diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpkfq" "b/data_all_eng_slimpj/shuffled/split2/finalzzpkfq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpkfq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe paper studies discrete time processes and their predictability and randomness\nin deterministic pathwise setting, without using probabilistic assumptions on the ensemble.\n\\par\nUnderstanding of the pathwise randomness leads to many applications in Monte-Carlo methods, cryptography, and control systems.\nThere are many classical works devoted to the concept of pathwise randomness and the problem of distinguishability\n of random sequences; see the references in \\cite{LV,Dow}. \\index{Li and Vitanyi\n(1993) and Downey (2004).} In particular, the approach from \\\n Borel (1909) \\cite{Bor} , Mises (1919) \\cite{Mis} , Church (1940) \\cite{Ch} was based on\n limits of the sampling proportions of zeros in the binary sequences and subsequences; Kolmogorov\n(1965) \\cite{Ko65} and Loveland (1966) \\cite{Lo} developed a different concept of the algorithmic randomness and compressibility;\nSchnorr (1971) \\cite{Sch} suggested approach based on predicability and martingale properties. So far, the exiting theory is devoted to the problem of distinguishability\n of random sequences and does not consider the problem of quantification of the degree of randomness.\nThis paper studies randomness in the sense of\nthe pathwise predicability and attempts to develop an approach for quantification and separation of the ``noise\" for the\nsequences that are deemed to be random. The estimation of the degree on randomness is a difficult problem, since the task of\ndetecting the randomness is nontrivial itself.\n\\par\nThe paper investigates randomness and noise for the\nsequences in a more special setting originated from the linear filtering and prediction of stochastic processes\nrather than algorithmic randomness in the sprit of Downey (2004). We suggest exploring the following straightforward pathwise\ncriterion: a class of sequences that is\npredictable or such that its missing value can be recovered without error from observations of remaining values\n is assumed to consist of non-random\nsequences.\n\n\n\nFor stationary discrete time processes, there\nis a criterion of predictability and recoverability in the frequency domain setting\ngiven by the classical minimality criterion \\cite{Ko41},\nTheorem 24, and the Szeg\\\"o-Kolmogorov theorem; see \\cite{Sz,V}\\index{\nSzeg\\\"o (1920, 1921), Verblunsky (1936), Kolmogorov (1941),\\index{\nTheorem 2}} and recent literature reviews in \\cite{Bin,S}.\\index{ Simon (2011) and Bingham\n(2012).} By this theorem, a stationary process is\npredictable if its spectral density is vanishing with a certain rate at a point of the\nunit circle $\\{z\\in{\\bf C}:\\ |z|=1\\}$. In particular, it holds if the spectral\ndensity vanishes on an arc of\nthe unit circle, i.e., the process is bandlimited. There are many works devoted to smoothing in frequency domain and sampling; see, e.g.,\n\\cite{Alem,rema, F94, FKR, Leef, TH,W} and the bibliography here.\n \\index{\\cite{jerry}, \\cite{pollock}, \\cite{PFG}, \\cite{PFG1}, \\cite{AU}.} In \\cite{D10,D12a,D12b,D12c,D12d,D16}, predictability was\nreaddressed in the deterministic setting for two-sided sequences for\nwith Z-transform vanishing in a point on ${\\mathbb{T}}$, and some predictors\nwere suggested. These results were based on frequency characteristics of the entire\ntwo-sided sequences, since the properties of the Z-transforms were used. Application of the two-sided Z-transform requires to select some\npast time at the middle of the time interval of the observations as the zero point for a model of the two-sided sequence; this could be inconvenient.\n In many applications, it is more convenient\n to represent data flow\nas one-sided sequences such that $x(t)$ represents outdated observations with diminishing significance as\n$t\\to -\\infty$.\nThis leads to the analysis of the one-sided sequences directed backward to the past. However,\nthe straightforward application of the one-sided\nZ-transform to the one-sided sequences does not generate Z-transform vanishing on a part of the unit circle even for a band-limited underlying sequence.\n\nThe paper suggests\nsome approaches to quantification of randomness based on frequency analysis of two-sided and one-sided sequences.\nIn addition, the paper suggests an extension of the notion\nof bandlimitiness on one-sided sequences and a procedure allowing to represent an one-sided sequence as a sum of left-bandlimited and predictable sequences and a non-reducible noise.\n\n\n\n\n\n\\section{Definitions and background}\nWe use notation ${\\rm sinc\\,}(x)=\\sin(x)\/x$ and ${\\mathbb{T}}=\\{z\\in{\\bf C}:\\ |z|=1\\}$, and we denote by ${\\mathbb{Z}}$ the set of all integers.\n\\par\n For a Hilbert space $H$, we denote by $(\\cdot,\\cdot)_{H}$ the\ncorresponding inner product. We denote by $L_2(D)$ the usual Hilbert space of complex valued\nsquare integrable functions $x:D\\to{\\bf C}$, where $D$ is an interval in ${\\bf R}$.\n\nLet $\\tau\\in{\\mathbb{Z}}\\cup\\{+\\infty\\}$ and $\\theta<\\tau$; the case where\n$\\theta=-\\infty$ is not excluded. We denote by $\\ell_r(\\theta,\\tau)$ the\nBanach space of complex valued sequences $\\{x(t)\\}_{t=\\theta}^\\tau$\nsuch that\n$\\|x\\|_{\\ell_r(\\theta,\\tau)}=\\left(\\sum_{t=\\theta}^\\tau|x(t)|^r\\right)^{1\/r}<+\\infty$\nfor $r\\in[1,\\infty)$ or $\\|x\\|_{\\ell_\\infty(\\theta,\\tau)}=\\sup_{t: \\theta-10$, there exists a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that $\\sup_{t\\in {\\mathbb{Z}}}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s\\le t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\item Let ${\\cal J}_1\\subset{\\cal J}$ be a set of $I$ such that $\\sup_{I\\in J_1}{\\rm mes\\,}(I)<2\\pi$. Let ${\\cal X}\\subset \\cup_{I\\in {\\cal J}_1} \\ell_2^{\\scriptscriptstyle BL}(I)$ be a bounded set in $\\ell_2$. Then, for any $\\varepsilon>0$, there exists a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that $\\sup_{t\\in {\\mathbb{Z}}}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s\\le t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\item\nLet ${\\cal J}_1$ be the set of $I\\in{\\cal J}$ such that $\\sup_{I\\in J}{\\rm mes\\,}(I)<2\\pi$, and ${\\cal X}\\subset \\cup_{I\\in {\\cal J}_1} \\ell_2^{\\scriptscriptstyle BL}(I)$ be a bounded set in $\\ell_2$ such\nthat $\\sum_{t\\le \\tau}|x(t)|^2\\to 0$ as $\\tau\\to +\\infty$ uniformly over $x\\in {\\cal X}$. Then, for any $\\varepsilon>0$, there exists $\\tau<0$ and a mapping $\\widehat k(\\cdot):{\\mathbb{Z}}\\to{\\bf R}$\n such that $\\sup_{t\\ge 1}\\|x(t)-\\widehat x(t)\\|\\le \\varepsilon$ for all $x\\in{\\cal X}$\nfor $\\widehat x(t)\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\ne^{i\\omega_I t} \\sum_{s=\\tau}^{t-1} \\widehat k(t-s)e^{-i\\omega_I s} x(s)$.\n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{lemmaPred}(iii) states that some predicability based on finite sets of observations also can be achieved if\nwe relax predicability requirement to cover times $t\\ge 1$ only; this would be a weaker version of predicability comparing with the one described in Theorem \\ref{lemmaPred} (ii).\n\nSome versions of this Theorem and some examples of predictable classes can be found in \\cite{D12a,D12b}.\n\nIn addition, it appears that the spectrum supporting sets $I$ can be estimated from the\nset of observations $\\{x(s)\\}_{s\\le \\tau}$ for any $\\tau<0$. More precisely, the following theorem holds.\n\\begin{theorem}\\label{ThO}\nLet ${\\cal X}\\subset \\ell_2$ be a set such that if $x\\in{\\cal X}$ then $x\\in \\ell_2^{\\scriptscriptstyle BL}(I)$ for some $I=I(x)\\in{\\cal J}$,\nand that $\\nu\\stackrel{{\\scriptscriptstyle \\Delta}}{=} 2\\pi-\\sup_{x\\in {\\cal X}}{\\rm mes\\,}(I(x))>0$. Let $\\widehat\\nu=\\nu\/3$. Then, for any $\\tau<0$,\n there exists a mapping $F:\\ell_2(-\\infty,\\tau)\\to (-\\pi,\\pi]$ such that,\n for $\\widehat\\omega_c=F(x(t)|_{t\\le\\tau})$, ${\\mathbb{T}}_c\n \\subset \\{e^{i\\omega},\\ \\omega\\in I^c\\}$, where\n \\begin{eqnarray*}\n{\\mathbb{T}}_c=\\left\\{e^{i(\\omega+\\pi)}:\\ \\omega\\in (-\\pi,\\pi],\\ \\min_{k=0,\\pm 1}|\\widehat\\omega_c-\\omega+2k\\pi|\\le\\widehat\\nu\\right\\}.\\end{eqnarray*}\nIn other words, if $x\\in {\\cal X}$, then $x\\in\\ell_2^{\\scriptscriptstyle BL}(\\widehat I)$ and $I\\subset \\widehat I$, where \\begin{eqnarray*}\n\\widehat I=\\left\\{\\omega\\in (-\\pi,\\pi]: e^{i\\omega}\\notin{\\mathbb{T}}_c\\right\\}.\n\\end{eqnarray*}\n\\end{theorem}\nThe set $\\widehat I$ in Theorem \\ref{ThO} can be regarded as an estimate of $I$ based on observations of $\\{x(t)\\}_{t\\le \\tau}$.\n\n\nLet ${\\cal X}\\subset \\ell_2\\cap \\ell_1$ be a class of processes such that $\\sigma(x)>0$ for $x\\in{\\cal X}$ and that, for $x\\in{\\cal X}$ and $X=Z x$, for any $m>0$, the\nfunctions $X\\left(e^{i\\o}\\right)$ and $|X\\left(e^{i\\o}\\right)|^{-1}$ are differentiable in $\\omega\\in{\\bf R}$ and that\n$\\sup_{x\\in{\\cal X}}\\sup_{\\omega\\in[-\\pi,\\pi]}|dX\\left(e^{i\\o}\\right)\/d\\omega|<+\\infty$. For the purpose of the\ninvestigation of the predictability for $x$, this smoothness and assumed without a loss generality: it is\nsufficient to replace $x$ by a faster vanishing processes with the\nsame predictability properties such that\n$x(t)\/(1+|t|^m)$, $m\\ge 2$. $\\sigma=\\min_{\\omega\\in[-\\pi,\\pi]}|X\\left(e^{i\\o}\\right)|>0$\n\n We want to\n represent each $x\\in {\\cal X}$ as\n\\begin{eqnarray*}\nx=y_{\\scriptscriptstyle BL}+n,\n\\end{eqnarray*}\nwhere $y_{\\scriptscriptstyle BL}$ is a band-limited predictable process such that the class ${\\cal Y}=\\{y_{\\scriptscriptstyle BL}\\}_{x\\in{\\cal X}}$,\nis predictable in the sense of Lemma \\ref{lemmaPred}. In this case, each\n $n= x-y_{\\scriptscriptstyle BL}$ is a non-predictable (random) noise.\n\n We suggest the following restrictions on the choice of $y_{\\scriptscriptstyle BL}$:\n \\begin{enumerate}\n \\item \\begin{eqnarray}\n\\|X\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}=\\|Y_{\\scriptscriptstyle BL}\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}+\\|N\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)},\\quad d=1,+\\infty, \\label{N}\\end{eqnarray}\n where $Y_{\\scriptscriptstyle BL}={\\cal Z} y_{\\scriptscriptstyle BL}$ and $N={\\cal Z} n'$.\n \\item\n $n$ does not allow a similar representation $n=y'_{\\scriptscriptstyle BL}+n'$, with a\nnon-random (predictable) non-zero $y'_{\\scriptscriptstyle BL}$ such that \\begin{eqnarray*}\n\\|N\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}=\\|Y'\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)}+\\|N'\\left(e^{i\\o}\\right)\\|_{L_d(-\\pi,\\pi)},\\quad d=1,+\\infty, \\label{N'}\\end{eqnarray*}\nwhere $Y'_{\\scriptscriptstyle BL}={\\cal Z} y'_{\\scriptscriptstyle BL}$ and $N'={\\cal Z} n'$.\n \\end{enumerate}\nIt appears that $n$ featuring these properties exists in some case\n and can be derived\nexplicitly from $X$. Let us show this.\n\n\n\nLet $\\o_0\\in(-\\pi,\\pi]$ be such\nthat $|X\\left(e^{i\\omega_I }\\right)| =\\sigma$, and\nlet \\begin{eqnarray} \\gamma\\left(e^{i\\o}\\right)=\\frac{\\sigma(x)}{|X\\left(e^{i\\o}\\right)|},\\quad Y\\left(e^{i\\o}\\right)=[1-\\gamma\\left(e^{i\\o}\\right)]X\\left(e^{i\\o}\\right),\\quad\nN\\left(e^{i\\o}\\right)=\\gamma\\left(e^{i\\o}\\right) X\\left(e^{i\\o}\\right). \\label{YN}\\end{eqnarray} Clearly,\n\\begin{eqnarray*}\nX=Y+N, \\quad Y\\left(e^{i\\omega_I }\\right)=0,\\quad |N\\left(e^{i\\o}\\right)|\\equiv\\sigma(x),\n\\end{eqnarray*}\nand\n (\\ref{N}) holds with $d=1$ and $d=\\infty$. By continuity of $X\\left(e^{i\\o}\\right)$ and\n$|X\\left(e^{i\\o}\\right)|^{-1}$, the function $Y\\left(e^{i\\o}\\right)$ is also continuous $\\omega$.\n\nIf $Y\\left(e^{i\\o}\\right)$ vanishes fast enough when $\\omega\\to\\o_0$ (see \\cite{D12a}), then $y={\\cal Z}^{-1}Y$ is predictable; in this case, the set\n$\\{n\\}_{x\\in{\\cal X}}$ can be considered as the set of pathwise noises; therefore, this gives a quantification $n$ as a norm of $n$ or $N$, such as\n\\begin{eqnarray}\n\\|N\\left(e^{i\\o}\\right)\\|_{L_1(-\\pi,\\pi)}=\\sigma(x).\n\\label{noise}\n\\end{eqnarray}\n\nHowever, it would be too restrictive to require that the set ${\\cal X}$ is such that (\\ref{YN}) leads to $Y\\left(e^{i\\o}\\right)$ that vanishes so fast as $\\omega\\to\\o_0$\nthat $y_{\\scriptscriptstyle BL}$ is predictable. To overcome this, we suggest to replace (\\ref{YN}) by\n\\begin{eqnarray} &&\\gamma_\\varepsilon\\left(e^{i\\o}\\right)=1\\quad \\hbox{if}\\quad |e^{i\\omega}-e^{i\\o_0}|\\le \\varepsilon,\\nonumber\\\\\n&&\\gamma_\\varepsilon\\left(e^{i\\o}\\right)=\\frac{\\sigma(x)}{|X\\left(e^{i\\o}\\right)|}\\quad\\hbox{if}\\quad |e^{i\\omega}-e^{i\\o_0}|> \\varepsilon,\\quad\n\\nonumber \\\\&& Y_\\varepsilon\\left(e^{i\\o}\\right)=[1-\\gamma\\left(e^{i\\o}\\right)]X\\left(e^{i\\o}\\right),\\qquad\nN_\\varepsilon\\left(e^{i\\o}\\right)=\\gamma_\\varepsilon\\left(e^{i\\o}\\right) X\\left(e^{i\\o}\\right),\n\\label{g}\\end{eqnarray}\nwhere $\\varepsilon\\to 0$. In this case,\n\\begin{eqnarray}\nx=y_\\varepsilon+n_\\varepsilon, \\quad y_\\varepsilon={\\cal Z}^{-1}Y_\\varepsilon\\in \\ell_2^{\\scriptscriptstyle BL}(I_\\varepsilon),\\quad n_\\varepsilon={\\cal Z}^{-1}N_\\varepsilon, \\label{Yg}\\end{eqnarray}\nwhere $I_\\varepsilon=\\{\\omega:\\ |e^{i\\omega}-e^{i\\o_0}|\\le \\varepsilon\\}$,\n\\begin{eqnarray} &&|N_\\varepsilon\\left(e^{i\\o}\\right)|= X\\left(e^{i\\omega}\\right),\\quad \\hbox{if}\\quad \\omega\\in I_\\varepsilon,\\nonumber\\\\\n&&|N_\\varepsilon\\left(e^{i\\o}\\right)|=|X\\left(e^{i\\o_0}\\right)|={\\rm const\\,}\\quad\\hbox{if}\\quad \\omega\\notin I_\\varepsilon,\\quad\n\\end{eqnarray}\nWe regard $n_\\varepsilon$ as approximation of the noise as $\\varepsilon\\to 0+$.\n\nTo justify this description of the noise, we have to show that the set of band-limited processes $\\{y_\\varepsilon\\}$ in (\\ref{g})-(\\ref{Yg}) is predictable in some sense. Theorem \\ref{lemmaPred}(i)-(ii) does not ensure predicability of this set, since it requires to know the values $\\o_0$.\nThis would require to know $\\omega_{I_\\varepsilon}$, which is inconsistent with the notion of predictability. However, Theorem \\ref{ThO}\nensures sufficient estimation of $I_\\varepsilon$ and $\\omega_{I_\\varepsilon}$ based on observations of $\\{x(t)\\}_{t\\le \\tau}$; we can take select $\\omega_{I_\\varepsilon}=\\widehat\\omega_{c}-\\pi$ if $\\widehat\\omega_c\\in (0,\\pi]$, and\n $\\omega_{I_\\varepsilon}=\\widehat\\omega_{c}+\\pi$ if $\\widehat\\omega_c\\in (-\\pi,0]$, in the notations of Theorem \\ref{ThO}.\n This leads to the following two step procedure:\nthe set $\\{x(s)\\}_{\\tau0}$ of a\nband-limited process $x_{\\scriptscriptstyle BL}$,\n are uniquely defined by the trace\n$x_{\\scriptscriptstyle BL}(t)|_{t\\le 0}$.\n This statement represent a reformulation in the deterministic setting\nof the classical Szeg\\\"o-Kolmogorov Theorem for stationary Gaussian processes\n\\cite{Ko65,Sz,Sz1,V}.\n\n\\subsection*{Existence of optimal\nband-limited approximation} Let $x\\in\\ell_2^-$ be a semi-infinite one-sided sequence representing available\nhistorical data, and let $I\\in{\\cal J}$.\n\\begin{theorem}\\label{Th1} There exists an unique optimal solution $\\widehat x$\nof the minimization problem \\begin{eqnarray} &&\\hbox{Minimize}\\quad \\sum_{t=-\\infty}^0|\\widehat\nx(t)-x(t)|^2 \\quad\\hbox{over}\\quad \\widehat x\\in \\ell_2^{-,\\LBL}(I) .\\label{min} \\end{eqnarray}\n\\end{theorem}\n\\par\nBy Lemma \\ref{lemmaPred}, there exists a unique band-limited process $x_{{\\scriptscriptstyle BL}}\\in\\ell_2^{\\scriptscriptstyle BL} (I)$\nsuch that $\\widehat x(t)|_{t\\le 0}= x_{{\\scriptscriptstyle BL}}(t)|_{t\\le 0}$. This offers a natural way to extrapolate a left\nband-limited solution $\\widehat x\\in \\ell_2^-$ of problem (\\ref{min}) on the future\ntimes $t>0$.\n\n\\index{It can\nbe interpreted as the optimal forecast (optimal given $\\Omega$ and\n$N$).}\n\\subsubsection*{The optimal solution}\nLet $I\\in{\\cal J}$ be given, and let ${\\rm mes\\,}(I)=2\\Omega$ for some $\\Omega\\in (0,\\pi)$.\n\n\nLet $I_0=(-\\Omega,\\Omega)$, i.e., $\\omega_{I_0}=0$.\n\nFor $\\omega\\in[-\\pi,\\pi)$, let the operator $p_{\\omega}: \\ell_2^-\\to\\ell_2^-$ be defined as $\\bar x(t)=e^{i\\omega t}x(t)$ for $\\bar x=p_\\omega x$.\n\n\nLet the operator ${\\cal Q}: {\\ell_2}\\to \\ell_2^{-,\\LBL}(I_0) $ be defined as $\\widehat x={\\cal Q} y={\\cal Z}^{-1}\\widehat X$, where\n\\begin{eqnarray}\n\\widehat X\\left(e^{i\\o}\\right) =\\sum_{k\\in\n{\\mathbb{Z}}}y_ke^{ik\\omega\\pi\/\\Omega}{\\mathbb{I}}_{\\{|\\omega|\\le\\Omega\\}},\n\\label{wX}\\end{eqnarray} for the corresponding $y=\\{y_k\\}\\in {\\ell_2}$.\nSimilarly to the classical sinc representation, we obtain that \\begin{eqnarray} \\widehat x(t)=\\frac{1}{2\\pi}\n\\int_{-\\Omega}^{\\Omega}\\left(\\sum_{k\\in {\\mathbb{Z}}}y_k e^{ik\\omega\\pi\/\\Omega}\\right)e^{i\\omega\nt}d\\omega\\nonumber\\\\ }\\def{\\nonumber\\\\&&} % Must be in the body after \\begin{documnet=\\frac{1}{2\\pi}\n\\sum_{k\\in {\\mathbb{Z}}}y_k\\int_{-\\Omega}^{\\Omega}e^{ik\\omega\\pi\/\\Omega+i\\omega t}d\\omega\\nonumber\\\\\n=\\frac{1}{2\\pi}\\sum_{k\\in {\\mathbb{Z}}}y_k \\frac{e^{ik\\pi+i\\Omega t}-\ne^{-ik\\pi-i\\Omega t}}{ik\\pi\/\\Omega+it}\\nonumber\\\\ }\\def{\\nonumber\\\\&&} % Must be in the body after \\begin{documnet=\\frac{\\Omega}{\\pi}\\sum_{k\\in\n{\\mathbb{Z}}_N }y_k {\\rm sinc\\,}(k\\pi+\\Omega t)=({\\cal Q} y)(t).\\label{sinc}\\end{eqnarray}\nIt follows that the ${\\cal Q}: {\\ell_2}\\to \\ell_2^{-,\\LBL}(I_0) $ is actually defined as\n\\begin{eqnarray*} \\widehat x(t)=({\\cal Q} y)(t)=\\frac{\\Omega}{\\pi}\\sum_{k\\in\n{\\mathbb{Z}}}y_k {\\rm sinc\\,}(k\\pi+\\Omega t).\\label{Qs}\\end{eqnarray*} Consider the operator ${\\cal Q}^*:\\ell_2^{-,\\LBL}(I_0) \\to {\\ell_2}$ being adjoint to the operator\n${\\cal Q}:{\\ell_2}\\to\\ell_2^{-,\\LBL}(I_0)$, i.e., such that\n\\begin{eqnarray}\n({\\cal Q}^*x)_k=\\frac{\\Omega}{\\pi}\\sum_{t\\in{\\cal T}}{\\rm sinc\\,}(k\\pi+\\Omega t)x(t).\n\\label{Q*}\\end{eqnarray}\n\nConsider a\nlinear bounded non-negatively defined Hermitian operator $R:{\\ell_2}\\to {\\ell_2}$ defined as\n\\begin{eqnarray*}\nR={\\cal Q}^*{\\cal Q}.\n\\end{eqnarray*}\nConsider operator $P_I=p_{\\omega_I}{\\cal Q} R^{-1} {\\cal Q}^* p_{-\\omega_I}:\\ell_2\\to\\ell_2^{-,{\\scriptscriptstyle LBL}}(I)$.\n\\begin{theorem}\n\\label{ThP}\n\\begin{itemize}\\item[(i)] The operator $R:{\\ell_2}\\to{\\ell_2}$ has a bounded inverse\n operator $R^{-1}:{\\ell_2}\\to{\\ell_2}$.\n\\item[(ii)] Problem (\\ref{min}) has a unique solution\n \\begin{eqnarray}\n\\widehat x=P_Ix.\\label{wx}\n\\end{eqnarray}\n\\end{itemize}\n\\end{theorem}\n\\begin{theorem}\\label{Th1n} For any $I\\in{\\cal J}$, there exists $n_I\\in \\ell_2^-$ such that $P_In_I=0$ and $n_I\\neq 0$.\n\\end{theorem}\nThe processes $n_I$ can be considered as the noise component with respect to smooth processes with the spectrum on $I$, for a given $I\\in{\\cal J}$.\n\\begin{corollary}\n\\label{corrxx}\nA process $x\\in\\ell_2^-$ is left-bandlimited with the spectrum $I$ if and only if $x=p_{\\omega_I}{\\cal Q} R^{-1} {\\cal Q}^* p_{-\\omega_I}x$.\n\\end{corollary}\n\\begin{remark}\nIt can be noted that $\\widehat x=p_{\\omega_I}{\\cal Q} {\\cal Q}^+ p_{-\\omega_I}x$, where ${\\cal Q}^+=R^{-1} {\\cal Q}^*:\\ell_2^-\\to{\\ell_2}$ is a Moore--Penrose pseudoinverse of the operator ${\\cal Q}:{\\ell_2}\\to\\ell_2^-$.\n\\end{remark}\nLet us elaborate equation (\\ref{wx}). The optimal process $\\widehat x$ can be expressed as \\begin{eqnarray*} \\widehat x(t)=e^{i\\omega_I t}\\frac{\\Omega}{\\pi}\\sum_{k\\in {\\mathbb{Z}} }\\widehat y_k {\\rm sinc\\,}(k\\pi+\\Omega t). \\label{wxx}\\end{eqnarray*}\nHere $\\widehat y=\\{\\widehat\ny_k\\}_{k\\in{\\mathbb{Z}}}$ is defined as \\begin{eqnarray} \\widehat y=R^{-1}{\\cal Q} p_{-\\omega_I} x.\\label{wy}\\end{eqnarray} The operator $R$ can be represented via a matrix\n$R=\\{R_{km}\\}$, where $k,m\\in{\\mathbb{Z}}$. In this\nsetting, $(Ry)_k=\\sum_{k=-\\infty}^\\infty R_{km}y_m$, and the components of the matrix $R$ are defined as \\begin{eqnarray*} R_{km}=\n\\frac{\\Omega^2}{\\pi^2}\\sum_{j=-\\infty}^0{\\rm sinc\\,}(m\\pi+\\Omega j)\\,{\\rm sinc\\,}(k\\pi+\\Omega j)\n.\\label{R}\\end{eqnarray*}\nRespectively, the components of the vector ${\\cal Q}^*x=\\{({\\cal Q}^*x)_k\\}_{k\\in{\\mathbb{Z}}}$ are defined as\n\\begin{eqnarray} ({\\cal Q}^*x)_{k}= \\frac{\\Omega}{\\pi} \\sum_{j=-\\infty}^0{\\rm sinc\\,}(k\\pi+\\Omega j)x(j) .\n\\label{r}\\end{eqnarray}\n\n\n\\subsection{A multi-step procedure for one-sided sequences}\\label{subsecMS}\nUnfortunately, the approach described in Section \\ref{Sec2Sided} does not lead toward a solution of the predictability problem, since it would require to know the entire sequence $\\{x(t)\\}_{t=-\\infty}^{+\\infty}$ to calculate\n$X={\\cal Z} x$ and quantitative characteristics suggested in Section \\ref{Sec2Sided}.\n\nOn the other hand, it is natural to use one-sided sequences interpreted as available past observations for predictability problems.\nIn this case, we have to use the notion of left bandlimitness for\none-sided sequences. We will use a modification of representation (\\ref{xxy}) that was stated for two-sided sequences.\n\nFor this, we suggest to replace the \"ideal\" projections $\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I} {\\cal Z} x)\\in \\ell_2$ for $x\\in\\ell_2$\nby their \"optimal\" one-sided substitutes\n$\\widehat x=P_I x\\in\\ell_2^-$; this substitution is optimal on $\\{t\\le 0\\}$ in the sense of optimization problem (\\ref{min}).\nUnfortunately, it may happen that\n\\begin{eqnarray*}\nx-\\widehat x\\notin \\ell_2^{-,{\\scriptscriptstyle LBL}}(I^c).\n\\end{eqnarray*}\n\n\nFor this, we suggest to replace the \"ideal\" projections $\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I} {\\cal Z} x)\\in \\ell_2$ for $x\\in\\ell_2$ and\n$y_{\\scriptscriptstyle BL}=x-\\widehat x_{\\scriptscriptstyle BL}={\\cal Z}^{-1}({\\mathbb{I}}_{I^c} {\\cal Z} x)$ by their \"optimal\" one-sided substitutes\n$\\widehat x=P_I x\\in\\ell_2^-$ and $\\widehat y=P_{I^c} (x-\\widehat x)\\in\\ell_2^-$; this substitution is optimal on $\\{t\\le 0\\}$ in the sense of optimization problem (\\ref{min}).\nUnfortunately, \\index{an analog of the connection between (\\ref{1}) and (\\ref{2}) is not valid for onesided sequences,\ni.e.,} it may happen that\n\\begin{eqnarray*}\n\\widehat y=P_{I^c} (x-\\widehat x)\\notin \\ell_2^{-,{\\scriptscriptstyle LBL}}(I^c).\n\\end{eqnarray*}\n\nWe suggest a multi-step procedure that to deal with this complication.\n\n\n\nAssume that we observe a semi-infinite one-sided sequence $\\{x(t)\\}_{t\\le 0}\\in \\ell_2^-$.\n\nConsider a sequence of sets $\\{I_k\\}_{k=0,1,2,..}\\subset{\\cal J}$, with the corresponding middle\npoints $\\omega_k\\in I_k$.\nFurther, let us consider the following sequences of elements of $\\ell_2^-$:\n\\begin{itemize}\n\\item Set \\begin{eqnarray*}\nx_0=x,\\qquad \\widehat x_0=P_{I_0}x_0,\\qquad y_0=x_0-\\widehat x_0,\\qquad \\widehat y_0=P_{I_0^c}y_0,\\quad x_1=y_0-\\widehat y_0.\n\\end{eqnarray*}\n\\item\nFor $k\\ge 1$, set\n \\begin{eqnarray*}\n\\widehat x_k=P_{I_k}x_k,\\qquad y_k=x_k-\\widehat x_k,\\qquad \\widehat y_{k}=P_{I_k^c}y_k,\\quad x_{k+1}=y_k-\\widehat y_k.\n\\end{eqnarray*}\n\\end{itemize}\nThe following lemma will be useful.\n\\begin{lemma}\\label{lemmaN} For any $I\\in{\\cal J}$ and $x\\in\\ell_2^-$, the following holds:\n\\begin{enumerate}\n\\item $\\|x\\|\\ge \\|x-P_{I}x\\|$, and \\item The equality in (i) holds if and only if $P_Ix=0$.\n\\end{enumerate}\n\\end{lemma}\n\\subsubsection*{Stopping upon arriving at a predictable process}\nIf there exists $k\\ge 0$ such that $y_k=0$ then\n\\begin{eqnarray}\nx=\\widehat x_0+y_0=\\widehat x_0+\\widehat y_0+x_1=\\widehat x_0+\\widehat y_0+\\widehat x_1 +y_1=...=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat x_k.\n\\label{xx}\\end{eqnarray}\nThis means that $x$ is a finite sum of left band-limited processes. These processes were calculated by the observer,\nand, in this sense, each of them can be deemed to be observed, with known (pres-selected) $I_k$;\nin particular, $x$ can be predicted without error. Similarly, if there exists $k\\ge 0 $ that $x_{k+1}=0$, then\n\\begin{eqnarray}\nx=\\widehat x_0+y_0=\\widehat x_0+\\widehat y_0+x_1=\\widehat x_0+\\widehat y_0+\\widehat x_1 +y_1=...=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat y_k.\n\\label{xy}\\end{eqnarray}\n This means that $x$ again\n is a finite sum of observed left band-limited processes. Again, $x$ can be predicted without error. \\par\nThe norms $\\|\\eta_k\\|_{\\ell_2^-}$ and $\\|\\bar \\eta_k\\|_{\\ell_2^-}$ can be used for quantification of the\nrandomness of one-sided semi-infinite sequences.\n \\subsubsection*{The case of never stopping procedure}\n It may happen that, for any $N>0$, there exists $k\\ge N$ such that either\n$\\|y_k\\|_{\\ell_2^-}+\\|x_k\\|_{\\ell_2^-}>0$. In this, the randomness can be quantified\nas\n\\begin{eqnarray*}\n\\max\\left(\\limsup_{k\\to +\\infty}\\|x_k\\|_{\\ell_2^-},\\limsup_{k\\to +\\infty}\\|y_k\\|_{\\ell_2^-} \\right).\n\\end{eqnarray*}\n \\subsubsection*{Arrival at a non-reducible noise}\n A process $x\\in\\ell_2^-$ is either left band-limited or not band-limited. Therefore, some processes cannot\n be represented as a finite sum of left bandlimited processes such as (\\ref{xx}) or (\\ref{xy}) with a finite $k$.\n In this case, the procedure will not be stopped according to the rule described above. It could be beneficial\n to stop procedure using the following rule.\n\\par\n Let \\begin{eqnarray*}\n\\delta_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=} \\|x_k\\|_{\\ell_2^-}-\\|x_k-\\widehat x_k\\|_{\\ell_2^-},\\qquad \\bar\\delta_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=} \\|y_k\\|_{\\ell_2^-}-\\|y_{k}-\\widehat y_k\\|_{\\ell_2^-},\n\\end{eqnarray*}\ni.e.,\n$\\delta_k=\\|x_k\\|_{\\ell_2^-}-\\|y_k\\|_{\\ell_2^-}$, $\\bar\\delta_k=\\|y_k\\|_{\\ell_2^-}-\\|x_{k+1}\\|_{\\ell_2^-}$,\n \\begin{eqnarray*}\n\\|x_{k}\\|_{\\ell_2^-}=\\|y_k\\|_{\\ell_2^-}+\\delta_k=\\|x_{k+1}\\|_{\\ell_2^-}+\\delta_k+\\bar\\delta_k, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\|y_{k}\\|_{\\ell_2^-}=\\|x_{k+1}\\|_{\\ell_2^-}+\\bar\\delta_k=\\|y_{k+1}\\|_{\\ell_2^-}+\\delta_k+\\bar\\delta_k, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\n By Lemma \\ref{lemmaN},\n it follows that $\\delta_k\\ge 0$ and $\\bar\\delta_k\\ge 0$ for all $k$, i.e., \\begin{eqnarray*}\n\\|x_{k}\\|_{\\ell_2^-}\\ge \\|y_k\\|_{\\ell_2^-}\\ge \\|x_{k+1}\\|_{\\ell_2^-}, \\quad k=0,1,...\\quad\n\\end{eqnarray*}\nBy Theorem \\ref{Th1n}, it may happen that $\\delta_k=0$, i.e., $\\|x_{k}\\|_{\\ell_2^-}=\\|y_k\\|_{\\ell_2^-}$.\nTo save the resources, the procedure should be stopped when this occurs, since further steps will not improve the result.\nOn this step, $x$ is presented as\n\\begin{eqnarray*}\nx=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat x_k+ y_k=x_{p}^{(k)} + \\eta_k.\n\\end{eqnarray*}\nwhere $x_{p}^{(k)}=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+...+\\widehat x_k$ is a predictable process since it is a finite sum of observed\nleft band-limited processes, and $\\eta_k=y_k$ is a noise. Given the selected set $\\{I_k\\}$, further reduction\nof the norm of this noise is impossible. Hence we can call $y_k$\na non-reducible noise.\n\nSimilarly, it may happen that $\\bar\\delta_k=0$ and $\\delta_k>0$, i.e. $\\|y_{k}\\|_{\\ell_2^-}=\\|x_{k+1}\\|_{\\ell_2^-}$.\nAgain, the procedure should be stopped when this occurs, since further steps will not improve the result. This means that the procedure have\nto stop on the step where $x$ is presented as\n\\begin{eqnarray*}\nx=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat y_k+ x_{k+1}=y_{{\\scriptscriptstyle BL}}^{(k)} +\\bar\\eta_k.\n\\end{eqnarray*}\nHere $y_{p}^{(k)}=\\widehat x_0+\\widehat y_0+\\widehat x_1 +\\widehat y_1+... +\\widehat y_k$ is a predictable process again, and $\\bar\\eta_k=x_{k+1}$\nis a non-reducible noise again.\n\n\\section{Proofs}\\label{SceP}\n For the case where $I_0=(-\\Omega,\\Omega)$, i.e.\n $\\omega_I=0$, the proofs of Theorem \\ref{lemmaPred}, Lemma \\ref{propU} and Theorems \\ref{Th1}-\\ref{ThP}, can be found in \\cite{D12a}. Let us extend these proofs on case where $\\omega_I\\neq 0$.\n\nLet us observe that $x\\in \\ell_2^{-,\\LBL}(I)$ and $X={\\cal Z} x\\in {\\mathbb{B}}_\\infty$ if and only if $x_0\\stackrel{{\\scriptscriptstyle \\Delta}}{=} p_{-\\omega_I}x\\in \\ell_2^{-,\\LBL}(I_0)$ and\n$X_0\\stackrel{{\\scriptscriptstyle \\Delta}}{=} {\\cal Z} x_0\\in {\\X}(I_0)$. In this case, $x=p_{\\omega_I}x_0$, and\n \\begin{eqnarray*} X\\left(e^{i\\o}\\right)=\\sum_{t=-\\infty}^{\\infty}x(t)e^{-i\\omega t}=\\sum_{t=-\\infty}^{\\infty}x_0(t)e^{i\\omega_I t}e^{-i\\omega t}= X_0\\left(e^{i(\\omega_I -\\omega )t} \\right), \\quad\n\\omega\\in[0,2\\pi). \\end{eqnarray*}\nThen the proof of Theorem \\ref{lemmaPred}(i)-(ii) and Lemma \\ref{propU} follows.\n\n\\par\n {\\em Proof of Theorem \\ref{lemmaPred}} (iii) follows from the robustness of the predictor used in \\cite{D12a} with respect to truncation\nof inputs from $\\ell_2$. $\\Box$\n\nFurther, we have that\n \\begin{eqnarray*} \\|\\widehat\nx-x\\|_{\\ell_2^-}=\\|p_{-\\omega_I}\\widehat\nx-p_{-\\omega_I} x\\|_{\\ell_2^-} \\quad\\hbox{for any}\\quad \\widehat x, x\\in \\ell_2^-.\\label{minn} \\end{eqnarray*}\nHence the problem\n\\begin{eqnarray} &&\\hbox{Minimize}\\quad \\|p_{-\\omega_I}\\widehat\nx-p_{-\\omega_I} x\\|_{\\ell_2^-} \\quad\\hbox{over}\\quad \\widehat x\\in \\ell_2^{-,\\LBL}(I) \\label{minnn} \\end{eqnarray}\nhas the same sets of solution as problem (\\ref{min}). Therefore, there is a bijection\nbetween the sets of optimal solutions for\nproblem (\\ref{min}) and for the problem\n\\begin{eqnarray} &&\\hbox{Minimize}\\quad \\|\\widehat\ny-y\\|_{\\ell_2^-} \\quad\\hbox{over}\\quad \\widehat y\\in \\ell_2^{-,\\LBL}(I_0),\\label{minnnn} \\end{eqnarray}\nwhere $y= p_{-\\omega_I} x $. This bijection has the form $\\widehat y= p_{-\\omega_I}\\widehat x $.\nTherefore, the proof for $\\omega_I\\neq 0$ follows from the proof for $\\omega_I=0$ from \\cite{D12a}.\nThen the proof of Theorem \\ref{Th1} and Theorem \\ref{ThP} follows. $\\Box$\n\n{\\em Proof of\nTheorem \\ref{ThO}}. \\index{Let $m$ the entire part of the number $2\\pi\/\\widehat\\nu +1$, and let $\\omega_k=-\\pi+2k\\pi\/m $, $k=1,...,m$.}\nIt is easy to see that there exists a finite set $\\{I_k\\}_{k=1}^M\\subset {\\cal J}$, $M<+\\infty$, such that ${\\rm mes\\,}(I_k)\\le \\nu\/3$,\n$\\cup_{k=1}^M I_k=(0,2\\pi]$, and that the intersections of two different $ I_k$ cannot contain two or more elements.\nLet $\\widehat I_k=(-\\pi,\\pi]\\setminus I_k$.\n\nLet $P_I$ be operators such as defined in Section \\ref{Sec1Sided}, with rather technical adjustment: we assume that the\nset of times\n $\\{t\\le 0\\}$ in Theorem \\ref{Th1} is replaced by the $\\{t\\le \\tau\\}$, and that $\\ell_2^-$ replaced by $\\ell_2(-\\infty,\\tau)$.\nAs is shown in Theorem \\ref{ThP}, the values $d_k\\stackrel{{\\scriptscriptstyle \\Delta}}{=}\\|{\\bf P}_{\\widehat I_k}x-x\\|_{\\ell_2(-\\infty,\\tau)}$ for $k=1,...,M$ can be found\nbased on observations of $\\{x(t)\\}_{t\\le\\tau}$. By the assumptions on $x$, there\nexists $m$ such that\n$d_m=0$. The set $\\widehat I=\\widehat I_m$ is such as described in the Theorem; the point $\\widehat\\omega_c$ can be defined as\nselect $\\widehat\\omega_{c}=\\widehat\\omega_{\\widehat I}-\\pi$ if $\\widehat\\omega_{\\widehat I}\\in (0,\\pi]$, and $\\widehat\\omega_{c}=\\widehat\\omega_{\\widehat I}+\\pi$ if $\\widehat\\omega_{\\widehat I}\\in (-\\pi,0]$.\nThen the proof of Theorem \\ref{ThO} follows. $\\Box$\n $\\Box$\n\n\n\n{\\em Proof of Theorem \\ref{Threc}}.\nLet $Y\\left(e^{i\\o}\\right) = \\sum_{k\\in{\\mathbb{Z}}\\setminus\\{m\\} } e^{-i\\omega k}x(k)$, $\\omega\\in(-\\pi,\\pi]$; this function to be observable. By the definitions, it follows that\n \\begin{eqnarray*}\n X\\left(e^{i\\o}\\right) -Y\\left(e^{i\\o}\\right) -e^{-im}x(m)\\equiv 0,\\quad \\omega\\in(-\\pi,\\pi].\n \\end{eqnarray*}\nHence\n\\begin{eqnarray*}\nx(m)=-e^{ im}Y\\left(e^{\\o_0}\\right)+ e^{ im}X\\left(e^{\\o_0}\\right)=-e^{ im}Y\\left(e^{\\o_0}\\right)+\\xi,\n \\end{eqnarray*}\nwhere $\\xi=e^{ im}X\\left(e^{\\o_0}\\right)$. Hence\n\\begin{eqnarray*}\n|x(m)+e^{ im}Y\\left(e^{\\o_0}\\right)|= |\\xi|=\\sigma.\n\\end{eqnarray*}\nLet us accept the value $\\widehat x(m)=-e^{ im}Y\\left(e^{\\o_0}\\right)$ as the estimate of the\nmissing value $x(m)$. For this estimator, the size of the recovery error is $\\sigma$ for any $x\\in{\\cal X}_\\sigma$.\nIf $\\sigma=0$ then the estimator is error-free. In a general case where $\\sigma\\ge 0$, we have that (\\ref{opt}) holds.\n\nLet us show that this estimator\nis optimal in the following sense:\n\\begin{eqnarray*}\n\\sigma=\\sup_{x\\in{\\cal X}_\\sigma}|\\widehat x(m)-x(m)|\\le \\sup_{x\\in{\\cal X}_\\sigma}|\\widetilde x(m)-x(m)|\n\\label{optrec2}\\end{eqnarray*}\nfor any other estimator $\\widetilde x(m)=F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, where $F:\\ell_2(-\\infty,m-1)\\times \\ell_2(m+1,+\\infty)\\to{\\bf R}$ is some mapping.\n\n\\index{Assume that $m=0$, $X_\\pm\\left(e^{i\\o}\\right) =\\pm \\sigma$, $x_\\pm={\\cal Z}^{-1}X_\\pm$, i.e. $x_\\pm(t)=\\pm \\sigma{\\mathbb{I}}_{\\{t\\neq 0\\}}$.\nClearly, $x_\\pm\\in{\\cal X}_\\sigma$ and $\\widetilde x_-=\\widetilde x_+$ for $\\widetilde x_\\pm= F\\left(x|_{t\\in{\\cal Z},\\ t\\neq 0}\\right)$, for any mapping $F$ such as described above.\nHence\n\\begin{eqnarray*}\n \\max(|\\widetilde x_-(0)-x_-(0)|,|\\widetilde x_+(0)-x_+(0)|)\\ge \\sigma.\n\\end{eqnarray*}}\n\nLet $m\\in {\\mathbb{Z}}$ be fixed, and let $X_\\pm\\left(e^{i\\o}\\right) =\\pm \\sigma e^{-im \\omega}$, $x_\\pm={\\cal Z}^{-1}X_\\pm$, i.e. $x_\\pm(t)=\\pm \\sigma{\\mathbb{I}}_{\\{t=m\\}}$.\nClearly, $x_\\pm\\in{\\cal X}_\\sigma$. Moreover, we have that $\\widetilde x_-=\\widetilde x_+$ for $\\widetilde x_\\pm= F\\left(x|_{t\\in{\\mathbb{Z}}\\setminus\\{m\\}}\\right)$, for any mapping $F$ such as described above.\nHence\n\\begin{eqnarray*}\n \\max(|\\widetilde x_-(m)-x_-(m)|,|\\widetilde x_+(m)-x_+(m)|)\\ge \\sigma.\n\\end{eqnarray*}\n Then (\\ref{optrec}) follows. This completes the proof of Theorem \\ref{Threc}. $\\Box$\n\n\\par\n{\\em Proof of Theorem \\ref{Th1n}}. It suffices to observe that $\\ell_2^-\\setminus {\\cal Q}(\\ell_2)\\neq \\emptyset$, for the operator ${\\cal Q}:\\ell_2\\to \\ell_2^-$, since\n${\\cal Q}(\\ell_2)=\\ell_2^{-,{\\scriptscriptstyle LBL}}$.\n Hence the kernel of the adjoint operator ${\\cal Q}^*:\\ell_2^-\\to\\ell_2$ contains non-zero elements.\n \\index{Since $R$ is invertible, we have that\n$R^{-1}{\\cal Q}^*x^\\bot\\neq 0_{\\ell_2}$. Clearly, ${\\cal Q} y\\neq 0_{\\ell_2^-}$ if $y\\neq 0_{\\ell_2}$. Hence ${\\cal Q} R^{-1}{\\cal Q}^*x^\\bot\\neq 0_{\\ell_2^-}}\n$\\Box$\n\n\\par\n{\\em Proof of Lemma \\ref{lemmaN}}. Statement (i) follows from the choice of $P_{I}x$ as a solution of\noptimization problem (\\ref{min}). To prove statement (ii), it suffices to show that if $\\|x\\|=\\|x-P_{I}x\\|$ then $P_Ix=0$.\nIf $\\|x\\|=\\|x-P_{I}x\\|$ then $\\|x-0_{\\ell_2^-}\\|=\\|x-P_{I}x\\|$. Hence both sequences $0_{\\ell_2^-}$ and $\\|P_{I}x\\|$\nare solutions of problem (\\ref{min}). We proved that the solution is unique, hence $\\|P_{I}x\\|=0_{\\ell_2^-}$.\nThis completes the proof. $\\Box$\n\\section{Possible applications and future development}\nThe approach suggested in this paper allows many modifications. We\noutline below some possible straightforward modifications as well as\n more challenging problems and possible applications that we leave for the future research.\n\\begin{enumerate}\n\\item It would be interesting to investigate sensitivity of the prediction results with respect to the choice\nof\n$\\{I_k\\}$. It would be interesting to find an optimal choice of the set $\\{I_k\\}$ such as\n \\begin{eqnarray*}\n \\hbox{Maximize}\\quad \\delta_k+\\bar \\delta_k\\quad \\hbox{over}\\quad I\\in {\\cal J}\n \\end{eqnarray*} for $k=1,2,..,$, with some constraints on the choice of $I_k$, for example, such that ${\\rm mes\\,}(I_k)$ is given.\n\\item It could be interesting to try another basis in $L_2(I_0)$ for expansion in (\\ref{wX}).\n\\item Optimization problem in (\\ref{min}) is based on optimal approximation in\n$L_2(I)$ for Z-transforms. This approximation in can be replaced by approximation in a weighted\n$L_2$-space on $I$. This leads to modification of the optimization\nproblem; the weight will represent the relative importance of the\napproximation on different frequencies.\n\\item It is unclear if an analog of property (\\ref{N}) can be obtained with $d=2$ instead of $d=1,+\\infty$.\n\\end{enumerate}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe discovery of superconformal theories (SCFTs) in six and five dimensions has been one of the most surprising results emerging from string theory in the past few decades. There are two types of 6d SCFTs, both of which are classified in terms of singular geometries: $\\mathcal N = (2,0)$ theories \\cite{Witten:1995zh} and $\\mathcal N =(1,0)$ theories \\cite{Heckman:2013pva,Heckman:2015bfa,Bhardwaj:2015xxa}. Given the surprising effectiveness of geometry in describing 6d SCFTs, a natural next step is to attempt to classify 5d SCFTs in terms of singular geometries.\nIn some ways, 5d SCFTs are more rigid as there is only a single type of 5d SCFT corresponding to the 5d ${\\cal N}=1$ (i.e. eight supercharges) superconformal algebra. Many examples of 5d SCFTs have been realized in string theory using brane probes \\cite{Seiberg:1996bd}, M-theory on local Calabi-Yau 3-folds \\cite{Morrison:1996xf,Douglas:1996xp,Intriligator:1997pq}, and type IIB $(p,q)$ 5-brane webs \\cite{Aharony:1997bh,Aharony:1997ju,Leung:1997tw,Bergman:2014kza}. \n\nThe classification of 6d $\\mathcal N = (1,0)$ theories led to a picture involving generalized `quiver-like' theories whose structures could by and large be anticipated from field theoretic reasoning.\nThere are of course exceptions to this idea and explicit geometric constructions in F-theory clarified which possible exceptions arise that evade field theoretic analysis \\cite{Heckman:2015bfa,Heckman:2013pva}. Similarly, in the 5d case, one might expect field theoretic reasoning to be a powerful, albeit incomplete guide. Indeed, as spearheaded in \\cite{Intriligator:1997pq} it has been clear for a long time that field theoretic tools combined with the constraints of supersymmetry provide an unexpectedly powerful method for deducing the existence of interacting UV fixed points. More recently it was found in \\cite{Jefferson:2017ahm} that relaxing some of the assumptions in \\cite{Intriligator:1997pq} can resolve the conflict between the gauge theoretic classification described in \\cite{Intriligator:1997pq} with low energy descriptions of some known stringy constructions, leading to a set of necessary (as opposed to sufficient) conditions for a 5d gauge theory to have a UV fixed point. However, it is unclear whether or not there are additional conditions needed to guarantee the existence of gauge theories\nas consistent 5d SCFTs. Moreover, there are known cases in which a 5d SCFT is not a gauge theory (for example, M-theory on a local $\\mathbb P^2$ embedded in a Calabi-Yau 3-fold)\\footnote{Despite the fact that these cases do not admit a Lagrangian description, they can nevertheless be obtained from a gauge theory by passing through phases where some non-perturbative degrees of freedom become massless.}.\nA reasonable follow-up to the field theoretic approach, then, is to try to check if the necessary gauge theoretic consistency conditions described in \\cite{Jefferson:2017ahm} are in fact also sufficient, by using other string constructions to engineer the same theories. The main aim of this paper is to use geometric constructions of 5d SCFTs, realized as M-theory compactified on local Calabi-Yau (CY) 3-fold (and cross checked with dual constructions involving ($p,q$) 5-brane webs), to devise a classification scheme for 5d SCFTs. As a byproduct of our efforts, we are led to either validate or exclude various candidate 5d SCFTs predicted by the perturbative gauge theoretic analysis.\n\nThe basic mathematical setup leading to 5d SCFTs from M-theory on CY 3-folds involves studying how all compact 4-cycles (compact complex surfaces) inside a non-compact 3-fold can be shrunk to a point at a finite distance in moduli space; we call CY 3-folds engineering 5d SCFTs in this manner `shrinkable' 3-folds. This geometric picture can be schematically represented by a graph whose nodes are 4-cycles (surfaces) and whose edges denote the resulting intersecting 2-cycles (curves).\nWe note that a systematic study of the consistency conditions needed to construct such geometries has not been undertaken in the mathematics literature. Starting from a collapsed set of 4-cycles, the condition that one can resolve the singularities and thereby bring the 4-cycles to finite volume restricts the admissible types of K\\\"ahler surfaces (i.e.\\ the nodes of the graph). We call the number of nodes of such a graph the \\emph{rank} of the 5d SCFT. In particular, we show that the nodes of the graph must be rational or ruled surfaces (possibly blown up at a positive number of points)\\footnote{Rational and ruled surfaces are equivalent to (respectively) $\\mathbb P^2$ and ruled surfaces over genus $g$ curves (which we argue can be restricted to $g=0$)---see Section~\\ref{sec:gtrans} for additional details.} in the rank 2 case, and further conjecture this to be true for arbitrary rank.\nThe Calabi-Yau condition and the requirement of positive volumes place further restrictions on the allowed intersections of the surfaces (i.e.\\ the edges of the graph; see Figure \\ref{fig:graph}). We thus devise a set of necessary critieria which must be satisfied for a 3-fold to engineer a 5d SCFT and conjecture that these criteria are sufficient to guarantee the existence of a 5d SCFT; this conjecture is supported by various cross checks using ($p,q$) 5-brane webs. Furthermore, we conjecture that all 5d SCFTs can be realized in M-theory on CY 3-folds satisfying these criteria. Similar to the 6d case, where F-theory compactified on elliptic 3-folds was used to classify $\\mathcal N = (1,0)$ theories and it was subsequently found that for a few exotic cases frozen singularities are necessary to realize $\\text{O7}^+$ planes in F-theory \\cite{Tachikawa:2015wka,Bhardwaj-progress}, we find that in the M-theory case it is also necessary to include frozen singularities to obtain a complete classification of 5d SCFTs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node[draw,circle] (a) at (0,0) {$S_5$};\n\t\t\t\\node[draw,circle] (b) at (2,0) {$S_3$};\n\t\t\t\\node[draw,circle] (e) at (2,2) {$S_4$};\n\t\t\t\\node[draw,circle] (f) at (2,-2) {$S_2$};\n\t\t\t\\node[] (c) at (4,0) {$\\cdots$};\n\t\t\t\\node[draw,circle] (g) at (4,-2) {$S_1$};\n\t\t\t\\node[draw,circle] (d) at (6,0) {$S_r$};\n\t\t\t\\draw (b) --(c) -- (d);\n\t\t\t\\draw (a) -- (b);\n\t\t\t\\draw (b) -- (e);\n\t\t\t\\draw (e) -- (a);\n\t\t\t\\draw (b)--(f);\n\t\t\t\\draw (f) -- (g) -- (d);\n\t\t\\end{tikzpicture}\n\t\\end{center}\t\n\t\\caption{Graphical representation of a rank $r$ K\\\"ahler surface $S = \\cup S_i \\subset X$ embedded in local Calabi-Yau 3-fold $X$. The nodes of the graph correspond to 4-cycles $S_i$, while the edges $C_{i,i+1} = S_i \\cap S_{i+1}$ correspond to 2-cycles along which the nodes intersect.}\n\t\\label{fig:graph}\n\\end{figure}\t\t\n\nA complete classification of such CY 3-folds appears to be a rather daunting task. For example, it is unknown whether or not the list of possible 5d SCFTs is finite for a given rank. Luckily, it turns out that the rank 2 case is finite, permitting an exhaustive classification of physically distinct SCFTs.\n\nBy classifying rank 2 SCFTs in terms of Calabi-Yau geometry, we learn that all rank 2 gauge theories predicted in \\cite{Jefferson:2017ahm}, except for one family, are realized.\\footnote{We conjecture that all SCFTs admit at least one Coulomb branch parameter at the CFT point. The missing family which is represented by $SU(3)$ at Chern-Simons level $k=8$ has no Coulomb branch parameter at the would-be CFT point\nand that is why we rule it out. This family would have led to a putative CFT which allows a Coulomb branch deformation only after a mass deformation (i.e. turning on $1\/g^2$).} Additionally, we are also able to pinpoint the non-perturbative physics missing in the gauge theoretic approach of \\cite{Jefferson:2017ahm} responsible for excluding this family of SCFTs. Furthermore, the geometric approach allows us to identify additional non-Lagrangian SCFTs whose existence motivates the existence of dual ($p,q$) 5-brane web configurations.\n\nGiven the significant practical challenges presented by this classification program, it is natural to ask if the insight we have gained from the rank 2 case can be used to streamline the classification of higher rank cases. Indeed, a careful examination of the list of rank 2 theories reveals a beautifully simple picture: rank 2 SCFTs in 5d can be organized into four distinct families, related and interconnected by RG flows triggered by mass deformations---see Figure \\ref{tree}. Each family of 5d SCFTs has a parent 6d SCFT, where the parent 6d SCFT is related to a 5d descendant by circle compactification, up to a choice of automorphism twist (see \\cite{Apruzzi:2017iqe} for work on classifying such automorphism twists, and see \\cite{DelZotto:2015isa} for a discussion of additional discrete data characterizing circle compactifications of 6d SCFTs.) Thus the rank 2 classification could have been anticipated entirely from the 6d perspective! This result echoes a well-known property of rank 1 SCFTs: rank 1 5d SCFTs belong to a single family which descends from the 6d E-string theory via circle compactification.\n\nWe thus conjecture that {\\it all 5d SCFTs arise from 6d SCFTs compactified on a circle, possibly up to an automorphism twist}.\nMore precisely, we anticipate that all 5d SCFTs can be organized into distinct families, each of which arises from a 6d theory. For a fixed rank in 5d, the possible 6d SCFT parents are rather limited. For example (ignoring the possible automorphism twist), the 6d SCFTs leading to rank $r$ 5d SCFTs will have $r-k$ dimensional tensor branches with rank $k$ gauge algebra. This suggests a practical method to classify 5d SCFT families starting with the 6d classification: compactifying a 6d SCFT on a circle produces a 5d theory with a Kaluza Klein (KK) tower of states. We call such theories `5d KK theories'; these theories are in some sense analogous to 6d little string theories. To obtain non-trivial 5d SCFTs from 5d KK theories we need to turn on holonomies suitably tuned to trigger an RG flow to a nontrivial 5d SCFT in the infrared. Aspects of the phase structure of 5d theories arising from circle compactifications of 6d SCFTs were analyzed in \\cite{DelZotto:2017pti}.\n\nThe organization of this paper is as follows. \nIn Section \\ref{sec:review} we discuss the preliminaries of 5d SCFTs, their effective gauge theory descriptions on the Coulomb branch, and their realizations in M-theory.\nIn Section \\ref{sec:algorithm} we discuss the mathematics of shrinkable 3-folds and explain the basic approach of our geometric classification program.\nIn Section \\ref{sec:classification} we repeat the classification of rank 1 5d SCFTs and extend the same methods to the rank 2 case. We also discuss the connection to 6d $\\mathcal N = (1,0)$ SCFTs. Some mathematical results essential for the rank 2 classification are collected in the appendices: Appendix \\ref{app:AG} contains an explicit description of the Mori cones of blowups of Hirzebruch surfaces; Appendix \\ref{app:bound} contains some numerical bounds constraining rank 2 shrinkable 3-folds; finally, Appendix \\ref{app:smooth} contains a detailed discussion of some smoothness assumptions which simplify the classification program.\n\n\\section{Effective Description of 5d SCFTs}\n\\label{sec:review}\n\nIn this section we discuss some of the preliminaries that set the stage for the classification of 5d SCFTs later in this paper. The following discussion involves two perspectives on 5d $\\mathcal N =1$ theories:\nthe gauge theoretic perspective, and the geometric perspective of M-theory compactified on a Calabi-Yau 3-fold. \n\n5d superconformal field theories (SCFTs) are strongly interacting systems with no marginal deformations \\cite{Cordova:2016emh} and no known Lagrangian description at the CFT fixed point. In order to study the physics of these conformal theories, one needs to use rather indirect approaches. 5d SCFTs admit supersymmetric relevant deformations which lead to several weakly interacting effective descriptions while preserving some amount of supersymmetry. Surprisingly, these effective descriptions can be powerful tools for studying the dynamics of the conformal point. There exist some CFT observables which are rigidly protected under the renormalization group (RG) flow triggered by these deformations. Many BPS quantities are such observables: for example, the spectrum of BPS operators, supersymmetric partition functions, effective Lagrangians on the Coulomb branch, the Coulomb branch of moduli space, etc. In particular, BPS observables are protected by supersymmetry and thus we expect BPS quantities appearing in the effective theories to be a reliable description of the corresponding observables at the CFT fixed point.\n\nString theory provides many effective descriptions of 5d SCFTs. Multiple D4-brane systems in Type IIA string theory and ($p,q$) 5-brane webs in Type IIB string theory can engineer various 5d SCFTs as singularities. Away from the singularity, when mass parameters and gauge couplings are turned on, these brane systems often permit a gauge theory description of the corresponding 5d theories.\n\n5d SCFTs can also be engineered in M-theory: M-theory on a singular non-compact Calabi-Yau 3-fold is described at long distances by an SCFT living on the five-dimensional spacetime transverse to the 3-fold. In familiar cases, the Calabi-Yau singularity can be resolved by means of various K\\\"ahler deformations, which correspond to mass and Coulomb branch deformations in the corresponding gauge theory. \n\n\n\n \n\\subsection{Gauge theory description}\n\nGauge theories in five dimensions are non-renormalizable and flow to free fixed points at low energy. As a result, these theories are typically believed to be `trivial' theories. However, a large class of 5d gauge theories, mostly engineered in string theory, turn out to have interacting CFT fixed points in the UV \\cite{Seiberg:1996bd}. In such cases, 5d gauge theories are rather interesting since they can provide low energy effective descriptions of the CFT.\n\nIn this paper, we focus primarily on gauge theories which have 5d SCFTs as their UV completions. These theories preserve $\\mathcal{N}=1$ supersymmetry, and their massless field content consists of vector multiplets with gauge algebra $G$ and hypermultiplets in a representation $\\textbf{R} = \\oplus \\textbf{R}_j$ of $G$. These gauge theories might be further specified by topological data $k$ corresponding to classical Chern-Simons level, as in the case of $G = SU(N \\geq 3)$, or discrete $\\theta$-angle as in the cases $G= Sp(N)$. We can also consider the cases with product gauge algebra $G=\\prod_i G_i$. Once the data $G,\\textbf{R},k$ is fixed, the low energy gauge theory Lagrangian is uniquely determined by supersymmetry. Our notation for describing 5d gauge theories is\n\\begin{align}\nG_k + \\sum_j N_{\\textbf{R}_j} \\textbf{R}_j,\n\\end{align} \nwhere $\\textbf{R}_j$ is the representation under which the $j$-th matter hypermultiplet is charged, $N_{\\textbf{R}_j}$ is the number of hypermultiplets in the representation $\\textbf{R}_j$.\n\n5d $\\mathcal{N}=1$ gauge theories possesses a rich vacuum structure. The moduli space of vacua is parametrized by expectation values of various local operators. In particular, we are interested in the Coulomb branch of vacua parametrized by vacuum expectation values of scalar fields $\\phi$ in the vector multiplets. Here the scalar field $\\phi$ takes values in the Cartan subalgebra of the gauge group $G$. So the dimension of the moduli space of the Coulomb branch is given by the rank of group $G$, $r={\\rm rank}(G)$. \nBy abuse of notation, we will denote both a scalar field in the vector multiplet and its expectation value by $\\phi$ from now on. \n\nThere are global symmetries acting on the hypermultiplets. The classical Lagrangian has global symmetry algebra $F$ rotating the perturbative hypermultiplets and also a topological $U(1)_I$ symmetry for each gauge group. The objects charged under the $U(1)_I$ are non-perturbative particles called `instantons'. Surprisingly, this classical global symmetry is often enhanced in the CFT fixed point by non-perturbative instanton dynamics \\cite{Seiberg:1996bd,Douglas:1996xp}. The flavor symmetry of the perturbative hypermultiplets can combine with the topological $U(1)_I$ instanton symmetry and enhance to an even larger symmetry algebra in the UV CFT. One can turn on mass parameters $m_i$ associated to the global symmetry. Doing so breaks some of the global symmetry. In particular, the mass deformation with parameter $g^{-2}$ along the $U(1)_I$ instanton symmetry leads to a gauge theory description with gauge coupling $g$ at low energy.\n\nAt a generic point in the Coulomb branch, the gauge symmetry $G$ is broken to the maximal torus $U(1)^{r}$. Thus the low energy dynamics on the Coulomb branch can be effectively described by abelian gauge theories.\nThe low energy abelian action is determined by a prepotential $\\mathcal{F}$. The prepotential is 1-loop exact and the full quantum result is a cubic polynomial of the vector multiplet scalar $\\phi$ and mass parameters $m_j$, given by \\cite{Witten:1996qb,Intriligator:1997pq}:\n\\begin{equation}\n\\label{eqn:pre}\n\t\\mathcal{F} = \\frac{1}{2g^2}h_{ij}\\phi_i \\phi_j + \\frac{k}{6} d_{ijk}\\phi_i\\phi_j\\phi_k + \\frac{1}{12}\\left(\\sum_{e\\in {\\rm root}} |e\\cdot \\phi|^2 -\\sum_j \\sum_{w\\in {\\bf R}_j}|w\\cdot\\phi+m_j|^3\\right) \\ ,\n\\end{equation}\nwhere by abuse of notation ${\\bf R}_j$ denotes the set of weights of the $j$-th hypermultiplet representation of $G$, $h_{ij}={\\rm Tr}(T_iT_j)$, and $d_{ijk}=\\frac{1}{2}{\\rm Tr}_{\\bf F}(T_i\\{T_j,T_k\\})$ with ${\\bf F}$ in the fundamental representation. The first two terms in the prepotential are from the classical Lagrangian and the last two terms are 1-loop corrections coming from integrating out charged fermions in the Coulomb branch. We remark that the prepotential may have different values in the different sub-chambers (or phases) of the Coulomb branch due to the absolute values in the 1-loop contributions.\n\nThe 1-loop correction to the prepotential renormalizes the gauge coupling. The effective coupling in the Coulomb branch is simply given by a second derivative of the quantum prepotential which also fixes the exact metric on the Coulomb branch:\n\\begin{equation}\n\\label{eqn:der}\n\t(\\tau_{\\rm eff})_{ij} = (g^{-2}_{\\rm eff})_{ij} = \\partial_i\\partial_j\\mathcal{F} \\ , \\qquad ds^2 = (\\tau_{\\rm eff})_{ij}d\\phi_id\\phi_j \\ .\n\\end{equation}\nInterestingly, the exact spectrum of magnetic monopoles on the Coulomb branch can be easily obtained from the quantum prepotential. Since monopoles are magnetically dual to electric gauge bosons, tensions of magnetic monopole strings can be computed as\n\\begin{equation}\n\\label{eqn:mono}\n\t\\phi_{Di} = \\partial_i\\mathcal{F} \\ , \\quad i=1,\\cdots r \\ .\n\\end{equation}\nOne can also compute Chern-Simons couplings:\n\t\\begin{align}\n\t\tk_{ijk} = \\partial_ i \\partial_j \\partial_k \\mathcal F. \n\t\\end{align}\nTherefore, we can use $\\mathcal F $ to exactly compute some quantum observables such as the Coulomb branch metric and monopole spectrum.\n\nIn \\cite{Intriligator:1997pq,Jefferson:2017ahm}, the above supersymmetry protected data is used to attempt a classification of possible 5d SCFTs admitting low energy gauge theory descriptions. The main idea in these classification programs is that the quantum metric on the Coulomb branch should be positive semi-definite in the CFT limit, as required by unitarity. In \\cite{Intriligator:1997pq}, the positivity condition of the metric was imposed throughout the `perturbative' Coulomb branch and all sensible gauge theories were subsequently identified using this constraint. In this classification, the `perturbative' Coulomb branch is determined by forcing only \\emph{perturbative} particles to have positive masses. Under this condition, the number and type of hypermultiplets are strictly constrained and quiver type gauge theories are ruled out; see \\cite{Intriligator:1997pq} for details. We refer to this classification as the `IMS classification'.\n\nHowever, it was pointed out later works \\cite{Aharony:1997ju,Bergman:2014kza,Hayashi:2015fsa,Gaiotto:2015una,Yonekura:2015ksa} that string theory can engineer many 5d gauge theories with non-trivial CFT fixed points not included among the theories in the IMS classification. It turns out that the condition of metric positivity throughout the entire perturbative Coulomb branch is too strong \\cite{Jefferson:2017ahm} and unnecessarily excludes many non-trivial 5d gauge theories. This suggests that the IMS classification is incomplete, and the gauge theories exceeding the IMS bounds lead us to revisit the problem of classifying 5d SCFTs. \n\nLet us briefly review the classification of \\cite{Jefferson:2017ahm}. One of the main results of this analysis is the observation that the `perturbative' Coulomb branch receives quantum corrections by light non-perturbative states \\cite{Aharony:1997ju}. It is possible that some of non-perturbative states can become massless somewhere in the perturbative Coulomb branch. These hyperplanes in the Coulomb branch where these light states become massless can be thought of as `non-perturbative' walls. Beyond such walls, the perturbative Coulomb branch breaks down. One way to see this is to note that the signature of the quantum metric on the Coulomb branch changes beyond these non-perturbative walls, which implies the metric cannot be trusted in these regions. However, the classification in \\cite{Intriligator:1997pq} imposes metric positivity on the whole perturbative Coulomb branch, even beyond non-perturbative walls. The result is that some theories are excluded because of the unreliability of the metric in these regions, and this leads to an incomplete classification. In order to obtain a complete classification, metric positivity should be applied only on the `physical' Coulomb branch, which can be computed by accounting for restrictions introduced by non-perturbative states.\n\nIn general, it is difficult to identify the correct physical Coulomb branch after taking into account non-perturbative effects since this necessarily involves studying the full non-perturbative spectrum. In particular, it is not easy to analyze the spectrum of gauge theory instantons. Only when we know a precise UV completion of the instanton moduli space, such as the ADHM construction, can we compute the exact spectrum using localization. For most gauge theories, such a convenient construction of the instanton moduli space is lacking.\n\nFortunately, the perturbative prepotential contains part of the exact spectrum of non-perturbative states. As noted in (\\ref{eqn:mono}), the full monopole spectrum can be obtained from the prepotential. We can use this information to identify some of the non-perturbative walls in the perturbative Coulomb branch. By relaxing the metric positivity constraint to apply only to the region interior to such non-perturbative walls, it was conjectured in \\cite{Jefferson:2017ahm} that all gauge theories having interacting CFT fixed points satisfy the metric positivity condition in the sub-locus of Coulomb branch where perturbative particles and monopole strings have positive masses. In \\cite{Jefferson:2017ahm}, it was also shown that a large class of known 5d gauge theories satisfy this criterion. It may be true that all the known 5d gauge theories having 5d SCFT fixed points satisfy this refined condition.\n\nIn addition, there are two more conjectures in \\cite{Jefferson:2017ahm} used to carry out the classification of 5d gauge theories with simple gauge algebras. The first conjecture is that if all perturbative particles and monopoles have positive masses \\emph{somewhere} in the Coulomb branch, the gauge theory has a UV CFT fixed point. The second conjecture is that perturbative prepotentials of all gauge theories with UV CFT fixed points are positive \\emph{everywhere} in the perturbative Coulomb branch.\nNote that the first conjecture is not sufficient to guarantee that all instanton particles have positive mass and also that the metric is positive in the same region. So this is simply a necessary condition. We will see later that certain theories predicted by this approach must be excluded because some non-perturbative particles acquire negative masses in the CFT limit.\nThe second conjecture is based on the convergence of the 1-loop sphere partition function of 5d CFTs, but there is neither physical nor mathematical motivation for this conjecture beyond its practical implications.\nUsing these two conjectures, non-trivial gauge theories with single gauge node were fully classified in \\cite{Jefferson:2017ahm}. This classification includes all known single gauge node theories and additionally predicts a large number of new gauge theories.\n\nIn this paper, we construct rank 1 and rank 2 CFTs using Calabi-Yau geometry. Rank 1 gauge theories arising from SCFTs were classified in \\cite{Seiberg:1996bd,Morrison:1996xf,Intriligator:1997pq,Katz:1996fh}; these theories have gauge algebra $SU(2)$ with $N_\\textbf{F}\\leq 7$. Geometrically, the rank 1 SCFTs can be engineered by del Pezzo surfaces embedded in a non-compact $3$-fold.\nThe families of rank 2 gauge theories predicted by the classification of \\cite{Jefferson:2017ahm} are displayed in Table \\ref{tb:rank2-gauge-theory-clssification}. The UV completions of the theories shown in Table \\ref{tb:rank2-gauge-theory-clssification} are all expected to be 6d theories, rather than 5d SCFTs; on the other hand, their descendants obtained by mass deformations are expected to have 5d CFT fixed points. \nMany of these theories in Table \\ref{tb:rank2-gauge-theory-clssification} are new theories, for example $SU(3)$ with $(N_{\\bf F},|k|)=(6,4),(3,\\frac{13}{2}),(0,9)$ in $(a)$.\n\n\nOne of the purposes of this paper is to check if the new rank 2 CFTs predicted in \\cite{Jefferson:2017ahm} (or descendants of theories in Table \\ref{tb:rank2-gauge-theory-clssification}) can be constructed geometrically. We will see that, surprisingly, almost all new theories in Table \\ref{tb:rank2-gauge-theory-clssification} admit geometric constructions, therefore their descendants indeed have interacting CFT fixed points. However, some theories do not correspond to geometries in their conformal limits due to subtle non-perturbative effects. Therefore, the geometric constructions of this paper indicate that the criteria described in \\cite{Jefferson:2017ahm} require additional non-perturbative corrections in order to be complete.\nWe hope to revisit the field theoretic approach of \\cite{Jefferson:2017ahm} in the near future with the benefit of our improved understanding.\n\n\\begin{table}\n\\centering\n\\begin{subtable}[t]{0.45\\linewidth}\n\\centering\n\\vspace{0pt}\n\\begin{tabular}{|c|c|c|}\n\t\\hline\n\t $N_{\\textbf{Sym}}$ & $N_{\\textbf F}$ & $|k|$ \\\\\n\t\\hline\n\t$1$ & $0$ & $\\frac{3}{2}$\\\\\n\t\\hline\n\t$1$ & $1$ & $0$ \\\\\n\t\\hline\n\t$0$ & $10$ & $0$\\\\\n\t\\hline\n\t$0$ & $9$ & $\\frac{3}{2}$\\\\\n\t\\hline\n\t$0$ & $6$ & $4$\\\\\n\t\\hline\n\t$0$ & $3$ & $\\frac{13}{2}$\\\\\n\t\\hline\n\t$0$ & $0$ & $9$\\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Marginal $SU(3)$ theories with CS level $k$, $N_{\\textbf{Sym}}$ symmetric and $N_{\\textbf F}$ fundamental hypermultiplets.}\n\t\\label{tb:SU3-classification}\n\n\\end{subtable}\\hfill\n\\begin{subtable}[t]{0.45\\linewidth}\n\\centering\n\\vspace{0pt}\n\\begin{tabular}{|c|c|}\n \\hline\n $N_{\\textbf{AS}}$ & $N_{\\textbf F}$ \\\\\n \\hline\n $3$ & $0$\\\\\n \\hline\n $2$ & $4$\\\\\n \\hline\n $1$ & $8$\\\\\n \\hline\n $0$ & $10$\\\\\n \\hline\n\\end{tabular}\n\\caption{Marginal $Sp(2)$ gauge theories with $N_{\\textbf{AS}}$ anti-symmetric, $N_{\\textbf F}$ fundamental hypermultiplets. The theory with $N_{\\textbf{AS}}=3$ can have $\\theta=0,\\pi$.}\n\\label{tb:Sp2-classification}\n\n\\vspace{0.5cm}\n\n\\begin{tabular}{|c|c|c|}\n \\hline\n $N_{\\textbf F}$ \\\\\n \\hline\n $6$ \\\\\n \\hline\n\\end{tabular}\n\\caption{A marginal $G_2$ gauge theory with $N_{\\textbf F}$ fundamental matters.}\n\\label{tb:G2-classification}\n\n\\end{subtable}\n\\caption{Rank 2 gauge theories.}\\label{tb:rank2-gauge-theory-clssification}\n\\end{table}\n\n\n\\subsection{M-theory compactifications}\n\\label{sec:Mth}\n\nString compactifications are an extraordinarily useful tool for realizing local, non-perturbative models of gauge sector physics in terms of brane dynamics. Consider in particular M-theory on a non-compact singular Calabi Yau variety $Y$, which is conjectured to be described at low energies by a 5d $\\mathcal N = 1$ SCFT. We are specifically interested in studying the Coulomb branch deformations of these 5d SCFTs. The heart of this analysis is the correspondence between the Coulomb branch $\\mathcal C$ and the extended K\\\"ahler cone $\\mathcal K(Y)$ of the singular threefold $Y$ \\cite{Witten:1996qb}:\n\t\\begin{align}\n\t\t\\mathcal C =\\mathcal K(Y).\n\t\\end{align}\t\n\nThe above correspondence is made more precise by establishing a dictionary between the geometry of the threefold and the BPS spectrum of the associated 5d theory, which we now describe in detail. Consider a smooth non-compact 3-fold $X$. The K\\\"ahler metric of $X$ depends on $h^{1,1}(X)$ moduli controlling the sizes of complex $p$ cycles in $X$. In order to decouple gravitational interactions, it is necessary to scale the volume of $X$ to be infinitely large while keeping the volumes of all 2- and 4-cycles at finite size; this has the effect of sending the 5d Planck mass to infinity. Given a basis $D_i \\in H^{1,1}(X)$, one may therefore express the K\\\"ahler form $J$ as the linear combination \n\t\\begin{align}\n\t\tJ = \\phi_i D_i,~~ i = 1, \\dots, h^{1,1}(X),\n\t\\end{align}\nwhere the K\\\"ahler moduli $\\phi_{i=1,\\dots, r}$ associated to (cohomology classes dual to) compact 4-cycles $D_i = S_i$ are identified with Coulomb branch moduli, while the K\\\"ahler moduli $\\phi_{r+j,\\dots, r+M}=m_{j=1,\\dots, M}$ associated to non-compact 4-cycles $D_{r+j} = N_j$ are interpreted as mass parameters of the 5d theory. To align the discussion with the 5d field theoretic interpretation, we find it useful to partition the K\\\"ahler moduli into $r$ Coulomb branch parameters and $M$ mass parameters:\n\t\\begin{align}\n\t\th^{1,1}(X) = r + M. \n\t\\end{align}\t\nNote that when the associated 5d field theory admits a description as a gauge theory, $r$ coincides with the rank of the gauge group. \n\nThe BPS states of the 5d theory include electric particles and (dual) magnetic strings. Geometrically these states correspond to M2 branes wrapping holomorphic 2-cycles and magnetic dual M5 branes wrapping holomorphic 4-cycles, and the masses and tensions of these BPS degrees of freedom are proportional to the volumes of the corresponding holomorphic cycles. At a generic point $\\phi \\in \\mathcal C$ the spectrum of BPS states is massive, and this is reflected by the fact that the 2- and 4-cycles of $Y$ have finite volume. Since the conformal point $\\phi = 0$ is characterized by the appearance of interacting massless and tensionless degrees of freedom, we interpret the threefold $Y$ as a singular limit of the smooth threefold $X$ in which some collection of compact 4-cycles have collapsed to a point. Said differently, $X$ is a desingularization of $Y$.\n\n\nThe above discussion suggests that the data of the massive BPS spectrum is encoded in the geometry of $X$. Indeed this is the case, the main connection to geometry being the interpretation of the 5d prepotential (\\ref{eqn:pre}) as the cubic polynomial of triple intersection numbers of 4-cycles in $X$:\n\t\\begin{align}\n\t\\mathcal F = \\text{vol}(X) =\\frac{1}{3!} \\int_X J^3 =\\frac{1}{3!} \\phi_i \\phi_j \\phi_k\\int_X D_i \\wedge D_j \\wedge D_k.\n\t\\end{align}\nIn the previous section, we saw that various data characterizing the massive BPS spectrum can be expressed as derivatives of $\\mathcal F$. This data equivalently characterizes the geometry of $X$. In particular, the tensions (\\ref{eqn:der}) of elementary monopole strings are the volumes of the compact 4-cycles $S_i$:\n\t\\begin{align}\n\t\t\\phi_{Di} = \\partial_i \\mathcal F =\\text{vol}(S_i)= \\frac{1}{2!} \\int_X J^2 \\wedge S_i,~~ 1 \\leq i \\leq r,\n\t\\end{align} \t\nthe matrix of effective couplings has as its components the volumes of various 2-cycles:\n\t\\begin{align}\n\t\t\\tau_{ij} = \\partial_i \\partial_j \\mathcal F= \\text{vol}(S_i \\cap S_j) = \\int_X J \\wedge S_i \\wedge S_j,~~ 1 \\leq i,j \\leq r,\n\t\\end{align}\nand the effective Chern-Simons couplings $k_{ijk}$ are triple intersection numbers:\n\t\\begin{align}\n\t\tk_{ijk} = \\partial_i \\partial_j \\partial_k \\mathcal F = \\int_X D_i \\wedge D_j \\wedge D_k. \n\t\\end{align}\nThe K\\\"ahler cone $\\mathcal K$ of the singularity $Y$ can also be specified quite easily; $\\mathcal K$ is simply the set of all positive K\\\"ahler forms (parametrized by the moduli $\\phi$):\n\t\\begin{align}\n\t\\mathcal K(X \\backslash Y) = \\{ J = \\phi_i D_i ~|~\\int_{C} J > 0 ~~\\text{for all holomorphic curves $C \\subset X$} \\}. \n\t\\end{align}\nThus, it is possible to study Coulomb branch deformations of 5d SCFTs purely in terms of the geometry of a smooth 3-fold $X$. Generically there are multiple smooth 3-folds $X_i$ which share a common singular limit $Y$, so the extended K\\\"ahler cone is simply the closure of the union of K\\\"ahler cones,\n\t\\begin{align}\n\t\t\\mathcal K(Y) = \\overline{\\cup \\mathcal K(X_i \\backslash Y)}.\n\t\\end{align}\nThe extended K\\\"ahler cone has the structure of a fan, with pairs of cones separated by hypersurfaces in the interior of $\\mathcal K(Y)$. The boundaries of $\\mathcal K(X_i \\backslash Y)$ correspond to loci where the 3-fold $X_i$ develops a singularity. The interior boundaries are regions where a holomorphic curve collapses to zero volume and formally develops negative volume in the adjacent K\\\"ahler cone, signaling a flop transition (see Section (\\ref{sec:gtrans}) for further discussion.) By contrast, the boundaries of $\\mathcal K(Y)$ are loci where one of the 4-cycles can collapse to a 2-cycle or a point. The SCFT point is the origin of $\\mathcal K(Y)$, and corresponds to the singularity $Y$ which is characterized by a connected union of 4-cycles shrinking to a point. \n\nIn some cases the 5d theory associated to a 3-fold $X$ admits a description as a gauge theory. In such cases, the abelian gauge algebra is $H^2(X,\\mathbb R) \/ H^2(X ,\\mathbb Z)$ and enhances to a non-abelian gauge algebra in the singularity $Y$. The simple coroots of the gauge algebra correspond to the classes $S_i \\in H^2(X,\\mathbb Z)$, whereas the simple roots are generic fibers $f_j$ contained in $H_2(X,\\mathbb Z)$. More precisely, the W-bosons of the 5d theory correspond to M2-branes wrapping holomorphic curves $f_j$, and so the Cartan matrix $A_{ij}$ is the matrix of charges\n\t\\begin{align}\n\t\\label{eqn:Cartan}\n\t\tA_{ij} = - \\int_{f_j} S_i.\n\t\\end{align}\n\t\nIn practice, we work in an algebro-geometric setting in which volumes of holomorphic cycles can be computed as intersection products. Thus the volumes of 2-cycles $C_i \\subset H_2(X,\\mathbb Z)$ and 4-cycles $S_i \\subset H_4(X,\\mathbb Z)$ are expressed in terms of the intersection products of numerical classes of (resp.) complex curves $[C]$ and surfaces $[D]$. That is, $\\text{vol}(C) = (J \\cdot [C])_X$ and $\\text{vol}(S) = (J \\cdot J \\cdot S_i)_{X}$. We abuse notation and use the same symbols to denote $p$-cycles, their homology classes, and their numerical equivalence classes whenever the context is clear.\n\n\\section{Classification Program}\n\\label{sec:algorithm}\n\n\\subsection{Physical equivalence classes of 3-folds}\n\\label{subsec:constructshrink}\n\nIn this section we propose a classification of CY 3-folds defining 5d SCFTs via M-theory compactification. One way to approach this problem is to study singular 3-folds for which there exist desingularizations that preserve the Calabi-Yau condition (i.e. \\emph{crepant resolutions}.) However, the problem of classifying singular 3-folds admitting crepant resolutions is notoriously difficult. Rather than attempting to classify singularities, we instead classify \\emph{physical equivalence classes} of singularities. We define a pair of 3-folds to be physically equivalent (i.e. leading to the same SCFT, up to decoupled sectors) if they are related by a finite change in K\\\"ahler and complex parameters.\nThere is a conjectural aspect to this definition which we now clarify.\n\nIt is immediate from the above definition that normalizable K\\\"ahler and complex deformations do not change the physical equivalence class of a 3-fold, since these deformations do not change the singular limit (and hence do not change the SCFT). However, we also find it useful to identify 3-folds that differ by non-dynamical large complex deformations. While the singular limits of such 3-folds are not identical, we claim they are nevertheless closely related in that their SCFTs differ at most by decoupled free states\n\nAs we will see, the notion of physical equivalence dramatically simplifies the problem of classification. \n\n\\subsection{Shrinkable 3-folds}\n\\label{sec:shrinkable}\n\nIn this section we specify the necessary criteria a smooth 3-fold must satisfy in order to define a 5d SCFT. Note that we assume all 5d SCFTs have a \\emph{maximal} Coulomb branch, meaning that there exists a phase in which the 5d theory has no dynamical massless hypermultiplets, possibly after turning on some mass parameters. Geometrically this means that we assume there exists a smooth 3-fold which has no normalizable (dynamical) complex structure deformations. The geometry of such a 3-fold is thus controlled by three types of parameters: normalizable K\\\"ahler (i.e. Coulomb branch) parameters, non-normalizable K\\\"ahler (i.e. mass) parameters, and non-dynamical non-normalizable complex structure deformation parameters (see Section \\ref{sec:transitions} for an example).\n\nBefore spelling out the necessary criteria, we recall the key features of the geometries which are the subject of our analysis. We are interested in smooth, non-compact CY 3-folds $X$ containing a finite number of compact 4-cycles $S_i$ and non-compact 4-cycles $N_j$. As discussed in the previous section the number of independent compact 4-cycles is equal to the number of Coulomb branch parameters, while the number of mass parameters is identified with the number of non-normalizable K\\\"ahler deformations. The 4-cycles $S_i \\subset X$ are irreducible projective algebraic surfaces, hence K\\\"ahler. Moreover, $X$ also contains compact 2-cycles which can either be isolated or part of a family of compact 2-cycles belonging to one of the 4-cycles. \n\nFrom the physics perspective the natural condition for CY 3-folds to lead to SCFTs is that we can tune non-normalizable K\\\"ahler parameters (mass parameters) so that at a finite distance in normalizable K\\\"ahler moduli space we can reach a singular CY 3-fold which has no finite volume cycles or surfaces. However, formulating this in algebro-geometric terms is not simple. Instead we formulate it in a somewhat different way which we believe is equivalent to this. Namely,\nin order for a 3-fold $X$ to define a 5d SCFT, $X$ must satisfy the property of being \\emph{shrinkable}, which we define below:\n\n\\medskip\\noindent\n{\\bf Definition.} \n\\label{def:shrinkability}\nLet $X$ be a smooth CY 3-fold modeled locally as the neighborhood of a connected union of compact K\\\"ahler surfaces $S = \\cup S_i$. We say $X$ is \\emph{shrinkable} if there exists an intersecting (possibly empty) union of non-compact surfaces $N=\\cup N_j$ and a limit $Y$ of K\\\"ahler metrics such that:\n\t\\begin{enumerate}\n\t\t\\item $S$ (and all curves $C \\subset S$) have zero volume in $Y$;\n\t\t\\item $Y$ is at finite distance from a metric $X_0$ for which $N$ has zero volume while $S$ has positive volume.\n\t\\end{enumerate} \nBy abuse of terminology, we say the surface $S$ is shrinkable if $S$ is contained in a shrinkable 3-fold $X$ as a maximal compact algebraic surface.\n\\medskip\n\n\n\n\n\n\nLet us now translate the above definition of shrinkability into a set of necessary geometric conditions. We consider first the limit where all non-normalizable K\\\"ahler moduli have been set to zero. In this limit we may have a singular 3-fold which is described by the K\\\"ahler class $J=\\phi_iS_i$. Our convention is to assume $\\phi_i\\ge0$ and compute volumes with respect to $-J$; thus, the volume of a curve $C$ is given by $\\mathrm{vol}(C)= -J\\cdot C$ and the volume of a divisor $D$ is $\\text{vol}(D) = J^2 \\cdot D$.\\footnote{This choice of sign is consistent with \nthe description of K\\\"ahler classes $J$ on compact CY 3-folds, as the expansion of $J$ (or any other ample divisor class) in terms of $S_i$ will have non-positive coefficients. A simple example illustrating this point is the rank~1 case, for which $S$ is a del Pezzo surface. Since $J\\cdot C = \\phi K_S \\cdot C$, it follows that $J$ has non-positive intersection with all curves $C \\in S$. We therefore have to change the sign in order for $J$ to be a limit of K\\\"ahler classes on $X$.} Since we require $-J$ to define a K\\\"ahler metric which assigns postive volumes to complex $p$-cycles in $X$, a necessary condition for shrinkablity is \n\\begin{equation}\n \\label{eq:shrinkability}\n \\mathrm{vol}(C)= -J\\cdot C\\ge0,~~\\forall C\\subset S.\n\\end{equation}\n\n\nWhat happens when the inequality (\\ref{eq:shrinkability}) is saturated? Suppose there exists a curve $C$, with $\\text{vol}(C)=0$. So far, we have only considered the case in which all non-normalizable K\\\"ahler moduli are set to zero. To give finite volume to $C$ requires a non-normalizable K\\\"ahler deformation, which in turn implies the existence of a non-compact 4-cycle $N$ attached to $S$ along $C$. Notice that since $C$ belongs to $N$, there may also be other compact curves $C'$ which are homologous to $C$ in $N$; in particular, the full set of curves homologous to $C$ can fiber over $N$. For each of these curves $C'$ it must be that $\\text{vol}(C')=0$, and thus $N$ can be said to have degenerated to a non-compact 2-cycle along its fibers.\\footnote{It would interesting to compare this defintion of shrinkability with the conjecture of \\cite{Xie:2017pfl} that canonical 3-fold singularities give 5d SCFTs, since it is known that the only noncompact 4-cycles in a Calabi-Yau (crepant) resolution of a canonical 3-fold singularity are ADE fibrations. However, we do not need this for the description in our classification.} By making a non-normalizable K\\\"ahler deformation, we can bring the curve $C = S \\cap N$ to finite volume, and we expect that we are again in a situation where the surface $S$ is contractible.\n\nWe believe that the above necessary criteria are in fact sufficient to define a shrinkable 3-fold:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Let $X$ be a smooth CY 3-fold modeled locally as the neighborhood of a connected union of compact K\\\"ahler surfaces $S= \\cup S_i$. Then $S$ is shrinkable provided that $- J \\cdot C \\geq 0$ for all curves $C \\subset S$ and that there is one $S_i$ with positive volume and the rest should have non-negative (possibly zero) volume.\n\\medskip\n\nElliptic Calabi-Yau 3-folds are immediately ruled out by these criteria. F-theory on an elliptic 3-fold engineers a 6d theory. In a 6d theory, cubic terms in the prepotential $\\mathcal{F}$ are trivial; they are non-trivial only when we compactify the 6d theory on a circle and turn on holonomies for gauge symmetries where the circle size is inversely proportional to a mass parameter (or a non-compact K\\\"ahler parameter). This means that the volumes of all 4-cycles in the associated 3-fold are zero when we turn off mass parameters (or equivalently, in the 6d limit). Therefore elliptic 3-folds are not shrinkable.\n\n\\subsection{Building blocks for shrinkable 3-folds}\n\\label{sec:buildingblocks}\nWe now argue in favor of a series of simplifying assumptions we make concerning the surfaces $S$ which are instrumental for our proposed classification of shrinkable rank 2 surfaces modulo physical equivalence. Observe that when the inequalities of (\\ref{eq:shrinkability}) are all strict, then $S$ is \\emph{contractible} \\cite{grauert}, so that $S$ can be contracted to an isolated singular point $p$ of a singular 3-fold $Y$. In more precise mathematical terms, this means there exists a holomorphic map $f:X \\to Y$ with $f(S)=p$ such that $f$ restricts to an isomorphism away from $S$, i.e. $f|_{X-S}:X-S \\cong Y-p$. Since $X$ is at finite distance from $Y$ in moduli space, it is evident that contractibility of $S \\subset X$ implies shrinkability of $X$. When a curve has zero volume, we expect that we can obtain a contractible surface by means of a non-normalizable K\\\"ahler deformation which involves bringing non-compact 4-cycles to finite volume. Hence, we conjecture that a holomorphic map $f$ exists when $S$ is shrinkable, as well:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Let $X$ be a shrinkable CY 3-fold modeled locally as a neighborhood of a connected union of compact K\\\"ahler surfaces $S= \\cup S_i$ meeting a (possibly empty) collection of non-compact surfaces $N = \\cup N_j$. Then there exists a holomorphic map $f:X \\to Y$ sending $S$ to a point $p$ and $N$ to a collection of curves $C$ such that $\\left. f\\right|_{X - S - N} : X - S - N \\to Y - C$ is an isomorphism.\n\\medskip\n\nThe existence of a holomorphic map $f$ as described above permits a number of simplifying assumptions for the following reasons. Replacing the singular 3-fold $Y$ by its normalization if necessary, we can assume that the singularities of $Y$ are normal. It follows that $Y$ has ``canonical singularities'', and moreover that $X$ is a crepant resolution of $Y$. But it is known the components of the resolutions of canonical threefold singularities $Y$ are rational or ruled \\cite{can3f}.\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe next argue that we can further restrict the types of possible building blocks by exploiting physical equivalence:\n\n\\medskip\n\\noindent\\emph{Conjecture}. Shrinkable surfaces are physically equivalent to a shrinkable surface $S=\\cup S_i$, where the irreducible components $S_i$ are either equal to $\\mathbb P^2$ or a blowup $\\text{Bl}_{p} \\mathbb F_n$ of a Hirzebruch surface at $p$ points intersecting one another (or self-intersecting) transversally. Moreover, there exist non-negative integers $p_{\\text{max}}(n)$ such that $p \\leq p_{\\text{max}}(n)$.\n\\medskip\n\n\nWe briefly discuss the content of the above conjecture, deferring a more detailed discussion of the first two points to Section \\ref{sec:transitions}. \nIn that section, we describe the rank~2 case only. For higher rank, we have to also consider the situation where three surfaces can intersect transversally.\\footnote{Since four or more surfaces in a threefold cannot intersect nontrivially and transversally, we only need to consider intersections of three surfaces at a time.} At such a point of intersection, called a triple point, the three intersecting surfaces have local equation $xyz=0$. As part of the argument in Section \\ref{sec:transitions}, we blow up a point where two surfaces intersect, at which the intersecting surfaces have local equation $xy=0$, so our construction will not apply at a triple point. To handle triple points, we simply supplement the argument in Section~\\ref{sec:gtrans} by noting that a complex structure deformation will keep a point to be blown up distinct from any of the triple points. \n\n\\smallskip\\noindent\n\\begin{enumerate}\n \n\n\n\\item Using a combination of complex structure and K\\\"ahler deformations, it is possible\nto map a 3-fold containing a ruled surface over a genus $g$ to a 3-fold containing a Hirzebruch surface. We defer a detailed discussion to Section~\\ref{sec:transitions}.\n\n\\item In all examples that we have investigated, we have been able to bypass non-transverse intersections in one of two ways: either by a complex structure deformation, or by a K\\\"ahler deformation in the form of a flop. The idea is that when we flop a curve (in $S_1$, say) which passes through a point of non-transversal intersection, the result is to blow up $S_2$ at that point, simplifying the singularity of the intersection curve and rendering it more transverse.\nWe therefore assume that a combination of complex and K\\\"ahler deformations will always suffice to produce a 3-fold containing transversally intersecting surfaces $S_i$.\n\n\\item We prove in Appendix \\ref{app:Mori} that if $p>p_{\\text{max}}(n)$ there are infinitely many generators for rational curves. The presence of infinitely many generators is expected to indicate the presence of an infinite dimensional global symmetry group. An example of this is $\\text{dP}_9$ (note $p_{\\text{max}}(1)=7$), in which case the symmetry group permuting these generators is the affine $E_8$ Weyl group. In such a case, the Weyl group is infinite dimensional, and can be interpreted as a finite symmetry group of a 6d theory viewed from the 5d perspective. As we discussed above, geometries associated to 6d theories are not shrinkable. Since a CFT should not have an infinite dimensional global symmetry group, we claim that surfaces $S_i$ with an infinite number of Mori cone generators cannot be building blocks for 5d SCFTs and are thus excluded.\n\\end{enumerate}\n\n\n\n\n\\subsection{Consistency conditions for shrinkable 3-folds}\n\\label{sec:consistency}\n\nThe condition that $S$ is contained in a CY 3-fold imposes constraints on the curves of intersection of the components of $S$, which will be exploited in a crucial way in our classification program. \n\nLet $S_1$ and $S_2$ be two smooth surfaces glued along a curve $C = S_1 \\cap S_2$. Now suppose that $S_1\\cup S_2$ is contained in a 3-fold $X$, and that the intersection of $S_1$ and $S_2$ is transverse in $X$.\nThen the normal bundle of $C$ in $X$ is given by $N_{C,X}=N_{C,S_1}\\oplus N_{C,S_2}$. The Calabi-Yau condition then implies\n\\begin{equation}\n \\label{eq:gluingcond}\n C^2_{S_1}\\oplus C^2_{S_2}=2g-2,\n\\end{equation}\nwhere $g$ is the genus of $C$ and the subscripts on the right-hand side denote the irreducible surface in which the self-intersection takes place. The gluing curves must satisfy the adjunction formula for each surface $S_i$:\n\t\\begin{align}\n\t\\label{eq:adjunction}\n\t\t(K \\cdot C)_{S_i} + C_{S_i}^2 = 2g - 2,\t\n\t\\end{align}\nwhere $K_{S_i}$ is the canonical class of the surface $S_i$. For the rank 2 case, which is the primary focus of this paper, we argue in Section \\ref{sec:rank2} that it suffices for our classification to assume that $g=0$.\n\nSuppose a compact connected holomorphic surface $S$ satisfies the above constraints on its curves of intersection. These constraints immediately imply that a CY 3-fold can be found containing a neighborhood in $S$ of the curves of intersection (for example, the total space of the normal bundle of $S_1 \\cap S_2$ in $X$ works, as the complement of $S_1 \\cap S_2 \\subset S$ is smooth). Moreover, we can also find local CY 3-folds containing the complement of the intersection curves $S_1 \\cap S_2$ in $S$ (for example, just take the total space of the canonical bundle as before). Therefore, it seems reasonable to expect that above two types of local models can be glued to form a local model of a CY 3-fold. In other words,\ngiven smooth holomorphic surfaces $S_1$ and $S_2$ glued along a smooth curve $C$ and satisfying (\\ref{eq:gluingcond}), \na smooth CY 3-fold $X$ can be found containing $S=S_1\\cup S_2$.\nWhile we have not proven that such an $X$ can always be found if (\\ref{eq:gluingcond}) and (\\ref{eq:adjunction}) are satisfied, these conditions are consistent with all known examples and it is presumably not too difficult to rigorously prove this.\n\nWe emphasize here that the above gluing condition is a local condition that has no bearing on the overall topology of the surface $S$, and therefore permits a variety of interesting configurations. In principle there is nothing preventing, for example, gluing two surfaces together along multiple irreducible curves. Another interesting configuration involves two curves belonging to a single surface $S_i$ being glued together. However, we will see that the only gluing configurations which play a role in the rank 2 classification are pairwise transverse intersections between the irreducible components $S_1$ and $S_2$.\n\nThe above discussion plays an essential role in our classification because we do not need to actually construct $X$ to proceed; rather, we only require the existence of $X$ and the existence of a surface $S$ can be used as a proxy for the existence of a local 3-fold. Thus the problem of classifying shrinkable 3-folds can be reduced to the problem of classifying embeddable, shrinkable surfaces $S$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n \n \n\n\n\n\n\\subsubsection*{A simple example: $S = \\mathbb F_0 \\cup \\mathbb F_2$}\n\nAn illustrative example of this construction is a simple complex surface $S=S_1 \\cup S_2$ with $S_1= \\mathbb F_0, S_2 = \\mathbb F_2$ as depicted in Figure \\ref{fig:F0F2}. Our rank 2 ansatz gives us \n\t\\begin{align}\n\t\\begin{split}\n\t\\label{eqn:geotrip}\n\t\tJ^3 &= S_1^3 \\phi_1^3 + S_2^3 \\phi_2^3 + 3 \\phi_1 \\phi_2 (J \\cdot S_1 \\cdot S_2) =K_{S_1}^2 \\phi_1^3 + K_{S_2}^2 \\phi_2^3 - 3 \\phi_1 \\phi_2 \\text{vol}(S_1 \\cap S_2). \n\t\\end{split}\n\t\\end{align}\n\nThe first order of business is to determine an appropriate gluing. Gluing these two surfaces together requires us to identify an irreducible, smooth curve $C = S_1 \\cap S_2$ belonging to the Mori cone of both surfaces, satisfying (\\ref{eq:gluingcond}). In the case of Hirzebruch surfaces $\\mathbb F_{n_i}$, the Mori cones are the positive linear spans $\\langle E_{i}, F_{i} \\rangle$, where the curve classes satisfy the intersections $F_i^2= 0, E_i \\cdot F_i =1, E_i^2 = -n_i$, so the range of possibilities is severely restricted. The gluing condition (\\ref{eq:gluingcond}) implies that the self intersection of one of the two gluing curves must be negative. Since the curve $E$ is the unique rational curve with negative self intersection \\cite{GH}, it therefore follows that we must select $C_{S_i} = E_{i}$ for one of the two surfaces, say $C_{S_2} =E_{2}$. The other curve must then satisfy\n\t\\begin{align}\n\t\tC_{S_1}^2 = 0. \n\t\\end{align}\nAs a trial solution let us take $C_{S_1} = a F_{1} + b E_{1}$, so that $C_{S_1}^2 = 2 ab = 0$. Therefore, either $a = 0$ or $b = 0$. From the adjunction formula (\\ref{eq:adjunction}), we know that $(C \\cdot E_1 + C \\cdot F_1)_{S_1}= a+b= 1$, and therefore the remaining nonzero coefficient must be set equal to unity. To be concrete, we choose\n\t\\begin{align}\n\t\tC_{S_1} = F_{1},~~~ C_{S_2} = E_{2}. \n\t\\end{align}\n\nNow that we have constructed the surface $S$, we must check that the local 3-fold $X$ associated to this surface is shrinkable. We parametrize a K\\\"ahler class $J$ as follows:\n\t\\begin{align}\n\t\tJ = \\phi_1 [\\mathbb F_0] + \\phi_2 [ \\mathbb F_2],\n\t\\end{align}\nwhere $[\\mathbb F]$ is the class associated to the 4-cycle $\\mathbb F \\subset X$. The Mori cone of $X$ is the union of the Mori cones of the component surfaces $S_i$, namely the positive span $\\langle E_{1}, E_{2}, F_{2} \n\\rangle$ (we omit $F_{1}$ because the gluing identifies $F_1$ and $E_2$.) Therefore, the shrinkability condition (\\ref{eq:shrinkability}) implies \n\t\\begin{align}\n\t\t(\\text{vol}(E_1), \\text{vol}(E_2), \\text{vol}(F_2) ) = (2 \\phi_1 -\\phi_2 , 2 \\phi_1, -\\phi_1 + 2\\phi_2 ) \\geq 0. \n\t\\end{align}\nSince that the above conditions can be satisfied for a nontrivial set of Coulomb branch parameters $\\phi_i$, we conclude that the geometry $X$ corresponds to a 5d SCFT on the Coulomb branch. \n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[scale=.5]{F0-F2.pdf}\n\\end{center}\n\\caption{Example of a gluing construction of the K\\\"ahler surface $S = \\mathbb F_0 \\cup \\mathbb F_2$. The gluing curves in both surfaces, $C_1, C_2$, are encircled by dashed lines in the left figure. The final geometry (on the right) is the result of identifying these two curves subject to the conditions described in Section \\ref{sec:algorithm}.}\n\\label{fig:F0F2}\n\\end{figure}\n\n \n\n\n\n\n\n\\subsection{Geometry of physical equivalences}\n\\label{sec:transitions}\n\nIn this section we discuss some important types of physical equivalences upon which our classification relies. Many of these equivalences identify 3-folds related by geometric transitions, i.e.\\ maps between smooth geometries which involve passing through an intermediate singularity. Another type of physical equivalence identifies 3-folds related by a ``large\" change in the complex structure of non-dynamical modes, which interpolates between two singular geometries---this is a Hanany-Witten transition \\cite{Hanany:1996ie}. We illustrate these two types of maps in turn.\n\n\\subsubsection{Geometric transitions}\n\\subsubsection*{Flop transitions}\n\\label{sec:gtrans}\nOne of the simplest and most thoroughly studied types of geometric transitions is a \\emph{flop transition}, which is a topology-changing transition $X \\rightarrow X'$ between two 3-folds $X, X'$ that is in practice typically realized by blowing down a $-1$ curve $C \\subset X$ and blowing up a different $-1$ curve $C' \\subset X'$ (see Figure \\ref{fig:flop}). A flop is a birational map $X\\dashrightarrow X'$ which is an isomorphism away from curves $C,C'$, with $K_X\\cdot C=K_{X'}\\cdot C'=0$. If $C$ and $C'$ are both isomorphic to ${\\mathbb P}^1$, the flop is called a simple flop. Simple flops were classified in \\cite{km}. \n\n\nIn field theoretic terms, a flop transition corresponds to a continuous change of the mass of a particular state in the matter hypermultiplet from positive to negative values; this change corresponds to a singular phase transition on the Coulomb branch.\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.7]{flop-trans.pdf}\n\t\\end{center}\n\t\\caption{A local illustration of a flop transition $X \\rightarrow X'$ between two CY 3-folds. The red lines in both diagrams correspond to the $-1$ curves in (respectively) $X$ and $X'$.}\n\t\\label{fig:flop}\n\\end{figure}\t\n\n\n\\subsubsection*{Genus reduction}\nWe saw in Section~\\ref{sec:buildingblocks} that the $S_i$ can be ruled surfaces over higher genus curves as well as genus 0. Here we argue that by our notion of physical equivalences we can restrict to $g=0$ using geometric transitions. This can be obtained by composing a complex structure deformation of a surface $S_i$ with a flop transition. This provides a map from a ruled surface over a curve of \ngenus $g$ to a self-glued Hirzebruch surface.\n\nThis type of geometric transition is particularly important because it exhibits the non-normalizable K\\\"ahler moduli of the local 3-fold defined by a ruled surface over a curve of genus $g$ as blowup parameters of the 3-fold defined by a self-glued surface $\\text{Bl}_{2g} \\mathbb F_n$. While we have not proven that the transition can always be achieved in the higher rank case due to the requirement that additional compact surfaces remain glued throughout the transition, we nevertheless believe this construction can be extended to higher rank surfaces with at most minor modifications.\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\\node(a) at (0,0) {$\n\\begin{tikzpicture}[yscale=.8,xscale=1.1]\n\\draw [thick] (0,0) to [out=90,in=180] (1,1) to [out=0,in=180] (2,.85) to [out=0,in=180] (3,1) to [out=0,in=90] (4,0) to [out=270,in=0] (3,-1) to [out=180,in=0] (2,-.85) to [out=180,in=0] (1,-1) to [out=180,in=270] (0,0);\n\\draw[thick] (.7,0) to [out=-20,in=180] (1.1,-.13) to [out=1,in=200] (1.5,0);\n\\draw[thick] (2.5,0) to [out=-20,in=180] (2.9,-.13) to [out=1,in=200] (3.3,0);\n\\draw[thick] (2.6,-.05) -- (3.2,-.05);\n\\draw[thick] (.8,-.05) -- (1.4,-.05);\n\\end{tikzpicture}\n$};\n\\node(b) at (6,0) {$\n\\begin{tikzpicture}[yscale=.8,xscale=1.1]\n\\draw [thick] (0,0) to [out=90,in=180] (1,1) to [out=0,in=180] (2,.85) to [out=0,in=180] (3,1) to [out=0,in=90] (4,0) to [out=270,in=0] (3,-1);\n\\draw[thick] (.8,-1) to [out=180,in=270] (0,0);\n\\draw[thick] (2.5,0) to [out=-20,in=180] (2.9,-.13) to [out=1,in=200] (3.3,0);\n\\draw[thick] (2.6,-.05) -- (3.2,-.05);\n\\draw[thick] (.9,-.09) -- (1.45,-.09);\n\\draw[thick](.7,-.05) to [out=-10,in=90] (1.2,-.45) to [out=270,in=0] (.8,-1);\n\\draw[thick] (1.6,-.03) to (1.4,-.13) to [out=200,in=90] (1.2,-.5) to [out=270,in=180] (1.6,-1) to (3,-1);\n\\end{tikzpicture}\n$};\n\\draw[big arrow] (a) -- (b);\n\\end{tikzpicture}\n\\end{center}\n\\caption{A genus $g= 2$ Riemann surface degenerating into a $g= 1$ Riemann surface with a nodal singularity as the result of identifying two points. By identifying $g$ pairs of points in this manner, it is possible for a smooth curve of genus $g$ to degenerate into a rational curve with $g$ nodal singularities.}\n\\label{fig:degen}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.6]{self-gluing.pdf}\n\t\\end{center}\n\t\\caption{A transition from a ruled surface over a $g=1$ curve to a Hirzebruch surface. The red point in the second figure is a blowup point on a nodal curve and the red lines in the third figure are the exceptional curves. Two proper transforms of the fiber $F$ in a blown up Hirzebruch surface are glued together along the nodal curve.}\n\t\\label{fig:selfgluing}\n\\end{figure}\n\nBefore giving a detailed description of this geometric transition, we recall that by the irreducibility of the moduli space $\\overline{M}_g$\nof stable curves of genus $g$ the complex structure of a smooth curve $C$ of genus $g$ can be\ndegenerated to a rational curve $C_0$ with $g$ nodes (see Figure \\ref{fig:degen}.) The curve $C_0$ can be \nconstructed directly by identifying $g$ pairs of points of ${\\mathbb P}^1$. Note that this construction immediately extends to give a degeneration of a ruled surface\n$S$ over $C$ to a ruled surface $S_0$ over the singular curve $C_0$. Conversely,\nthe degeneration of the ruled surface can be described by starting with ${\\mathbb P}^1$-bundle over\n${\\mathbb P}^1$ (i.e.\\ a Hirzebruch surface ${\\mathbb F}_n$) and identifying $g$ pairs of fibers\n$F \\subset \\mathbb F_n$. \n\nHowever, this description of $S_0$ is not completely satisfactory, as $S_0$ cannot be embedded into a CY 3-fold for the following reason. Let $F\\subset S_0$ be one of the singular fibers obtained by identifying $g$ pairs of fibers. Locally, $S_0$ has two branches near $F$ with equation $xy=0$ (pulled back\nfrom the local equation $xy=0$ of a node of $C_0$). Being a fiber, $F$ has self-intersection 0 in each branch, So if $S_0$ were contained in a smooth\nthreefold, the normal bundle of $F$ would be ${\\cal O}_F\\oplus{\\cal O}_F$. Fortunately, the geometric transition naturally rectifies this problem by introducing blowups, in a manner which we describe below.\n\nConsider again the degeneration point of view, which can be described by a holomorphic map $\\pi:{\\cal S}\\to \\Delta$. Here ${\\cal S}$ is a smooth\\footnote{Requiring ${\\cal S}$ to be smooth is not a problem; its local equation near a point of $F$ can be taken as $xy=t$, which is smooth. This is the same local calculation which shows that $\\overline{M}_g$ is smooth at the nodal curves (in the orbifold sense).} threefold, $\\Delta$ is a disk, $\\pi^{-1}(0)\\simeq S_0$, and $\\pi^{-1}(t)$ is diffeomorphic to\n$S$ for $t\\ne0$. We now pick a point $p\\in F\\subset S_0\\subset {\\cal S}$ and blow up $p$ to get $\\phi:\\widetilde{{\\cal S}}\\to {\\cal S}$. Via $\\pi\\circ\\phi$ we can view $\\widetilde{\\cal S}$ as a family over $\\Delta$. However, $\\widetilde{\\cal S}$ and ${\\cal S}$ are isomorphic over $\\Delta-0$, so this gives another degeneration of $S$. The singular limit is $(\\pi\\circ\\phi)^{-1}(0)$, which we now describe.\n\nBlowing up a point $p$ in a smooth threefold creates an exceptional divisor $E$ isomorphic to ${\\mathbb P}^2$, and blows up $S_0$ to a surface $\\widetilde{S_0}$. We have $(\\pi\\circ\\phi)^{-1}(0)=\\widetilde{S}_0\\cup {\\mathbb P}^2$. It remains to describe $\\widetilde{S}_0$ and how ${\\mathbb P}^2$ is attached to it.\n\nSince $S_0$ has local equation $xy=0$ at $p$, the exceptional curve of $\\widetilde{S}_0\\to S_0$ has $xy=0$ as its equation. In this latter instance, the equation $xy=0$ is understood as a homogeneous equation in the exceptional\n${\\mathbb P}^2$ of the blown-up threefold. In other words, ${\\mathbb P}^2$ meets $\\widetilde{S_0}$ in two intersecting projective lines $L,L'$; each of these ${\\mathbb P}^1$'s can be thought of as arising from the blowup of $p$ in a corresponding branch of $S_0$ near $p$.\n\nThe point of intersection $q = L \\cap L'$ also intersects the proper transform $\\widetilde{F}$ of the original singular fiber $F$. The curve $\\widetilde{F}$ is still singular in $\\widetilde{S_0}$ and still has two branches in a local\ndescription, but now the blowup has reduced the self-intersection from $0$ to $\\widetilde F^2 = -1$ in each branch. So if $\\widetilde{S_0}$ is contained in a smooth threefold, then the normal bundle of $\\widetilde{F}$ is ${\\cal O}_F(-1)\\oplus {\\cal O}_F(-1)$ and the threefold can be Calabi-Yau!\n\nWe can apply this construction to all of the $g$ singular fibers. Since $\\widetilde{F}$ has self-intersection $-1$ in each branch, we can view it as the gluing of a pair of exceptional ${\\mathbb P}^1$'s. \nTherefore the\nresulting $\\widetilde{S}_0$ is a blown up Hirzebruch surface with $g$\npairs of exceptional curves identified. Each singular fiber consists of a double curve with self-intersection $-1$ in each branch, glued at a common point $q$ to curves $L,L'$ of self-intersection $-1$ in each of the respective local branches\n(the surface $\\widetilde{S}_0$ is smooth along $L\\cup L'-\\{q\\}$).\n\nIn the degeneration described above, we also need to attach $g$ copies of ${\\mathbb P}^2$. However, we are only concerned with the rank~2 case, so in our examples\nthese ${\\mathbb P}^2$'s can replaced by noncompact cycles containing $L\\cup L'$ and safely ignored.\n\nThe final step is to flop the $g$ curves $\\widetilde{F}_1,\\ldots\\widetilde{F}_g$, where we have added a subscript to $\\widetilde{F}$ to distinguish\nthese curves. Let us investigate the birational transform of $\\widetilde{S_0}$ after the flops.\nWhen the curves $\\widetilde{F}_i$ are contracted, the points of intersection $q_i = L_i \\cap L_i'$ become conifolds. When we complete the flops, new ${\\mathbb P}^1$'s appear in place of the $q_i$ and the curves $L_i,L'_i$ get separated. These curves\nbecome identified with fibers of a ruled surface over the desingularization $\\widetilde{C}_0$ of $C_0$, the fibers over the pairs of points of $\\widetilde{C}_0$ which get identified to form a node of $C_0$. Since $\\widetilde{C}_0$ is isomorphic to ${\\mathbb P}^1$, the result\nis a Hirzebruch surface in general with blowups.\n\n\\subsubsection*{An example of genus reduction: $G_2 + N_\\textbf{F} \\textbf{F}$ }\n\nAn illustrative example of complex deformations that exchange ruled surfaces over a curve of genus $g >0$ for self-glued Hirzebruch surfaces blown up at $2g$ points is the family of shrinkable 3-folds engineering $G_2 + N_{\\textbf{F}} \\textbf{F}$, as described in \\cite{Diaconescu:1998cn}. \n\nWe begin by recalling the form of the gauge theoretic 1-loop prepotential for $G_2 + N_{\\textbf{F}} \\textbf{F} + N_{\\textbf{adj}} \\textbf{adj}$:\n\t\\begin{align}\n\t\\begin{split}\n\t\\label{eqn:G2nomass}\n\t\t6 \\mathcal F_{\\text{1-loop}} &= ( 8 - 8 N_\\textbf{F} -8 N_\\textbf{adj}) \\phi_1^3 + ( 8 - 8 N_{\\textbf{adj}}) \\phi_2^3\\\\\n\t\t&~+ 3 \\phi_1 \\phi_2 [ (6 +3 N_\\textbf{F} - 6 N_\\textbf{adj} )\\phi_1 + (8 N_{\\textbf{adj}} - N_\\textbf{F} - 8 ) \\phi_2 ].\n\t\\end{split}\n\t\\end{align}\nWe set $N_{\\textbf{adj}} =0$ to be consistent with $\\mathcal N = 1$ supersymmetry. By giving a nonzero value to mass parameters in the hypermultiplet contributions to the prepotential, one can study the RG flow from $N_{\\textbf{F}}$ to $N_{\\textbf{F}}-1$ flavors. In order to decouple a massive hypermultiplet, the theory must pass through three phase transitions. These four phases have the following prepotentials (we omit mass parameter terms for brevity): \n\t\\begin{align}\n\t\t\\begin{split}\n\t\t\\label{eqn:G2RG}\n\t\t\t6 \\mathcal F_{}^{(1)} &=(8- 8N_\\textbf{F}) \\phi_1^3 + 8 \\phi_2^3 + 3 \\phi_1 \\phi_2 [\\phi _1 \\left(3 N_{\\textbf F}+6\\right)-\\phi _2 \\left(N_{\\textbf F}+8\\right)]\\\\\n\t\t\t6 \\mathcal F_{}^{(2)} &=(16-8N_\\textbf{F}) \\phi_1^3 + 7 \\phi_2^3 + 3 \\phi_1 \\phi_2 [ \\phi _1 \\left(3 N_{\\textbf F}+2\\right)-\\phi _2 \\left(N_{\\textbf F}+6\\right)]\\\\\n\t\t\t6 \\mathcal F_{}^{(3)} &=(15 - 8 N_\\textbf{F}) \\phi_1^3 + 8 \\phi_2^3 + 3 \\phi_1 \\phi_2 [ \\phi _1 \\left(3 N_{\\textbf F}+3\\right)-\\phi _2 \\left(N_{\\textbf F}+7\\right) ]\\\\\n\t\t\t6 \\mathcal F_{}^{(4)} &=6 \\mathcal F_{N_\\textbf{F}-1}^{(1)}.\n\t\t\\end{split}\n\t\\end{align}\n\t\nWe determine a shrinkable K\\\"ahler surface $S$ that engineers this theory by setting the triple intersection polynomial (\\ref{eqn:geotrip}) equal to prepotential (\\ref{eqn:G2nomass}) and demanding that there exist an intersection matrix $f_i \\cdot S_j = (A_{G_2})_{ij}$ for some choice of fiber classes $f_i \\subset S_i$. Restricting the possible building blocks to be blowups of rational and ruled surfaces \\emph{without self-gluing}, the only solutions to these conditions are the geometries shown in Table \\ref{tab:G2geo}. For all of these surfaces we have $9n_2+6a=2g-2+n_1$, as required by (\\ref{eq:gluingcond}). A key point here is that the surface $S_1$ must be a ruled surface of a curve of genus $g = N_{\\textbf{F}}$. This is precisely the geometric setup described in \\cite{Diaconescu:1998cn}. \n\n\\begin{table}\n\t\\begin{center}\n$\n\t\t\t\\begin{array}{|c|c|c|c|}\n\t\t\\hline\n\t\t\tg & a & (n_1,n_2) \\\\\\hline\n\t\t\t0 & 1 & (8,0) \\\\\\hline\n\t\t\t1 & 0 & (9,1) \\\\\\hline\n\t\t\t2 & 2 & (10,0) \\\\\\hline\n\t\t\t3 & 1 & (11,1) \\\\\\hline\n\t\t\t4 & 0 & (12,2) \\\\\\hline\n\t\t\t4 & 3 & (12,0) \\\\\\hline\n\t\t\t5 & 2 & (13,1) \\\\\\hline\n\t\t\t6 & 4 & (14,0) \\\\\\hline\n\t\t\\end{array}\n\t\t$\n\t\\end{center}\n\t\\caption{Shrinkable surfaces $S = \\mathbb F^{g}_{n_1} \\cup \\mathbb F_{n_2}$ engineering $G_2 + N_\\textbf{F} \\textbf{F}$ gauge theories. The surface $\\mathbb F^g_{n_1}$ is a ruled surface over a curve $E$ with $g(E) = N_{\\textbf{F}}$ and satisfying $E^2 = -n_1$. The gluing curve $C = S_1 \\cap S_2$ is given by $C_{S_1} = E$ and $C_{S_2} = a F + 3 H$. The fiber classes are given by are $f_i = F_i$.}\n\t\\label{tab:G2geo}\n\t\\end{table}\n\n\n\nWe now demonstrate that we can engineer the same family of theories described above by replacing $S_1$ with the surface $S_1' = \\text{Bl}_{ 2g} \\mathbb F_{n_1}^{(g)} $, where again $g = N_{\\textbf{F}}$ and the superscript notation indicates $S_1'$ is obtained by identifying $g$ pairs of exceptional curves in $\\text{Bl}_{2g} \\mathbb F_{n_1}$ (i.e. self-gluing; see Appendix \\ref{app:math} for some mathematical background.) This shrinkable surface not only reproduces the prepotential (\\ref{eqn:G2nomass}) and $G_2$ Cartan matrix, but also has the merit of exhibiting the RG flow (\\ref{eqn:G2RG}) in a very natural manner. The four phases, related by flops, have the following geometries: \n \\begin{enumerate}\n\t\t\\item $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g} \\cup \\mathbb F_{n_2}$, where the blowups are all at special points\\footnote{Note that while we consider blowups at special points $F \\cap E \\subset \\mathbb F_n$ here for convenience, since we do not introduce any additional irreducible curves with self intersection less than $-1$, we can without loss of generality view a blowup of $\\mathbb F_n$ at $p$ special points as a blowup of $\\mathbb F_{n+p}$ at $p$ general points. We explore the distinction between special and general points in more depth in Section \\ref{sec:rank2}.} $F \\cap E$.\n\t\t\\item $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{8- g} \\cup \\text{Bl}_1 \\mathbb F_{n_2}$.\n\t\t\\item $\\text{Bl}_{2g-1} \\mathbb F^{(g-1)}_{8- g} \\cup \\mathbb F_{n_2\\pm{} 1}$.\n\t\t\\item $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{9- g} \\cup \\mathbb F_{n_2 \\pm{} 1}$. \n\t\\end{enumerate}\n\n\tThe first phase is $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g} \\cup \\mathbb F_{n_2}$, where we introduce $g$ self-gluings of $\\text{Bl}_{2g} \\mathbb F_p$ along the pairs of exceptional divisors $X_{2i}, X_{2i-1}, i = 1, \\dots, g$,\\footnote{Here and in the sequel, we use the notation $X_i$ to denote the exceptional divisor of the $i$-th blowup, since we reserve the more standard notation $E_i$ for sections of Hirzebruch surfaces.}the where the gluing curve is defined by $C_{S_1} = E - \\sum_{i=1}^{2 g} X_i$ and $C_{S_2} = F + 3 H$, so that $a=1$ in the notation adopted in the caption of Table~\\ref{tab:G2geo}. Since the canonical class\\footnote{More precisely, the dualizing sheaf of the singular surface $ \\text{Bl}_{2g} \\mathbb F^{(g)}_{8-g}$, pulled back to its natural desingularization $ \\text{Bl}_{2g} \\mathbb F_{8-g}$.} is given by $K_{\\mathbb F_{8 - g}} + 2\\sum_{i=1}^{N_\\textbf{F}} (X_{2i-1}+X_{2i})$, we find a perfect match with the first line of (\\ref{eqn:G2RG}), using the adjunction relation $9n_2+6-(8+g)=2g-2$.\n\t\n\tWe now describe the flop to the second phase. The matter curve with volume $2\\phi_1 - \\phi_2$ which shrinks is one of the self-gluing exceptional divisors, say $X_1$. Blowing down $X_1$ forces us to also blow down $X_2$.\nWe can blow up $\\mathbb F_{n_2}$ at a generic point $F_2 \\cap H_2$ if we eventually want to decrease $n_2$ to $n_2 -1$, or at a special point $F_2 \\cap E_2$ if we want to increase $n_2$ to $n_2 +1$ in the third phase.\n\t\nThe geometry of the second phase is $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{8- g} \\cup \\text{Bl}_1 \\mathbb F_{n_2}$, where $C_{S_1} = E -\\sum_{i=1}^{2g -2} X_i$ and $C_{S_2}=a F + 3 H - 2 Y_1$. Since the blowup of ${\\mathbb F}_{n_2}$ is at the double point of $E$ introduced by gluing $X_{2g-1}$ to $X_{2g}$, the coefficient of $Y$ in $C_{S_2}$ is $-2$. \n\nThe matter curve with volume $\\phi_2 - \\phi_1$ which we blow down is $F_2 - Y_1 \\subset \\text{Bl}_1 \\mathbb F_{n_2}$. Because $F - Y_1$ meets $C$ in one point, we must introduce an exceptional divisor $Y_2$ in the surface $S_1$, leading us to the third phase.\n\t\n\tThe geometry of the third phase is $\\text{Bl}_{2g-1} \\mathbb F^{(g-1)}_{8- g} \\cup \\mathbb F_{n_2\\pm{} 1}$, where $C_{S_1} = E - \\sum_{i=1}^{2g-2} X_i - Y_2$. Concerning the gluing curve class $C \\subset \\mathbb F_{n_2 \\pm{} 1}$, there are two possible cases. In the case of a generic blowup, the proper transforms of $H, F \\subset S_2$ are $H- Y_1, Y_1$, so we set $C_{S_2} = (a +1) F + 3 H$, where now $ H^2_{S_2} = n_2 -1$. It follows that $C^2_{S_2} =((a +1) F + 3 H)_{S_2}^2 = 6 (a + 1) + 9 (n_2-1) = 3 g + 3$, which is a nontrivial check that this geometry is consistent with the phase structure of the $G_2$ theory. On the other hand, in the case of a special blowup, the difference is that the proper transform of $H \\subset S_2$ is $H$, so that $C_{S_2} = H + (a - 2) F$, where now $H_{S_2}^2 = n_2 +1$. We again confirm that $C^2_{S_2} = ((a -2) F + 3 H)_{S_2}^2 = 6 (a - 2) + 9 (n_2 +1) = 3 g+ 3$.\n\t\t\n\t\tIn order to reach the fourth and final phase, the matter curve with volume $\\phi_1$ which we blow down is $F - Y_2 \\subset S_1$. The geometry of the fourth phase is $\\text{Bl}_{2g-2} \\mathbb F^{(g-1)}_{9- g} \\cup \\mathbb F_{n_2 \\pm{} 1}$. Keeping in mind the previous identity $n_1 = 8- g$ along with the fact that we blow down the curve $F - Y_2 \\subset S_1$, we compute the canonical class:\n\t\t\\begin{align}\n\t\t\\begin{split}\n\t\t\tK_{S_1} &= -2 H + (n_1 -2) F + 2\\sum_{i=1}^{g-1} (X_{2i-1} + X_{2i} ) + Y_2\\\\\n\t\t\t&= -2 H + ((n_1+1) -2) F + 2 \\sum_{i=1}^{g-1} (X_{2i-1} + X_{2i} ) .\n\t\t\\end{split}\n\t\t\\end{align}\n\tNote also that the self-intersection of $H \\subset S_1$ shifts from $8-g$ to $9-g$.\n\n\n\n\\subsubsection{Hanany-Witten transitions and complex deformations}\nThe next type of transition we will discuss is a \\emph{complex structure deformation}.\nIn particular, we concern ourselves with two types of complex structure deformations that preserve the rank of the 3-fold. The first type of complex structure deformation is a Hanany-Witten (HW) transition \\cite{Hanany:1996ie}. This type of transition is most easily understood in the setting of $(p,q)$ 5-brane webs, and involves interchanging the relative position of a $(p,q)$ 7-brane and a $(p,q)$ 5-brane. After the transition, despite the fact that the brane webs look different, in the low-energy decoupling limit the corresponding SCFTs describe the same physics up to decoupled free sectors. The example displayed in Figure \\ref{fig:HW} describes a geometric (or HW) transition from a local 3-fold $X$ with $S= \\mathbb F_2$ to another 3-fold $X'$ with $S' = \\mathbb F_0$. Therefore, $X$ and $X'$ are physically equivalent.\n\n\\begin{figure}\n\\begin{center}\n\t\\includegraphics[scale=.5]{F0-F2-HW.pdf}\n\\end{center}\n\\caption{Hanany-Witten transition from $\\mathbb F_2$ to $\\mathbb F_0$. The $\\otimes$ symbol denotes the location of a transverse $(0,1)$ 7-brane, and the dashed line denotes the location of the 7-brane monodromy cut.}\n\\label{fig:HW}\n\\end{figure}\n\nThis example can be geometrically described as follows: ${\\mathbb F}_2$ is physically equivalent to ${\\mathbb F}_0$ by a (non-normalizable) complex structure deformation. One way to see this is to first contract the curve $E$ in ${\\mathbb F}_2$ (with $E^2=-2$) to an $A_1$ singularity, which can be identified with the quadric cone $x^2+y^2+z^2=0$ in ${\\mathbb P}^3$. A complex structure deformation takes this to a smooth quadric surface (e.g.\\ $w^2+x^2+y^2+z^2 =0$), which is isomorphic to ${\\mathbb P}^1\\times{\\mathbb P}^1={\\mathbb F}_0$.\n\nAnother type of complex structure deformation involves changing special type blow ups (i.e. blow ups on top of blow ups) to generic blow ups, where the blow up points are not on top of one another, unless the blow up curve is part of the identification between $S_i$'s. We will show that in the rank 2 case this can be avoided and we can always assume general point blow ups.\n \n \n\n\n\n\n\n\n\\section{Classifications}\n\\label{sec:classification}\n\n\nLet $S=\\cup S_i$ be a connected union of surfaces contained in a CY 3-fold $X$. We classify all shrinkable $S$ for rank 1 and rank 2 according to the conjectures and algorithm described in Section \\ref{sec:algorithm}.\nWe first summarize the rank 1 and rank 2 classification results and in the next two subsections we present details of the classification. \n\nAll rank 1 and rank 2 shrinkable geometries (or SCFTs) belong to one or more families of geometric RG-flows, and the geometries in each RG-flow family are related by rank-preserving mass deformations (or blowdowns of -1 curves in geometric terminology), up to physical equivalence. The ideas of geometric RG-flow and rank-preserving mass deformations will be discussed later.\nBased on these ideas, we can start from a ``top'' geometry, which corresponds to a 5d CFT or a 6d CFT on a circle (equivalently, a 5d Kaluza-Klein (KK) theory), and obtain all other geometries in the same family by a finite sequence of geometric transitions or mass deformations. This UV geometry is at the top of the RG-flow in a given family and can therefore be a representative of the entire RG-flow family. We conjecture that all descendants of the top UV geometry engineer 5d SCFTs. When shrinkable, the top UV geometry itself also engineers a 5d SCFT.\n\nFor rank 1 geometries, we have only one RG-flow family corresponding to a local elliptic 3-fold defined by the del Pezzo surface $\\text{dP}_9$. All other rank 1 geometries are obtained by blowing down exceptional curves. The RG-flow family of $\\text{\\text{dP}}_9$ involves other del Pezzo surfaces $\\text{dP}_n$ with $n\\le 8$ and a Hirzebruch surface $\\mathbb{F}_0$; it is believed that these are the complete set of geometries leading to rank 1 5d SCFTs. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|}\n\t\\hline\n\t $S=S_1\\cup S_2$ & $G$ \\\\\n\t\\hline\n\n\n\t$(\\mathbb{F}_6\\cup \\text{dP}_4)^*$ & $Sp(2)_{\\theta=0} + 3\\textbf{AS}$ \\\\\n\t\\hline\n\t$(\\mathbb{F}_2\\cup \\text{dP}_7)^*$ & $SU(3)_4 + 6\\textbf{F}$ \\\\\n\t & $Sp(2)+ 4 \\textbf{F} + 2\\textbf{AS}$ \\\\\n\t & $G_2 + 6 \\textbf{F}$ \\\\\n\t\\hline\n\t$(\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0)^*$ & $SU(3)_{\\frac{3}{2}} + 9 \\textbf{F}$ \\\\\n\t & $Sp(2)+8\\textbf{F} + \\textbf{AS}$ \\\\\n\t\\hline\n\t$(\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0)^*$ & $SU(3)_0+10 \\textbf{F}$ \\\\\n\t & $Sp(2)+ 10\\textbf{F}$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Rank 2 geometries with maximal $M$. In the above table, $S$ is the rank 2 K\\\"ahler surface, while $G$ is the corresponding gauge theory description. These geometries denoted as $(\\cdot)^*$ are not shrinkable and correspond to 5d KK theories.}\n\t\\label{tb:rank2-classification}\n\\end{table}\nSimilarly, the top rank 2 geometries are summarized in Table \\ref{tb:rank2-classification}. We have identified four geometric RG-flow families represented by these top geometries. These geometries are not shrinkable; rather, we expect that these geometries have 6d UV completions and thus they engineer 5d KK theories. However, their descendants, obtained by blowing down $-1$ curves, are shrinkable and therefore give rise to 5d SCFTs.\nFor example, the geometry $\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0$ is ruled out from our CFT classification because its building block $\\text{Bl}_9\\mathbb{F}_4$ has an infinite number of Mori cone generators as explained in Appendix~\\ref{sec:mori}, violating our criterion in Section \\ref{sec:buildingblocks}. However, a geometric RG-flow from this geometry by blowing down an exceptional curve as well as a number of flop transitions leads to the geometry $\\text{Bl}_8\\mathbb{F}_3\\cup \\text{dP}_1$ which is now shrinkable and engineers a 5d SCFT. Similarly, other geometries in Table \\ref{tb:rank2-classification} are associated to KK theories, but their descendants are shrinkable. Therefore, we find that all rank 1 and 2 smooth 3-fold geometries engineering 5d SCFTs are mass deformations of 5d KK theories. See Section \\ref{sec:rank2} for further discussion.\n\nThis result confirms the existence of many new rank 2 SCFTs predicted in \\cite{Jefferson:2017ahm} which are listed in Table \\ref{tb:rank2-gauge-theory-clssification}.\nFor example, the $SU(3)_7$ gauge theory is predicted to exist in Table \\ref{tb:SU3-classification}. This theory turns out to have a geometric realization as $\\mathbb{F}_0\\cup \\mathbb{F}_8$ which is a descendant of $\\mathbb{F}_2\\cup \\text{dP}_7$. This implies that the gauge theory approach in \\cite{Jefferson:2017ahm}, which analyzes the magnetic monopole and perturbative BPS spectrum, is quite powerful and capable of predicting new interacting 5d SCFTs.\n\nOur study also reveals that there are no smooth 3-fold geometries associated to the following gauge theories:\n\\begin{align}\n\\begin{split}\n\\label{eqn:ruleout}\n\t&SU(3)_{\\frac{1}{2}} + 1\\bf{Sym} \\ , \\\\\n\t& SU(3)_{7} +2 {\\bf F} \\ \\rightarrow \\ SU(3)_{\\frac{15}{2}}+1{\\bf F} \\ \\rightarrow \\ SU(3)_8 \\ .\n\\end{split}\n\\end{align}\nThese theories are expected to have interacting CFT fixed points by the perturbative gauge theory analysis in \\cite{Jefferson:2017ahm}. See Table \\ref{tb:SU3-classification}.\nThe SCFT of the first gauge theory indeed exists---this theory is a mass deformation of the $SU(3)_0$ theory with $N_{\\bf Sym}=1,N_{\\bf F}=1$ whose brane construction is given in \\cite{Bergman:2015dpa,Hayashi:2015vhy}. Our study of smooth 3-folds fails to capture this theory. The reason for this failure is because the corresponding geometry involves a `frozen' singularity. For example, the brane construction in \\cite{Bergman:2015dpa,Hayashi:2015vhy} contains O7$^+$-planes; indeed, constructions involving O7$^+$ planes are dual to frozen singularities involving non-geometric monodromies and a fractional M-theory 3-form background as discussed in \\cite{Tachikawa:2015wka}. Therefore, we do not expect that our analysis can capture this type of singularity, and hence the geometric classification in this paper is incomplete in this sense.\nWe nevertheless conjecture that our classification includes all 5d SCFTs coming from {\\it smooth} Calabi-Yau threefolds which do not involve frozen singularities dual to brane constructions involving O7$^+$ planes.\nIn the following sections, we classify smooth rank 1 and rank 2 3-fold geometries engineering 5d SCFTs in their singular limits.\n\nOn the other hand, we predict that there are no SCFTs corresponding to three gauge theories belonging to the RG flow in the second line of (\\ref{eqn:ruleout}). As we discuss in Section \\ref{sec:rank2}, despite the fact that these gauge theories can be realized geometrically using our algorithm, they are shrinkable only when we attach a number of non-degenerate non-compact 4-cycles to the compact surface $S$. Introducing these non-compact 4-cycles entails non-normalizable K\\\"ahler deformations which in the field theory setting corresponds to introducing nonzero mass parameters. We find that these mass parameters cannot be set to zero in the CFT limit---at small nonzero values, the corresponding geometries develop at least one 2-cycle with negative volume and therefore their singular limits do not engineer well-defined CFT fixed points. \nThis computation excludes the three gauge theories in the second line of (\\ref{eqn:ruleout}) as possible candidates for interacting 5d SCFTs. This is also an indication that the classification criteria described in \\cite{Jefferson:2017ahm} are necessary, but not sufficient to identify 5d SCFT fixed points. The criteria of \\cite{Jefferson:2017ahm} must be modified to account for non-perturbative BPS states (such as instantons in gauge theories) in order to be both necessary and sufficient. \n\nWe also remark that a single 3-fold $X$ can admit multiple gauge theory descriptions.\nThis is possible because some geometries admit more than one distinct choice of fiber class associated to charged gauge bosons. The existence of multiple gauge theoretic descriptions corresponding to a single geometry suggests that the gauge descriptions are dual to one another. Starting with the the ``top'' UV geometries in Table \\ref{tb:rank2-classification}, we predict the following dualities:\n\\begin{align}\n\\begin{split}\n\\label{eqn:dual}\n\t SU(3)_{5-\\frac{N_{\\bf F}}{2}} + N_{\\bf F} {\\bf F} ~&\\cong ~Sp(2)+N_{\\bf F}{\\bf F} \\ , \\quad N_{\\bf F} \\le 10 \\\\\n\tSU(3)_{6-\\frac{N_{\\bf F}}{2}}+ N_{\\bf F}{\\bf F} ~&\\cong ~Sp(2)+1{\\bf AS}+(N_{\\bf F}-1){\\bf F} \\ , \\quad 1 \\le N_{\\bf F} \\le 9 \\\\\n\t SU(3)_{7-\\frac{N_{\\bf F}}{2}}+N_{\\bf F}{\\bf F} ~&\\cong ~ G_2 + N_{\\bf F}{\\bf F} ~ \\overset{2\\le N_{\\bf F}}{\\cong}~ Sp(2)+ 2{\\bf AS}+(N_{\\bf F}-2){\\bf F} \\ , \\quad N_{\\bf F} \\le 6\n\\end{split}\n\\end{align}\nThe first and the second dualities in (\\ref{eqn:dual}) were conjectured already in \\cite{Gaiotto:2015una} and in \\cite{Jefferson:2017ahm}, respectively. So our construction provides concrete geometric evidence for these duality conjectures. On the other hand, the third duality is a new duality discovered by an explicit geometric construction in this section.\n\n\\subsection{Rank 1 classification}\n\\label{sec:rank1}\n\nWe warm up by starting with rank 1, recovering the result that all rank 1 5d SCFTs are geometrically engineered by local 3-folds containing a del Pezzo surface. More precisely, our algorithm identifies del Pezzo surfaces as shrinkable, but also identifies additional shrinkable surfaces; however, each of these turns out to be physically equivalent to a del Pezzo surface.\n\n\nRecall that a del Pezzo surface $S$ is defined to be a smooth algebraic surface whose anticanonical bundle $-K_S$ is ample---this means that $-K_S \\cdot C > 0$ for all effective curves $ C \\subset S$. The classification of del Pezzo surfaces is well known: $S$ is either $\\text{dP}_n$ for $0\\le n\\le 8$ or ${\\mathbb P}^1\\times{\\mathbb P}^1={\\mathbb F}_0$. Such a surface satisfies (\\ref{eq:shrinkability}) as well as $K_S^2>0$, so is shrinkable. We now set out to systematically classify rank 1 shrinkable surfaces up to physical equivalence.\n\n\n\n\nTo apply (\\ref{eq:shrinkability}), we need to know $K_S$, the generators of the Mori cone of curves on $S$, and the intersection numbers of the curves in $S$. Our algorithm leads us to consider ${\\mathbb P}^2$, ${\\mathbb F}_n$, and their generic blowups. \n\n${\\mathbb P}^2$ is del Pezzo, but it is instructive to check shrinkability anyway. For ${\\mathbb P}^2$, the Mori cone is generated by the class $\\ell$ of a line, $\\ell^2=1$, and \n$K_{{\\mathbb P}^2}=-3\\ell$. So $K_{{\\mathbb P}^2}^2=9>0$ and $K_{{\\mathbb P}^2}\\ell=-3<0$, so ${\\mathbb P}^2$ is shrinkable.\n\nNext, we consider ${\\mathbb F}_0$,\\ ${\\mathbb F}_1$ and ${\\mathbb F}_{n \\geq 2}$ separately. Since ${\\mathbb F}_1$ is the blowup of ${\\mathbb P}^2$ at a point, ${\\mathbb F}_1$ and its generic blowups are just the generic blowups of ${\\mathbb P}^2$. Similarly, ${\\mathbb F}_0$ is del Pezzo, and the blowup of ${\\mathbb F}_0$ at a point is isomorphic to the blowup of ${\\mathbb P}^2$ at two points \\cite{GH}. So the possibilities for $S$ can be reduced to either generic blowups of ${\\mathbb P}^2$, or ${\\mathbb F}_{n \\geq 2}$.\n\nAs usual, we denote by $\\text{dP}_n$ the blowup of ${\\mathbb P}^2$ at general points\n$p_1,\\ldots,p_n$. Let $X_1,\\ldots, X_n$ denote the corresponding\nexceptional ${\\mathbb P}^1$'s,\\footnote{As noted earlier, we reserve the more customary notation\n $E$ for the curves on Hirzebruch surfaces described in Appendix~\\ref{app:Mori}.} and we let $\\ell$ denote\nthe class of the total transform in $\\text{dP}_n$ of a line in ${\\mathbb P}^2$. The intersection numbers are\n\\begin{equation}\n \\label{eq:intp2}\n \\ell^2=1,~~\\ X_i\\cdot X_j = -\\delta_{ij},~~\\ \\ell\\cdot X_i=0 \n\\end{equation}\nand $K_{\\text{dP}_n}=-3\\ell+\\sum_{i=1}^nX_i$. Then $K_{\\text{dP}_n}^2=9-n>0$ for $n\\le 8$.\n\nWe first observe that $\\text{dP}_n$ is not shrinkable for $n\\ge9$. To see this, we simply observe that $K_{\\text{dP}_n}^2\\le0$ for $n\\ge9$ which implies that the string tensions are not positive.\n\nAgain, we can cite known results simply say that $\\text{dP}_n$ is shrinkable for $n\\le8$, but it is instructive to work out details without assuming this fact.\nWe adopt a convenient shorthand to describe the generators of the Mori cone: Any curve\n$C\\subset \\text{dP}_n$ other than the $X_i$ will project to a curve $D\\subset {\\mathbb P}^2$ of some degree $d>0$.\nLet $m_i$ be the multiplicity of $D$ at $p_i$, so that $m_i=0$ if $p_i\\not\\in D$, $m_i=1$\nif $p$ is a nonsingular point of $D$, $m_i=2$ if $p$ is a node or cusp of $D$, etc. Then \nthe class of $C$ is $d\\ell-\\sum_{i=1}^n a_i X_i$. It is customary to abbreviate this class as\n$(d;m_1,\\ldots,m_n)$, as well as to omit any $m_i$ which are zero. Then the\nMori cone of $\\text{dP}_n$ is generated by the classes\\footnote{Strictly speaking, we have only written the Mori generators for $n=8$.\nFor $n<8$, we modify (\\ref{eq:moridp}) by removing those generators which need more than $n$ exceptional divisors to define them. In addition, for\n$n=1$, we include $(1;1)$ as a generator.}\n\\begin{equation}\nX_i,\\ (1;1^2),\\ (2,1^5),\\ (3,2,1^6),\\ (4,2^3,1^5),\\ (5,2^6,1^2),\\ (6;3,2^7)\n \\label{eq:moridp}\n\\end{equation}\nup to permuting the order of the $p_i$. It follows from the adjunction formula (\\ref{eq:adjunction}) that each of the curve\nclasses $C$ in (\\ref{eq:moridp}) satisfies $K_{\\text{dP}_n}\\cdot C=-1$,\\footnote{ For $n=1$, we also check that $K_{\\text{dP}_1}\\cdot(\\ell-X_1)=-2$.} so $\\text{dP}_n$ is shrinkable. \n\n\nNext, consider the Hirzebruch surfaces $S={\\mathbb F}_n$. \nUsing the notation in Appendix~\\ref{app:Mori}, there are two disjoint toric sections $E,H$\nand the fiber class $F$. These classes satisfy\n\\begin{equation}\n \\label{eq:intfn}\n H^2=n,\\ E^2=-n,\\ H \\cdot E=0,\\ H \\cdot F=E\\cdot F=1,\\ F^2=0,\\ H=E+nF.\n\\end{equation}\nThe canonical bundle of ${\\mathbb F}_n$ is $K_{{\\mathbb F}_n}=-2H+(n-2)F$ and so\n $K_{{\\mathbb F}_n}^2=8>0$. Furthermore, \n the Mori cone of effective curves is generated by $E$ and $F$. While $K_{{\\mathbb F}_n} \\cdot F=-2<0$,\n we also have $K_{{{\\mathbb F}}_n}\\cdot E=n-2$, which is strictly negative for $n<2$, zero for $n=2$, but\n strictly positive for $n>2$. Thus ${\\mathbb F}_2$ is shrinkable. However, as discussed in section 3, this is physically equivalent to ${\\mathbb F}_0$.\n The same reasoning combined with the earlier observation that $\\text{Bl}_1{\\mathbb F}_0\\simeq \\text{dP}_2$ shows that $\\text{Bl}_p \\mathbb F_2$ is physically equivalent to $\\text{dP}_{p+1}$.\n\n In conclusion, all rank 1 shrinkable surfaces are physically equivalent to $\\text{dP}_n$ for some $n$ or ${\\mathbb F}_0$.\n\n\\subsection{Rank 2 classification}\n\\label{sec:rank2}\n\nThe main result of this paper is a full classification of shrinkable rank 2 geometries up to physical equivalence. We preface our result by arguing some further simplifying assumptions we make about the surface $S$ in order to make the classification into a manageable problem.\n\n\\subsubsection*{Three simplifications}\nIn this section we show that we can utilize the following three simplifying assumptions for classifying shrinkable rank 2 surfaces:\n\n\\begin{itemize}\n\\item $S_1 \\cap S_2$ is an irreducible curve.\n\\item $S_1 \\cap S_2$ is a rational curve. \n\\item The surfaces $S_i$ are equal to ${\\mathbb P}^2$ or Hirzebruch surfaces and their blowups at general points.\n\\end{itemize}\nWe now discuss these three simplifications in order.\n\nFirst, we argue that in the case of a rank 2 surface $S = S_1 \\cup S_2$, we can assume that $S_1$ is not glued to $S_2$ along multiple curves. Namely, there exists a single edge between two nodes.\nSuppose we glue two surfaces along $C_1,C_2$ with appropriate identifications. Since $S_1$ and $S_2$ should intersect transversally, we have $(C_1 \\cdot C_2)_{S_1} = (C_1 \\cdot C_2)_{S_2}= 0$. This means that $C_1, C_2$ do not intersect. \nWe claim there always exists an effective curve $D=d_1+d_2$ such that ${\\rm vol}(D) \\le 0$. If ${\\rm vol}(D)<0$, then $S$ is not shrinkable, so it suffices to consider the situation where ${\\rm vol}(D)=0$. But in that case, we will further show below that we can arrange for the curve $D$ to be elliptic (i.e. $g(D) = 1$), which would contradict our conjectures. Therefore, the full surface is not shrinkable implying that we cannot glue two surfaces along two or more curves.\n\nIn order to show this, we first prove that there always exist curves $d_i \\subset S_i$ with $K_{S_i}\\cdot d_i\\ge-2$ that intersect both $C_1$ and $C_2$. These classes $d_1$ and $d_2$ are identified as follows. \nFirst, if both $C_1$ and $C_2$ are not fiber classes, we can always find a curve $d_1$ satisfying these conditions among $\\{F, \\, F-X_i,\\, H-X_i-X_j \\}$\\footnote{For general $n$ we choose $d_1=F-X_i$ if $C_1=X_i$ or $C_2=X_i$, otherwise $d_1=F$. When $n=2$ and $C_1=X_1,C_2=X_2$, we choose $d_1=H-X_1-X_2$.} in $\\text{Bl}_p\\mathbb{F}_n$, where $X_i$ are exceptional curves associated to the blowups of $\\mathbb F_n$ at $p$ general points. When $n>2$, $C_1 =E$, otherwise the volume of the curve $E$ will be negative.\nNext, suppose $C_1$ or $C_2$ is a fiber class. This is possible only when $S_1=\\text{Bl}_p\\mathbb{F}_1$ or $\\text{dP}_n$, otherwise the class $E$, which has $E\\cdot C_1\\neq0$ or $E\\cdot C_2 \\neq 0$, will have negative volume thus preventing the surface $S$ from being shrinkable. In the case that $S_1=\\text{Bl}_p\\mathbb{F}_1$, when $C_1$ is a fiber class $F_1$, $C_2$ must be one of $X_i$'s, due to the assumption of transversal intersection. Then we can take $d_1=H-X_i$ with $H^2=1$.\nWith any choice of $d_1$ given here, we find that ${\\rm vol}(d_1)=m\\phi_1 - n\\phi_2$ with $m=1,2$ and $n\\ge2$ where $\\phi_i \\geq 0$.\nWe can choose $d_2 \\subset S_2$ in the same manner and then show that ${\\rm vol}(d_2)=m'\\phi_1 - n'\\phi_2$ with $m'=1,2$ and $n'\\ge2$.\n\nThis proves ${\\rm vol}(D) \\le 0$ for an effective curve $D=d_1+d_2$. Now we will assume ${\\rm vol}(C_i)\\ge 0$ for all other curves $C_i$ because otherise the surface is not shrinkable and already ruled out. As already noted above, it is clear that the total surface is not shrinkable when ${\\rm vol}(D)<0$. Moreover, when ${\\rm vol}(D) = 0$, i.e. when $m=m'=n=n'=0$, the curves $d_1$ and $d_2$ are both fiber classes $F_i\\subset S_i$. In this case, the curve $F_1$ and $F_2$ can be deformed so that $F_1\\cap C_i=F_2\\cap C_i$ for $i=1,2$. Then the curve $D=F_1+F_2$ is the union of two rational curves intersecting in two points, hence elliptic. By further complex structure deformation if necessary, we can arrange that all fibers $F_1$ of $S_1$ meet all fibers $F_2$ of $S_2$ in two points, or in other words, that $S=S_1\\cup S_2$ is elliptically fibered.\n\nWe argue that we can deform the complex structure of $X$ if necessary so that $X$ is also elliptically fibered. To see this, let $E$ be an elliptic fiber of $S$. Since $E$ is part of an elliptic fibration of $S$, we have that $N_{E\/S}\\simeq\\mathcal{O}_E$. Furthermore, $\\det(N_{E\/X})$ is trivial by the Calabi-Yau condition and the ellipticity of $E$. Then the normal bundle sequence\n\\begin{equation}\n \\label{eq:nbs}\n0 \\to N_{E\/S} \\to N_{E\/X} \\to N_{S\/X}|_E \\to 0 \n\\end{equation}\nis identified with\n\\begin{equation}\n \\label{eq:Atiyah}\n 0 \\to \\mathcal{O}_E \\to N_{E\/X} \\to \\mathcal{O}_E \\to 0.\n\\end{equation}\nHowever, since $H^1(\\mathcal{O}_E) \\ne 0$, (\\ref{eq:Atiyah}) generically does not split\\footnote{The non-splitting of (\\ref{eq:Atiyah}) identifies $N_{E\/X}$ as the Atiyah bundle on $E$.} and dim $H^0(N_{E\/X})=1$. The uniqueness of a normal direction says that $E$ moves in a 1-parameter family, enough deformations to fiber $S$ but not enough to fiber $X$.\n\nHowever, we can choose a complex structure deformation of $X$ so that (\\ref{eq:Atiyah}) splits, and then $N_{E\/X}\\simeq\\mathcal{O}_E^2$. In this situation, $E$ moves in two independent directions and fibers $X$.\n\n This justifies our claim, hence $S$ is not shrinkable. The same argument holds for cases with more than two edges (i.e. gluing curves) between $S_1$ and $S_2$. Therefore rank 2 geometries formed by two surfaces glued along two or more different curves are not shrinkable.\n\nSecond, we claim that the gluing curves must be rational. Suppose $C = S_1 \\cap S_2$ has $g>0$. In Appendix~\\ref{app:Mori} we explain that we must have finitely many Mori cone generators in each $S_i$ (which implies a bound on the number of blowups), hence we have finitely many Mori cone generators in $X \\supset S = S_1 \\cup S_2$. We argue that this implies $C_{S_i}^2 \\ge 0$ as follows. \nWe assume $C_{S_i}^2<0$ and derive a contradiction. Since $C_{S_i}^2+C \\cdot K_{S_i}=2g-2\\ge0$, we have $C\\cdot K_{S_i}>0$. \nAnticipating the next bulleted claim that the building blocks are generic blowups of Hirzebruch surfaces at a bounded number of points, we show in Appendix~\\ref{app:Mori} that $C_{S_i} \\cdot K_{S_i}>0$ implies $C_{S_i}=E$. This is a contradiction, since $g>0$.\nAlthough this argument is slightly circular in its current form depending as it does on the next bulleted claim, we believe that with further care we can independently justify $C_{S_i}^2\\ge0$. Furthermore, an extensive computer search has revealed no counterexamples.\n\n\nLet us now return to the claim that the gluing curves are rational. Recalling equations (\\ref{eq:gluingcond}) and (\\ref{eq:adjunction}), we have\n\\begin{equation}\n\t C^2_{S_1} + C^2_{S_2} = C_{S_i}^2 + K_{S_i}\\cdot C = 2g-2 \\ .\n\\end{equation}\nThese conditions tell us that $K_{S_i} \\cdot C\\ge0$. This implies that the volume of the intersection curve, ${\\rm vol}(C)=-\\phi_1 K_{S_1}\\cdot C -\\phi_2 K_{S_2}\\cdot C$, is negative unless $C^2_{S_1}=C_{S_2}^2=0$ and $g=1$, i.e. unless $C$ is an elliptic curve. This proves that rank 2 geometries containing two surfaces meeting in a curve with genus $g>0$ are not shrinkable.\n\n\nThird, we observe that many of the building blocks in our classification program are related to one another by maps (for instance, isomorphisms and complex deformations) which at the level of 5d SCFT physics constitute physical equivalences. Therefore, we observe that the full number of rank 2 surfaces that can be constructed from our list of building blocks dramatically overcounts the number of unique CFT fixed points, and hence we can reduce the complexity of the problem at the outset by restricting our attention to a minimal representative set of configurations capturing the full list of physical equivalence classes. We will argue in particular that we need only consider configurations $S = S_1 \\cup S_2$ for which $S_1$ is a blowup of $\\mathbb F_{n> 0}$ at $p$ generic points\\footnote{By ``generic point'', we mean a point not contained in any exceptional divisors, i.e.\\ rational curves with self intersection $-1$.} and $S_2$ is $\\text{dP}_m$ or $\\mathbb F_0$. We summarize our simplifications by stating that {\\it every rank 2 shrinkable CY 3-fold can be realized locally as a neighborhood of} $S = S_1 \\cup S_2${\\it , for which \t}$S_1 = \\text{Bl}_{p} \\mathbb F_{n_1 > 0} $\\, {\\it and} $S_2 = \\text{dP}_{n_2}$ {\\it or }$\\mathbb F_0$. {\\it Moreover, the surfaces }$S_1, S_2$ {\\it are glued along a single smooth rational curve} $C =S_1 \\cap S_2$.\n\nWe argue the third simplification as follows. First, observe that all of the curves $C'$ with self intersection $C'{}^2 < -2$ which do not intersect the gluing curve $C$ have negative volume. Therefore, the only curves $C' \\neq C$ with negative self-intersection should have $C'{}^2 \\geq -2$. Suppose $C'{}^2 = -2$ and the surface $S$ is shrinkable. Then, it should follow that such a geometry is related via complex deformation to a physically-equivalent surface for which the only curves $C'$ of negative self-intersection have $C'{}^2 = -1$. The idea is essentially identical to the description of a transitions already described in Section~\\ref{sec:transitions}: we perform a conifold transition. Strictly speaking, this is only true up to physical equivalence, but that is good enough for us. Hence, we may assume that the only component surfaces $S_i$ appearing in our representative classes are those for which all curves $C' \\ne C$ satisfy $C'{}^2 \\geq -1$. This already places a significant constraint on the possible configurations $S_1 \\cup S_2$. \n\nNext, recall that our list of possible building blocks includes $\\mathbb P^2$ and $\\text{Bl}_p \\mathbb F_n$, where the configuration of $p$ points can be special or generic. The gluing condition (\\ref{eq:gluingcond}) implies that one of the two gluing curves $C_{S_1}$ or $C_{S_2}$ must have negative self-intersection. Therefore, we are forced to fix one of the two surfaces, say $S_1 = \\text{Bl}_p \\mathbb F_{n_1}$. Observe that any blowup of $\\mathbb F_n$ at $p$ points $F \\cap E$ is always isomorphic to the blowup of $\\mathbb F_{n+p}$ at $p$ generic points, so (redefining $n$) we can always assume that $S_1$ is a blowup of $\\mathbb F_{n_1}$ at $p$ points away from the curve $E$ with self intersection $E^2 = -n_1$. \n\nAssume that $n \\geq 2$ and suppose we take such a surface $S_1$ and glue it to $S_2$ along some curve $C_{S_1} \\ne E$. Then this violates the condition that all curves $C' \\ne C_1$ satisfy $C'{}^2 \\geq -1$, in particular for $C' = E$. Hence, we are forced to set $C_{S_1} = E$, and moreover we are confined to surfaces $S_1 = \\text{Bl}_p \\mathbb F_{n_1}$ for which the configuration of points $p$ is a generic configuration (a special configuration of points would produce curves with self-intersection less than $-1$). \n\nLet us focus on $S_2$. If $n_1 \\geq 2$, then $S_2$ must be glued to $S_1$ along a curve $C_{S_2}$ with non-negative self intersection, $C_{S_2}^2 \\geq 0$. Since we may again assume that all $C' \\ne C_{S_2}$ satisfy $C'{}^2 \\geq -1$, it follows that $S_2 = \\text{dP}_{n_2}$ or $S_2 = \\mathbb F_0$. Returning to the remaining cases $n _1< 2$, we find these cases consist of gluing configurations for which $S_i = \\text{dP}_{n_i}$ glued along curves $C_{S_i}$ with $C_{S_i}^2 = -1$. However, $\\text{dP}_n \\cong \\text{Bl}_{n-1} \\mathbb F_1$, and therefore in order to avoid overcounting we assume that our configuration is again of the form conjectured above.\n\nFinally, we turn our attention to the case where one of the component surfaces $S_i$ is a ruled surface over a curve of genus $g >0$. As explained in Section \\ref{sec:transitions}, a ruled surface over a curve with genus $g >0$ is physically equivalent to a blowup of $\\mathbb F_n$ at $2g$ generic points with $g$ self-gluings. Notice that when $S_1$ is the $\\text{Bl}_{2g}\\mathbb{F}_n$ with $g$ self-gluings, the gluing curve $C_{S_1}$ should be the section $E$ (with $E^2=-n$) since otherwise $E$ has negative volume or leads to an elliptic fiber class. This implies due to the shrinkability condition that the second surface $S_2$ is again ${\\rm dP}_m$ or $\\mathbb{F}_0$. The self-gluing curves must always be exceptional curves, and hence we perform a flop transition in which we blow these curves down at the expense of blowing up another curve inside the surface $S_2$. Provided we always perform enough blow downs to completely eliminate the self-glued curves, we can always exchange a configuration involving a self-glued blowup of $\\mathbb F_n$ with one of the configurations described in the above conjecture. This completes our argument concerning the representative configurations for rank 2 surfaces $S=S_1 \\cup S_2$.\n\n\\subsubsection*{Endpoint classification: 0 and 1 mass parameters}\n\\label{subsec:endpoint}\n\nIn this section we show that we can first classify geometries which are blown down `as much as possible'; we refer to these as `endpoint geometries'. The general\nclassification then follows by classifying endpoints and subsequently classifying their possible blowups.\n \nSuppose a SCFT admits mass deformations for its global symmetry. Then we can take a large mass limit and integrate out all the heavy degrees of freedom. This triggers an RG flow and it is expected that the SCFT below energy scales set by the masses flows to another SCFT with a lower rank global symmetry group commuting with the mass deformations of the UV SCFT. In general, such mass deformations can reduce the rank of the resulting theory. Another possibility is for the IR theory to be a trivial free theory. \n\nWe pay attention to a particular class of mass deformations which leads to interacting SCFTs while preserving the rank of the UV SCFT. Equivalently, we restrict our attention to mass deformations which do not change the dimension of the Coulomb branch. One can typically obtain a new interacting SCFT with the same rank by means of such `rank-preserving mass deformations'. We expect that RG flows of the UV SCFT triggered by such mass deformations can generate a family of SCFTs with the same rank but different global symmetries. SCFTs in the family are distinguished by their global symmetries (i.e. the number of mass parameters), as well as topological data such as the classical Chern-Simons level $k$ or ${\\mathbb Z}_2$-valued $\\theta$ angle.\n\nThese types of RG flows terminate in a class of interacting SCFTs which we will call `endpoint SCFTs'. An endpoint SCFT is defined to be a theory which does not admit any rank-preserving mass deformations. Thus these theories are `endpoints' of RG flows and they cannot flow to other SCFTs via rank-preserving deformations. Endpoint geometries engineer endpoint SCFTs.\n\nRank-preserving mass deformations and endpoint geometries are mathematically well-defined notions. We define distinct endpoint geometries to be surfaces which cannot be related to another smooth surface of the same rank via a large mass deformation. Rank-preserving mass deformations are defined as follows: suppose $S$ is shrinkable and $C\\subset S_j$ is a $-1$ curve which does not intersect any $S_k$ for $k\\ne j$. Then $S$ can be blown down to a surface $S'=\\cup S'_i$ with $S'_j$ the blowdown of the $-1$ curve of $S_j$ and $S'_k\\simeq S_k$ for $k\\ne j$. This type of blowdown is the geometric realization of a rank-preserving mass deformation.\n\n\nWe will now show that {\\it if }$S$ {\\it is shrinkable, then its endpoint geometry }$S'$ {\\it is also shrinkable.}\n If $C'\\subset S'_i$, let $C\\subset S_i$ be its proper transform. We have $K_{S'_i}^2=K_{S_i}^2+1$. If $i\\ne j$\nwe have $K_{S_j}\\cdot C=K_{S'_j} \\cdot C'$, so we need only consider the case $i=j$. \nLet $p\\in S'_j$ be the point that the $-1$ curve in $S_j$ blows down to, and suppose that $C'$ has multiplicity $m$ at $p$. Then\n$K_{S'_i}\\cdot C'=K_{S_i}\\cdot C-m$. The desired conclusion follows immediately.\n\nEndpoint SCFTs are interesting due to the following reasons. First, these theories are the simplest theories in their family of RG flows. Their parameter spaces are smaller, so they are comparatively easier to understand than other theories belonging to the same family.\nThe classification of endpoint SCFTs is therefore a much easier problem than the full classification, as we will see below.\nWe can thus regard the endpoint classification as a tutorial on our classification algorithm.\nSecond, all other SCFTs in the family of RG flows in principle can be obtained from endpoint theories by increasing the number of mass parameters. Namely, we can undo mass deformations, and retrace the RG flow to obtain an entire family of UV SCFTs. This could sound puzzling: we know that RG flow is irreversible. So it may be hard to accept the idea that we can restore UV theories starting from an IR theory. However, this turns out to be the case among 5d supersymmetric theories.\nSince 5d $\\mathcal{N}=1$ SCFTs are so strongly constrained by supersymmetry, one can control their RG flows by tuning discrete data such as (for theories with gauge theory descriptions) gauge algebra, matter representations, classical CS level, and discrete $\\theta$ angle. We expect that this allows us to build a family of SCFTs starting from an endpoint theory.\n\n\nFrom the geometric standpoint, these constraints can be understood as arising from the Calabi-Yau condition. Mass deformations of a 3-fold correspond to blowups or blowdowns of exceptional curves. As discussed above, a large mass deformation corresponds to blowing down a $-1$ curve which is isolated from gluing curves and is in fact a reversible geometric transition---one can just as easily blow up the same curve to recover the original 3-fold. This means that by starting from an endpoint geometry, it is possible to obtain a family of local (smooth) 3-folds by blowing up all possible exceptional curves. In this sense, the study of endpoint geometries is a good starting point for the classification of 5d SCFTs.\n\nLet us now classify all rank 2 endpoint geometries by employing our classification algorithm. We learned above that rank 2 geometries are constructed by gluing $S_1=\\text{Bl}_p\\mathbb{F}_{m_1}$ and $S_2={\\rm dP}_{m_2}$ or $\\mathbb{F}_0$. This implies that endpoint geometries with $M=0,1$ will take the form $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$ or $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$. For being an endpoint geometry with $M>1$, there must be no irreducible exceptional curve which does not intersect with the gluing curves and no flop transitions introducing such exceptional curve away from the gluing curves. This is possible only for ${\\rm dP}_2 \\cup {\\rm dP}_2$ with $C_1 = \\ell \\!-\\!X_1 \\!-\\!X_2$ and $C_2=\\ell \\!-\\!X_1\\!-\\!X_2$ which is shrinkable. We thus find that ${\\rm dP}_2 \\cup {\\rm dP}_2$ is the only endpoint geometry with $M>1$ \\footnote{We thank Sung-Soo Kim for pointing out that this geometry has no rank-preserving mass deformation}. Therefore the endpoint classification reduces to a simple classification of two types of geometries, $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$ for $M=0$ and $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ for $M=1$, other than ${\\rm dP}_2 \\cup {\\rm dP}_2$ with $M=3$.\n\nWe first classify geometries of the type $\\mathbb{P}^2 \\cup \\mathbb{F}_{n}$. We can choose a curve class $C_{S_1}=C_1=a \\ell$ in $\\mathbb{P}^2$ with a positive integer $a$ and $C_{S_2}=C_2=E$ in $\\mathbb{F}_n$ satisfying the gluing condition (\\ref{eq:gluingcond}). Since $C$ should be rational, the integer in $C_1$ is fixed to be either $a=1$ or $a=2$. Accordingly, the second surface is fixed to be $\\mathbb{F}_3$ or $\\mathbb{F}_6$ respectively. Hence we find only two geometries of this type:\n\\begin{align}\n\\begin{split}\n\t& \\mathbb{P}^2 \\cup \\mathbb{F}_3 \\quad {\\rm with} \\quad C_1 = \\ell \\ , \\ C_2 = E_3 \\ , \\\\\n\t &\\mathbb{P}^2 \\cup \\mathbb{F}_6 \\quad {\\rm with} \\quad C_1 = 2\\ell \\ , \\ C_2 = E_6 \\ .\n\\end{split}\n\\end{align}\nThese two geometries have brane constructions as depicted in Fig \\ref{fig:rank2-branes1}. These geometries have no mass parameter. Therefore we do not expect any gauge theory descriptions associated to these CFTs.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.4]{rank2-branes-m0.pdf}\n\t\\end{center}\n\t\\caption{Brane configurations of rank 2 SCFTs with zero mass.}\n\t\\label{fig:rank2-branes1}\n\\end{figure}\n\nThe second type of endpoint geometry can be classified in the same manner. Due to the gluing condition (\\ref{eq:gluingcond}), a gluing curve in one of two Hirzebruch surfaces should have negative self-intersection. We choose $C_2=E_2$ in the second surface $\\mathbb{F}_{n_2}$. Then the gluing curve $C_1$ in the first surface $\\mathbb{F}_{n_1}$ needs to be a rational irreducible curve with self-intersection $n_2-2$. The curve $C_1$ takes the form of $C_1 = aF_1+bH_1$ with $a,b\\ge0$ or $C_1=E_1$, and must satisfy\n\\begin{equation}\n\tC_1^2 = n_2-2 \\ , \\quad C_1 \\cdot S_1 = -n_2 \\ .\n\\end{equation}\nWe now need to check shrinkability conditions. In both irreducible components $S_i = \\mathbb F_{n_i}$, the curve classes generating Mori cone are $E_i, F_i$. When these curve classes have non-negative volumes with respect to the K\\\"ahler class $-J=-\\phi_1 S_1-\\phi_2 S_2$, the local 3-fold defined by $S$ is shrinkable and thus engineers a 5d SCFT. In this case, the criteria for shrinkability are\n\\begin{eqnarray}\n\t&&{\\rm vol}(E_1) = (2-n_1)\\phi_1-a \\phi_2 \\ge 0 \\ , \\quad {\\rm vol}(F_1) = 2\\phi_1-b\\phi_2 \\ge 0 \\ , \\nonumber \\\\\n\t&&{\\rm vol}(E_2) = (2a+2b-bn)\\phi_1+(2-n)\\phi_2 \\ge 0 \\ , \\quad {\\rm vol}(F_2) = -\\phi_1+2\\phi_2 \\ge 0 \\ ,\n\\end{eqnarray}\nwith $\\phi_1,\\phi_2>0$.\nWe can easily solve these conditions and the gluing condition (\\ref{eq:gluingcond}). Each solution will give a shrinkable geometry and thus a SCFT. The full list of shrinkable surfaces $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ (denoted by $(n_1,n_2)$) is given in Tables \\ref{tb:endpoint-F-F} and \\ref{tb:shirinkable-F-F}. Some of these geometries have brane constructions given in Figure \\ref{fig:rank2-branes2}. We find that only the six geometries in Table \\ref{tb:endpoint-F-F} are independent endpoint geometries.\n\\begin{table}\n\\centering\n\\begin{subtable}{.8\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\t\\hline\n\t $S_1\\cup S_2$ & $C_{S_1}$ & $C_{S_2}$\\\\\n\t\\hline\n\t$\\mathbb{P}^2\\cup \\mathbb{F}_3$ & $\\ell$ & $E$ \\\\\n\t\\hline\n\t$\\mathbb{P}^2 \\cup \\mathbb{F}_6$ & $2\\ell$ & $E$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Endpoint geometries with $M=0$.}\n\t\\label{tb:endpoint-P-F}\n\\end{subtable}%\n\\vspace{.5cm}\n\\begin{subtable}{.9\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c||c|c|c|}\n\t\\hline\n\t $(n_1,n_2)$ & $C_{S_1}$ & $G$ & $(n_1,n_2)$ & $C_{S_1}$ & $G$\\\\\n\t\\hline\n\t$(0,2)$ & $F$ & $SU(3)_1$ & $(0,8)$ & $F+3H$ & $SU(3)_7,G_2$ \\\\\n\t\\hline\n\t$(0,4)$ & $F+H$ & $SU(3)_3$ & $(1,1)$& $E$ & $SU(3)_0$ \\\\\n\t\\hline\n\t$(0,6)$ & $F+2H$ & $SU(3)_5,Sp(2)_{\\pi}$ & $(1,7)$& $2F+H$ & $SU(3)_6$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Endpoint geometries with $M=1$. Here $C_{S_2}=E$. These geometries have gauge theory descriptions with gauge group $G=SU(3)_k,Sp(2)_\\theta,G_2$ where $k$ is the classical CS level and $\\theta$ is the $\\mathbb Z_2$-valued $\\theta$ angle.}\n\t\\label{tb:endpoint-F-F}\n\\end{subtable}%\n\n\\vspace{.5cm}\n\\begin{subtable}{.8\\textwidth}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\t\\hline\n\t$(n_1,n_2)$ & $C_{S_1}$ & $G$ & Endpoint \\\\\n\t\\hline\n\t$(1,2)$ & $F$ & $SU(2)\\hat{\\times}SU(2)$ & $\\mathbb{P}^2\\cup \\mathbb{F}_3$ \\\\\n\t\\hline\n\t$(1,3)$& $H$ & $SU(3)_2$ & $\\mathbb{P}^2\\cup \\mathbb{F}_3$\\\\\n\t\\hline\n\t$(1,5)$& $F+H$ & $SU(3)_4$ & $\\mathbb{P}^2\\cup \\mathbb{F}_6$\\\\\n\t\\hline\n\t $(1,6)$ & $2H$ & $Sp(2)_{0}$ & $\\mathbb{P}^2\\cup \\mathbb{F}_6$\\\\\n\t \\hline\n\t \\hline\n\t$(2,4)$ & $H$ & $SU(3)_1$ & $\\cdot$\\\\\n\t\\hline\n\t$(0,10)$ & $F+4H$ & $SU(3)_9$ & $\\cdot$ \\\\\n\t\\hline\n\\end{tabular}\n\t\\caption{Other geometries of $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$. The first four are not endpoints and flow to geometries in (a) by mass deformations. $(2,4)$ is an endpoint, but is also equivalent to $(0,4)$ by a HW transition. $(0,10)$ is an endpoint, but not shrinkable.}\n\t\\label{tb:shirinkable-F-F}\n\\end{subtable}\n\n\\caption{Classification of all rank 2 geometries with $M=0,1$.}\\label{tb:rank2-F-F-clssification}\n\\end{table}\n\n\n\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=.35]{rank2-branes.pdf}\n\t\\end{center}\n\t\\caption{Brane configurations of rank 2 SCFTs with $M=1$.}\n\t\\label{fig:rank2-branes2}\n\\end{figure}\n\nIn fact, all the endpoint geometries in Table \\ref{tb:endpoint-F-F} have gauge theory descriptions with simple gauge group $G$. As explained in Section \\ref{sec:Mth}, a distinguished property of geometries corresponding to gauge theories is that the matrix of intersection numbers (\\ref{eqn:Cartan}) of holomorphic fiber classes $f_i$ with the surfaces $S_i$ is equal to (minus) the Cartan matrix of the gauge algebra. We remark here that the Hirzebruch surface $\\mathbb{F}_0$ has a base-fiber duality exchanging the base curve class $H$ and the fiber curve class $F$. Geometrically, this is an isomorphism between two geometries related by the exchange of $H$ and $F$. It is possible that the dual geometry often has different gauge theory realization from the gauge theory of the original geometry. In this case, the geometric duality leads to a duality between two different gauge theories. \n\nAside from studying the Cartan matrices, we can also compare the triple intersection polynomial $J^3$ to the perturbative expression for the prepotential given in (\\ref{eqn:pre}). For the geometries in Table \\ref{tb:endpoint-F-F} and \\ref{tb:shirinkable-F-F}, the prepotentials are \n\\begin{equation}\n\t6\\mathcal{F} = J^3 = 8\\phi_1^3 + 3\\phi_1\\phi_2(-n_2\\phi_1+(n_2-1)\\phi_2) + 8 \\phi_2^3\\ .\n\\end{equation}\nWe can compare these prepotentials against known gauge theory prepotentials as a means to identify the corresponding gauge theories.\n\nLet us first select the respective fibers $H,F$ for $\\mathbb{F}_{0}\\cup \\mathbb{F}_{n_2}$, and $F,F$ for $\\mathbb{F}_{1}\\cup \\mathbb{F}_{n_2}$. The Cartan matrix $A_{ij}$ of the following geometries computed using these fiber classes is that of the gauge algebra $SU(3)$ as\n\\begin{equation}\n\t(A_{SU(3)})_{ij} ~:~(n_1,n_2) ~=~ (0,2) \\, , \\ (0,4) \\,, \\ (0,6) \\,, \\ (0,8) \\,, \\ (1,1) \\,, \\ (1,7) \\ ,\n\\end{equation}\nfor the choices of degrees $(n_1,n_2)$ of $\\mathbb F_{n_1} \\cup \\mathbb F_{n_2}$.\nMoreover, their triple intersections agree with gauge theory prepotentials of $SU(3)_k$ listed in Table \\ref{tb:endpoint-F-F}. Therefore, we expect that these endpoint geometries have $SU(3)_k$ gauge theory realizations.\n\n\nThe geometries $(0,6)$ and $(0,8)$ are particularly interesting, as they have two different gauge theory descriptions related by the base-fiber exchange of $\\mathbb{F}_0$. \nWhen we consider the fibers classes to be $F,F$, the two geometries $(0,6),(0,8)$ exhibit (respectively) $Sp(2),G_2$ Cartan matrices. On the other hand, if we choose fiber classes $H,F$, the geometries exhibit the $SU(3)$ Cartan matrix in both cases.\n\nStudying triple intersection numbers gives us a means to narrow down the precise gauge theory that corresponds to these geometries. The triple intersection polynomial $J^3$ of the geometry $(0,6)$ is identical to the prepotentials of both pure $SU(3)_5$ gauge theory and also pure $Sp(2)_\\theta$ theory, which can have either $\\theta=0$ or $\\theta=\\pi$. However, the prepotential cannot distinguish two $Sp(2)$ cases. We can instead determine the $\\theta$ angle using the known duality between $SU(3)$ and $Sp(2)$. In \\cite{Gaiotto:2015una}, it was conjectured that $SU(3)_5$ is dual to $Sp(2)_\\pi$. This suggests that the geometry $(0,6)$ corresponds to $Sp(2)_\\pi$ while $(1,6)$ corresponds to $Sp(2)_0$. Thus, the geometric construction provides yet additional evidence supporting the duality between the $SU(3)_5$ and $Sp(2)_{\\pi}$ gauge theories.\n\n\nAs another example of a duality between gauge theories, the triple intersections of $(0,8)$ agree with the prepotentials of $SU(3)_7$ and $G_2$ gauge theories. We thus conjecture that $SU(3)_7$ and $G_2$ theories are dual and describe the low energy physics of the SCFT corresponding to $\\mathbb{F}_0\\cup \\mathbb{F}_8$.\n\n\n\nAdditional (not necessarily endpoint) geometries of type $\\mathbb{F}_{n_1}\\cup \\mathbb{F}_{n_2}$ are displayed in Table \\ref{tb:shirinkable-F-F}. The first five geometries in Table \\ref{tb:shirinkable-F-F} are shrinkable. However, the first four geometries of these are not endpoints. They all can be obtained from other endpoint geometries, $\\mathbb{P}^2\\cup \\mathbb{F}_3$ or $\\mathbb{P}^2\\cup \\mathbb{F}_6$, by blowing up a point and performing flop transitions; see Figure \\ref{fig:P2-F-transition} for more details. We find that these geometries but $(1,2)$ have gauge theory descriptions as listed in Table \\ref{tb:shirinkable-F-F}. The geometry $(1,2)$ has gauge algebra $SU(2)\\hat{\\times}SU(2)$ where $\\hat{\\times}$ denotes that we gauge the $SU(2)$ global symmetry of another $SU(2)$ gauge theory which arises from the $U(1)_I$ instanton symmetry in the IR gauge theory.\n\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[scale=.45]{P2-F-transitions.pdf}\n\t\t\n\t\\caption{Geometric transitions from $\\mathbb{P}^2\\cup \\mathbb{F}_3$ and $\\mathbb{P}^2\\cup \\mathbb{F}_6$ to $\\mathbb{F}_1\\cup \\mathbb{F}_n$'s with $n=2,3,5,6$.}\n\t\\label{fig:P2-F-transition}\n\\end{figure}\n\nThe geometry $(2,4)$ in Table \\ref{tb:shirinkable-F-F} is an endpoint geometry admitting no additional rank preserving mass deformations. However, this geometry is equivalent to another endpoint geometry $(0,4)$ by a complex structure deformation, or a Hanany-Witten transition. Thus these two geometries belong to the same physical equivalence class.\n\nLastly, the geometry $(0,10)$ is not shrinkable. This geometry satisfies all other shrinkablity conditions, but we find that no 4-cycles have nonzero volume at any point in the K\\\"ahler cone. Thus $(0,10)$ is not shrinkable unless we make a non-normalizable K\\\"ahler deformation. This means the corresponding field theory possesses an intrinsic energy scale set by the K\\\"ahler parameter of the non-compact 4-cycle. Therefore, we do not expect that this geometry corresponds to a 5d SCFT. Indeed, in Section \\ref{sec:rank2}, we will argue that this geometry gives a 5d KK theory.\n\nWe have finished the full classification of rank 2 endpoint geometries (thus rank 2 endpoint SCFTs), which have $M=0,1$. The result is rather surprising---we observe that all rank 2 SCFTs are actually realized by gauge theories and their mass deformations. Note that geometries $\\mathbb{P}^2\\cup\\mathbb{F}_3$ and $\\mathbb{P}^2\\cup\\mathbb{F}_6$ corresponding to non-Lagrangian theories can also viewed as deformations of geometries which admit gauge theory descriptions, for example (respectively) $\\mathbb{F}_1\\cup\\mathbb{F}_2$ and $\\mathbb{F}_1\\cup\\mathbb{F}_5$. This seems to suggest that gauge theory descriptions are generally quite useful, even for 5d SCFTs of higher rank.\n\nFurthermore, all geometries in Table \\ref{tb:rank2-F-F-clssification} except for $(1,2)$ were already predicted in \\cite{Jefferson:2017ahm} using perturbative gauge theory analysis. In fact these geometric constructions confirm all predictions with $r=2$ and $M=1$ in \\cite{Jefferson:2017ahm} except for $SU(3)_8$. \nIt was conjectured in \\cite{Jefferson:2017ahm} that the $SU(3)_8$ theory exists and has an interacting UV fixed point. However, the existence of this theory appears to be ruled out by our geometric classification.\n\nLet us briefly discuss the geometry of the $SU(3)_8$ gauge theory.\nThis theory in fact has a geometric realization as the local 3-fold with K\\\"ahler surface $\\mathbb{F}_1\\cup \\mathbb{F}_9$, where we identify the 2-cycles $C_{S_1}=3F_1+H_1$ and $C_{S_2} = E_2$. However, this geometry is not shrinkable because at least one 2-cycle contained in $S$ has negative volume. For example, the volumes\n\\begin{equation}\n\t\\text{vol}(E_1) = \\phi_1 - 3\\phi_2 \\ , \\quad \\text{vol}(F_2) = 2\\phi_2 - \\phi_1 \\ \n\\end{equation}\nwith $\\phi_1,\\phi_2>0$ cannot be both non-negative. Therefore the Coulomb branch of this geometry is trivial and this geometry is not shrinkable.\nIn order to make the geometry shrinkable we need to attach a non-compact 4-cycle with non-zero K\\\"ahler parameter corresponding to bare gauge coupling constant $1\/g^2$. This K\\\"ahler parameter cannot be tuned to zero while maintaining positivity of the K\\\"ahler metric. So even though the IR gauge description with $1\/g^2\\not=0$ makes sense geometrically, we cannot take the $1\/g^2=0$ limit without taking the Coulomb branch parameter to $0$. This means that if the point $1\/g^2=0$ is a CFT point, then it has no Coulomb branch deformation, and thus in conflict with a SCFT from this gauge theory based on our assumptions. Thus we do not expect that this geometry has a CFT limit. The gauge theory analysis in \\cite{Jefferson:2017ahm} uses only the perturbative spectrum and monopole tensions and thus cannot capture the spectrum of M2-branes wrapping the curve $E_1 \\subset \\mathbb F_1$ (which correspond to instantons in the gauge theory). Missing non-perturbative states such as these are crucial for assessing whether or not a geometry is shrinkable. This again shows that the perturbative constraints used in \\cite{Jefferson:2017ahm} are necessary but not sufficient to guarantee the existence of CFT fixed points.\n\n\\subsubsection*{Full rank 2 classification}\n\nWe showed in the previous section that our classification program can be reduced to a classification of the following types of geometric configurations: $\\text{Bl}_{p_1} \\mathbb F_n \\cup \\text{dP}_{p_2}$ and $ \\text{Bl}_{p_1} \\mathbb F_n \\cup \\mathbb F_0$. As already discussed $p_2$ and $p_1$ are bounded above by $p_{\\text{max}}(n)$, which we note depends upon both the degree $n$ and the type of gluing configuration. However, we are still faced with the problem of restricting the range of (non-negative) integer $n$ for which there exist shrinkable configurations. It turns out that some necessary conditions of shrinkability allows us to derive a crude bound on $n$. From a physical perspective, the existence of such a bound is not surprising as it is closely tied to the existence of only a finite number of 5d interacting fixed CFT points for a fixed rank. \n\nAppropriate bounds on $n$ can be determined in the two separate cases of $S_2 = \\text{dP}_{p_2}$ or $S_2= \\mathbb F_0$. For both cases, we need only consider $n \\geq 2$, since setting $n=0,1$ produces a geometric configuration isomorphic to $\\text{dP}_{p_1+1} \\cup \\text{dP}_{p_2}$. In the case of $S_2 = \\text{dP}_2$, we find that $n \\leq 7$, while in the case of $S_2 = \\mathbb F_0$, we find that $n \\leq 8$. See Appendix \\ref{app:bound} for proofs of these bounds.\n\nWe present our classification of rank 2 K\\\"ahler surfaces associated to 5d UV interacting fixed points in Figures \\ref{fig:11}-\\ref{fig:0}. These results are organized by the number of mass parameters $M$, with $0 \\leq M \\leq 11$. Given $M >0$ mass parameters, a shrinkable geometry with $M-1$ mass parameters may be obtained by performing a blowdown of an exceptional divisor (possibly after a sequence of flops) in the surface $S$; in the associated field theory, blowing down an exceptional curve corresponds to integrating out a massive matter hypermultiplet. \n\n\nIn each figure, we list the K\\\"ahler surface $S= S_1 \\overset{C_{S_2}}{\\cup} S_2$, where $C_{S_2}=( S_1 \\cap S_2)_{S_2}$ is the curve along which the two surfaces are glued, restricted to the \\emph{second} surface $S_2$. Geometries marked with $( \\cdot )^*$ correspond to 5d KK theories. Beneath each geometry, we also list the associated gauge theory; geometries with no associated gauge system indicated do not admit a known description as a gauge theory. \n\nOur method for identifying gauge theoretic descriptions involves comparing the triple intersection $J^3$ with the gauge-theoretic prepotential $6\\mathcal F$ in (\\ref{eqn:pre}) for given gauge group and matter content in the K\\\"ahler cone, as well as identifying a geometric realization of the Cartan matrix of associated to the gauge algebra.\n\n\nThe Cartan matrices are determined up to sign by a choice of fibers\\footnote{In the present discussion, a \\emph{fiber} is a rational curve $f$ with self intersection $f^2= 0$.} $f_1\\subset S_1, f_2 \\subset S_2$ satisfying \n\t\t\\begin{align}\n\t\t\t(f_i \\cdot S_j)_{S_i} = - (A_G)_{ij}.\n\t\t\\end{align}\t\nGeometrically, these fibers are rational curves over which M2-branes may be wrapped to give rise to charged BPS vectors in the 5d spectrum. In Figures \\ref{fig:11}-\\ref{fig:0}, we indicate to the left of each gauge description a possible choice of fibers giving rise to stated gauge algebra. We merely list all possible gauge theory descriptions and do not attempt to list all possible configurations of fibers. When there is more than one choice of fiber leading to different Cartan matrices (and hence different gauge symmetries), there are dualities between the associated gauge theory descriptions. For $\\text{dP}_{p_2 <8}$, the possible fibers are (using the same notation as in \\ref{eq:moridp})\n\t\\begin{align}\n\t\t(1;1)~,~(2;1^4)~,~(3;2,1^6)~,~(4;2^3,1^4)~,~(5;2^6,1).\n\t\\end{align}\nThe list of possible fibers in $\\text{Bl}_{p_1} \\mathbb F_n$ is significantly more complicated; see Appendix \\ref{app:fiber}.\n\n\n\tWe also note that the double arrows connecting pairs of different geometries $S$ indicate flop transitions mapping the geometries into one another. Each figure contains several clusters of geometries connected by arrows, with each cluster belonging to the same birational, and thus physical, equivalence class. Arrows decorated with the symbol $\\phi_1 \\leftrightarrow \\phi_2$ indicate that the flop transition requires us to reverse our identifications $S_1 \\leftrightarrow S_2$, and flip the sign of the Chern-Simons level, $k \\to - k$. \n\n\n\tFinally, we remark that the gluing curves $C_{S_2} \\in \\text{dP}_{p_2 \\geq 3}$ are only listed up to the action of the Weyl group $W(E_{p_2})$. Said differently, each choice of gluing curve displayed in the figures is a single element in the Weyl orbit. We now briefly describe the Weyl group action in $\\text{dP}_{p_2}$ and explain why in most cases we only need to distinguish geometric configurations whose gluing curves belong to the same Weyl orbit in a given surface. Given a simple root $\\alpha_i = X_i - X_{i+1}, i = 1, \\dots, p_2-1$, and an effective curve \n\t\t\\begin{align}\n\t\t\tC= d \\ell - m_i X_i,\t\t\n\t\t\\end{align}\n\tthe Weyl reflections $w_{\\alpha_i}$ act by transposing exceptional divisors, $X_i \\leftrightarrow X_{i+1}$, while the reflection $w_{\\alpha_{p_2}}$ associated to the root $\\alpha_{p_2} = \\ell - \\sum_{i=1}^3 X_i$ acts on $C$ as follows:\n\t\t\\begin{align}\n\t\t\\begin{split}\n\t\t\tw_{\\alpha_{p_2}}(C) &= (2 d - m_1 - m_2 -m_3) \\ell - ( d - m_2 - m_3 ) X_1 - ( d - m_1 - m_3) X_2 \\\\\n\t\t\t&- (d - m_1 - m_2) X_3 - \\sum_{i > 3} m_i X_i.\n\t\t\\end{split}\n\t\t\\end{align}\n\tAs was shown in \\cite{Iqbal:2001ye}, the action of $W(E_{p_2})$ on a rational curve $C \\in \\text{dP}_{p_2}$ for $p_2 \\geq 4$ and degree $d_C \\equiv - K \\cdot C = C^2 + 2 = n$ in all cases studied in this paper is transitive. Therefore, since the Weyl action $w_{\\alpha}: C \\mapsto C + (C \\cdot \\alpha) C$ preserves intersection products, \n\t\t\\begin{align}\n\t\t\tC \\cdot C' = ( C + (C \\cdot \\alpha) \\alpha) \\cdot ( C' + (C'\\cdot \\alpha) \\alpha),\n\t\t\\end{align}\n\tit is sufficient to set the gluing curve $C_{S_2}$ equal to a single element of the Weyl orbit in order to understand the full intersection structure, as the intersection numbers are identical up to permutation for any two elements belonging to the same Weyl orbit. For $p_2 <3$, the Weyl group either has multiple orbits (as in the case of $p_2 =3$) or is otherwise undefined (as in the case of $p_2 <3$), and so for $p_2<4$ we only list gluing curves $C_{S_2}$ up to cyclic permutations of the exceptional divisors $X_i$.\n\nUpon mass deforming these SCFTs and flowing to the IR we get a tree of relations between these conformal theories which is summarized in the RG flow tree diagram in Figure \\ref{tree}. The top theories of the RG families are related to 5d KK theories which are discussed in the next section.\n\n\\begin{figure}\n\\begin{center}\n\\noindent\\makebox[\\textwidth]{\n$\n\\begin{array}{c}\n\\begin{tikzpicture}[]\n\t\\node[yscale=1.2,xscale=1.1] at (0,0) {\\includegraphics[scale=.5]{rank2.pdf}};\n\\end{tikzpicture}\n\\end{array}\n$\n}\n\\end{center}\n\\caption[]{The diagram above shows the RG flow among rank 1 and rank 2 SCFTs obtained by mass deformations. The first and the second rows in each box correspond to the geometric and the gauge theoretic descriptions respectively of a 5d theory \\footnotemark. The parent theory in each branch is a 5d KK theory related to a 6d theory on $S^1$.}\n\\label{tree}\n\\end{figure}\n\\footnotetext{\\label{foot:GZ}\nWe note that while $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ has no gauge theory description, it is nonetheless related to $[SU(2)+5{\\bf F}]\\times SU(2)_0$ by a flop transition: a flop of $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ leads to the geometry $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$, which has gauge theory description $[SU(2)+5{\\bf F}]\\times SU(2)_0$. However, $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$ is not shrinkable, which implies that the BPS spectrum of the gauge theory will develop a negative mass before reaching a CFT fixed point. Nevertheless, this gauge theory theory makes sense as an effective description of the CFT from $\\text{Bl}_8\\mathbb{F}_3\\cup \\mathbb{P}^2$ through a flop transition to $\\text{Bl}_7 \\mathbb{F}_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1$ when mass parameters are turned on. We are greatful to Gabi Zafrir for pointing out that the CFT related to the $[SU(2)+5{\\bf F}]\\times SU(2)_0$ gauge theory should exist since an associated $(p,q)$ 5-brane system exists.}\n\n\n \\begin{figure}\n \\begin{center}\n $\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.4]\n\t\t\\node[](a) at (-4,2) {$ \\begin{array}{c} (\\text{Bl}_{10} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1)^* \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 10 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_1 &\\hat A_1\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,0) {$\\begin{array}{c} (\\text{Bl}_{9} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2)^* \\\\\\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_1 & SU(3)_0 + 10 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 10\\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_2& \\hat A_1 \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](b2) at (4,2) {$\\begin{array}{c} (\\text{Bl}_{10} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0)^* \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 10\\textbf{F} \\\\\\hline F, E & SU(3)_0 + 10 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i ,F& \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (0,-2.5) {$\\begin{array}{c} (\\text{Bl}_{8} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3)^* \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_0 + 10 \\textbf{F} \\\\\\hline F,\\ell - X_3 & Sp(2) + 10\\textbf{F}\\\\\\hline H + 2 F - \\sum X_i, \\ell - X_3 & \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (0,-5) {$\\begin{array}{c} (\\text{Bl}_{7} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4)^*\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_0 + 10 \\textbf{F} \\\\\\hline F,\\ell - X_4 & Sp(2) + 10\\textbf{F}\\\\\\hline H + 2 F - \\sum X_i, \\ell -X_4 & \\hat A_1 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (0,-7.5) {$\\begin{array}{c} (\\text{Bl}_{6} \\mathbb F_2 \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5)^* \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_0 + 10 \\textbf{F} \\\\\\hline F,\\ell - X_5 & Sp(2) + 10\\textbf{F}\\\\\\hline H- X_1 - X_2, 2 \\ell-\\sum_{i=1}^4 X_i & [SU(2) + 4 \\textbf{F}] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline H + 2 F - \\sum X_i,\\ell-X_5 & \\hat A_1 \\\\\\hline \\end{array} $} \\end{array}$};\n\t\t\\node[](f) at (0,-10) {$\\begin{array}{c} (\\text{Bl}_{5} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6)^*\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_0 + 10 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 10\\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=2}^5 X_i & [SU(2) + 4 \\textbf{F}] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline f_1 \\cdot E= 2, \\ell- X_6 & \\hat A_1 \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n$\n\\end{center}\n\\caption{$M=11$ geometries.}\n\\label{fig:11}\n \\end{figure}\n \n \n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (7.5,10) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (5.5,8) {$\\begin{array}{c} \\text{Bl}_{8} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_1 & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 9\\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](b2) at (3.5,10) {$\\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 9\\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,6) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_{1} & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline F, \\ell - X_3 & Sp(2) + 9\\textbf{F}\\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](d) at (5.5,4) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline F, \\ell - X_4 & Sp(2) + 9\\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (5.5,1.5) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_{1} & SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline F,\\ell - X_5 & Sp(2) + 9\\textbf{F}\\\\\\hline H - X_1-X_2, 2 \\ell- \\sum_{i=1}^4 X_i & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (-2,1.5) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF, \\ell- X_5 &SU(3)_{\\frac{1}{2}} + 9 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 9\\textbf{F}\\\\\\hline f_1 \\cdot E = 0, \\ell-X_6 & [SU(2) + 3 \\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node[](a3) at (-2,11.5) {$ \\begin{array}{c} \\text{Bl}_{10} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\t\\node[] at (5.5,12) {$ \\begin{array}{c} (\\text{Bl}_{9} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0)^* \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline F,E& SU(3)_{-\\frac{3}{2}} + 9 \\textbf{F} \\\\\\hline H + 2 F - \\sum_{i=1}^8 X_i ,E & Sp(2) + 8 \\textbf{F}+ 1\\textbf{AS} \\\\\\hline \\end{array}$} \\end{array} $};\n\t\t\\node[](b3) at (-2,10) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_1 & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F} \\\\\\hline H + 2 F- \\sum X_i , \\ell- X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c3) at (-2,8) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F} \\\\\\hline H + 2 F- \\sum X_i , \\ell- X_1 & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d3) at (-2,6) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_{1} & Sp(2) + 9 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e3) at (-2,4) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{-\\frac{1}{2}}+ 9 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i, \\ell - X_{1} & Sp(2) + 9 \\textbf{F} \\\\\\hline H - X_1 -X_2 , 2 \\ell - \\sum X_i & [SU(2) + 4 \\textbf{F}] \\times [ SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\draw[big arrow] (a3) -- (b3);\n\t\t\\draw[big arrow] (e3) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (e3);\n\t\t\\draw[big arrow] (b3) -- (c3);\n\t\t\\draw[big arrow] (c3) -- (d3);\n\t\t\\draw[big arrow] (d3) -- (e3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (a3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (b3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (c3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e3) -- (d3);\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=10$ geometries.}\n \\label{fig:10}\n \\end{figure}\n \n\n \n \\begin{figure}\n \t \\begin{center}\n\\noindent\\makebox[\\textwidth]{ $\n \\begin{array}{c}\n \\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (9,-4) {$ \\begin{array}{c} \\text{Bl}_{9} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (5,-4) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (6,-6) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{0}+ 8 \\textbf{F} \\\\\\hline H- X_1 - X_2 , 2\\ell - \\sum X_i & [SU(2) + 3 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (6,-8.5) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t F,\\ell-X_1 & SU(3)_0 + 8 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2l- \\sum_{i=1}^4 X_i & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t \\node[] at (4,-1.5) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-2} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum_{i=1}^7 X_i , \\ell - X_1 & Sp(2) + 7 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\\\\ \\\\\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (5,4.5) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (5,2) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF, \\ell - X_1 & SU(3)_{1} + 8 \\textbf{F} \\\\\\hline F,\\ell - X_2 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (5,-.5) {$\\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 8 \\textbf{F} \\\\\\hline F,E & SU(3)_{1} + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (.5,2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{1} + 8 \\textbf{F} \\\\\\hline F, \\ell - X_3 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (.5,-.5) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{1} + 8 \\textbf{F} \\\\\\hline F,\\ell - X_4 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (-6.5,-.5) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{1} + 8 \\textbf{F} \\\\\\hline F,\\ell - X_5 & Sp(2) + 8 \\textbf{F} \\\\\\hline H- X_1 - X_2 , 2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 2 \\textbf{F} ] \\times [ SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](f) at (-6.5,2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{1} + 8 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 8 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & [SU(2)+ 2 \\textbf{F} ] \\times [SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node(z) at (.5,6.5) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , F & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(y) at (.5,4.5) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell- X_1 & Sp(2) + 8 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(x) at (-6.5,4.5) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-1} + 8 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_{2} & Sp(2) + 8 \\textbf{F} \\\\\\hline H-X_1 -X_2 ,\t\\ell-X_1 & [SU(2) +4 \\textbf{F} ] \\times [ SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.7em}] (f) --node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (x);\n\t\t\t\\draw[big arrow] (x)-- (f);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (x) -- (y);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (x) -- (y);\n\t\t\t\\draw[big arrow] (y) -- (x);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (y) -- (z);\n\t\t\t\\draw[big arrow] (z) -- (y);\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$}\n\\end{center}\n \\caption{$M=9$ geometries.}\n \\label{fig:9}\n \\end{figure}\n \n\n \n \\begin{figure}\n \t \\begin{center}\n\\noindent\\makebox[\\textwidth]{ $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.4]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\scalebox{.65}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\\\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 7 \\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline F,\\ell - X_3 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline F,\\ell - X_4 & Sp(2) + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline F, \\ell - X_5 & Sp(2) + 7 \\textbf{F} \\\\\\hline H - X_1 - X_2 ,2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 1\\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}]\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](f) at (0,4) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 7 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2l - \\sum_{i=1}^4 X_i & [SU(2) + 1 \\textbf{F}] \\times [ SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](a4) at (-5,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline H + 2 F - \\sum X_i , \\ell - X_1 & Sp(2) + 7\\textbf{F}\\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\node[](a1) at (-5,2) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_2 & SU(3)_{-\\frac{3}{2}} + 7 \\textbf{F} \\\\\\hline H+ 2F - \\sum X_i , \\ell-X_2 & Sp(2) + 7 \\textbf{F} \\\\\\hline H -X_1 - X_2 , \\ell- X_1 & [SU(2) + 4\\textbf{F}]\\times [SU(2) + 1 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow] (a1)-- node[left,midway]{$\\phi_1 \\leftrightarrow \\phi_2$}(f);\n\t\t\\draw[big arrow] (a1)-- (a4);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a4) -- (a1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (a1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-2,0) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\n\t\t\\node[](b) at (-2,-2) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\\node[](c) at (-2,-4) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\n\t\t\\node[](d) at (-2,-6) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\\\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\\node[](e) at (-2,-8) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} &SU(3)_{\\frac{1}{2}}+ 7 \\textbf{F} \\\\\\hline H-X_1 -X_2 , 2\\ell - \\sum X_i & [SU(2) + 2 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (7,-8) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|} \\hline\n\tF,\\ell- X_1 & SU(3)_{\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & [SU(2) + 2 \\textbf{F}] \\times [ SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\t\\node[](a2) at (2.5,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_2 \\overset{ E}{\\cup} \\mathbb F_0 \\\\\\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-\\frac{5}{2}} + 7 \\textbf{F} \\\\\\hline H-X_1 -X_2 , E & [SU(2) + 5 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline H + 2F - \\sum_{i=1}^6 X_i , F & Sp(2) + 6 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node at (2.5,-3) {$ \\begin{array}{c} \\text{Bl}_{8} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\t\\node (b3) at (7,-1) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c3) at (7,-3) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.65}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d3) at (7,-5) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.65}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 7 \\textbf{F} \\\\\\hline H-X_1 - X_2 ,\\ell- X_1 & [SU(2) + 3 \\textbf{F} ] \\times [SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (c3);\n\t\t\t\\draw[big arrow] (c3) -- (b3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (d3);\n\t\t\t\\draw[big arrow] (d3) -- (c3);\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (f) -- node[left,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (d3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n$}\n\\end{center}\n \\caption{$M=8$ geometries. (See Footnote \\ref{foot:GZ} for a comment about $\\text{Bl}_{8} \\mathbb F_3 \\cup \\mathbb P^2$.)}\n \\label{fig:8}\n \\end{figure}\n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{2} + 6 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 6 \\textbf{F} \\\\\\hline F, E & SU(3)_{2} + 6 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{2} + 6 \\textbf{F} \\\\\\hline F,\\ell - X_3 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{2} + 6 \\textbf{F} \\\\\\hline F, \\ell - X_4 & Sp(2) + 6 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} & SU(3)_{2} + 6 \\textbf{F} \\\\\\hline F,\\ell - X_5 & Sp(2) + 6 \\textbf{F} \\\\\\hline H- X_1 - X_2 , 2 \\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [SU(2) + 4 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (5.5,4.5) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} &SU(3)_{2} + 6 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0 , 2\\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [SU(2) + 4 \\textbf{F} ] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\n\t\t\t\\node(g) at (-2,4.5) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-2} + 6 \\textbf{F} \\\\\\hline H+2F - \\sum X_i ,F & Sp(2) + 6 \\textbf{F} \\\\\\hline H-X_1-X_2, E & [SU(2) + 4 \\textbf{F}] \\times SU(2)_{\\pi} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(A) at (7.5,6) {$\\begin{array}{c} \\text{Bl}_7 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(B) at (3,6) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline H-X_1-X_2 , \\ell - X_1 & [SU(2) + 4 \\textbf{F} ] \\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(C) at (-3,6) {$ \\begin{array}{c} \\text{Bl}_5 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell-X_{1} & [SU(2) + 4 \\textbf{F}]\\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (A) -- (B);\n\t\t\\draw[big arrow] (B) -- (A);\n\t\t\\draw[big arrow](C) -- (B);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}](B) -- (C);\n\n\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow] (f) -- node[above,pos=.5]{$\\phi_1 \\leftrightarrow \\phi_2$} (g);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (g) -- (f);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=7$ geometries.}\n \\label{fig:7a}\n \\end{figure}\n \n\n \\begin{figure}\n \\begin{center}\n \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node (e) at (2.5,-1) {$\\begin{array}{c} (\\mathbb F_2 \\overset{\\ell - X_1}{\\cup} \\text{dP}_7 )^*\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline F, \\ell- X_2 & SU(3)_4 + 6 \\textbf{F}\\\\\\hline F, 2\\ell- \\sum_{i=2}^5 X_i & Sp(2) + 4 \\textbf{F} + 2 \\textbf{AS} \\\\\\hline F,4\\ell - \\sum_{i=1}^4 X_i - 2 \\sum_{j=5}^7 X_j& G_2 + 6 \\textbf{F} \\\\\\hline F,5\\ell - X_1 - 2 \\sum_{i=2}^7 X_i &A^{(2)}_2 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\\\\\\n\t\\begin{array}{c}\n\t\\begin{tikzpicture}[yscale=1,xscale=1.2]\n\t\t\\node[](a) at (3.2,0) {$ \\begin{array}{c} \\text{Bl}_{7} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} & SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{1}+ 6 \\textbf{F} \\\\\\hline H-X_1-X_2 , 2\\ell - \\sum X_i & [SU(2)+ 1 \\textbf{F}] \\times [SU(2) + 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-8.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\tF,\\ell-X_1 & SU(3)_{1} + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2 \\ell - \\sum_{i=1}^4 X_i & [SU(2) + 1 \\textbf{F} ] \\times [ SU(2)\\times 3 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array}$};\n\t \\node[](a3) at (0,-11) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_2 & SU(3)_{-1} + 6 \\textbf{F} \\\\\\hline H-X_1-X_2, \\ell-X_1 & [SU(2) + 3 \\textbf{F}] \\times [ SU(2)+ 1 \\textbf{F}] \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[](a4) at (0,-13) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-1} + 6 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[] at (7.5,0) {$ \\begin{array}{c} \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_7 \\\\\\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{3} + 6 \\textbf{F} \\\\\\hline F,\\ell-X_2 & Sp(2) + 5 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node (b3) at (7.5,-3) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,E & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c3) at (7.5,-5) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\\\scalebox{.7}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_{1} & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d3) at (7.5,-7) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_{2} & SU(3)_{0} + 6 \\textbf{F} \\\\\\hline H-X_1-X_2 , \\ell -X_1 & [SU(2) + 2 \\textbf{F} ] \\times [SU(2) \\times 2\\textbf{F}] \\\\\\hline\\end{array} $}\\end{array}$};\n\t\t\t\\node(e3) at (7.5,-9) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_{3} & SU(3)_0 + 6 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, \\ell- X_{1} & [SU(2) + 2 \\textbf{F} ] \\times [SU(2) \\times 2\\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b3) -- (c3);\n\t\t\t\\draw[big arrow] (c3) -- (b3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c3) -- (d3);\n\t\t\t\\draw[big arrow] (d3) -- (c3);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d3) -- (e3);\n\t\t\t\\draw[big arrow] (e3) -- (d3);\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (a3) -- (f);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (a3) -- (a4);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (a3);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a4) -- (a3);\n\t\t\\end{tikzpicture}\n\t\t\\end{array}\n\t\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=7$ geometries, cont.}\n\\label{fig:7b}\n \\end{figure}\n \n \\clearpage\n \n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 5 \\textbf{F} \\\\\\hline F, E & SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline F, \\ell - X_3 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline F,\\ell - X_4 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline F,\\ell - X_5 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](f) at (-1,2) {$\\begin{array}{c} \\mathbb F_1 \\overset{2 \\ell-\\sum_{i=1}^5 X_i }{\\cup} \\text{dP}_6\\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}} + 5 \\textbf{F} \\\\\\hline F, \\ell - X_6 & Sp(2) + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (f) -- (e);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (4,-2) {$ \\begin{array}{c} \\text{Bl}_{6} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_{1} & SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline H - X_1 - X_2 , 2\\ell - \\sum X_i & SU(2)_\\pi \\times [SU(2) + 3 \\textbf{F} ] \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-10.8) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{.7}{$ \\begin{array}{|c|c|} \\hline\n\t F,\\ell - X_1 &SU(3)_{\\frac{3}{2}}+ 5 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, 2\\ell - \\sum_{i=1}^4 X_i & SU(2)_\\pi \\times [ SU(2) + 3 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\n\t\t\n\t \\node(g) at (0,-13) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F, F & SU(3)_{-\\frac{3}{2}} + 5 \\textbf{F} \\\\\\hline H-X_1 - X_2 ,E& [SU(2) + 3\\textbf{F}] \\times SU(2)_\\pi \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow] (f) -- (g);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (g) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (f);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t \t \\node(A) at (8,-8) {$\\begin{array}{c} \\text{Bl}_6 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(B) at (8,-10) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|} \\hline H-X_1-X_2 , \\ell - X_1 & [SU(2) + 3 \\textbf{F} ] \\times SU(2)_0 \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(C) at (8,-12) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\scalebox{.7}{$\\begin{array}{|c|c|}\\hline F,\\ell-X_{1} & [SU(2) + 3 \\textbf{F}] \\times SU(2)_{0} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (A) -- (B);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (C) -- (B);\n\t\t\t\\draw[big arrow] (B) -- (A);\n\t\t\t\\draw[big arrow] (B) -- (C);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=6$ geometries.}\n \\label{fig:6a}\n \\end{figure}\n \n\n \\begin{figure}\n \\begin{center}\n \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\n\t\t\t\\node (e) at (1.5,-2.5) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_6 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell - X_6 & SU(3)_{\\frac{9}{2}} + 5 \\textbf{F}\\\\\\hline F, 2\\ell - \\sum_{i=3}^6 X_{i} & Sp(2) + 3 \\textbf{F} + 2 \\textbf{AS} \\\\\\hline F, 3 \\ell - \\sum_{i=1}^5 X_i - 2 X_6 & G_2 + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (f) at (9,-2.5) {$\\begin{array}{c} \\mathbb F_2 \\overset{\\ell - X_1}{\\cup} \\text{dP}_6 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline F,\\ell-X_2 & SU(3)_{\\frac{7}{2}} + 5 \\textbf{F}\\\\\\hline F,2 \\ell- \\sum_{i=3}^6 X_i & Sp(2) + 4 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\\\\\n\t\\begin{array}{c}\n\t\\begin{tikzpicture}\n\t\t\\node (bn) at (8.1,-1) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (cn) at (8.1,-3) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (dn) at (8.1,-5.5) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline H- X_1 - X_2 , \\ell-X_1 & [SU(2) + 1 \\textbf{F} ] \\times [SU(2) +2 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(en) at (8.1,-8) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,\\ell -X_4 & SU(3)_{\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline f_1 \\cdot E =0 , \\ell-X_1 & [SU(2) + 1 \\textbf{F} ] \\times [SU(2) +2 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\node[](a1n) at (8.1,-13) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node[](a2n) at (8.1,-10.5) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_2 \\overset{ \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{-\\frac{1}{2}} + 5 \\textbf{F} \\\\\\hline H- X_1 -X_2 ,\\ell-X_1 & [SU(2) + 2 \\textbf{F}] \\times [ SU(2) + 1 \\textbf{F}]\\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (bn) -- (cn);\n\t\t\t\\draw[big arrow] (cn) -- (bn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (cn) -- (dn);\n\t\t\t\\draw[big arrow] (dn) -- (cn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (dn) -- (en);\n\t\t\t\\draw[big arrow] (en) -- (dn);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a1n) -- (a2n);\n\t\t\t\\draw[big arrow] (a2n) -- (a1n);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a2n) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$} (en);\n\t\t\t\\draw[big arrow] (en) -- (a2n);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=6$ geometries, cont.}\n\\label{fig:6b}\n \\end{figure}\n \n \n \\clearpage\n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\tF,\t\\ell - X_1 & SU(3)_{3} + 4 \\textbf{F} \\\\\\hline F,\\ell - X_2 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 4 \\textbf{F} \\\\\\hline F, E & SU(3)_{3} + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F} \\\\\\hline F,\\ell - X_3 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F} \\\\\\hline \\ell - X_4 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\\node[](e) at (5.5,2) {$\\begin{array}{c} \\mathbb F_2\t \\overset{2 \\ell-\\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{3} + 4 \\textbf{F} \\\\\\hline F, \\ell - X_5 & Sp(2) + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{5} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,\\dots,3} &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,\\dots,4} &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](f) at (0,-10) {$ \\begin{array}{c} \\mathbb F_1 \\overset{2 \\ell - \\sum_{i=1}^5 X_i}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t F,\\ell - X_1 &SU(3)_{2}+ 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\t\t\\draw[big arrow] (e) -- (f);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- (e);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=5$ geometries.}\n \\label{fig:5a}\n \\end{figure}\n \n \\clearpage \n\n \\begin{figure}\n \\begin{center}\n \\noindent\\makebox[\\textwidth]{ \t$\n\t\\begin{array}{c}\n\t\t\t\\begin{array}{c}\n\t\t\t\\begin{tikzpicture}\n\t\t\t\t\\node (b) at (0,-3) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\node (c) at (0,-5) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{1} + 4 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d) at (0,-7) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3 \\\\\\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_2 & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline H-X_1-X_2 , \\ell - X_1 & SU(2)_\\pi \\times [SU(2) + 2 \\textbf{F}]\\\\\\hline\\end{array}$} \\end{array}$};\n\t\t\t\\node(e) at (0,-9.5) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell- X_3 & SU(3)_{1} + 4 \\textbf{F} \\\\\\hline f_1 \\cdot E =0, \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 2 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$}; \n\t\t\t\\node(f) at (0,-12.5) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{-1} + 4 \\textbf{F} \\\\\\hline H-X_1-X_2 ,E&[SU(2) + 2 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline \\end{array}$} \\end{array}$};\n\n\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (e);\n\t\t\t\\draw[big arrow] (e) -- (d);\n\t\t\t\\draw[big arrow] (e) --(f);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (f) -- node[right,midway]{$\\phi_1 \\leftrightarrow \\phi_2$}(e);\n\t\t\t\\end{tikzpicture}\n\t\t\t\\end{array} ~~~~~~~ \\begin{array}{c}\\begin{tikzpicture} \\node(A) at (6,-7) {$\\begin{array}{c} \\text{Bl}_5 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\\node(B) at (6,-9) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|} \\hline H-X_1-X_2,\\ell-X_1 &[SU(2) + 2 \\textbf{F}] \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\\node(C) at (6,-11) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell-X_{1} & [SU(2) + 2 \\textbf{F}] \\times SU(2)_{0} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (A) -- (B);\n\t\t\\draw[big arrow] (B) -- (A);\n\t\t\\draw[big arrow] (B) -- (C);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (C) -- (B);\n\t\t\\end{tikzpicture}\\end{array}\\\\ \\\\\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (e) at (-2,-7.5) {$\\begin{array}{c} \\mathbb F_4 \\overset{2\\ell-X_1-X_2}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell- X_1 & SU(3)_{5} + 4 \\textbf{F}\\\\\\hline F, \\ell - X_5 & Sp(2) + 2 \\textbf{F} + 2 \\textbf{AS} \\\\\\hline F, 2\\ell - \\sum_{i=2}^5 X_i & G_2 + 4 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (f) at (5,-7.5) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_5 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F, \\ell - X_1 & SU(3)_{4} + 4 \\textbf{F}\\\\\\hline F, 2\\ell - \\sum X_i & Sp(2) + 3 \\textbf{F} + 1 \\textbf{AS} \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (-4,-4.5) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{0} + 4 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array} $};\n\t\t\t\\node (l) at (3.5,-4.5) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline F,\\ell-X_2 & SU(3)_{0} + 4 \\textbf{F} \\\\\\hline H-X_1-X_2 , \\ell-X_1 & [SU(2) +1 \\textbf{F}] \\times[ SU(2) +1 \\textbf{F}] \\\\\\hline \\end{array} $}\\end{array}$};\n\t\t\t\\node (m) at (3.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline F,\\ell - X_3 & SU(3)_{0} + 4 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0,\\ell- X_1 & [SU(2) +1 \\textbf{F}] \\times[ SU(2) +1 \\textbf{F}] \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\end{array}\n\t$}\n\\end{center}\n\\caption{$M=5$ geometries, cont.}\n\\label{fig:5b}\n \\end{figure}\n \n \\clearpage \n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.3]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{7}{2}} + 3 \\textbf{F} \\\\\\hline F, \\ell - X_2 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\\\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,F & Sp(2) + 3 \\textbf{F} \\\\\\hline F,E &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F} \\\\\\hline F, \\ell - X_3 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\\node[](d) at (5.5,0) {$\\begin{array}{c} \\mathbb F_3 \\overset{2 \\ell-\\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_4\\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{7}{2}} + 3 \\textbf{F} \\\\\\hline F,\\ell - X_4 & Sp(2) + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{4} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](c) at (0,-4) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](d) at (0,-6) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F}\\\\\\hline \\end{array}$}\\end{array}$};\n\t\t\\node[](e) at (0,-8) {$ \\begin{array}{c} \\mathbb F_2 \\overset{2 \\ell - \\sum_{i=1}^4 X_i}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{1} &SU(3)_{\\frac{5}{2}}+ 3 \\textbf{F} \\\\\\hline \\end{array}$}\\end{array}$};\n\t\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\t\t\\draw[big arrow] (d) -- (e);\n\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (e) -- (d);\n\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t~~~~~~~~~~~~~~~~~~~\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (b) at (1,-3) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{3}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (c) at (1,-5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F,\\ell - X_1 &SU(3)_{\\frac{3}{2}} + 3 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node (d) at (1,-7) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3 \\\\ \\scalebox{1}{$ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{3}{2}} + 3 \\textbf{F}\\\\\\hline \\end{array}$} \\end{array}$};\n\t\t\t\\node(e) at (1,-9) {$\\begin{array}{c} \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_4 \\\\ \\scalebox{1}{$\\begin{array}{|c|c|}\\hline F, \\ell - X_3 &SU(3)_{\\frac{3}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}$} \\end{array}$}; \n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (e);\n\t\t\t\\draw[big arrow] (e) -- (d);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=4$ geometries.}\n \\label{fig:4a}\n \\end{figure}\n \n \\clearpage \n \n\n \\begin{figure}\n \\begin{center}\n \t \\noindent\\makebox[\\textwidth]{ $\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node(a) at (-2,2) {$\\begin{array}{c} \\text{Bl}_4 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (-2,0) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline H-X_1 -X_2 , \\ell - X_1 &[SU(2) + 1\\textbf{F}] \\times SU(2)_0\\end{array} \\\\\\hline \\end{array}$};\n\t\t\t\\node(C) at (7,0) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|}\\hline F,\\ell-X_{j=1,2} & [SU(2) + \\textbf{F}] \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\draw[big arrow] (b) -- (C);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (C) -- (b);\n\t\t\n\t\t\t\\node (e) at (6,-7.5) {$\\begin{array}{c} \\mathbb F_4 \\overset{2\\ell-X_1-X_2}{\\cup} \\text{dP}_4 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell -X_1 & SU(3)_{\\frac{9}{2}} + 3 \\textbf{F}\\\\\\hline F,\\ell - X_3 & Sp(2) + 2 \\textbf{F} + 1\\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (-2,-7.5) {$\\begin{array}{c} \\mathbb F_5 \\overset{2\\ell - X_1}{\\cup} \\text{dP}_4 \\\\ \\begin{array}{|c|c|} \\hline F , \\ell - X_1 & SU(3)_{\\frac{11}{2}} + 3 \\textbf{F}\\\\\\hline F, \\ell - X_1 & Sp(2) + \\textbf{F} + 2 \\textbf{AS} \\\\\\hline F , 2\\ell - \\sum X_i & G_2 + 3 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (3,2.5) {$\\begin{array}{c} (\\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_4)^* \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 & Sp(2)_{ 0} + 3 \\textbf{AS} \\\\\\hline F, 2\\ell - \\sum X_i & A^{(2)}_2 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (-2.5,-4.5) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (-2.5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell -X_2 &SU(3)_{\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline H-X_1 - X_2 , \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 1 \\textbf{F}] \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node (m) at (6.5,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_3 &SU(3)_{\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline f_1 \\cdot E =0, \\ell - X_1 & SU(2)_\\pi \\times [ SU(2) + 1 \\textbf{F}] \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\t\\node(n) at (6.5,-4.5) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F, F & SU(3)_{-\\frac{1}{2}} + 3 \\textbf{F} \\\\\\hline H - X_1 - X_2, E & [SU(2) + 1 \\textbf{F}] \\times SU(2)_\\pi \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\t\\draw[big arrow] (m) -- (n);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (n) -- node[right,midway]{$\\phi_1\\leftrightarrow \\phi_2$} (m);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t$}\n\\end{center}\n\\caption{$M=4$ geometries, cont.}\n\\label{fig:4b}\n \\end{figure}\n \n \\clearpage \n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-4.5,-2) {$ \\begin{array}{c} \\text{Bl}_{3} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\tF,\t\\ell - X_1 & SU(3)_{3}+ 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{3}+ 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](d) at (5.5,-4.5) {$ \\begin{array}{c} \\mathbb F_3 \\overset{2 \\ell - \\sum_{i=1}^3 X_i}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF, \\ell - X_{1} &SU(3)_{3}+ 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\t\t\\draw[big arrow] (c) -- (d);\n\n\t\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (d) -- (c);\n\n\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\\\\ \n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (0,0) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](b1) at (0,-2) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{4} + 2 \\textbf{F} \\\\\\hline F,\\ell - X_2 & Sp(2) + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](b2) at (-5,-2) {$\\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|} \\hline \n\tF,\tF & Sp(2) + 2 \\textbf{F} \\\\\\hline F, E &SU(3)_{4} + 2 \\textbf{F}\\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](c) at (5.5,-2) {$\\begin{array}{c} \\mathbb F_4 \\overset{2 \\ell-\\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_{1} &SU(3)_{4} + 2 \\textbf{F} \\\\\\hline F, \\ell - X_3 & Sp(2) + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (b1);\n\t\t\\draw[big arrow] (b1) -- (c);\n\t\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (b1) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b1);\n\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node (b) at (-5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (c) at (-.5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 &SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (d) at (4.5,-2.5) {$ \\begin{array}{c} \\mathbb F_2 \\overset{\\ell- X_1}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{2} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (d);\n\t\t\t\\draw[big arrow] (d) -- (c);\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=3$ geometries.}\n \\label{fig:3b}\n \\end{figure}\n \n \\clearpage \n\n \\begin{figure}\n \\begin{center}\n \t$\n\t\t\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node(C2) at (4,5) {$\\begin{array}{c} \\text{dP}_2 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|}\\hline \\ell- X_1, \\ell-X_{1} & SU(2)_{0} \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(a) at (-2,5) {$\\begin{array}{c} \\text{Bl}_3 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (-2,3) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline H-X_1-X_2, \\ell - X_1 & SU(2)_\\pi \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(C1) at (6,3) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline f_1 \\cdot E = 0, \\ell-X_{1} & SU(2)_{\\pi} \\times SU(2)_0 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow] (b) -- (C1);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (C1) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={xshift=.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\node(d1) at (-2.5,1) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{0} + 2 \\textbf{F} \\\\\\hline H - X_1 - X_2 , E & SU(2)_\\pi \\times SU(2)_\\pi \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d2) at (5.5,1) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{0} + 2 \\textbf{F} \\\\\\hline f_1 \\cdot E = 0, \\ell - X_2 & SU(2)_\\pi \\times SU(2)_\\pi \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (-3.5,-3) {$\\begin{array}{c} \\mathbb F_5 \\overset{2\\ell - X_1}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{5} + 2 \\textbf{F}\\\\\\hline F,\\ell - X_{2} & Sp(2) + 1\\textbf{F} + 1\\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (2,-3) {$\\begin{array}{c} \\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_3\\\\ \\begin{array}{|c|c|} \\hline F,\\ell -X_{1} & Sp(2)_{0}+ 2 \\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node[](k) at (-3,-1) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{1} + 2 \\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (2,-1) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell-X_2 &SU(3)_{1} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (m) at (7,-1) {$\\begin{array}{c} \\mathbb F_1 \\overset{\\ell-X_1-X_2}{\\cup} \\text{dP}_3 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_3 &SU(3)_{1} + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (g) at (7,-3.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{3 \\ell - 2 X_1 - X_2 }{\\cup} \\text{dP}_3\\\\ \\begin{array}{|c|c|} \\hline F, \\ell-X_2 &Sp(2)_{\\pi} + 2 \\textbf{AS} \\\\\\hline F,\\ell - X_1 & SU(3)_6 + 2 \\textbf{F} \\\\\\hline F, \\ell- X_3 &G_2 + 2 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (l) -- (m);\n\t\t\t\\draw[big arrow] (m) -- (l);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (d1) -- (d2);\n\t\t\t\\draw[big arrow] (d2) -- (d1);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t$\n\\end{center}\n\\caption{$M=3$ geometries, cont. Note that for the geometry $\\text{dP}_2 \\cup \\text{dP}_2$ at the top, the gluing curves in \\emph{both} surfaces are $C = \\ell - X_1 - X_2$, in contrast to the other geometries.}\n\\label{fig:3b}\n \\end{figure}\n \n \\clearpage \n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c} \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}[yscale=1]\n\t\t\\node[](a) at (-5,-2) {$ \\begin{array}{c} \\text{Bl}_{2} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b) at (0,-2) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{\\frac{7}{2}}+1 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](c) at (5,-2) {$ \\begin{array}{c} \\mathbb F_4 \\overset{2 \\ell - \\sum_{i=1}^2 X_i}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_{j=1,2} &SU(3)_{\\frac{7}{2}}+ 1 \\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\n\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b);\n\t\t\\draw[big arrow] (b) -- (c);\n\n\n\t\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (a);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (c) -- (b);\n\n\n\n\t\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a) at (5,0) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & Sp(2) + 1\\textbf{F} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\node[](b1) at (0,0) {$\\begin{array}{c} \\mathbb F_5 \\overset{2 \\ell-X_1}{\\cup} \\text{dP}_2 \\\\\\begin{array}{|c|c|} \\hline \n\t\tF,\\ell - X_1 & SU(3)_{\\frac{9}{2}} + 1 \\textbf{F} \\\\\\hline F,\\ell - X_2 & Sp(2) + 1 \\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\node[](b2) at (-5,0) {$\\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{F + 2 E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|} \\hline \n\tF,\tF & Sp(2) + 1\\textbf{F} \\\\\\hline F,E &SU(3)_{\\frac{9}{2}} + 1\\textbf{F}\\\\\\hline \\end{array} \\end{array}$};\n\t\n\n\t\n\n\t\n\t\n\t\n\t\n\t\t\\draw[big arrow] (a) -- (b1);\n\n\t\n\n\t\n\t\t\\draw[big arrow] (b1) -- (b2);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b2) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a);\n\n\n\n\t\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \\\\\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\node (b) at (-2.5,-2.5) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_4 \\overset{F+E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F,F & SU(3)_{\\frac{5}{2}} + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (c) at (2.5,-2.5) {$ \\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 &SU(3)_{\\frac{5}{2}} + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\n\t\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (c);\n\t\t\t\\draw[big arrow] (c) -- (b);\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\\\\\n\\begin{tikzpicture}\n\t\t\t\\node(a) at (-1,3) {$\\begin{array}{c} \\text{Bl}_2 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2\\end{array}$};\n\t\t\t\\node(b) at (3,3) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\end{array}$};\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (a) -- (b);\n\t\t\t\\draw[big arrow] (b) -- (a);\n\t\t\t\\node(d1) at (.5,1.5) {$\\begin{array}{c} \\text{Bl}_1 \\mathbb F_2 \\overset{E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|}\\hline F, F & SU(3)_{\\frac{1}{2}} + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d2) at (5.5,1.5) {$\\begin{array}{c} \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{1}{2}} + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node(d3) at (7,3) {$\\begin{array}{c} \\mathbb F_1 \\overset{\\ell - X_1 - X_2}{\\cup} \\text{dP}_2 \\end{array}$};\n\t\t\t\\draw[big arrow] (d3) -- (b);\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b) -- (d3);\n\t\t\n\t\t\n\t\t\t\\node (f) at (-2,-2.5) {$\\begin{array}{c} \\mathbb F_7 \\overset{3\\ell -2 X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|}\\hline F,\\ell - X_1 & SU(3)_{\\frac{13}{2}} + 1\\textbf{F}\\\\\\hline F, \\ell - X_2 & G_2 + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f) at (3,-2.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{2\\ell}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 &Sp(2)_0 + 1\\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\t\\node (f1) at (8,-2.5) {$\\begin{array}{c} \\mathbb F_6 \\overset{3\\ell - 2 X_1 - X_2}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1& SU(3)_{\\frac{11}{2}} + 1\\textbf{F}\\\\\\hline F,\\ell - X_2 & Sp(2)_\\pi +1 \\textbf{AS} \\\\\\hline \\end{array} \\end{array}$};\n\t\t\n\t\t\t\\node[](k) at (0,-.5) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_3 \\overset{ \\ell}{\\cup} \\text{dP}_1 \\\\\\begin{array}{|c|c|}\\hline F, \\ell - X_1 & SU(3)_{\\frac{3}{2}} + 1\\textbf{F} \\\\\\hline \\end{array}\\end{array} $};\n\t\t\t\\node (l) at (5,-.5) {$\\begin{array}{c} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_2 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell -X_2 &SU(3)_{\\frac{3}{2}} + 1\\textbf{F} \\\\\\hline \\end{array} \\end{array}$};\n\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (k) -- (l);\n\t\t\t\\draw[big arrow] (l) -- (k);\n\t\n\t\n\t\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (d1) -- (d2);\n\t\t\t\\draw[big arrow] (d2) -- (d1);\n\t\t\\end{tikzpicture}\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=2$ geometries.}\n \\label{fig:2a}\n \\end{figure}\n \n \n \\clearpage\n \n\n \\begin{figure}\n \t \\begin{center}\n $\n \\begin{array}{c}\n \\begin{array}{c}\n \t\\begin{tikzpicture}[yscale=1.2]\n\t\t\\node[](a1) at (-4,-3) {$ \\begin{array}{c} \\text{Bl}_{1} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\mathbb P^2 \\end{array} $};\n\t\t\\node[](b1) at (-.5,-3) {$ \\begin{array}{c} \\mathbb F_5 \\overset{2 \\ell - X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 & SU(3)_{4} \\\\\\hline \\end{array}\\end{array}$};\n\t\t\\draw[big arrow] (a1) -- (b1);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (b1) -- (a1);\n\t\t\\node[](a) at (3,-3) {$ \\begin{array}{c} \\text{Bl}_1 \\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2 \\end{array}$};\n\t\t\\node(ee) at (6.5,-3) {$\\begin{array}{c} \\mathbb F_2 \\overset{\\ell-X_1}{\\cup} \\text{dP}_1 \\end{array}$};\n\t\t\\node[](b2) at (1.5,.5) {$\\begin{array}{c} \\mathbb F_{2b} \\overset{F + (b-1) E}{\\cup} \\mathbb F_0 \\\\ \\begin{array}{|c|c|c|} \\hline \n\t\tb=1 & F,E & SU(3)_1 \\\\\\hline b=2 &F,F & SU(3)_3 \\\\\\hline b=3 &F, E & SU(3)_5 \\\\\\hline b = 3 &F,F & Sp(2)_{\\pi} \\\\\\hline b = 4 & F,E & SU(3)_7 \\\\\\hline b = 4 &F,F &G_2 \\\\\\hline b = 5 & F,E & SU(3)_9\\\\\\hline b = 5 & F, F & A^{(2)}_2 \\\\\\hline \\end{array} \\end{array}$};\n\t\t\\draw[big arrow] (a) -- (ee);\n\t\t\\draw[big arrow,transform canvas={yshift=-.5em}] (ee) -- (a);\n\t\\end{tikzpicture}\n\t\\end{array}\n\t\\\\ \n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\t\\node at (-2.5,0) {$\\begin{array}{c} \\mathbb F_3 \\overset{\\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell - X_1 & SU(3)_2 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node[](a) at (1,0) {$ \\begin{array}{c} \\mathbb F_6 \\overset{2 \\ell}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline\n\t\tF,\\ell - X_1 &Sp(2)_0\\\\\\hline \\end{array}\\end{array}$};\n\t\t\t\\node at (4.5,0) {$\\begin{array}{c} \\mathbb F_1 \\overset{X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F,\\ell - X_1 & SU(3)_0 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\t\\node at (8,0) {$\\begin{array}{c} \\mathbb F_7 \\overset{3\\ell - 2X_1}{\\cup} \\text{dP}_1 \\\\ \\begin{array}{|c|c|} \\hline F, \\ell - X_1 & SU(3)_6 \\\\\\hline\\end{array} \\end{array}$};\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\begin{array}{c}\n\t\t\\begin{tikzpicture}\n\t\t\n\t\t\n\t\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{tikzpicture}\n\t\\end{array}\t\n\t\\end{array}\n$\n\\end{center}\n \\caption{$M=1$ geometries.}\n \\label{fig:1}\n \\end{figure}\n \n\n \\begin{figure}\n \\centering\n $\n \t\\begin{array}{ccc}\n\t\t\\mathbb F_3 \\overset{\\ell}{\\cup} \\mathbb P^2 \n\t& ~~~~&\n\t\\mathbb F_6 \\overset{2\\ell}{\\cup} \\mathbb P^2\n\t\\\\ \\\\\n\t\\end{array}\n$\n \\caption{$M=0$ geometries.}\n\\label{fig:0}\n \\centering\n \\end{figure}\n \n\n\n\n\n\n\\subsubsection*{6d Theories on a Circle}\nIn this section we show that the complicated web of theories we have uncovered are actually unified from the perspective of 5d Kaluza-Klein (KK) theories arising from 6d SCFTs compactified on a circle (up to possible automorphism twists and holonomies).\n\nAs discussed in Section \\ref{sec:rank1}, shrinkable rank 1 geometries are classified by del Pezzo surfaces $\\text{dP}_{n\\leq 8}$ and $\\mathbb{F}_0$ up to physical equivalence. Interestingly, all of them can be obtained via geometric RG flows from $\\text{dP}_9$ (equivalently, $\\frac{1}{2}$K3). The local $\\text{dP}_9$ model is an elliptic 3-fold engineering the 6d SCFT called the `E-string theory'. Therefore all rank 1 5d SCFTs are descendants (i.e. related by rank preserving mass deformations) of the 6d E-string theory compactified on a circle.\n \nWe also find that all rank 2 5d SCFTs have 6d origin, but the rank 2 case is significantly more elaborate than the rank 1 case. Geometric constructions produce 5d SCFTs belonging to the four distinct families displayed in Table \\ref{tb:rank2-classification}. The geometries of type $(\\cdot)^*$ are not shrinkable but rather 5d KK theories~\\footnote{These theories are also called \\emph{marginal} theories \\cite{Jefferson:2017ahm}.}. We expect that these geometries correspond to 6d SCFTs compactified on a circle, possibly with automorphism twists.\n\n\nOne distinguished property of geometries corresponding to 5d KK theories is that there must exist an elliptic curve class whose volume is not controlled by normalizable K\\\"ahler moduli. The M2-branes wrapping this elliptic class correspond to KK momentum states. For example, the canonical class $-K_{\\text{dP}_9} \\subset\\text{dP}_9$ is an elliptic class with zero volume associated to the KK momenta of the E-string theory compactified on a circle. Another important property is that some KK geometries contain fiber classes forming an affine gauge algebra. Namely, we can find fiber classes $f_i$ such that\n\\begin{equation}\n\t-f_i \\cdot S_j = (A_{\\hat{G}})_{ij},\n\\end{equation}\nwhere $\\hat{G}$ denotes an affine gauge algebra. This signals that the corresponding geometry is an elliptic geometry realizing a 5d KK theory. We will now identify 6d origins of the geometries in Table \\ref{tb:rank2-classification} using these properties.\n\nWe begin with $\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0$. This geometry has two gauge theory descriptions, namely $SU(3)_0 + 10 \\textbf{F}$ and $Sp(2)+ 10 \\textbf{F}$. The 6d origin of these gauge theories is discussed in \\cite{Yonekura:2015ksa,Hayashi:2015fsa,Gaiotto:2015una,Hayashi:2016abm}. These theories are a circle reduction of the 6d $(D_5,D_5)$ conformal matter theory introduced in \\cite{Heckman:2013pva,DelZotto:2014hpa}. The geometry $\\text{Bl}_{10}\\mathbb{F}_6\\cup \\mathbb{F}_0$ realizes the circle compactification of this 6d theory. This theory has another duality frame in which an affine gauge algebra is manifest. To see this, choose the fiber classes $f_1=H+2F-\\sum_{i=1}^{10}X_i $ and $f_2= F$. These fiber classes indeed form the affine $\\hat{A}_1$ Cartan matrix:\n\\begin{equation}\n\t-(f_i \\cdot S_j) = \\begin{pmatrix} 2 & -2 \\\\ -2 & 2 \\end{pmatrix} .\n\\end{equation}\n\nAnother geometry $\\mathbb{F}_2\\cup \\text{dP}_7$ is interesting for similar reasons. This geometry admits three different gauge theory descriptions corresponding to the following choices of fiber classes:\n\\begin{align}\n\\begin{split}\n f_1&= F,~~ f_2= \\ell - X_2 \\ \\ \\rightarrow \\ \\ SU(3)_4+6{\\bf F} \\ , \\\\\n f_1&= F,~~ f_2= 2\\ell - \\sum_{i=2}^5X_i \\ \\ \\rightarrow \\ \\ Sp(2)+2{\\bf AS}+4{\\bf F} \\ , \\\\ \n f_1&= F,~~ f_2= 3\\ell - \\sum_{i=2}^6X_i -2X_7 \\ \\ \\rightarrow \\ \\ G_2+6{\\bf F} \\ .\n \\end{split} \n\\end{align}\nHere, the two surfaces are glued along the curves $C_{S_1}=E$ and $C_{S_2}=\\ell-X_1$. This implies new dualities between these three gauge theories and their descendants obtained by RG-flows induced by relevant mass deformations. In addition, we find another distinct duality frame:\n\\begin{equation}\n\tf_1 = F \\,, \\ f_2 = 5\\ell -X_1-2\\sum_{i=2}^7X_i \\ .\n\\end{equation}\nThe fiber classes in this last frame form the affine Cartan matrix $ A^{(2)}_2$:\n\\begin{equation}\n\t-(f_i\\cdot S_j) = \\left(\\begin{array}{cc} 2 & -1 \\\\ -4 & 2 \\end{array}\\right) \\ .\n\\end{equation}\nThis algebra $ A^{(2)}_2$ is obtained by an outer automorphism twist of the affine $A^{(1)}_2=\\hat{A}_2$ algebra which identifies ${\\bf 3}$ and $\\bar{\\bf 3}$ representations in $A_2\\subset \\hat{A}_2$. Therefore, one can expect that this geometry is also a KK geometry corresponding to a 6d $SU(3)$ gauge theory compactified on a circle with an outer automorphism twist. The unique 6d theory satisfying these properties is the 6d $\\mathcal{N}=(1,0)$ SCFT with $SU(3)$ gauge group and $N_\\textbf{F}=12$ fundamental hypermultiplets. Circle compactification of this 6d theory with an outer automorphism twist of the $SU(3)$ gauge algebra leads to a 5d theory with affine $A^{(2)}_2$ gauge algebra and 6 flavors. This interpretation agrees with the geometric model $\\mathbb{F}_2\\cup \\text{dP}_7$. Therefore, we conclude that $\\mathbb{F}_2\\cup \\text{dP}_7$ is a `KK geometry' engineering the circle compactification of the 6d $SU(3)$ theory with $N_\\textbf{F} = 12$.\n\n$\\mathbb{F}_6\\cup \\text{dP}_4$ is also a KK geometry. When one chooses the fiber classes $f_1=F_1,f_2=\\ell-X_1$ (with the gluing curve $C_{S_2}=2\\ell$), this geometry has a gauge theory description as $Sp(2)_{0}+3{\\bf AS}$. However, if we choose the fiber classes $f_1=F,f_2=2\\ell-\\sum_{i=1}^4X_i$, their intersections with the irreducible components $S_i$ form the affine $A^{(2)}_2$ Cartan matrix, up to sign. This suggests that $\\mathbb{F}_6\\cup \\text{dP}_4$ is a KK geometry. Indeed we find that the 6d $SU(3)$ gauge theory with $N_\\textbf{F}=6$ can give rise to the 5d KK theory associated to this geometry upon circle reduction with an outer automorphism twist.\n\n$\\mathbb{F}_{10}\\cup \\mathbb{F}_0$ is yet another KK geometry constructed by our building blocks. This geometry admits two dual descriptions related to the base-fiber exchange symmetry of $\\mathbb{F}_0$. One description is $SU(3)_9$, while the other is the $ A^{(2)}_2$ gauge theory description without matter hypermultiplets. We anticipate that this affine $ A^{(2)}_2$ gauge theory is the 5d KK theory coming from the 6d theory $\\mathcal{O}(-3)$ minimal SCFT with $SU(3)$ gauge group compactified on a circle with an outer automorphism twist of the $SU(3)$ gauge algebra.\n\nLastly, $\\text{Bl}_9\\mathbb{F}_4\\cup \\mathbb{F}_0$ is a KK geometry. This geometry is formed by gluing two surfaces along $C_{S_1}=E$ in $\\text{Bl}_9\\mathbb{F}_4$ and $C_{S_2}=F+H$ in $\\mathbb{F}_0$. We find that this geometry involves an elliptic fiber class given by $E+2X$ (with $E^2=-4,X^2=-1,E\\cdot X=2$) in $\\text{Bl}_9\\mathbb{F}_4$ which signals that this geometry is an elliptic CY 3-fold. \nIn the 5d reduction, this geometry has two gauge theory descriptions as predicted in \\cite{Jefferson:2017ahm}: $SU(3)_{\\frac{3}{2}}$ with $N_{\\bf F}=9$ and $Sp(2)$ with $N_{\\bf AS}=1,N_{\\bf F}=8$. This geometry is associated to the 6d rank 2 E-string theory on a circle. This becomes clearer after a flop transition with respect to the exceptional curve $X$. The flop transition described in Section \\ref{sec:transitions} leads to $\\text{dP}_9\\cup \\mathbb{F}_0^{g=1}$ geometry where we glue the anticanonical class in $\\text{dP}_9$ to the elliptic class $E$ (with $E^2=0$) in $\\mathbb{F}_0^{g=1}$. This is the rank 2 generalization of $\\text{dP}_9$ (or the 6d rank 2 E-string theory).\n\n\nAll top geometries in Table \\ref{tb:rank2-classification} come from 6d SCFTs. \nWe also claim that all smooth rank 2 3-folds engineering 5d SCFTs belong to one of the RG-flow families exhibited in Table \\ref{tb:rank2-classification}.\nTherefore, we deduce the following conclusion: \\emph{All rank 2 5d SCFTs realized by smooth non-compact 3-folds have 6d SCFT origins.}\n\nThis is one of the most important lessons from our classification of rank 2 5d SCFTs.\nThe same conclusion may hold also for singular geometries involving $\\text{O7}^+$-planes. As mentioned earlier, the classification of smooth 3-folds misses a single geometry corresponding to the theory $SU(3)_{\\frac{1}{2}} + 1\\textbf{Sym}$, despite the fact that this theory is known to have a brane construction involving $\\text{O7}^+$-planes \\cite{Hayashi:2015vhy}. This theory may be the only rank 2 SCFT which cannot be engineered by a smooth 3-fold. But, we also know that this theory can be obtained from a KK theory with 6d origin, so we have found no counterexamples to the notion that all rank 2 5d SCFTs come from 6d SCFTs.\n\nThe above discussion motivates classifying automorphisms of 6d SCFTs which lead to 5d KK theories, as in \\cite{Apruzzi:2017iqe}. Given the fact that 6d SCFTs are already classified (not counting frozen singularities involving $\\text{O7}^+$ planes), the possible automorphisms can be deduced from symmetries of the tensor branch diagrams of 6d SCFTs dressed by gauge symmetries which respect the automorphisms.\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Ron Donagi, Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Dave Morrison, Kantaro Ohmori and Gabi Zafrir for useful comments and discussions. We also like to thank SCGP summer workshop 2017 for hospitality during part of this work. The research of P.J. and H.K. and C.V. is supported in part by NSF grant PHY-1067976. S.K. is supported by NSF grant DMS-1502170.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nTranslocation features of polymers through natural and artificial\npores is a current active research topic in biophysics and\nnanotechnology~\\cite{KasPNAS96,RMP,NL}. Motivated by many broad\ninterest experimental results, different models have been introduced\nto describe and study in a simple way this and related problems. For\ninstance, single barrier potentials \\cite{Pizz}, as well as flashing\nratchet models \\cite{linke}, have been studied to describe the polymer\ntranslocation and polymer transport dynamics. The passage of small\nmolecules through passive cell channels can be also modeled by\nstochastic and rachetlike forces \\cite{shulten}. In some cases the\ntransport phenomena involves not translocation through pores, but also\nmolecular motors, whose complex action has been recently addressed at\nhigh attention \\cite{Bust,Bust09}. In addition, nanotechnological\napplications try to emulate the complex biological process related to\nthe translocation dynamics \\cite{mickler, starikov}.\n\nRecently, we have studied different models for the 1d translocation of\na spring-bead polymer helped by a motor using a sinusoidal\nforce~\\cite{fjf-sine}. The introduction of a time dependent driving\nforce imposes a new time scale on the system, and provides new and\nricher phenomenology: for sinusoidal driving, the translocation time\nshows an oscillatory behavior as a function of the frequency.\n\nIn order to introduce stochasticity in the motor action and motivated\nby the relevant role played by dichotomous noise in biological\nproblems, in this manuscript we consider the case of a polymer driven\nby a two-state force: constant force which pushes the polymer chain in\none direction during the activity of the motor, and zero force which\nleaves the polymer to diffuse freely otherwise. This pure dichotomous\nmechanism constitutes a first approach in describing a machine working\ndichotomously between two on-off states~\\cite{gomez,fjf-damn}.\n\nThe motor modeled in \\cite{gomez,fjf-damn} acts during a fixed time,\nwhile the waiting times are exponentially distributed with a mean time\ndepending on the ATP concentration. In the present work a simpler\ndichotomous mechanism which can well point out, by contrast, the\nspecific behavior of the ATP based machines is studied.\n\nOn the other hand, pure dichotomous driving makes sense in the\nnanotechnological context as well as in the biophysical one. In the\nfirst case the passage of a polymer can be induced through a graphene\npore or solid state channeling \\cite{han99,luan2010} by applying a\ndichotomous force between the two sides of the layer. In the second\ncase, the model can describe the translocation of a linear molecule\nthrough a cell membrane gate having a chemical potential difference\nbetween their two sides. The driving is in this case induced by the\ntypical open\/close mechanism which follow the purely dichotomous\nswitching largely used in literature \\cite{shulten,Millonas,kargol}.\n\nThe purpose of our work is to model phenomenologically the possible\nphysical systems described above. We want to stress here the\nqualitative specific results connected to the purely dichotomous\ndriving.\n\nThus, differently from the sinusoidal case, no special behavior is\nobserved in the mean translocation time of the polymer for the case\nhere studied. However, for this problem, another observable parameter\ncan be studied. In fact, single molecule experiments are able to\ndetect and use the instantaneous velocity in order to quantify the\ntranslocation process in forced systems \\cite{Bust}. Remarkably, we\nfind a non trivial behavior of the polymer translocation velocity as a\nfunction of the mean frequency $\\nu$ of the driving with the presence\nof a maximum, even if the translocation time shows only a monotonic\nbehavior. This difference reveals the importance of dealing with\nseveral measures to explore the complex behavior of the polymer\ntranslocation.\n\nThe dependence of the stall force $F_{stall}$ of the machine is also\ncalculated. We find, again, a strong nonmonotonic behavior of\n$F_{stall}$ with the frequency, similar to the one found in\n\\cite{fjf-sine}.\n\nThe paper is organized as follows: first we present the model for\npolymer and the properties of the stochastic driving force. The main\nproperties of the translocation process are then calculated:\ntranslocation time, mean velocity and stall force. Finally, we analyze\nthe dependence of the above properties with the chain stiffness.\n\n\\section{The model}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=8.2cm]{SchemaRTN.eps}\n\\includegraphics[angle=-90,width=9.5cm]{RTN.eps}\n\\caption{Scheme of a linear chain driven by a dichotomous force\n restricted to a small space region (width $L_M$). $T$ is the mean\n time during which the force maintains a same value\n $\\{0,F_M\\}$.}\n \\label{schema}\n\\end{figure}\n\nThe polymer is modeled as a unidimensional chain of $N$\ndimensionless monomers connected by harmonic springs \\cite{Rouse}.\n\n1d models are suitable in order to describe the dynamics of polymers\nconstrained to move in confined channels \\cite{luan2010}. Also, in\nmany experimental situations [7] the polymer is stretched, thus\nremoving the dimensionality dependence of the measured\nquantities. Moreover, in this work, we want to fix our attention to\nthe motor activity in the translocation more than the delay given by\nother effects, such as entropic contributions.\n\nThe total potential energy is\n \\begin{equation}\n V_{\\rm har}=\\frac{k}{2}\\sum_{i=1}^{N} (x_{i+1}-x_i-d_0)^2,\n \\label{v-har}\n \\end{equation}\n\\noindent where $k$ is the elastic constant, $x_i$ the position of the\n$i$-th particle, and $d_0$ the equilibrium distance between adjacent\nmonomers.\n\nThe translocation is helped by the presence of a motor which is\nactivated dichotomously. The machine has a spatial working width $L_M$\nand the position $x=0$ represents the right edge of its action (see\nFig.~\\ref{schema}). Thus the monomers $i$ such that $x_i \\in [-L_M,0]$\nexperience a force made by the motor. We define $\\eta(x,t)$ to\nrepresent the dichotomous force, which fluctuates between two values\n$0$ (no force) and $F_M$. Thus \\begin{equation} {\\langle \\eta(t)\n \\rangle}=\\frac{F_M}{2} \\; \\; \\, \\; {\\rm and} \\; \\; \\; {\\langle\n \\eta(t) \\eta(t') \\rangle}=\\frac{F_M^2}{4}(1+ e^{-2\\frac{(t'-t)}{T}})\n \\label{eta-corr}\n \\end{equation}\nHere $T$ gives the mean residence time in each state. With\n respect to the spatial dependence\n\\begin{equation} \\eta(x) = \\left\\{ \\begin{array} {lr}\n F_M & x \\in [-L_M,0] \\vspace{0.2cm}\\\\\n 0 & otherwise\n \\end{array} \\right.\n \\label{Potential-s}\n \\end{equation}\n\n\n\nThe dynamics of the $i^{\\rm th}$ monomer of the chain is then\ndescribed by the following overdamped Langevin equations:\n \\begin{equation}\n \\dot{x}_i = -\\frac{\\partial{V_{har}}}{\\partial{x_i}} + \\eta(t,x_i) + \\xi_{i}(t)\n \\label{lang}\n \\end{equation} where the viscosity parameter for each monomer is included in the\n normalized time units. $\\xi_{i}(t)$ stands for Gaussian uncorrelated\n thermal fluctuation and follows the usual statistical properties\n $\\langle\\xi_i(t)\\rangle=0$ and $\\langle\\xi_i(t)\\xi_j(t')\\rangle = 2 D\n \\delta_{i j}\\delta(t'-t)$.\n\n\n\\section{Results}\n\nWe performed a set of $N_{exp}=20,000$ numerical experiment with a\nstochastic Runge-Kutta algorithm, using a time step of $dt=0.01$. The\npolymer is compound by $N$ monomers and starts with all the spring at\nthe rest length ($d_0=1$), and the last monomer of the chain lies at\n($x_N=0$), just in the final action range of the dichotomous\nforce. The noise intensity is held fixed at the value $D=0.001$,\n$L_M=5.5$, and $N=12$. The choice of the number of monomers $N$, or\nequivalently the length $L$, is arbitrary and this small number has\nbeen used for computational convenience. We note that in a previous\nwork \\cite{fjf-sine}, also with 1d chain, it was found that $\\tau$\nscales with $L^2=(N-1)^2$. Similarly we find that $v$ scales with\n$1\/N$.\n\nIn this first part, the elastic constant $k$ is held equal to 1, a\nmeaningful choice that corresponds to a not too rigid approximation\nfor the polymer. We will study the main observables of the system as a\nfunction of the mean frequency transition $\\nu=1\/T$.\n\n\\subsection{Translocation times}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{tauRTN-OD.eps}\n\\caption{Polymer translocation time $\\tau$ as a function of the\n frequency $\\nu$ of the fluctuating force. The solid line shows the\n theoretical prediction of Eq.~\\ref{theor}. In the inset is shown the\n standard deviation of the exit times distribution.}\n\\label{RA}\n\\end{figure}\n\nThe translocation time $\\tau$ is computed as a mean first passage time\nof the center of mass of the polymer: the average over the $N_{exp}$\nrealizations of the time spent by the center of mass of the chain to\nreach the position $x=0$. In Fig.~\\ref{RA} we see the value of $\\tau$\nplotted as function of the mean frequency of the driving, and, in the\ninset, the standard deviation $\\sigma$ whose values are of the same\norder of magnitude than the mean time values, as expected. We find\nthat $\\tau$ is a monotonic function of $\\nu$. This result is different\nfrom that for a periodic force (sinusoidal or square wave ones) where\nit is observed a minimum in the translocation for $\\nu \\sim 10^{-2}$\nand an oscillating behavior for higher frequencies~\\cite{fjf-sine}.\n\nIn contrast with the behavior of the translocation time, as we will\nsee, the velocity is not a monotonic function of $\\nu$ and a maximum\nis found in this function for $\\nu \\sim 10^{-2}$ (see\nFig.~\\ref{vRTN}). Both effects (minimum translocation\ntime~\\cite{fjf-sine} or maximum mean velocity) reveal some interesting similarities with the resonant activation phenomenon~\\cite{Pizz2,RA}.\n\nWe can make a simple analytical prediction for $\\tau(\\nu)$ in the low\nfrequency region which however is found to be valid in a broad\nfrequency range (see solid line in Fig.~\\ref{RA}). Let $\\tau_{on}$ be\nthe value of the exit time when a constant force $F_M$ is applied\nduring all the dynamics. In the $\\nu \\to 0$ limit we have to\ndistinguish between two cases depending on the initial value of the\nforce, $F_M$ or $0$. In the first case the translocation time is\n$\\tau_{on}$ corrected in a first approximation by a long waiting time\n$T$ if the system switches to the {\\em off} state before $\\tau_{on}$,\nwhich occurs with a probability $p_s=1-e^{\\tau_{on}\/T}$. This\ncorrection gives a contribution of $\\tau_{on} (1-p_s)+ (T+\\tau_{on})\np_s$ to the total time. In the second case there is an additional time $T$ in the off state for escaping. Thus the total translocation time\nis\n \\begin{equation}\n \\tau \\simeq \\frac{1}{2}(\\tau_{on} + T (1-e^{\\tau_{on}\/T})) +\n \\frac{1}{2} (\\tau_{on} + T + T (1-e^{\\tau_{on}\/T}))\n \\label{theor1}\n \\end{equation}\nSince this equation is derived in the low frequency limit where\n$1-e^{\\tau_{on}\/T} \\simeq \\tau_{on}\/T$ we have\n \\begin{equation}\n \\tau \\simeq 2 \\tau_{on} + T\/2\n \\label{theor}\n \\end{equation}\nThe intermediate frequency region is characterized by the presence of\nthe constant force alternated by the absence of the force (diffusive\ndynamics) with an average time ratio between them different for\ndifferent values of the mean frequency. Surprisingly,\nEq.~(\\ref{theor}) also describes in a good way that frequency region.\nThe third region is instead characterized by a high frequency\nswitching rate between the two force states. There the translocation\ntime is much smaller than $T$, the polymer experiences a mean force\n$F_M\/2$, and $\\tau \\simeq 2 \\tau_{on}$. A careful observation of our\nnumerical results show that in this high frequency regime\n \\begin{equation}\n \\tau \\simeq 2 \\tau_{on} + T ,\n \\label{theorH}\n \\end{equation} not observable in Fig.\\ref{RA}.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{vRTN-OD.eps}\n \\caption{Mean velocity and number of monomers inside the motor while\n in its active state (inset) as a function of the mean frequency\n $\\nu$ of the fluctuating force.}\n \\label{vRTN}\n\\end{figure}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{Pt2.eps}\n\\caption{First passage time (left) and velocity (right) probability\n distributions at four frequency values.}\n\\label{P}\n\\end{figure}\n\n\\subsection{Mean velocity}\n\nFig.~\\ref{vRTN} shows the mean velocity $v$ of the polymer as a\nfunction of $\\nu$. The inset of the figure shows the average number of\nmonomers inside the motor during the active states, $n_{mot}$. We can\nsee that this number is not constant for different values of the mean\nfrequency $\\nu$, at least for the value of the elastic constant $k=1$\nused in those calculations.\n\nThe main result in the velocity curves is the presence of a well\npronounced maximum, which put in evidence the qualitative difference\nbetween the calculation of the mean first passage time and the mean\nvelocity. In fact, the velocity is computed as $v_{cm}=1\/N_{exp}\n\\sum_i^{N_{exp}} L\/t_i$, where $t_i$ is the escape time in the $i$-th\nrealization.\n\nAs visible in Fig.~\\ref{P}, the exit times distribution changes in\nshape by changing the mean driving frequency $\\nu$. For high values of\n$\\nu$, the time distribution is very narrow around its mean value\n$\\tau$. The corresponding probability distribution function for the\nvelocity is also a narrow function. Decreasing the value of $\\nu$, the\ndistributions are more asymmetric and the width increases. The maximum\nof the time distribution moves toward lower values of time, but the\nasymmetry changes and higher and higher values of translocation times\nare involved. That's why in that region the mean first passage time\n$\\tau$ increases, although the time of the $P(t)$'s maximum\ndecreases. The distribution of the velocity change as well; but\nbecause of the increased width of the time distribution, the mean\nvalue of the velocity in that region does not follow the relation\n$v_{med}=1\/\\tau$ and increases with respect to the high frequency\nlimit, in opposite direction as the one expected from the time\nbehavior. The reason of this effect is that in the average, the\nsmaller times have a higher weight in the inverse $1\/t_i$ than bigger\nones. Thus the mean velocity increases up to a maximum. Decreasing\n$\\nu$ in the low frequency region, the average of the times continues\nrising up, because the distribution involves higher and higher\ntimes. The velocity, however, now decreases since the very high times\nescapes do not contribute importantly to the mean velocity.\n\nIn a first approach the translocation velocity in the high frequency\nlimit is given by \\begin{equation} v_l=\\frac{F_M}{2} \\frac{n_{mot}}{N}, \\end{equation} a\nfraction of monomers given by $n_{mot}\/N$ experience a force\n$F_M\/2$. Then, the corresponding translocation time is $2 \\tau_{on}$\n(remind that $\\tau_{on}$ is the escape time if the motor is always\nworking). On the contrary, in the low frequency limit one half of the\nrealizations give a very long escape time (and velocity goes to zero)\nand another half give $\\tau_{on}$. Thus for low frequencies we obtain\nthe same value of the velocity that for high frequencies.\n\nHowever, we can see in Fig.~\\ref{vRTN} that the low frequency limit of\nthe mean velocity does not satisfy the relationship just derived,\nbeing lower than the high frequency value $v_l$. This happens because\nthe force exerted on the polymer is affected by the number of monomers\ninside the motor which, as shown in the inset of the figure, also\ndepends on $\\nu$. We will see below a confirmation of the given\nrelation by using a strong elastic constant between the monomers,\nwhich guarantees a constant number of monomers inside the motor\nnevertheless the dynamical conditions are (see Fig.~\\ref{ka}).\n\nFrom Eq.~(\\ref{lang}) it is easy to derive the following equation for\nthe mean velocity \\begin{equation} v=\\frac{1}{N} \\frac{1}{N_{exp}}\n\\sum_{i=1}^{N_{exp}} \\langle \\eta_i(t) \\rangle_T = \\frac{F_M}{N}\n\\frac{1}{N_{exp}} \\sum_{i=1}^{N_{exp}} \\frac{n_{\\rm mot, i}^{\\rm\n on}(\\nu) t^{on}_i} {t_i},\n \\label{v_t}\n \\end{equation}\nwhere, for each experiment $i$, $n_{\\rm mot, i}^{\\rm on} (\\nu)$ is the\naverage number of particles inside the motor when the motor is $on$,\n$t^{on}_i$ is the total motor working time, and $t_i$ is the\ntranslocation time of each realization.\n\nAt high frequency, $t_i^{on}=\\frac{1}{2}t_i$. However, decreasing the\nfrequency for most of the cases, $t_i^{on}>\\frac{1}{2}t_i$, during the\ntranslocation the motor spends more time activated that deactivated\nsince most translocations happen during the activation stage of the\nmotor. Thus both, the translocation time and the mean velocity\nincrease\\footnote{This is not the case at low values of $k$, where\n the strong change in $n_{mot}$ with the frequency dominates the\n overall behavior and suppress the velocity maximum as shown in\n the inset of Fig.~\\ref{ka} for $k=0.1$.}. This behavior changes when $1\/\\nu \\sim\n\\tau_{on}$. Then $t_i^{on}$ remains constant in Eq.~(\\ref{v_t}), $t_i$\nincreases when $\\nu$ decreases and the velocity also decreases towards\nthe expected $v_l$ value moderate by the mean number of monomers in\nthe motor in the low frequency limit. This explains the presence of\nthe maximum in the velocity.\n\nA rough estimation of $n_{\\rm mot}$ is given by the fixed value\n$n_{\\rm mot}=5.5$, corresponding to the distribution of monomer inside\nthe motor in the case that they maintain the same relative distance,\nequal to the rest, over all the dynamics. This condition will be\ncompletely satisfied for high values of the elastic constant $k$\n(rigid chain limit), when $n_{mot}$ becomes independent on $\\nu$. As\nwe will see below, both the high and low frequency limits for the mean\nvelocity take in that case the same value (see the inset of\nFig.~\\ref{ka})\n\n \\[v_{l, theor}= \\frac{0.2 \\cdot 5.5}{2 \\cdot 12}= 0.04583,\\]\nwhich is slightly higher than the limit value $v_l$ shown in the inset\nof Fig.~\\ref{RA} because $n_{mot}(k=1)<5.5$.\n\n\n\\subsection{Stall Force}\n\nThe stall force $F_{stall}$ is the force that we need to apply against\nthe motor in order to stop the polymer translocation. It is a measure\nof the strength of the motor and, in this model, it depends on the\nfrequency of the driving.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=9.5cm]{SchemaMotorePullCM.eps}\n\\caption{A force pull is applied at the first monomer to measure the\n motor stall force.} \\label{stall2}\n\\end{figure}\nA set of simulations have been performed by applying a pull force\n$F_p$ (see Fig.~\\ref{stall2}) on the left extremum of the chain, in\nopposite direction to the motor driving force. \u00a0The initial condition\nfor the chain has been fixed with the polymer center of mass in the\ncenter of the motor. Then, the velocity of the center of mass is\nmeasured waiting for the exit on the left or on the right of the motor\nregion. That way, the force for which the mean velocity is zero gives\n$F_{stall}$.\n\nFig.~\\ref{StallForce} shows the stall force as a function of the\nfrequency. As shown in the lower inset, for a given frequency the mean\nvelocity decreases linearly with $F_p$. The upper inset, shows that\nfor pull forces of the order of the stall force the velocity presents\na minimum, contrary to the behavior at $F_p=0$ (Fig.~\\ref{vRTN}). Then\nthe stall force, which presents a similar trend, shows a clear minimum\nin the same frequency region.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{f_v_OD.eps}\n\\caption{Stall force as a function of the frequency $\\nu$ of the\n driving. The upper inset shows the mean velocity as a function of\n $\\nu$ for the three pull force values $F_p=0.447, 0.454, 0.465$. The\n bottom \u00a0inset shows the linear behavior of the mean velocity as a\n function of the pull force for the three frequency values\n $\\nu=0.032, 0.02, 0.1$. The other parameters are the same of\n Fig.~\\ref{RA}.}\n\\label{StallForce}\n\\end{figure}\n\n\nAs in the oscillating case \\cite{fjf-sine}, the scale variation of the\nstall force is small (around $7.5\\%$), and an experimental\nverification of its behavior with the mean frequency the minimum could\nbe not immediately simple to perform.\n\n\\subsection{Elastic constant dependence}\n\nFinally, we investigate the dependence of translocation time and\nvelocity on the elastic constant $k$ of the polymer. A magnitude that\nstrongly depends on $k$ is the mean number of monomer inside the motor\nduring the pushing cycle, $n_{mot}$. This number modules the velocity\nas it is show in Eq~(\\ref{v_t}). Results are plotted in Fig.~\\ref{ka},\nwhere the translocation time and velocity (in the inset) are presented\nfor different value of $k$.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[angle=-90, width=9.5cm]{ka.eps}\n\\caption{Translocation time and mean velocity (in the inset) of the\n polymer chain for different values of the elastic constant $k$.}\n\\label{ka}\n\\end{figure}\nWe can see that notable differences (especially visible in the mean\nvelocity plot) are evident by changing the value of $k$. For the case\n$k=0.1$ the velocity looses the maximum, which is always present for\nthe higher values of $k$. As expected, a clear saturating behavior of\nthe whole curve is evident by increasing $k$ when the chain behaves\nlike a rigid bar. As announced before in the text, in this limit the\nmean velocity in the cases of both high and low switching frequency\ngives the expected value $v_{l,theor}$ given above. This limit is\nalready fulfilled for $k=5$.\n\n\n\\section{Conclusions}\nThe interest in the introduction of simple models is that they can\ncapture the more relevant features of different processes. In that\nway, they can result to be very useful for a coarse-grain description\nof different systems.\n\nThe model described here studies the translocation process of a\npolymer driven by a simple motor which exerts a dichotomous force. We\nanalyze the dependence of the translocation time with the mean\nfrequency of the driving field, and find an analytical expression for\nthe low frequency regime. In spite of the monotonic behavior of the\ntranslocation time, the velocity presents a clear maximum at a\nresonant value of the mean frequency. We argue that this maximum comes\nfrom the optimization of the \"on states\" duration of the driving\nforces with the corresponding translocation time. The detection of\nthis maximum, (also seen in the periodic case) could be tackled with\nthe recent single molecule experimental techniques.\n\nThe stall force able to stop the polymer translocation against the\nmotor has been also evaluated, finding in our calculations results\nvery close to the oscillating driving, previously studied. The stall\nforce show a very clear minimum at a resonant mean frequency of the\ndriving.\n\nThe model can have application in artificial nanotechnological devices\ndriven by dichotomously fluctuating fields, as well as biological pore\nmembrane with intrinsic noise.\n\n\n\n\\vspace{0.5cm}\nThis work has been supported by the project\nFIS2008-01240 of the Spain MICINN.\n\n\\vspace{0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Acknowledgment} %\nThe research presented in this paper was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - EXC 2075 - 390740016.\nThis work was possible due to kind funding by the Ministerium f\u00fcr Wissenschaft, Forschung und Kunst Baden-W\u00fcrttemberg via the project SiVeGCS-MWK.\nThe authors gratefully acknowledge the support and the computing time on ``Hawk'' provided by the HLRS through the project ``hpcdg'' and the support by the Stuttgart Center for Simulation Science (SimTech).\nMoreover, the authors gratefully acknowledge the SmartSim developers for their support.\n\n\n\n\n\\bibliographystyle{elsarticle-num}\n\n\\section{Reinforcement Learning - A Brief Outline}\\label{sec:rl}\n\nThe following section gives a brief outline of the general reinforcement learning (RL) paradigm.\nHowever, this summary is by no means exhaustive and provides only the bare fundamentals required to motivate our software implementation, which is introduced in \\secref{sec:software}.\nFor a more thorough discussion of RL, the reader is referred to \\cite{sutton2018reinforcement}.\nIn the RL paradigm, an agent trains by interacting with an environment, as illustrated in \\figref{fig:MDP}.\nIn each point in time $t$, the environment is in some state $s_t$, based on which the agent's (possibly parametrized) policy $\\pi_{\\theta}\\left(a\\:|\\:s_t\\right)$ prescribes which action $a_t$ the agent should perform.%\n\\footnote{In principal, the policy $\\pi_{\\theta}\\left(a\\:|\\:s=s_t\\right)$ is a random variable, which describes the conditional probability distribution of performing action $a$, given the state $s_t$. To keep the notation short, we will use $\\pi_{\\theta}\\left(a\\:|\\:s_t\\right) \\equiv \\pi_{\\theta}\\left(a\\:|\\:s=s_t\\right)$ and refer the reader again to \\cite{sutton2018reinforcement} for more details. The same holds for the transition function $\\mathcal{T}\\left(s_{t+1}\\:|\\:a_t,s_t\\right)\\equiv\\mathcal{T}\\left(s_{t+1}\\:|\\:a=a_t,s=s_t\\right)$.}\nThis action causes the environment to change its state to a new state $s_{t+1}$, which is determined by the environment's transition function $\\mathcal{T}(s_{t+1}\\:|\\:a_t,s_t)$.\nThe transition function thus encodes the dynamics of the environment, which could be for instance the spatial and temporal integration of the discretized Navier-Stokes equations.\nAlongside the new state $s_{t+1}$, the agent receives a reward $r_{t+1}=\\mathcal{R}(s_{t+1})$, which quantifies how favorable the transition is with respect to some performance metric.\nThe reward function $\\mathcal{R}(s)$ is highly problem-specific and has to be designed by a domain expert.\nBased on the new state $s_{t+1}$, the agent performs another action, until the environment reaches a final state $s_n$.\nEventually, such an episode results in a trajectory $\\tau$ of states, actions and rewards:\n\\begin{equation}\n \\tau = \\left\\{ \\left(s_0,a_0\\right),\\left(s_1,a_1,r_1\\right),\\:......\\;,\\left(s_{n},a_{n},r_{n}\\right)\\right\\}.\n \\label{eq:trajectory}\n\\end{equation}\nThis problem formulation is typically framed as a Markov decision process (MDP), which again can be interpreted as a discrete-time control task.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{fig\/RL_MDP.pdf}\n \\caption{General outline of the Markov decision process (MDP). At step $t$, the environment is in state $s_t$. Following its policy $\\pi_{\\theta}(a\\:|\\:s_t)$, the agent performs an action $a_t$. In deep RL, the policy is a deep artificial neural network (ANN) with parameters $\\theta$. The action causes the environment to transition from state $s_t$ to a new state $s_{t+1}$ that is prescribed by the environment's transition function $\\mathcal{T}(s\\:|\\:a_t,s_t)$. Based on how desirable the new state is, the agent receives a reward, which is determined by the reward function $r_{t+1}=\\mathcal{R}(s_{t+1})$.}\n \\label{fig:MDP}\n\\end{figure}\n\nThe behavior of the agent is described by its policy $\\pi_{\\theta}$, which determines the action the agent performs for each of the possible states of the environment.\nIn deep RL, this policy is represented by an ANN with the weights $\\theta$.\nThe key quantity to distinguish between favorable and unfavorable actions is the expected future return along a trajectory $\\tau$ with $n$ steps\n\\begin{equation}\n R(\\tau) = \\sum_{t=1}^{n} \\gamma^t r_{t} .\n\\end{equation}\nHere, $\\gamma \\leq 1$ is the discount factor, which balances the importance of short-term and long-term rewards.\nThe overall goal of the RL algorithm is then to find the optimal policy, i.e. the set of optimal model parameters, which maximizes the expected return on all possible initial states.\n\nThe key purpose of an RL algorithm is to state an optimization task that allows optimization of the policy based on sampled interactions of the agent with the environment.\nRL algorithms thus differ primarily in the way those interactions are sampled and how these interactions are used to update the policy.\nThroughout this work, we employ the clipping version of proximal policy optmization (PPO) \\cite{schulman2017proximal} as our RL algorithm of choice.\nWe highly recommend \\cite{notter2021hierarchical} for a clear and concise summary of the PPO method.\n\n\n\\section{Software Architecture}\\label{sec:software}\n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=\\linewidth]{fig\/Framework_paper.pdf}\n \\caption{General architecture of Relexi. Before entering the training loop, the SmartSim IL is used to launch the SmartSim Orchestrator, which provides a database to exchange data between Relexi and the HPC workload. At the beginning of each training loop, the SmartSim IL starts a batch of FLEXI simulations and distributes them to the \\textit{worker} nodes. The simulations are initialized on a randomly drawn initial state and are then evolved in time by the flow solver. For our application in turbulence modeling, FLEXI sends each time interval $\\Delta t_{RL}$ its current state to the agent and receives a new action, i.e. a new set of parameters for the turbulence model. Since a syncronous RL algorithm is applied here, Relexi waits until all simulations have terminated. The collected episodes $\\tau^{(i)}$ are then used by the gradient ascent algorithm to improve the model weights $\\theta$. Here, $\\nabla_{\\theta} J(\\tau^{(i)})$ is the gradient of the loss function, evaluated on the sampled experience. The training loop is then repeated with the new set of weights, i.e. the new policy, until the policy converges.}\n \\label{fig:relexi}\n\\end{figure*}\nIn this work, we present a novel and modular RL framework named Relexi\\footnote{Code available at: \\href{https:\/\/github.com\/flexi-framework\/relexi}{https:\/\/github.com\/flexi-framework\/relexi}} that implements an RL training loop for applications in the field of scientific computing, as illustrated in \\figref{fig:relexi}.\nAs discussed beforehand, coupling HPC codes with modern ML libraries is oftentimes tedious.\nThis holds especially for RL, where the interaction between the HPC application and the RL algorithm becomes much more intricate.\nA considerable amount of these difficulties stem from the differences in the required hardware and the different preferences in terms of programming languages.\nMany approaches in coupling HPC and ML tried to either rewrite basic ML capabilities in languages used for HPC or writing an HPC solver in Python from scratch.\nConsidering the enormous codebases on both sides and the enormous pace of progress in the ML community, this approach is, from our perspective, doomed to fail.\nIn addition, both families of methods require significant computational effort:\nWhile the parameter update of the policy requires access to the collected trajectories to compute the gradient vectors through backpropagation, running CFD simulations mandates high parallel efficiency, in particular for problems of practical interest.\nThus, compromising the performance of any one component, e.g. through re-writing, is not advisable.\nRelexi tries to bridge this gap differently by employing the SmartSim library to couple a state-of-the-art RL library with modern HPC solvers.\nSpecial focus is put on the modularity of the framework, which allows the exchange of RL algorithms and simulation environments with only minimal changes of the underlying code. \nRelexi is designed with application on HPC system in mind, scaling applications up to thousands of cores on modern HPC systems.\nRelexi comprises the following building blocks, which will be discussed in more detail in the following sections.\n\\begin{itemize}\n \\item \\textbf{SmartSim}: The SmartSim library \\cite{partee2021using} fulfills two major tasks. Firstly, the library is used to start and manage the MPI-parallelized HPC workloads on the allocated hardware resources. Moreover, it implements the communication between the HPC workload and Relexi by deploying an in-memory database and providing communication clients for several programming languages.\n \\item \\textbf{FLEXI}: In this work, the flow solver FLEXI \\cite{krais2021flexi} provides simulations of turbulent flow as a training environment for the agent. FLEXI is used here for illustrative purposes only and can be easily replaced by other simulation codes.\n \\item \\textbf{TensorFlow\/TF-Agents}: Relexi is build upon TensorFlow \\cite{tensorflow2015-whitepaper}, which provides a framework for efficient ML workflows and allows for distributed training on multiple GPUs in parallel. In addition, its RL extension TF-Agents \\cite{TFAgents} provides implementations of several RL algorithms and allows for custom environments that can be integrated in a straight-forward fashion.\n\\end{itemize}\n\n\n\\subsection{SmartSim}\\label{subsec:smartsim}\nThe SmartSim library \\cite{partee2021using} is a workflow library that simplifies the convergence of traditional HPC workloads and ML.\nThe library comprises two components: the SmartSim Infrastructure Library (IL) and SmartRedis.\nThe IL provides extensive functionalities to start, manage and distribute workloads in HPC environments as well as submitting workloads automatically as batch jobs to the job scheduler. \nIn Relexi, we employ the IL to repeatedly start the MPI-parallelized simulation instances on demand for each training iteration inside of a single allocated batch job.\nThe simulations can be either started individually or multiple simulations can be lauched within a single call by using the multiple-program-multiple-data (MPMD) paradigm.\nWe also use another key aspect of the SmartSim IL, which is its ability to configure and launch an in-memory, Redis-based datastore referred to as the Orchestrator. \nThis database serves as a data intermediary between the simulations and the main program.\nFor Relexi, we used a non-clustered Orchestrator, which is launched on the \\textit{head} node.\nSmartSim also supports the distribution of the Orchestrator across multiple nodes, but a single instance on the \\textit{head} node was sufficient for our application.\nThe simulation communicates with the Orchestrator using the SmartRedis clients (available in Python, Fortran, C, and C++) which can send and receive data from the orchestrator or trigger actions within the Orchestrator, i.e. running scripts or ML model inference previously loaded into the database.\nIn our case, during training, FLEXI sends its current flow state via the Fortran SmartRedis client to the Orchestrator.\nIn addition, a scalar flag is sent which indicates whether FLEXI has reached its final state and will terminate.\nRelexi then uses the Python SmartRedis client to read this data from the Orchestrator.\nSubsequently, the agent's actions are sent by Relexi to the database via its Python SmartRedis Client and are read by FLEXI via its Fortran client.\nWhile SmartSim itself employs a Redis database by default, we used the multi-threaded fork of Redis called KeyDB, which provided significantly more performance for our application.\n\n\\subsection{FLEXI}\\label{subsec:flexi}\nThe open-source flow solver FLEXI \\cite{krais2021flexi} is based on the discontinuous Galerkin (DG) method, which can be seen as a hybrid of the finite element and the finite volume methods.\nFor the DG method, the computational domain is partitioned into individual elements.\nIn each element, the solution is represented by a polynomial basis of degree $N$ and the individual elements are then coupled by consistent surface fluxes at the element faces.\nThis results in a small communication stencil, which allows FLEXI to scale perfectly on hundreds of thousands of cores using a pure distributed-memory parallelization with MPI.\nA detailed description of FLEXI can be found in \\cite{krais2021flexi}.\nThe communication between FLEXI and the database is implemented by means of the SmartRedis clients provided by the SmartSim library.\nTo this end, SmartRedis is linked to FLEXI during compile time.\nImplementing the data transfer in FLEXI is straight-forward and requires only a few lines of additional code.\nSince FLEXI is parallelized with a distributed-memory approach, every rank contains only a chunk of the global flow state, which represents the environment's state $s_t$ in the RL formulation.\nTo communicate with the database, the flow state is thus first gathered across all ranks of the respective FLEXI instance before it is written to the database by the root rank.\nAnalogously, the predictions sent by Relexi are retrieved from the database by the root rank and are then scattered across the other ranks of the respective FLEXI instance.\nIt seems important to stress once again that FLEXI is used primarily for demonstration purposes and can be easily replaced by other simulation codes by using the SmartRedis Clients.\n\n\n\\subsection{Relexi}\\label{subsec:relexi}\nRelexi implements the main RL training loop by means of the TF-Agents \\cite{TFAgents} library, as illustrated in \\figref{fig:relexi}.\nIn TF-Agents, custom environments can be implemented by subclassing the provided environment class and implementing a few mandatory methods like initialization and time-stepping.\nSince the TF-Agents library interacts with the environment via these pre-defined function interfaces, all intricacies of the coupling with the HPC workload are ``hidden'' inside of the environment class.\nThis has the advantage that all RL algorithms and tools provided by TF-Agents can interact natively with the custom environment and thus, with the HPC workload.\nThe hardware resources are distributed as follows.\nRelexi itself is started as a single thread on a node termed \\textit{head}.\nThe \\textit{head} node executes all ML-specific work and thus should be equipped with a GPU.\nRelexi also supports setups with multiple GPUs on the \\textit{head} node by using the distribution strategies provided by TensorFlow.\nThe SmartSim Orchestrator is also started on the \\textit{head} node at the beginning of Relexi.\nDuring training, the HPC workload is repeatedly started with MPI on the available \\textit{worker} nodes.\nTo ensure that each MPI rank is placed correctly on the available hardware and to avoid double occupancy, Relexi generates rankfiles on-the-fly based on the available hardware resources.\nThese rankfiles are then passed to MPI to ensure the correct placement of the MPI ranks.\n \nAfter the simulations are launched, each simulation instance writes its initial states to the database.\nRelexi reads these states, provides the respective actions based on the current policy and writes them back to the database.\nEach FLEXI instance reads its prescribed actions from the database and proceeds with its simulation to obtain the next state. %\nIn the meantime, Relexi polls until the new state becomes available and reads it from the database.\nWith the new state, the agent can compute the reward and get the new actions from the current policy.\nThese steps are repeated until the required amount of experience is sampled, on which the policy then can be trained.\nThis algorithmic outline of Relexi is also summarized in \\algoref{alg:relexi}.%\n\\footnote{For the sake of clarity, \\algoref{alg:relexi} only gives the algorithm for a single environment. To generalize it to multiple environments in parallel, each of the lines 7 to 11 is simply executed for each individual environment.}\n\n\\begin{algorithm}\n \\caption{Relexi}\n \\begin{algorithmic}[1]\n \\State Initialization\n \\State Launch SmartSim Orchestrator\n \\For{$i = 1,i_{max}$} \\Comment{Run $i_{max}$ iterations}\n \\State Start FLEXI instances\n \\State Read $s_{0}$\n \\For{$t=1,n$} \\Comment{Run simulation for $n$ steps}\n \\State $a_t \\leftarrow \\pi_{\\theta}(a\\:|\\:s_t)$\n \\State Write $a_t$\n \\State Polling for $s_{t+1} $\\Comment{Here, the HPC solver runs}\n \\State Read $s_{t+1}$\n \\State $r_{t+1}\\leftarrow \\mathcal{R}(s_{t+1})$\n \\EndFor\n \\State $\\tau \\leftarrow \\{ (s_0,a_0), (s_1,a_1,r_1), ..)\\}$\n \\For{$j=1,n_{epochs}$} \\Comment{Train ANN for $n_{epochs}$}\n \\State $\\theta \\leftarrow \\theta + \\alpha \\, \\nabla_{\\theta} J(\\tau)$\n \\EndFor\n \\EndFor\n \\State Shutdown SmartSim Orchestrator\n \\end{algorithmic}\n \\label{alg:relexi}\n\\end{algorithm}\n\n\nA potential bottleneck we identified is the overhead introduced by repeatedly starting hundreds of parallel environments with thousands of MPI ranks.\nFor some configurations, the time required for starting the simulations exceeded the actual simulation time.\nTo tackle this issue, we implemented two major improvements.\nFirst, we employed the multiple-program-multiple-data (MPMD) functionality provided by OpenMPI's implementation of MPI, which is also supported by SmartSim.\nWith MPMD, all simulations can be started with individual commandline arguments within a single call of MPI.\nSecondly, we implemented a functionality to copy all files required by the simulation, e.g. parameter files and restart files, to local drives located in the random access memory (RAM) of each node.\nThis reduced the access times compared to using a parallel file system like Lustre significantly.\nWith these improvments in place, the performance penality of launching large amounts of environments became negligible.\n\n\n\n\\section{Hardware Configuration}\\label{sec:hardware}\n\n All benchmarks and experiments are performed on the HPE Apollo 9000 supercomputer (Hawk) and the Hawk-AI extension, a HPE Apollo 6500 Gen10 Plus at the High-Performance Computing Center Stuttgart (HLRS).\n In the following, the hardware of both systems is given in detail.\n\n \\subsection{Hawk -- HPE Apollo 9000}\n\n Hawk consists of 5,632 dual socket nodes with 256 GiB of main memory each.\n Each node is equipped with two 64-core AMD EPYC 7742 (Rome) processors with a base frequency of 2.25 GHz.\n The nodes are connected via an enhanced 9D-hypercube.\n For the node to node interconnect, the high-performance interconnect InfiniBand HDR200 is used.\n This leads to a homogeneous massively parallel system with 720,896 compute cores and approximately 1.37 TiB of main memory.\n Hawk has a theoretical peak-performance of 25.1 Pflop\/s.\n The system reached 19.334 Pflop\/s in the LINPACK-benchmark and 334.65 TFlop\/s in the HPCG benchmark.\n Hawk is connected to a Lustre file system with a capacity of about 25 PiB.\n\n \\subsection{Hawk-AI -- HPE Apollo 6500 Gen10 Plus}\n The Hawk-AI extension consists of 24 dual socket nodes, which are each equipped with two 64-core AMD EPYC 7702 processors, 1 TiB of main memory, a local hard disk of 15 TiB and eight Nvidia A100 GPUs.\n 20 Nodes are equipped with Nvidia A100 with 40 GiB of memory and four nodes are equipped with Nvidia A100 with 80 GiB of memory.\n The nodes are also connected via the InfiniBand HDR200 interconnect.\n The Hawk-AI extension is connected directly to the enhanced 9D-hypercube of Hawk and to the same Lustre filesystem.%\n\n\\section{Conclusions}\\label{sec:conclusion}\nSupervised learning is generally suitable for learning in situations when input--output pairs can be defined a priori.\nIn contrast, in reinforcement learning, training data is gathered in an online process, in which the policy is sampled for the current state of the dynamical system.\nThis makes RL more suitable for learning optimal behaviors in dynamical systems, e.g. those described by the equations of fluid dynamics, where an a priori definition of an admissible and complete training data space is illusive and would, at best, be cumbersome.\nFlow control problems thus lend themselves naturally to an RL approach, however, other modeling tasks can be expressed in this context as well.\nHere, we have chosen to interpret the task of finding an optimal eddy viscosity in space and time as a control problem, and have the RL-trained agent predict a strategy.\nOur results show that this approach outperforms existing, established models and indeed returns a near optimal behavior in the chosen reward norm.\nThis highlights the potential of combining CFD and RL into an inclusive optimization framework.\n\nHowever, before we can explore or leverage this potential, we need to enable training and deploying such algorithms at scale, recognizing that they pose different challenges for software development and HPC.\nIn RL, trajectory data consisting of both actions and states needs to be gathered along the solution evolution during training and made available to the gradient update.\nThe RL agent and the CFD scheme thus need to be coupled during training, and they need to exchange large datasets continuously. In addition, the current policy must be explored efficiently, meaning that many parallel runs of the environments are required.\nThus, in this work, we propose Relexi as a novel framework for coupling RL algorithms with essentially generic solvers for partial differential equations on heterogeneous hardware.\nThe framework is made up of three components:\nThe PDE solver of choice, in our case, the LES solver FLEXI, TensorFlow for the ML definition and training, and the SmartSim library, which handles job allocation and management and provides an in-memory database for communication and intermediate storage of solution trajectories and policy commands.\nThe PDE solver and TensorFlow can run independently on their chosen hardware, thus exploiting their full parallel potential.\nWe show two methods for improving the overall time-to-solution of the RL problem with Relexi:\nFirst of all, the weak scaling across the gradient estimator, that is increasing the amount of runs of the environment for a given policy.\nMore parallel runs explore the current policy more efficiently and thus allow an overall better gradient update and thus quicker convergence.\nOur scaling results here indicate that the framework scales up to hundreds of parallel environments and thousands of compute cores with very good performance.\nFor the second approach, we scaled the individual simulation runs across more MPI ranks, i.e. we exploit the strong scaling characteristics of the standalone PDE solver.\nHere, we recover the expected behavior up until the individual core load becomes too small.\nIn combination, these results demonstrate that the Relexi framework is capable of efficient training on HPC systems at scale and can enable RL-methods for CFD for complex flow cases.\nFurther work will focused on applying Relexi to more complex cases and to other combinations of RL and PDE solvers.\n\n\\section{Introduction}\nIn recent years, there have been increasing efforts to transfer advances in machine learning (ML) to the field of computational fluid dynamics (CFD) in order to enhance simulations for a myriad of different applications \\cite{brunton2020machine}, which cover the fields of turbulence modeling \\cite{beck2021perspective,kurz2022machine,beck2019deep,maulik2019subgrid}, shock detection \\cite{ray2019detecting,beck2020neural}, the formulation of turbulent inflow conditions \\cite{fukami2019synthetic,kim2020deep} or applications in flow control \\cite{paris2021robust}.\nAs of today, most of these advances are based on the supervised learning (SL) paradigm.\nIn SL, the ML model is trained based on a dataset that is obtained a priori from CFD simulations or experiments.\nDuring training, the parameters of the ML model are optimized to approximate the functional relationship between the input and output quantities of the dataset.\nSince the training dataset is fixed and does not interact with the predictions of the ML model, the training itself is thus, in a sense, static and offline.\nThis oftentimes leads to inconsistencies, if, during inference, the SL-trained model is confronted with a varying and dynamic environment.\nIn this case, the model's prior predictions affect how the system evolves and thus which input states the model will see in the future.\nEnsuring that all potential states of a non-linear dynamical system are sufficiently represented within the training dataset is by no means trivial and for many applications elusive.\nThe reinforcement learning (RL) paradigm addresses this issue by training ML models not on a static dataset, but instead trains models by letting the model interact with the actual environment it will later be deployed in.\nThus, the training process not only aims at predicting outputs, but it does so by taking dynamically generated inputs into account.\nThe goal of RL is to find an optimal strategy for moving forward from the current, observed state.\nIn the course of the training process, predictions of the model change the state of the environment and the model will be rewarded based on how beneficial this transition is.\nTraining based on such sequences of actions and transitions allows incorporation of the long-term implications of the model's predictions into the training process.\nThus, the training itself becomes dynamic and requires ``online'' joint runs of the environment and the model.\n\nCFD simulations are typically computationally expensive and thus rely heavily on high-performance computing (HPC) systems.\nCoupling such HPC flow solvers with the more recently emerged ML libraries is tough, since both rely on different hardware, algorithms and overall programming paradigms.\nFor instance, most HPC codes are still written and optimized for CPU architectures and are parallelized with the Message Passing Interface (MPI), while ML libraries run most efficiently on GPUs.\nThe HPC environment thus has to provide sufficient hardware resources for both components of the application and efficient communication between them.\nIn addition, most HPC codes are written in compiled languages like C\/C++ or Fortran.\nOn the other hand, ML libraries are oftentimes also written in compiled languages but are exposed to the user via a Python interface.\nAs these languages typically lack a well-defined interface, bridging the language gap in an efficient manner for HPC is tough.\nThese problems are less pronounced for SL tasks, since the generation of the training dataset with HPC codes and the training itself are mostly decoupled and can be performed separately on appropriate hardware.\nThe HPC solver and the ML model only have to be coupled for inference, i.e. when the trained ML model is evaluated in actual simulations.\nFor this, Maulik et al. \\cite{maulik2021deploying} proposed to link the TensorFlow library directly to the flow solver and execute the model via TensorFlow's C-API.\nOtt et al. \\cite{ott2020fortran} proposed the Fortran-Keras-Bridge, which allows convenient execution of trained Keras models from Fortran code, and is based on the Neural-Fortran library by Curcic \\cite{curcic2019parallel}.\nFor RL however, the integration of ML and HPC workloads is much more involved, since the training process itself requires running simulations and thus needs to run HPC simulations and the optimization of the ML model in parallel.\nAn efficient RL framework for HPC workloads needs to manage the simulations in the HPC environment and has to implement efficient communication between the simulation codes and the employed ML library.\nIn \\cite{novati2021automating}, Novati et al. used the smarties library to couple a flow solver and TensorFlow in order to train a data-driven turbulence model for large eddy simulation (LES) on an HPC system.\nBae and Koumoutsakos applied a similar framework to the wall-modeling of wall-bounded flows in \\cite{bae2021scientific}.\nPawar and Maulik \\cite{pawar2021distributed} developed a distributed RL framework called PAR-RL, which they applied to optimize the time-stepping of a CFD simulation.\nRabault and colleagues developed an RL framework for flow control \\cite{rabault2019accelerating,rabault2019artificial,tang2020robust}, which was also coupled with a spectral flow solver by Li and Zhang in \\cite{li2022reinforcement}.\nSimilarly, Fan et al. \\cite{fan2020reinforcement} coupled a spectral element LES solver with TensorFlow to control the flow around a cylinder.\n\nHowever, most of those applications are limited in the problem sizes that can be investigated, since the simulations cannot take advantage of the parallel computing resources provided by modern HPC systems.\nIn this paper, we present a scalable RL framework that overcomes the gap between numerical simulation and ML workflows on HPC systems providing both components with its required specialized hardware.\nMoreover, we demonstrate the prospects of the reinforcement learning paradigm in scientific computing by applying our framework to derive data-driven turbulence models for large eddy simulation. \n\nThe paper is structured as follows.\n\\secref{sec:rl} gives a brief outline of the reinforcement learning paradigm.\nIn \\secref{sec:software}, we present the different software components of our framework and how they are integrated.\nThe hardware environment for our benchmarks is presented in \\secref{sec:hardware}.\nIn \\secref{sec:numerics}, we outline how our framework can be applied to derive turbulence models for large eddy simulations.\nThe results of our scaling studies and the performance of the derived data-driven turbulence models are presented in \\secref{sec:results}.\n\\secref{sec:conclusion} summarizes and concludes the paper.\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Scaling}\\label{sec:scaling}\n\n\n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure0.pdf}\n \\caption{Scaling behavior of the Relexi framework on up to 16 Hawk compute nodes (2048 MPI ranks) and one Hawk-AI node for the HIT test case with 24 DOF and 32 DOF for 2, 4, 8 and 16 MPI ranks per FLEXI instance. The black line indicates perfect scaling.}\n \\label{fig:scaling_hit}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure1.pdf}\n \\caption{Strong scaling of FLEXI within the Relexi framework for the HIT test case with 24 DOF and 32 DOF for 2, 4, 8 and 16 MPI ranks per FLEXI instance. For clarity, only the runs for 2, 8, 32 and 128 parallel FLEXI environments are shown. The black line indicates perfect scaling.}\n \\label{fig:strong_scaling_hit}\n\\end{figure*}\n\nTo analyze the scaling behaviour of our framework, we use the HIT test case for the two different configurations listed in \\tabref{tab:testcases}.\nFor the benchmarks, we used a single Hawk-AI node of the HPE Apollo 6500 Gen10 Plus as \\textit{head} node and up to 16 Hawk compute nodes of the HPE Apollo 9000 as \\textit{worker} nodes in a single batch job.\nThis results in up to 2048 compute cores used for generating training data.\nIn the scaling analysis, we investigate two types of scaling.\nFirst, we examine how the framework scales if the number of parallel environments is increased, while keeping the amount of MPI ranks per environment constant.\nThis corresponds to a weak scaling approach, since the parallel load and the available resources are increased simultaneously, while keeping the load per rank constant.\nSince the used resources, and thus also the simulation time, for a FLEXI simulation should theoretically stay unchanged, eventually emerging losses in parallel speedup can be attributed solely to the communication overhead, the limited throughput of the database, the cost of the data managment in Relexi, the increasing amount of policy evaluations and the start as well as the termination of the simulations.\nThe measured executation time includes launching the simulations and running the simulation with the policy until all simulations terminated.\nAs a performance metric, we use the ``Speedup'', which is computed as the quotient of the time needed for sampling the $n_{envs}$ parallel environments and the time that would be needed to run $n_{envs}$ environments sequentially, i.e. the speedup of running the environments in parallel instead of sequentially.\nFor each configuration, we ran Relexi in two separate jobs for 6 iterations each to account for fluctuations in hardware and communication performance and computed the mean of the 12 measurements.\nThe scaling is performed with either 2, 4, 8 or 16 MPI ranks for each FLEXI run.\nFurther, we start with two parallel FLEXI instances and double them until the 16 Hawk compute nodes are fully occupied.\nIt seems important to note that the results are not compared directly to the performance of standalone FLEXI simulations, since the performance differences in running a single FLEXI instance within Relexi and running it standalone were negligible.\n\nThe results in \\figref{fig:scaling_hit} demonstrate that the framework can scale efficiently up to a thousand parallel environments on thousands of cores.\nTwo major trends can be identified in the scaling results.\nAs mentioned before, the observed decrease in parallel efficiency when running more environments in parallel can be attributed mainly to the sequential work done by Relexi.\nThe framework should scale better if the FLEXI instance gets more time-consuming, since then, this sequential work becomes less relevant and the perfect scaling abilities of FLEXI can be recovered.\nIn contrast, if the necessary time to compute the FLEXI simulation decreases, i.e. FLEXI gets more ranks, the sequential work of Relexi becomes more dominant, which decreases the scaling efficiency.\nThis causes the runs with fewer ranks per FLEXI instance to scale better than the runs using more ranks.\nThe second interesting behavior is that the decrease in performance when switching from a single to two parallel environments is most pronounced for the FLEXI environment with only two cores, which is counterintuitive. \nWe attribute this to the limited memory bandwith and the hierarchical architecture of the processors used.%\n\\footnote{The used EPYC CPUs comprise several dies, which contain 8 cores each. All cores on a single die share the available memory bandwidth.}\nIf a single FLEXI instance with two ranks is spawned on a compute node, this instance gets all available memory bandwidth.\nIf however, a second FLEXI is spawned, both instances have to share the available bandwidth, which slows down the simulation as well as the interaction with the database.\nThis effect vanishes with an increasing amount of used cores.\nThe observed loss of parallel efficiency especially for the last data points, which correspond to using all 2,048 available cores, can in parts be attributed to single simulations running significant slower than the average.\nThese outliers can probably be attributed to fluctuations in the load of the interconnect. %\nThis issue is subject of current investigations.\n\n\nFor the strong scaling, we examine the configurations with 2, 8, 32 and 128 parallel FLEXI instances.\nThe results of the strong scaling given in \\figref{fig:strong_scaling_hit} match the general behavior observed in the weak scaling benchmark. %\nIn cases where the computational cost of the HPC workload is dominant, i.e. if only few MPI ranks are used per environment, the optimal scaling of FLEXI can be recovered.\nIf the amount of MPI ranks is increased, the time spent for the simulation decreases and the work done by Relexi becomes the limiting factor.\nThis causes the parallel efficiency to decrease for low simulation loads per core.\nIt seems important to note that for both cases, using 16 MPI runs per simulation falls quite below the optimal load per core for FLEXI.\nFor the more realistic cases of up to 8 ranks per FLEXI, most of the FLEXI performance can be recovered.\n\n\n\n\n\n\n\n\n\\subsection{Turbulence Modeling}\n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/fig\/draft-figure2.pdf}\n \\caption{Training results of the 24 DOF configuration with 16, 32 and 64 sampled episodes per training update. Top left: Collected return averaged over all training runs and normalized with the maximum achievable return. The minimum and maximum return in each iteration is indicated by the shaded area. Top right: Normalized return on the unseen testing state, which was evaluated every 10 iterations. Bottom left: Spectra at the end of the simulation $t_{end}$ on the unseen test state after 4,000 training iterations for the configuration running 64 parallel environments. The resulting spectra of Smagorisky's model and the implicit model are shown for comparison. The shaded area around the time-averaged DNS spectrum indicates the maximum and minimum observed amplitudes during the DNS. Bottom left: Distribution of the model's predictions for $C_s$ during the simulation on the test state.}\n \\label{fig:results_rl_24DOF}\n\\end{figure*}\n\nTo investigate the performance of the RL-based turbulence model against established analytical closure models, we trained the agent on the 24 DOF configuration for 4,000 Iterations, which corresponds to 20,000 gradient ascent steps for the policy overall.\nTo quantify the benefits of gathering more episodes per policy update, the training was carried out using 8 MPI ranks per environment and 16, 32 and 64 parallel FLEXI instances, respectively.\nFor the 16 and 64 runs configurations, the overall training required 20 and 30 hours, respectively.\nSampling the trajectories took 15 and 18 seconds per iteration, respectively, while updating the policy on a single GPU took 0.5 and 2 seconds, respectively.\nThe results in \\figref{fig:results_rl_24DOF} indicate that increasing the number of parallel simulations does indeed improve the training performance in several ways.\nFirstly, the training runs with more episodes converge faster, i.e. need less training iterations to achieve a given episode return. \nMoreover, the collected return of the 16 episode configuration increases less consistently after around 2,000 iterations than the training configurations with more parallel episodes.\nThis confirms our prior assumption that the gradient estimator becomes more reliable if more episodes are sampled, which again leads to more efficient training updates and thus to faster convergence.\nIn the same vein, the training runs using more episodes tend to converge to a higher total return.\nThis notion holds especially for the return collected during training, where the return is computed on a variety of different initial states.\nWhile the return on the unseen initial state for testing gives similar results, the reward of the 16 episodes configuration fluctuates between 2,000 and 3,000 iterations and temporarily exceeds the return of the 32 episodes configuration before converging to a stable return.\nThis behavior probably stems from the high variance of the 16 episodes training which causes the less consistent improvement on the unseen test state.\nThe results in \\figref{fig:results_rl_24DOF} also clearly demonstrate that for our application, the RL-based model outperforms both Smagorinsky's model and the implicit modeling strategy with regard to the energy spectrum.\nEspecially surprising is that the RL-enhanced model agrees with the high-fidelity data almost up to $k_{max}$ with deviations only at $k=6$.\nThis indicates that our novel data-driven model does not only replicate the vital flow physics related to the energy cascade, but also pushes the resolution limits of the underlying discretization.\nThe predictions of the initial untrained model appear to be almost normally distributed, as is expected from the initialization of the policy ANN.\nInterestingly, the trained model modifies this distribution heavily by predicting tiny $C_s$ parameters for the vast majority of the flow field while increasing the parameter only in selective elements. \nThe model also takes advantage of the entire admissable range of $C_s\\in[0,0.5]$.\nThe reported results demonstrate that the application of the RL paradigm to turbulence modeling (and control tasks in CFD in general) can contribute to major advancements of the state-of-the-art given that the necessary resources can be used efficiently on massively parallel systems.\n\n\n\\section{Application to Turbulence Modeling}\\label{sec:numerics}\n\n\\subsection{Turbulence Modeling}\nTurbulent flows are notoriously hard to resolve accurately, since turbulence is a multiscale phenomenon.\nThe resolution requirements for numerical simulations to resolve the wide range of active length scales are usually intractable.\nInstead, reduced order descriptions of turbulence can be derived, which only resolve the large energy-containing flow scales, while employing a turbulence model to account for the influence of the non-resolved fine scales.\nFrom a mathematical standpoint, this is equivalent to applying a low-pass filter to the flow field.\nThis approach is commonly referred to as large eddy simulation (LES).\nA myriad of different LES turbulence models have been proposed in literature.\nHowever, such models typically contain empirical parameters, which have to be tuned to the employed numerical discretization and the specific test case.\nAs a consequence, no universal or overall \\emph{best} model has been found to this date.\n\nThe most important property of the LES model is to mimic the energy drain from the large to the small flow scales.\nBased on this reasoning, a popular modeling strategy is to approximate the unresolved scales by introducing an additional turbulent viscosity $\\nu_t$ that is added to the physical viscosity.\nThe common model by Smagorinsky \\cite{smagorinsky1963general} computes this turbulent viscosity as\n\\begin{equation}\n \\nu_t= \\left(C_s \\Delta\\right)^2 \\sqrt{2\\:\\overline{S}_{ij}\\overline{S}_{ij}} \\qquad \\text{with} \\quad \\overline{S}_{ij} = \\frac{\\partial \\overline{u}_i}{\\partial x_j}.\n \\label{eq:smagorinsky}\n\\end{equation}\nHere, $\\overline{S}_{ij}$ is the rate-of-strain tensor of the coarse-scale velocity field $\\overline{u}_i$ with respect to the coordinates $x_j$ with $i,j=1,2,3$.\nThe filter width $\\Delta$ is a measure of the employed spatial resolution and $C_s$ is a model coefficient, which has to be tuned for each specific test case.\nAnother common modeling approach is the implicit modeling paradigm, which assumes that the numerical dissipation error introduced by the numerical scheme acts as an implicit LES model. The implicit LES model can obviously be seen as a special case of Smagorinsky's model with $C_s=0$.\n\nWith our RL framework, we strive to improve the performance of Smagorinsky's model given in \\eqref{eq:smagorinsky} by employing an RL algorithm to tune the model coefficient $C_s$ dynamically in space and time during the simulation.\nIn this regard, we show that RL represents a promising approach to enhance current simulation workloads and that our novel RL framework is capable of handling computationally intensive simulation environments on modern HPC architectures.\n\n\\subsection{Stating the Reinforcement Learning Task}\nAs test case, i.e. as training environment, we perform LES of homogeneous isotropic turbulence (HIT) at a Reynolds number of $Re_{\\lambda}\\approx 200$ with respect to the Taylor microscale.\nThe cubic domain of side length $2\\pi$ is equipped with periodic boundary conditions and is discretized by a Cartesian mesh with the resolutions given in \\tabref{tab:testcases}.\nThis test case of \\emph{turbulence in a box} can be seen as the building block of turbulence and describes freely decaying turbulence in the absence of boundaries.\nIn order to obtain a quasi-static solution, an isotropic linear forcing is employed as proposed by \\cite{lundgren2003linearly,de2015anisotropic} to balance the dissipation of the turbulence model.\nThis results in a quasi-stationary distribution of the turbulent kinetic energy in the system, which is mainly characterized by the energy drain from the large to the small scales.\nSince this energy cascade is a fundamental property of turbulence that the LES should reproduce, we define our optimization target for the RL task in terms of this energy spectrum.\nThe reward for the RL algorithm is thus computed based on the error of the instantaneous energy spectrum of the LES $E_{LES}(k)$ compared to the mean energy distribution of the underlying ground truth solution $E_{DNS}(k)$, which was obtained beforehand from a high-fidelity simulation.\nFor this, we use the mean relative error across the wavenumbers $k$ up to $k_{max}$, which can be computed as\n\\begin{equation}\n \\ell = \\underset{k}{\\mathrm{mean}}\\left[\\left(\\frac{E_{DNS}(k)-E_{LES}(k)}{E_{DNS}\\left(k\\right)}\\right)^2\\right], \\quad k\\in\\left[1,k_{max}\\right].\n \\label{eq:reward_error}\n\\end{equation}\nTo ensure that the reward is normalized to $r_t\\in[-1,1]$, the reward is eventually computed as\n\\begin{equation}\n r_t = 2 e^{\\left(\\ell \/ \\alpha \\right)}-1,\n \\label{eq:reward}\n\\end{equation}\nwith $\\alpha$ as a scaling parameter.\n\nWith the reward function in place, the optimization task for the RL algorithm is framed as follows.\nThe state of the environment observed by the agent is the current coarse-scale velocity field $\\overline{u}_i$.\nThe agent predicts as action a single $C_s$ coefficient for each element in the computational mesh solely based on the local flow state in the respective element.\nThe environment's state is then evolved with the flow solver for some time $\\Delta t_{RL}$ before requesting new predictions for the elementwise $C_s$.\nThe time inteval $\\Delta t_{RL}$ is generally chosen much larger than the computational timestep of the simulation.\nThe reward for the agent is computed based on the differences in energy distribution compared to the ground truth, as given in \\eqref{eq:reward}.\nThis loop is repeated until some final time $t_{end}$ is reached.\n\n\n\n\\subsection{Computational Setup}\nIn this work, we perform LES at different resolutions, which are listed in \\tabref{tab:testcases}.\nAll simulations are run up to $t_{end}=5$ and actions are performed with in a time interval of $\\Delta t_{RL}=0.1$, which corresponds to 50 predictions per simulation.\nThe initial state for each simulation run is drawn randomly from a set of flow states that are computed by filtering different realizations of the high-fidelity solution.\nA single initial state is kept hidden to evaluate the model performance on unseen test data.\nBased on these setups, two different approaches can be followed to reduce the required computational time for the training by using parallel HPC resources.\nFirst, the number of MPI ranks per environment can be increased to reduce the simulation time of the environments by exploiting the strong scaling capabilities of FLEXI.\nSecond, the number of simulated environments per training update can be increased, which can be seen as a weak scaling approach for the full framework.\nSince an increased number of sampled episodes might result in a better estimate of the gradient for the optimizer, this approach can reduce the number of iterations required for convergence and thus the necessary training time.\nSince both perspectives highlight crucial scaling properties of the underlying framework, both will be investigated in \\secref{sec:scaling}.\nHere, we employ up to 16 cores per FLEXI environment and up to 1024 parallel environments per training iteration using a maximum of 2,048 cores.\n\\begin{table}\n \\centering\n \\begin{tabular}{lccccc}\n \\hline \n & N & \\#Elems & \\#DOF & $k_{max}$ & $\\alpha$ \\\\\n \\hline \n 24 DOF & 5 & $4^3$ & 13,824 & 9 & 0.4 \\\\\n 32 DOF & 7 & $4^3$ & 32,768 & 12 & 0.2 \\\\\n \\hline \n \\end{tabular}\n \\caption{Investigated configurations for the LES of the HIT test case. The runs are named by the number of degrees of freedom (DOF) per spatial direction, which result from the given polynomial degree $N$ and the number of elements \\#Elems. The total number of DOF can be computed by $\\text{\\#DOF} = \\text{\\#Elems}\\:(N+1)^3$. The hyperparameters $k_{max}$ and $\\alpha$ refer to the maximum wavenumber and the scaling factor of the reward function, respectively.}%\n \\label{tab:testcases}\n\\end{table}\n\nAs RL algorithm, we use the clipping variant of the proximal policy optimization (PPO) \\cite{schulman2017proximal} algorithm as already discussed in \\secref{sec:rl}.\nThis algorithm is synchronous, which means that the algorithm performs the steps of sampling experience and updating the policy in a sequential fashion.\nThis means that first, FLEXI simulations are performed based on the current policy and thereafter, the policy is updated based on the collected experience to maximize the reward on future runs.\nFor all experiments we used a discount factor of $\\gamma=0.995$ and a learning rate of $10^{-4}$ with the Adam optimizer \\cite{kingma2014adam} to train the policy for 5 epochs per iteration.\nFor the PPO algorithm, we used a clipping parameter of 0.2 and set the entropy coefficient to zero.\nThe employed policy ANN comprises around 3,300 parameters with its architecture given in \\tabref{tab:ann}.\n\n\\begin{table}\n \\centering\n \\begin{tabular}{lrrrr}\n \\hline \n Layer & Kernel & Filters & Padding & Dimension \\\\\n \\hline \n Input & & & & $6\\times 6\\times 6\\times 3$ \\\\\n Conv3D & 3 & 8 & zero & $6\\times 6\\times 6\\times 8$ \\\\\n Conv3D & 3 & 8 & none & $4\\times 4\\times 4\\times 8$ \\\\\n Conv3D & 3 & 4 & none & $2\\times 2\\times 2\\times 4$ \\\\\n Conv3D & 2 & 1 & none & $1\\times 1\\times 1\\times 1$ \\\\\n Scale & & & & $ 1$ \\\\\n \\hline \n \\end{tabular}\n \\caption{Architecture of the policy ANN for $N=5$ with the dimensions of each layer's output. The model's input dimensions follow from the $(N+1)^3$ solution points in each element times the three velocity components $\\overline{u}_i$, which are used as input features. The ANN is build from three-dimensional convolutional layers with a specific kernel size and number of filters, each. The first layer uses zero-padding, while the rear layers employ no padding to convolve the high-dimensional input to a single scalar. All layers except the last convolutional layer employ the rectified linear unit (ReLU) as activation function. The final scaling layer uses the operation $y=\\frac{1}{2}\\sigma_s(x)$ to scale the input to the interval $[0,0.5]$ with $\\sigma_s(x)$ denoting the sigmoid activation function.}\n \\label{tab:ann}\n\\end{table}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nGiven a passage (context) and a question about it, a reading comprehension system should be able to read the passage and answer the question. While not a hard task for a human, it requires that the system both understand natural language and have knowledge about the world. \nBecause of the renaissance of neural networks and accessibility of large-scale datasets, great progress has recently been made in reading comprehension. For example, according to the leaderboard of SQuAD 1.0 \\cite{DBLP:conf\/emnlp\/RajpurkarZLL16}, over $80$ systems have been submitted, and human performance has been left behind.\nIn experiments, the reimplementation and comparison of these solutions are necessary but not easy tasks, because researchers usually build their blocks from scratch and in different environments. Meanwhile, the efficient construction of original prototypes is not possible, although reading comprehension models often share similar components and architectures.\n\nIn this paper, we present the Sogou Machine Reading Comprehension toolkit\\footnote{\\url{https:\/\/github.com\/sogou\/SMRCToolkit}}, which has the goal of allowing the rapid and efficient development of modern machine comprehension models, including both published models and original prototypes.\nFirst, the toolkit simplifies the dataset reading process by providing dataset reader modules that support popular datasets. Second, the flexible preprocessing pipeline allows vocabulary building, linguistic feature extraction, and operations to work in a seamless way.\nThird, the toolkit offers frequently used neural network components, a trainer module, and a save\/load function, which accelerates the construction of custom models. Last, but not the least, some published models are implemented in the toolkit, making model comparison and modification convenient. The toolkit is built based on the Tensorflow\\footnote{\\url{https:\/\/github.com\/tensorflow\/tensorflow}} library \\cite{DBLP:conf\/osdi\/AbadiBCCDDDGIIK16}.\n\n\\begin{figure*}[htbp] \n\\centering\n\\includegraphics[width=\\textwidth]{architecture.pdf}\n\\caption{Toolkit Architecture}\n\\label{architecture} \n\\end{figure*}\n\n\\section{Toolkit Framework}\nAs shown in Figure 1, the architecture of our toolkit mainly contains four modules: the\nDataset Reader, Data Preprocessing, Model Construction, and Model Training modules. These four modules\nare designed as a pipeline flow and can be used for most machine reading comprehension tasks. In the following, we\nwill introduce each part in detail.\n\\subsection{Dataset Reader}\nOne reason that machine reading comprehension has made rapid progress, that cannot be ignored, is the release of a variety of large-scale and high-quality question answering datasets. In addition, preprocessing and evaluating are essential steps when doing research on these datasets.\n\n\\noindent \\textbf{Reader} \\indent To avoid repeating the development of dataset reading codes, the toolkit provides reader modules for some typical datasets SQuAD 1.0 \\cite{DBLP:conf\/emnlp\/RajpurkarZLL16}, SQuAD 2.0 \\cite{DBLP:conf\/acl\/RajpurkarJL18} and CoQA \\cite{DBLP:journals\/corr\/abs-1808-07042}. To enhance the language diversity, we also support a Chinese dataset, CMRC2018 \\cite{DBLP:journals\/corr\/abs-1810-07366}.\nThe reader modules first tokenize texts and generate labels (e.g., start\/end positions), and then transform data instances into nested structure objects, the fields of which are uniformly named. This makes data serialization\/deserialization convenient and helps in error analysis. By inheriting the base reader, users can develop custom readers for other datasets.\n\n\\noindent \\textbf{Evaluator} \\indent Most datasets offer official evaluation scripts. To ease the model validation and early stopping, we integrate these evaluation scripts into the toolkit and simplify the evaluation in the model training process.\n\n\\subsection{Data Preprocessing}\nTo prepare data for the training model, we need to build a vocabulary, extract linguistic features, and map discrete features into indices. The toolkit provides modules for fulfilling these requirements.\n\n\\noindent \\textbf{Vocabulary Builder} \\indent By scanning the training data from the dataset reader, Vocabulary Builder maintains a corresponding vocabulary of words (and characters if needed). Adding any special tokens or setting the whole vocabulary is allowed as well. Another important function of Vocabulary Builder is creating an embedding matrix from pretrained word embeddings. If you feed a pretrained embedding file, for example Glove\\footnote{\\url{https:\/\/nlp.stanford.edu\/projects\/glove\/}} \\cite{DBLP:conf\/emnlp\/PenningtonSM14}, to the Vocabulary Builder, it will produce a word embedding matrix for its inner vocabulary.\n\n\\noindent \\textbf{Feature Extractor} \\indent Linguistic features are used in many machine reading comprehension models such as DrQA \\cite{DBLP:conf\/acl\/ChenFWB17}, FusionNet \\cite{DBLP:journals\/corr\/abs-1711-07341} and have been proven to be effective. The Feature Extractor supports commonly used features, e.g., part-of-speech (POS) tags, called entity recognition (NER) tags, along with normalized term frequency (TF) and word-level exact matching. By simply adding new feature fields, Feature Extractor does not break the serializability and readability of data instance objects. Meanwhile, Feature Extractor also builds vocabularies for discrete features like POS and NER, which will be used in the next steps for index mapping.\n\n\\noindent \\textbf{Batch Generator} \\indent The last step of the preprocessing is to pack all of the features up and modify them to fit the form of the model input. In Batch Generator, we first map words and tags to indices, pad length-variable features, transform all of the features into tensors, and then batch them. To make these steps efficient, we implement Batch Generator based on the Tensorflow Dataset API\\footnote{\\url{https:\/\/www.tensorflow.org\/api_docs\/python\/tf\/data\/Dataset}}, which parallelizes data transformation and provides fundamental functions like dynamic padding and data shuffling, which make it behave consistently with the \"generator\" in Python. Batch Generator is designed to be flexible and compatible with the feature types frequently used in machine reading comprehension tasks.\n\n\\subsection{Model Construction}\nThe core part of the machine reading comprehension task is constructing an effective and efficient model for generating answers from given passages. The toolkit provides two methods: build your own model or use a built-in model. For the first one, we implement frequently used neural network components in the machine reading comprehension task. We follow the idea of functional API and wrap them as MRC specific supplements of Tensorflow layers.\n\n\\noindent \\textbf{Embedding} \\indent Besides a vanilla embedding layer, the toolkit also provides \\textit{PartiallyTrainableEmbedding}, as used in \\cite{DBLP:conf\/acl\/ChenFWB17} \\cite{DBLP:journals\/corr\/abs-1711-07341}, and pretrained contextualized representation layers, including \\textit{CoVeEmbedding}, \\textit{ElmoEmbedding}, and \\textit{BertEmbedding}.\n\n\\noindent \\textbf{Recurrent} \\indent \\textit{BiLSTM} and \\textit{BiGRU} are basic recurrent layers, and their CuDNN version \\textit{CudnnBiLSTM} and \\textit{CudnnBiGRU} are also available.\n\n\\noindent \\textbf{Similarity Function} \\indent Functions are available for calculating the word-level similarities between texts, e.g., \\textit{DotProduct}, \\textit{TriLinear}, and \\textit{MLP}.\n\n\n\\noindent \\textbf{Attention} \\indent Attention layers are usually used together with the Similarity Function, e.g., \\textit{BiAttention}, \\textit{UniAttention}, and \\textit{SelfAttention}.\n\n\\noindent \\textbf{Basic Layer} \\indent Some of the basic layers are used in machine reading comprehension models, e.g., \\textit{VariationalDropout}, and \\textit{Highway}, \\textit{ReduceSequence}.\n\n\n\\noindent \\textbf{Basic Operation} \\indent These are mainly masking operations, e.g., \\textit{masked\\_softmax}, \\textit{mask\\_logits}. By inheriting the base model class and combining the components above, developers should be able to construct most mainstream machine reading comprehension models. To build a custom model, a developer should define the following three member methods,\n\n\\begin{enumerate}\n \\item \\textit{\\_build\\_graph}: Define the forward process of the model\n \\item \\textit{compile}: Schedule the optimization of the model like the learning rate decay and gradient clipping\n \\item \\textit{get\\_best\\_answer}: Transform the model output (probability) to answer text\n\\end{enumerate}\n\n\n\\noindent Training functions (\\textit{train\\_and\\_evaluate}, \\textit{evaluate}, and \\textit{inference}) should also be inherited if needed.\n\nThe toolkit also provides simple interfaces for using the built-in models. We will introduce the details in Section \\ref{built-in}.\n\\subsection{Model Training}\nWhen training a model, we usually care about how the metrics change on the train\/dev set, when to perform early stopping, how many epochs the model needs to converge, and so on. Because most models share a similar training strategy, the toolkit provides a Trainer module, with main functions that include baby-sitting the training, evaluation and inference processing, saving the best weights, cooperating with the exponential moving average, and recording the training summary. Each model also provides interfaces for saving and loading the model weights.\n\\section{Using Built-In Models}\\label{built-in}\n\n\\subsection{Have a Try}\n\nWe will show an example of running the BiDAF model on the SQuAD 1.0 dataset in this section.\n\nFirst, the data file of SQuAD 1.0 is loaded using SquadReader. Meanwhile, we also create an evaluator for validation.\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{reader.pdf}\n\\end{figure}\n\n\nSecond, we build a vocabulary and corresponding word embedding matrix.\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{vocab.pdf}\n\\end{figure}\n\nThird, data instances are fed to Batch Generator for the necessary preprocessing and batching.\n\n\\begin{figure}[H] \n\\centering\n\\includegraphics[width=\\textwidth]{batch.pdf}\n\\end{figure}\n\n\nLast, we use the built-in BiDAF model and compile it with default hyperparameters. \\textit{train\\_and\\_evaluate} will handle the training process and save the best model weights for inference.\n\\begin{figure*}[!htbp] \n\\centering\n\\includegraphics[width=\\textwidth]{model.pdf}\n\\end{figure*}\n\n\nWith our toolkit, users can try different machine reading comprehension models in a neat and fast way.\n\n\\subsection{Model Zoo}\nIn the section, we will briefly introduce the machine reading comprehension models implemented in this toolkit. \n\n\\noindent \\textbf{BiDAF} was introduced by \\cite{DBLP:journals\/corr\/SeoKFH16}. Unlike the attention mechanisms in previous work, the core idea of BiDAF is bidirectional attention, which models both the query-to-context and context-to-query attention.\n\n\\noindent \\textbf{DrQA} was proposed by \\cite{DBLP:conf\/acl\/ChenFWB17} and aims at tackling open-domain question answering. DrQA use word embedding, basic linguistic features, and a simple attention mechanism, and proves that simple models without sophisticated architectural designs can also achieve strong results in machine reading comprehension.\n\n\\noindent \\textbf{FusionNet} Based on an analysis of the attention approaches in previous work, \\cite{DBLP:journals\/corr\/abs-1711-07341} proposed FusionNet, which extends the attention from three perspectives. They proposed the use of the \"history of word\" and fully aware attention, which let the model combine the information flows from different semantic levels. In addition, the idea was also applied to natural language inference.\n\n\\noindent \\textbf{R-Net} The main contribution of R-Net was the self-matching attention mechanism. After the gating matching of the context and question, passage self-matching was introduced to aggregate evidence from the whole passage and refine the passage representation.\n\n\\noindent \\textbf{QANet} The architecture of QANet \\cite{DBLP:journals\/corr\/abs-1804-09541} was adapted from the Transformer \\cite{DBLP:conf\/nips\/VaswaniSPUJGKP17} and only contains the convolution and self-attention. By not using the recurrent Layers, QANet gains a 3\u201313-fold speed increase in the training time and 4\u20139-fold increase for the inference time.\n\n\\noindent \\textbf{IARNN} In our toolkit, two types of Inner Attention-based RNNs (IARNNs) \\cite{DBLP:conf\/acl\/WangL016} are implemented, which are advantageous for sentence representation and efficient in the answer selection task. IARNN-word weights the word representation of the context for the question before inputting into the RNN models. Unlike IARNN-word, which only achieves input word embedding, IARNN-hidden can capture the relationships between multiple words by adding additional context information to the calculation of the attention weights.\n\n\\noindent \\textbf{BiDAF++} \\cite{DBLP:conf\/acl\/GardnerC18} originally introduced a model for multi-paragraph machine reading comprehension. Based on BiDAF, BiDAF++ adds a self attention layer to increase the model capacity. We also apply the model to CoQA \\cite{DBLP:journals\/corr\/abs-1809-10735} for conversational question answering.\n\n\\noindent \\textbf{BERT} Pretrained models like BERT \\cite{DBLP:journals\/corr\/abs-1810-04805} and ELMo\\cite{DBLP:conf\/naacl\/PetersNIGCLZ18} have shown great efficacy in many natural language processing tasks. In our toolkit, we use BERT, ELMo, and Cove\\cite{DBLP:conf\/nips\/McCannBXS17} as embedding layers to provide a strong contextualized representation. Meanwhile, we also include the BERT model for machine reading comprehension, as well as our modified version. The results of the models in our toolkit are listed in Section \\ref{experiments}.\n\\section{Experiments}\\label{experiments}\n\nWe conducted experiments on a supported dataset with the models in the toolkit. By following the experimental settings in the original papers, we attempted to reproduce the results of the models on a different dataset. It is worth mentioning that slight modifications were applied when necessary, and the scripts and hyperparameters for producing the results shown below are included in the toolkit.\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score on SQuAD 1.0 dev set}\\label{tab:squad1}\n\\begin{tabular}{ | l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF & 77.3\/67.7 & 77.3\/67.7 \\\\ \\hline \n\tBiDAF+ELMo & 81.0\/72.1 & - \\\\ \\hline\n\tIARNN-Word & 73.9\/65.2 & - \\\\ \\hline\n\tIARNN-hidden & 72.2\/64.3& - \\\\ \\hline \n\tDrQA & 78.9\/69.4 & 78.8\/69.5 \\\\ \\hline \n DrQA+ELMO&83.1\/74.4 & - \\\\ \\hline\n\tR-Net & 80.0\/71.6 & 79.5\/71.1 \\\\ \\hline \n\tBiDAF++ & 78.6\/69.2 & - \\\\ \\hline \n\tFusionNet & 81.0\/72.0 & 82.5\/74.1 \\\\ \\hline \n\tQANet & 80.8\/71.8 & 82.7\/73.6 \\\\ \\hline \n\tBERT-Base & 88.3\/80.6 & 88.5\/80.8 \\\\ \\hline\n\n\\end{tabular}\n\\end{table}\n\nIn Table \\ref{tab:squad1}, we report the results of the implemented models on the development set of SQuAD 1.0. A sophisticated and effective attention mechanism is necessary for building a high-performance model, according to the table. In addition, pretrained models like ELMo and BERT give reading comprehension a big boost and have become a new trend in natural language processing. Our toolkit also wraps commonly used attention and pretrained models in a high-level layer and allows flexible combinations.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score on SQuAD 2.0 dev set}\\label{tab:squad2}\n\\begin{tabular}{ | l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF & 62.7\/59.7 & 62.6\/59.8 \\\\ \\hline \n\tBiDAF++ & 64.3\/61.8 & 64.8\/61.9 \\\\ \\hline \n\tBiDAF++ + ELMo & 67.6\/64.8& 67.6\/65.1 \\\\ \\hline\n\tBERT-Base & 75.9\/73.0 & 75.1\/72.0 \\\\ \\hline\n\\end{tabular}\n\n\\end{table}\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1 score on CoQA dev set}\\label{tab:coqa}\n\\begin{tabular}{| l | c | c |}\n\t\\hline\n\tModel & toolkit implementation & original paper\\\\ \\hline\n\tBiDAF++ & 71.7 & 69.2 \\\\ \\hline \n\tBiDAF++ + ELMo & 74.5 & 69.2 \\\\ \\hline \n\tBERT-Base & 78.6 & - \\\\ \\hline\n\tBERT-Base+Answer Verification & 79.5 & - \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nBecause SQuAD 2.0 and CoQA are different from SQuAD 1.0 in a variety of respects, the models are not directly transferrable between these datasets. Following \\cite{DBLP:conf\/conll\/LevySCZ17} and \\cite{DBLP:journals\/corr\/abs-1809-10735}, we implement several effective models.\nMoreover, our implemented BiDAF achieves Exact Match $35.0$ and F1 $57.01$ on the CMRC dataset, providing a strong baseline.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{F1\/EM score of different embedding}\\label{tab:embedding}\n\\begin{tabular}{ | c | c | c |}\n\t\\hline\n\tEmbedding & BiDAF & DrQA \\\\ \\hline\n\trandom & 70.3\/59.3 & 72.4\/62.5 \\\\ \\hline\n\tword2vec & 77.1\/67.7 & 78.2\/68.8 \\\\ \\hline\n\tglove & 77.3\/67.7 & 78.9\/69.4 \\\\ \\hline\n\tfast-text-wiki & 77.1\/67.7 & 75.4\/66.0 \\\\ \\hline\n\tfast-text-crawl & 76.9\/67.4 & 77.0\/67.2 \\\\ \\hline\n\tELMo & 79.9\/71.1 & 82.7\/74.3 \\\\ \\hline \n\\end{tabular}\n\\end{table}\n\nTo investigate the effect of the word representation, we selected two popular models and tested their performances with different embeddings. Table \\ref{tab:embedding} suggests that DrQA is more sensitive to the word embedding and ELMo helps improve the score consistently (here, when ELMo was used, no word embedding was concatenated).\n\n\\section{Conclusion and Future Work}\nIn the paper, we present the Sogou Machine Reading Comprehension toolkit, which has the goal of allowing the rapid and efficient development of modern machine comprehension models, including both published models and original prototypes.\n\nIn the future, we plan to extend the toolkit, and make it applicable to more tasks, e.g. multi-paragraph and multi-document question answering, and provide more available models.\n\\bibliographystyle{acl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}