission seen in AGN.\nIn order to keep our treatment as model independent as possible, we do not \nspecify the physical origin of the primary VHE $\\gamma$-ray spectrum. The\nsimplest assumption consistent with most models of $\\gamma$-ray emission\nin blazars is a simple power-law, which we use as input spectrum in our\nsimulations. The shape of the resulting pair cascade emission is only \nvery weakly dependent on the exact shape of the incident VHE $\\gamma$-ray \nspectrum. This is illustrated in Figure \\ref{PLExpcomparison}, in which\nwe run a cascade simulation, once with a straight power-law, once with\na powerlaw + exponential cut-off (as a more realistic representation of\na physical blazar high-energy spectrum), with identical environmental\nparameters, as listed in the figure caption. While the overall normalization\nof the cascade spectrum, obviously, depends on the flux of absorbed\nVHE $\\gamma$-rays (which is higher in the pure power-law case), the\ncascade spectra at energies below the $\\gamma\\gamma$ absorption\ntrough are virtually identical. In the cases relevant for this study,\nonly a small fraction of the $\\gamma$-ray power is absorbed and re-processed\ninto cascades. Therefore, feedback between the primary $\\gamma$-ray production\nregion and the cascade emission may be neglected, and the cascade development\ncan be treated as a process separate from the (unspecified) VHE $\\gamma$-ray\nproduction mechanism.\n\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f2.eps}\n\\caption{\\label{PLExpcomparison}Comparison of the Compton emission\nfrom cascades for two different input VHE $\\gamma$-ray spectral\nshapes: Blue (thin) lines indicate the emission for a pure power-law\ninput spectrum; red (thick) lines for a power-law + exponential cut-off.\nEnvironmental parameters are the same as in Figure \\ref{standardfig}\n(see below). Different line styles correspond to different viewing\nangles, $\\mu = \\cos\\theta_{\\rm obs}$, with respect to the jet axis,\nas indicated in the legend. Parameters: $B_x = B_y = 1$~$\\mu$G, \n$\\theta_B = 45^o$; $u_{\\rm ext} = 10^{-6}$~erg~cm$^{-3}$, \n$R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K. \nThe angular bin $0.8 \\le \\mu \\le 1$ contains the forward direction. \nThe (unabsorbed) primary $\\gamma$-ray input spectra are shown by the \ndot-dot-dashed lines.\n}\n\\end{figure}\n\nOur code evaluates $\\gamma\\gamma$ absorption and pair production using the\nfull analytical solution to the pair production spectrum of \\cite{bs97} under\nthe assumption that the produced electron and positron travel initially along\nthe direction\nof propagation of the incoming $\\gamma$-ray. The trajectories of the particles are\nfollowed in full 3-D geometry. Compton scattering is evaluated using the head-on\napproximation, assuming that the scattered photon travels along the direction of\nmotion of the electron\/positron at the time of scattering. The Compton\nenergy loss to the electron is properly accounted for at the time of each scattering.\n\nFor simplicity, the magnetic field in our simulations is treated as \nhomogeneous, oriented at an angle $\\theta_B$ with respect to the jet axis.\nThis may be considered an appropriate proxy for a helical magnetic field\nwith the ratio of toroidal ($B_{\\rm tor}$) and poloidal ($B_x$) magnetic\nfields given by $\\tan\\theta_B = B_{\\rm tor}\/B_x$.\nThe code calculates the synchrotron energy loss of cascade particles in the\nfollowing way: The energy of electrons\/positrons is decreased by $\\Delta E_{\\rm sy}\n= \\dot E_{\\rm sy} \\frac{l_c}{c}$ between successive Compton scatterings, where\n$l_c$ is the Monte Carlo generated distance traveled to the next scattering\nand $ \\dot E_{\\rm sy} = - 2 c \\sigma_{T} u_B \\gamma^2 \\sin^2 \\psi$ with\n$\\psi$ being a pitch angle between the particle momentum and the magnetic field.\nWe assume that the trajectory of the particles between two Compton scatterings\nis not affected by synchrotron radiation which is valid for $ u_B \\lesssim u_{\\rm ext}$.\nThen for $10$ random points between two successive Compton scatterings, we determine\nthe position and direction of motion of the particles at these points and write\nthe spectral power in synchrotron radiation $P_{\\nu}$ into a synchrotron output\nfile for the angular bin corresponding to the electron's\/positron's direction of\nmotion. The synchrotron power $P_{\\nu}$ of a single $e^{\\pm}$ is approximated as:\n\n\\begin{equation}\nP_{\\nu} = 2 \\frac{c \\sigma_T}{\\Gamma(\\frac{4}{3})} u_B \\beta^2\\gamma^2\n\\frac{\\nu^{1\/3}}{\\nu_c^{4\/3}} e^{-\\nu\/\\nu_c}\n\\label{sy_asymptotic}\n\\end{equation}\n\n\\citep{boettcher12} where the critical frequency \n$\\nu_c = \\frac{3 q B}{4\\pi m_e c}\\gamma^2 \\sin\\psi = \n4.2\\times10^6 \\sin\\psi B_G \\gamma^2$~Hz with $B_G$ being the magnetic \nfield in units of Gauss. \nThe approximation (\\ref{sy_asymptotic}) represents the synchrotron\nspectrum to whithin a few \\% at all frequencies.\n\n\n\\section{\\label{parameterstudy}Numerical Results}\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f3.eps}\n\\caption{\\label{standardfig}\nCascade emission at different viewing angles ($\\mu = \\cos\\theta_{\\rm obs}$). \nParameters of the target photon field are the same as for Figure 1. The \ninput photon spectrum is a pure power-law with $\\alpha = 2.5$, \n{$E_{\\gamma, {\\rm max}} = 5$~TeV}. The green solid line represents \nthe target photon field.\n}\n\\end{figure}\n\nWe have used the cascade Monte-Carlo code described in the previous section\nto evaluate the angle-dependent Compton and synchrotron spectra from VHE\n$\\gamma$-ray induced pair cascades for a variety of generic\nparameter choices. Figure \\ref{standardfig} illustrates the viewing angle\ndependence of the cascade emission. For this simulation, we assumed a\nmagnetic field of $B = \\sqrt{2} \\mu$G, oriented at an angle $\\theta_B = 45^o$\nwith respect to the X axis ($B_x = 1 \\, \\mu$G, $B_y = 1 \\, \\mu$G). The\nexternal radiation field is a thermal blackbody with $u_{\\rm ext} =\n10^{-6}$~erg~cm$^{-3}$, extended over a region of radius $R_{\\rm ext} = \n10^{18}$~cm, with a blackbody temperature of $ T = 10^3$~K (corresponding \nto a peak of the blackbody spectrum at a photon energy of $E_s^{\\rm pk}= \n0.25$~eV). This leads to a $\\gamma\\gamma$ absorption cut-off at an energy\n$E_c = (m_e c^2)^2 \/ E_s \\sim 2$~TeV. The incident $\\gamma$-ray spectrum \nhas a photon index of $\\alpha = 2.5$ and extends out to $E_{\\gamma, {\\rm max}} \n= 5$~TeV.\n\nFor any given viewing angle $\\theta$ with respect to the direction of \npropagation of the primary $\\gamma$-rays a critical electron energy for \nwhich the deflection angle over a Compton length equals the observing \nangle, i.e., $\\theta \\sim \\lambda_{\\rm IC} \/ r_g$ can be defined. This \nyields the characteristic electron energy $E_{\\rm e, br} = \\gamma _c \nm_e c^2$ corresponding to a given observing angle $\\theta$:\n\n\\begin{equation}\n\\gamma_c =\\sqrt{\\frac {3 e B}{4 \\sigma_T u_{\\rm ext}\\theta}}\\sim 7.2\\times10^5\nB_{-6}^{1\/2} u_{-3}^{-1\/2} \\theta^{-1\/2}\n\\end{equation}\nwhere $B_{-6} = B\/\\mu$G and $u_{-3} = u_{\\rm ext}\/(10^{-3}$~erg~cm$^{-3}$).\n\nThis expression has been derived assuming that the Compton cooling\nlength can be calculated in the Thomson regime, which is valid for\n$\\gamma \\lesssim 2 \\times 10^6 T_3^{-1}$ for a thermal target photon\nfield with temperature $T = 10^3 \\, T_3$~K, or $\\gamma \\lesssim 5 \\times \n10^4$ for a Ly$\\alpha$-dominated target photon field.\nIf these electrons radiate their energy by synchrotron radiation and Compton\nupscattering with the soft photon field in the Thomson regime, we can find the\ncorresponding spectral breaks for synchrotron radiation and Compton scattering\nas a function of viewing angle:\n\n\\begin{equation}\nE_{\\rm sy,br}\\cong \\gamma_c^2 B m_e c^2\/B_{cr}= \\frac{ 3m_e c^2 e B^2}{4\n\\sigma_T u_{\\rm ext}\n\\, \\theta \\ B_{\\rm cr}}\\sim 6.15 B_{-6}^2 u_{-3}^{-1}\n\\theta^{-1}{\\rm meV}\n\\label{ViewingangleSy}\n\\end{equation}\n\n\\begin{equation}\nE_{\\rm IC, br}\\cong\\gamma_c^2 E_s = {3 \\, e \\, B \\over 4 \\, \\sigma_T \\, u_{ext}\n\\, \\theta}\n\\, E_s \\sim 5.4\\times 10^2 \\, E_{s,1}B_{-6} u_{-3}^{-1} \\theta^{-1}\\; {\\rm GeV}.\n\\label{ICbreak}\n\\end{equation}\nwhere the $B_{\\rm cr}= \\frac{m_e^2 c^3}{e \\hbar} = 4.4 \\times10^{13}~G$ and\n$E_{s,1} = E_s\/(1$~eV).\n\nTherefore, the ratio of the Compton to the synchrotron peak frequency is given by:\n\\begin{equation}\n\\frac{E_{\\rm IC, br}}{ E_{\\rm sy,br}} = \\epsilon_s \\frac{B_{\\rm cr}}{B}\n\\label{RatioComToSyn}\n\\end{equation}\nwhere $\\epsilon_s = \\frac{E_s}{m_e c^2}$.\n\nFigure \\ref{standardfig} shows that with increasing viewing angle, the\nspectral peaks of both the synchrotron and Compton emission shift to\nlower energy. This is because the Compton cooling length of the high\nenergy particles is much smaller than their Larmor radius $\\lambda_{IC}\\ll\nr_g$, so they are emitting while traveling in the forward direction. Instead,\nfor low energy particles, $\\lambda_{IC}\\geq r_g$, so that they are deflected\nbefore they are emitting. For the Compton emission this effect was already\ndiscussed in \\cite{rb10,rb11}.\n\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f4.eps}\n\\caption{\\label{ufig}The effect of a varying external radiation energy density.\nParameters: $B_x = B_y = 10^{-6}$~G; $R_{\\rm ext} = 10^{16}$~cm, $T = 10^3$~K;\n$\\alpha = 2.5$, $E_{\\gamma, {\\rm max}} = 5$~TeV. The cascade emission in the \nangular bin $0.2 \\leq \\mu \\leq 0.4$ is shown.}\n\\end{figure}\n\nFigure \\ref{ufig} shows the cascade spectra for different values\nof the external radiation field energy density $u_{\\rm ext}$. In accordance\nwith equations \\ref{ViewingangleSy} and \\ref{ICbreak}, for higher values of\nthe external radiation field, the spectral breaks of both radiation components\nshift to lower energies. Figure \\ref{ufig} also shows that the synchrotron\nluminosities of the cascades decrease with increasing $u_{\\rm ext}$ while\nthe Compton luminosities of the cascades increase. For a larger value of\n$u_{\\rm ext}$ and fixed blackbody temperature the soft target photon number\ndensity increases and $\\tau_{\\gamma\\gamma}$ becomes larger so that the number\nof VHE photons which will be absorbed increases and the photon flux of Compton\nemission from the cascades becomes larger. For very large values of $u_{\\rm ext}$,\n$\\tau_{\\gamma\\gamma}\\gg 1$ for photons above the pair production threshold so\nthat essentially all VHE photons will be absorbed and the Compton flux from\nthe cascade becomes independent of $u_{\\rm ext}$ \\cite[]{rb10,rb11}. The ratio\nof emitted power in Compton to synchrotron radiation in the linear regime\n$(\\tau_{\\gamma\\gamma} \\lesssim 1)$ is given by:\n\\begin{equation}\n\\frac{P_{sy}}{P_{IC}}=\\frac{B^2 \/ 8 \\pi}{u_{ext}}\n\\label{RatioCom}\n\\end{equation}\nif Compton scattering occurs in the Thomson regime.\nThe flux ratio $\\frac{F_{sy}}{F_{IC}} \\varpropto\nu_{\\rm ext}^{-1}$, so that by increasing the $u_{\\rm ext}$ the synchrotron\nflux decreases.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f5.eps}\n\\caption{\\label{Bfig}The effect of a varying magnetic field strength for a\nfixed angle of $\\theta_B = 45^o$ between jet axis and magnetic field. Parameters:\n$u_{\\rm ext} = 10^{-6}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K,\n$\\alpha = 2.5$, {$E_{\\gamma, {\\rm max}} = 5$~TeV}. The cascade emission in the\nangular bin $0.2\\leq\\mu\\leq0.4$ is shown. The solid green line represents \nthe target photon field.\n}\n\\end{figure}\n\nFigure \\ref{Bfig} illustrates the effect of a varying magnetic field strength\nfor fixed magnetic field\norientation ($\\theta_B = 45^o$). We see that the synchrotron peak energy\nincreases proportional to the square of the magnetic field strength as expected\nfrom Eq. \\ref{ViewingangleSy}. As already discussed in \\cite{rb10,rb11} the\nCompton energy break is proportional to the magnetic field strength. as long\nas it occurs below the $\\gamma\\gamma$ absorption cut-off energy. The synchrotron\nflux is proportional to the square of the magnetic field strength.\nThe flux ratio is $\\frac{F_{sy}}{F_{IC}} \\varpropto B^2$ until the fluxes become\ncomparable, at which point our treatment of synchrotron losses breaks down.\n\n\\begin{figure}[ht]\n\\vskip 1cm\n \\centerline{\n \\includegraphics[width=8cm,height=8cm]{f6a.eps}\n \\includegraphics[width=8cm,height=8cm]{f6b.eps}}\n \\caption{\\label{BfigF}The effect of a varying magnetic field orientation\nfor a fixed magnetic field strength of $B = 1 \\, \\mu$G, $u_{\\rm ext} =\n10^{-6}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{18}$~cm, $T = 1000$~K, $\\alpha = 2.5$,\n{$E_{\\gamma, {\\rm max}} = 5$~TeV}. Left figure: angular bin $0.8 \\leq\\mu\\leq 1.0$\n(dominated by the forward direction, i.e., the blazar case). Right figure: angular \nbin $0.2 \\le \\mu \\le 0.4$, representative of radio galaxies.\nThe green solid lines represent the target photon fields.\n}\n \\end{figure}\n\nFigure \\ref{BfigF} illustrates the effects of a varying magnetic-field orientation\nwith respect to the jet axis, for fixed magnetic-field strength $B = 1 \\, \\mu$G\nfor different angular bins. The results for the Compton component have been\ndiscussed \\cite{rb11}. The figure illustrates that primarily the perpendicular\n($B_y$) component of the magnetic field is responsible for synchrotron radiation.\n\n\n\\begin{figure}[ht]\n\\vskip 1cm\n \\centerline{\n \\includegraphics[width=8cm,height=8cm]{f7a.eps}\n \\hskip 1cm\n \\includegraphics[width=8cm,height=8cm]{f7b.eps}}\n \\caption{\\label{fitCenAandNGC1275}Compton and synchrotron radiation form\n the cascades. Left figure: (Fit to the SED of Cen~A.); Right figure:\n (spectrum of NGC~1275 with a simulated cascade spectrum from a mis-aligned\n blazar, along with the cascade spectra at larger viewing angles)}\n \\end{figure}\n\nFigures \\ref{fitCenAandNGC1275} illustrate that the cascades emissions from\nthe synchrotron radiation for parameters used in \\cite[]{rb10,rb11} are\nnegligible compared to the cascade Compton emission and much smaller than\nthe synchrotron radiation from the jet itself. This confirms that neglecting\nsynchrotron radiation in our previous works was justified.\n\n\\section{\\label{degeneracy}Magnetic Field degeneracy}\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=12cm]{f8.eps}\n\\caption{\\label{Degeneracy1}\nSynchrotron and Compton emission form the cascades for NGC~1275\n($0.6 \\leq\\mu\\leq 0.8 $). Parameters: $\\theta_B = 11^o$;\n$u_{\\rm ext} = 5\\times10^{-2}$~erg~cm$^{-3}$, $R_{\\rm ext} = 10^{16}$~cm,\n$E_s=E_{L\\alpha}$,\n$\\alpha = 2.5$, {$E_{\\gamma, {\\rm max}} = 5$~TeV}.\n}\n\\end{figure}\n\nIn \\cite{rb10}, we presented a fit to the \\emph{Fermi} spectrum of the radio\ngalaxy NGC~1275. We now show that there is a degeneracy of the magnetic field,\nboth orientation and strength, if only the high energy output from the cascades\nis considered. Figure \\ref{Degeneracy1} shows this effect for NGC~1275. In this\nplot, the external radiation field is\nparameterized through $u_{\\rm ext} = 5 \\times 10^{-2}$~erg~cm$^{-3}$ with photon\nenergy $E_s = E_{Ly\\alpha}$ and $R_{\\rm ext} = 10^{16}$~cm. This size scale is\nappropriate for low-luminosity AGN as observed in NGC~1275 \\citep[e.g.][]{kaspi07},\nand the parameters combine\nto a BLR luminosity of $L_{\\rm BLR} = 4 \\pi R_{\\rm ext}^2 \\, c \\, u_{\\rm ext} =\n1.9\\times 10^{42}$~erg~s$^{-1}$, in agreement with the observed value for NGC~1275.\nThe magnetic field orientation is at an angle of $\\theta_B = 11^o$. \nThe mass of the black hole in NGC~1275 is uncertain, and estimates \nrange from a few times $10^6 \\, M_{\\odot}$ \\citep{levinson95} to \n$\\sim 10^8 \\, M_{\\odot}$ \\citep{wilman05}. Assuming a characteristic\nfraction of $0.1$ of the accretion-disk luminosity to be re-processed\nin the BLR, the accretion-disk luminosity may be estimated to be $L_D\n\\sim 10^{43}$~erg~s$^{-1}$. Figure \\ref{diskabs} shows the $\\gamma\\gamma$\nabsorption depth due to the disk radiation field for $L_D = 10^{43}$~erg~s$^{-1}$\nfor the two possible extreme values of the black-hole mass, as a function\nof height $z$ of the emission region above the accretion disk. It \nshows that for $M_{\\rm BH} = 10^6 \\, M_{\\odot}$, the $\\gamma\\gamma$\nopacity drops below one at $\\sim 10^3 \\, r_g \\sim 10^{14}$~cm from \nthe black hole, while for $M_{\\rm BH} = 10^8 \\, M_{\\odot}$ $\\gamma\\gamma$\nabsorption becomes negligible at $\\sim 10^2 \\, r_g \\sim \\sim 10^{15}$~cm \nfor primary $\\gamma$-rays of $E_{\\gamma} = 1$~TeV, and much earlier for\nlower-energy photons. Therefore, throughout most of our simulation \nvolume ($R_{\\rm ext} = 10^{16}$~cm), $\\gamma\\gamma$ absorption in\nthe disk radiation field can be safely neglected.\n\nThe cascade spectrum shown in Figure \\ref{Degeneracy1} pertains\nto the angular bin $0.6 < \\mu < 0.8$ (corresponding to $37^o \\lesssim\n\\theta \\lesssim 53^o$), appropriate for the known orientation of NGC~1275. In\n\\cite[]{rb10,rb11}, we have shown that for magnetic field values of $B\\geq 1$~nG\nand for energy density $u_{ext} \\geq 10^{-3}$~erg~cm$^{-3}$, there is no pronounced\nbreak in the cascade spectrum and the cascade is independent of magnetic field.\nIn general we expect no break in the cascade Compton emission if $E_{\\rm IC,br}\n\\gtrsim \\frac{(mc^2)^2}{E_s}$, which leads to the condition:\n\n\\begin{equation}\nB \\gtrsim \\frac{{(m_e c^2)}^2 4\\sigma_T u_{\\rm ext} \\theta}{3 e (E_s)^2} \\sim 5 \\,\nu_{ext,-3} E_{s,1}^{-2} \\theta \\; {\\rm nG}\n\\label{relation}\n\\end{equation}\n\nFigure \\ref{Degeneracy1} shows that while the high energy emission due to\ndeflection of the cascade up to the $\\gamma\\gamma$ absorption trough remains\nthe same for the different magnetic fields, the synchrotron emission from the\ncascade changes. Therefore, determining the B field requires knowledge of the\nsynchrotron emission.\n\nIn the regime where $E_{\\rm IC, br}$ is independent of the magnetic field,\n$\\nu_{\\rm sy}\\varpropto B$ according to Eq. \\ref{RatioComToSyn} and the synchrotron\npower is proportional to the square of magnetic field in agreement with figure\n\\ref{Degeneracy1}.\n\nSince the synchrotron\/Compton flux ratio $\\frac{F_{\\rm sy}}{F_{\\rm IC}}\n\\varpropto B^2$, we expect that for sufficiently high magnetic fields,\nwe will reach the regime where the Compton flux from the cascades is\nequal to or smaller than the synchrotron flux in which case our numerical\nscheme is no longer applicable.\n\n\n\\section{\\label{3C279}The Big Blue Bump}\n\nThe spectral Energy distribution of AGN in the ultraviolet (UV) to soft X-ray\nband ($ \\sim 10$~eV-$1$~keV) is notoriously difficult to observe because of\ndust and gas in our galaxy and the AGN environment. The SEDs of many blazars\nexhibits a UV soft X-ray excess, called the big blue bump (BBB)\n\\citep[]{pian99,palma11,raiteri05,raiteri06,raiteri07}. It is often\nattributed to the thermal emission from the accretion disk. In blazars,\nits signature is often particularly hard to detect because of dominant\nnon-thermal emission from the jet. Understanding the origin of the BBB\nis important since this provides information on the central engine of\nthe AGN.\n\n3C~279 was among the first blazars discovered as a $\\gamma$-ray source with\nthe Compton Gamma-Ray Observatory \\citep[]{hartman92}. In 2007 it was detected\nas a VHE $\\gamma$-ray source with the MAGIC I telescope, making it the most\ndistant known VHE $\\gamma$-ray source at a redshift of $0.536$ \\citep[]{HB93}.\nIts relativistic jet is oriented at a small angle to the line of the sight\nof $< 0.5^0$ \\citep[]{J04}. It is also detected by \\emph{Fermi} \\citep[]{abdo09c}\nwith photon spectral index $2.23$. There is evidence of a spectral break of\naround a few GeV to a photon spectral index of $2.50$. It is strongly believed\nthat the radio to optical emission is due to synchrotron radiation by relativistic\nparticles in the jet. However, the origin of the high energy emission is still not\nwell understood \\citep[see, e.g.,][]{br09}.\n\n\\cite{pian99} monitored 3C~279 in the ultraviolet, using IUE, and combined\ntheir data with higher-energy observations from ROSAT and EGRET from 1992 December\nto 1993 January. During this period, the source was in a very low state, allowing\nfor the detection of a UV excess (the BBB), which is typically hidden below a\ndominant power-law continuum attributed to non-thermal emission from the jet.\n\\cite{pian99} proposed that the $\\gamma$-ray emission in the SED of 3C~279\nis produced by the external Compton mechanism, and suggested that the observed\nUV excess might be due to thermal emission from an accretion disk.\n\nAs an alternative to thermal emission from the accretion disk, \\cite{S97} proposed\nthe bulk Compton mechanism as a possible explanation of a UV\/X-ray excess in quasar\nSEDs. If the jet contains a substantial population of cold (i.e., thermal,\nnon-relativistic or mildly relativistic) electrons, they could scatter\nexternal optical\/UV photons with the bulk Lorentz factor of $\\Gamma \\thicksim 10$,\nresulting in bulk Compton radiation in the far UV or soft X-ray range.\n\nHere we suggest an alternative contribution to the BBB feature from\ncascade synchrotron emission. Figures 2 -- 4\nillustrate that the synchrotron emission from cascades may peak in the UV\/X-ray\nrange, thus mimicking a BBB for sufficiently strong magnetic fields\n($B \\gtrsim 1$~mG). Figure \\ref{fit3C279} illustrates \nthe contribution that synchrotron emission from VHE $\\gamma$-ray\ninduced pair cascades can make to the BBB in 3C~279.\nThe primary HE $ \\gamma$-ray spectrum with a photon spectral index of\n$\\alpha = 2.5$ matcheds the Fermi spectrum of 3C~279.\nThe external radiation field is parameterized through $u_{\\rm ext} =\n10^{-4}$~erg~cm$^{-3}$ and $R_{\\rm ext} = 5\\times10^{17}$~cm,\nand the parameters combine to the luminosity of $L = 4 \\pi R_{\\rm ext}^2 \\, c\n\\, u_{\\rm ext}\n\\sim 10^{43}$~erg~s$^{-1}$ corresponding to a $\\nu F_{\\nu}$ peak\nflux of $\\thicksim 10^9$~JyHz, about $2$ orders of magnitude below the observed\nIR\/optical -- UV flux level. The magnetic field is $B = 10^{-2}$~G, oriented at\nan angle of $\\theta_B = 85^o$. The incident $\\gamma$-ray spectrum extends out\nto $E_{\\gamma, {\\rm max}} = 5$~TeV, and the external radiation field is modeled\nas a blackbody with a temperature of $ T= 2000$~K (corresponding to a peak of the\nblackbody spectrum at a photon energy of $E_s^{\\rm pk}= 0.5$~eV). This leads\nto a $\\gamma\\gamma$ absorption cut-off at an energy $E_c = (m_e c^2)^2 \/ E_s\n\\sim 1$~TeV.\n\n\\begin{figure}[ht]\n\\vskip 1.5cm\n\\centering\n\\includegraphics[width=10cm]{f9.eps}\n\\caption{\\label{fit3C279}Illustration of a possible BBB in 3C~279 from\ncascade synchrotron emission.\nParameters: $B= 10^{-2}$~G, $\\theta_B = 85^0$;\n $R_{\\rm ext} = 5\\times10^{17}$~cm, $T = 2000$~K,\n$\\alpha = 2.37$, $u_{\\rm ext} = 10^{-4}$~erg~cm$^{-3}$, {$E_{\\gamma, {\\rm max}}\n= 5$~TeV}. Data from \\cite{abdo10}.\n}\n\\end{figure}\n\nWe suggest that synchrotron emission from VHE $\\gamma$-ray induced pair\ncascades can enhance the BBB feature in the SEDs of several\nblazars such as 3C~279. An observational test of this hypothesis may be\nprovided through spectropolarimetry. A BBB due to (unpolarized) thermal\nemission from an accretion disk will produce a decreasing percentage of\npolarization with increasing frequency throughout the optical\/UV range.\nIn contrast, if the BBB is produced as synchrotron emission from cascade\npairs in globally ordered magnetic fields, it is also expected to be polarized.\nTherefore, we predict that a BBB due to cascade synchrotron emission would result\nin a degree of polarization showing only a weak dependence on frequency over the\noptical\/UV range. As an example, in recent observations of the high-redshift\n$\\gamma$-ray loud quasar PKS~0528+134, \\cite{palma11} found a decreasing\ndegree of polarization with increasing frequency throughout the optical range,\narguing for an increasing contribution from thermal emission towards the blue\nend of the optical spectrum.\n\n\n\\section{\\label{summary}Summary}\n\nWe investigated the magnetic-field dependence and synchrotron emission\nsignatures of Compton-supported pair cascades initiated by the interaction\nof nuclear VHE $\\gamma$-rays with arbitrary external radiation fields, \nfor a model-independent, generic power-law shape of the primary\nVHE $\\gamma$-ray emission.\nWe\nfollow the spatial development of the cascade in full 3-dimensional geometry\nand study the dependence of the radiative\noutput on various parameters pertaining to the external radiation field and\nthe magnetic field in the cascade region.\nWe confirm that synchrotron radiation from the cascades is negligible in\nNGC~1275 and Cen~A for the parameters we used in our previous works. We\ndemonstrated that the magnetic field can not be well constrained by considering\nthe high-energy (Compton) output from the cascade emission alone, without\nobservational signatures from their synchrotron emission. This was illustrated\nfor the case of NGC~1275, for which we could produce equally acceptable fits\nto the Fermi spectrum for a variety of magnetic-field values, which resulted\nin substantially different synchrotron signatures.\n\nWe have shown that synchrotron emission from VHE $\\gamma$-ray induced pair\ncascades may produce UV\/X-ray signatures resembling the BBB observed in the\nSEDs of several blazars, in particular in their low states. \nWe used the example of 3C~279 to illustrate that cascade synchrotron\nemission may make a substantial contribution to the BBB feature.\nWe point out that spectropolarimetry may serve\nas a possible observational test to distinguish a thermal from a non-thermal\n(cascade) origin of the BBB.\n\n\\acknowledgements{ This work was supported by NASA through Fermi Guest\nInvestigator Grants NNX09AT81G and NNX10AO49G. We thank the anonymous \nreferee for valuable suggestions. }\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}