A graded quiver with superpotential is a quiver whose arrows are assigned degrees

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3$ would correspond to a ``field theory'' in $d=6-2m<0$, with $n_c$ distinct types of matter fields, and with some ``superpotential'' interactions amongst them.} they still have a natural interpretation as describing fractional branes at a CY$_{m+2}$ singularity, as we now explain. ]]>

3$~\cite{toappear1}.We collectively refer to them as \emph{generalized dimers}. Via graph dualization, they are in one-to-one correspondence with periodic quivers on $\mathbb{T}^{m+1}$ which, likewise, fully encode both the quivers and the superpotentials of the ``field theories.'' As we will explain in section~\ref{subsec: toric quivers and pms}, given one of these brane setups, finding the corresponding ${\bf X}_{m+2}$ is reduced to a combinatorial problem, which is a huge simplification with respect to alternative approaches. Conversely, there are various efficient procedures for constructing generalized dimers --- equivalently, quiver theories with superpotentials --- starting from the corresponding toric ${\bf X}_{m+2}$. One way to do this is by using mirror symmetry. This method was developed for $m=1$ in~\cite{Feng-ml-2005gw} and for $m=2$ in~\cite{Futaki-ml-2014mpa,Franco-ml-2016qxh}, where its extension to higher $m$ was also outlined. In this paper, we focus on toric varieties. For each infinite family of examples, we present a convenient toric method to derive graded quivers with superpotential for ${\bf X}_{m+2}$, and discuss some of their interesting properties. We then proceed to check those results with an explicit B-brane computation, following the three steps above. The B-model computation provides a strong check of those recently devised toric methods. This paper is organized as follows. In section~\ref{sec: 2}, we review the relevant aspects of graded quivers and of the B-brane category, and we spell out the relation between the two approaches. In section~\ref{sec: Cn}, we illustrate our methods in the simplest example, that of flat space $\C^{m+2}$. In section~\ref{section_family_orbifolds}, we consider an orbifold singularity, $\C^{m+2}/\Z_{m+2}$. In section~\ref{sec: Y10Pm}, we consider a family of singularities, dubbed $Y^{1,0}(\mathbb{P}^m)$, which reduces to the conifold singularity for $m=1$. In section~\ref{sec: F0m}, we consider a third family of singularities, dubbed $\mathbb{F}_0^{(m)}$, which reduces to an orbifold of the conifold for $m=1$. Appendix~\ref{App: alg geom} contains a pedagogical summary of the algebraic geometry techniques that we will need for our B-model computations. Appendix~\ref{app: mutations} reviews order $m + 1$ mutations of $m$-graded quivers. ]]>

0$ in each closed path. On the other hand, the number of chiral multiplets $\Phi^{(0)}$ is unbounded, \emph{a priori}. For instance, at low $m$ we have: \beas m&=1\; :\quad &W&= W(\Phi^{(0)})\,, \cr m&=2\; :\quad &W&= \Phi^{(1)}J(\Phi^{(0)})+ \b\Phi^{(1)} E(\Phi^{(0)})\,, \cr m&=3\; :\quad &W&= \Phi^{(1)}\Phi^{(1)} H(\Phi^{(0)})+ \Phi^{(2)} F(\Phi^{(0)})\,, \eeas schematically. The functions $W(\Phi^{(0)})$, $J(\Phi^{(0)}), E(\Phi^{(0)})$ and $H(\Phi^{(0)}), F(\Phi^{(0)})$ are holomorphic functions of the chiral fields. They correspond to the 4d $\CN=1$, 2d $\CN=(0,2)$, and 0d $\CN=1$ superpotentials, respectively. This obviously generalizes to any $m$: \be W= \Phi^{(c_1)} \cdots \Phi^{(c_k)} F_{c_1, \cdots, c_k}(\Phi^{(0)})\,, \qquad c_1+ \cdots+ c_k= m-1\,, \ee schematically,\footnote{In general, we can have distinct paths of degree-zero chiral fields connecting each field of higher degree in the closed loop.} though there is no supersymmetric field theory interpretation for $m{>}3$. \paragraph{Kontsevitch bracket condition.} There is an important condition we should impose on $W$, which can be written as: \be \label{WWzero} \{ W, W\}=0\,, \qquad \Leftrightarrow \qquad \sum_\Phi {\d W\ov \d \Phi} {\d W\ov \d \b \Phi}=0\,, \ee where the sum is over all the fields $\Phi$, for a given polarization~\eqref{double arrows gen}. Here, $\{f, g\}$ denotes the Kontsevitch bracket on the path algebra. It is defined as: \be \{ f, g \}= \sum_\Phi \left( {\d f\ov\d \Phi}{\d g\ov \d\b\Phi} +(-1)^{(|f|+1)|\b\Phi|+(|g|+1)|\Phi|+ |\Phi||\b\Phi|+1} {\d f\ov\d \b\Phi}{\d g\ov\d \Phi} \right)~. \ee Let us note that the condition~\eqref{WWzero} holds for any choice of polarization. The Kontsevitch bracket is a natural generalization of the Poisson bracket on a graded path algebra that admits a polarization. \paragraph{Differential and superpotential.} Given the superpotential above, one can define a differential, $\bd$, of degree $-1$, acting on paths. We have the Leibniz rule: \be \bd (f g) = (\bd f) g + (-1)^{|f||g|} f \bd g\,, \ee with $|f|$ denoting the degree of the path $f$. The differential is given explicitly on the quiver fields by: \beas\label{bd explicit on X} \bd e &= - e \otimes e\,, \cr \bd \Phi &= {\d W\ov \d \b \Phi} + (-1)^{|\Phi|} \Phi \otimes e- e\otimes \Phi\,, \cr \bd \b \Phi &= {\d W\ov \d \Phi} + (-1)^{|\b \Phi|} \b \Phi \otimes e- e\otimes \b \Phi\,, \cr \bd \b e&=\sum_ \Phi (-1)^{|\b \Phi|} \left(\b \Phi \otimes \Phi - \Phi \otimes \b \Phi \right)+ (-1)^{m+1} \b e\otimes e - e\otimes \b e~. \eeas This is obviously of degree $-1$ since $W$ has degree $m-1$ and $|\b \Phi|= m- | \Phi|$. One can check that this is a differential: \be\label{d2=0 rel} \bd^2 =0\,, \ee provided that~\eqref{WWzero} is satisfied. \paragraph{Representations of the quiver algebra and anomaly-free constraint.} Given a quiver algebra, we may want to study its representations. Recall that a quiver representation consists of a vector space $V_i \cong \C^{N_i}$ assigned to each node $i$, and of explicit homomorphisms $\Phi^{(0)}_{ij} \; : V_i \rightarrow V_j$ (that is, fixed $N_i \times N_j$ matrices such that all the quiver relations are satisfied). In physics, the positive integers $N_i$ are the ranks of the unitary gauge group~\eqref{gauge group Q} in a quiver gauge theory. The choice of homomorphism $\Phi^{(0)}$ is a choice of ``vacuum expectation values (VEVs)'' for the chiral multiplets. Not every choice of rank is physically acceptable. There are certain constraints on the allowed choices of ranks, the \emph{generalized anomaly cancellation} conditions~\cite{Franco-ml-2017lpa}, which we will review in section~\ref{subsec: anomaly free} below. It is always a good idea to distinguish between the algebra and its representations. In this work, most of our discussion will be focused on the general ``abstract'' quiver, not on a particular representation. In the B-model, a particular quiver representation corresponds to a particular bound state of D-branes, and the anomaly cancellation condition is a tadpole cancellation condition for the RR flux (at least in the physical setup with $m \leq 3$). ]]>

0\,, \ee Note that, in particular, $m_1$ is a differential --- that is, $(m_1)^2=0$, and $m_2$ is an associative product. The $\Ext$ algebra $A$ is a \emph{minimal} $A_\infty$ algebra, meaning that $m_1=0$ identically. There also exists a natural trace map: \be \gamma: A\rightarrow\C\,, \ee of degree $-m-2$. This is used, in particular, to map to top $\Ext$ elements of degree $m+2$ to elements of $\Ext^0 \cong \Hom$. The multi-products $m_k$ on the $\Ext$ algebra can be computed in the following manner~\cite{Aspinwall-ml-2004bs, Closset-ml-2017yte}. Given any $A_{\infty}$ algebra $\t A$, let us denote by $H^{\bullet}(\t A)$ to be the cohomology of $m_1$. If $\t A$ has no multiplications beyond $m_2$, it turns out that one can define an $A_{\infty}$ structure on $H^{\bullet}(\t A)$ in such a way that there exists an $A_{\infty}$ map~\cite{kad, Aspinwall-ml-2004bs}: \be f: H^{\bullet}(\t A) \rightarrow \t A\,, \ee with $f_1$ equal to a particular representation $H^{\bullet}(\t A) \hookrightarrow \t A$, in which cohomology classes map to (noncanonical) representatives in $\t A$, and such that $m_1=0$ in the $A_{\infty}$ algebra on $H^{\bullet}(\t A)$. One can then use the consistency conditions satisfied by elements of an $A_{\infty}$ map to solve algebraically for the higher products on $H^{\bullet}(\t A)$. In the B-brane description, the algebra $\t A$ is the algebra of complexes of coherent sheaves, with chain maps between complexes. In that construction, $m_1$ is identified with the BRST charge of the B-model. The ``physical'' open string states then live in the cohomology $H^\bullet(\t A)$, which gives us the derived category ${\bf D}^b({\bf X})$ --- see~\cite{Aspinwall-ml-2004jr} for a thorough review. The minimal $A_\infty$ algebra: \be A\equiv H^\bullet(\t A) \ee \looseness=-1 is precisely the $\Ext$ algebra. In the examples discussed in this paper, each B-brane will correspond to a single coherent sheaf, which can be represented in the derived category by a locally-free resolution. The $\Ext$ elements can then be represented by chain maps between resolutions, modulo chain homotopies. The $m_2$ products in $A$ are given by chain map composition. The higher products can be computed by the procedure that we just outlined. In appendix~\ref{App: alg geom}, we explain more thoroughly how to perform these computations explicitly. ]]>

1;i+j < m+2}\pdv{W}{\bar{\Phi}^{(m+1-j;m+2-j)}_{i+j,i}}\pdv{W}{\Phi_{i,i+j}^{(j-1;j)}} ~ . \end{equation} To simplify the resulting expression we use that fact that all terms in $\{W,W\}$ have degree $m-2$ and $m+2$ global symmetry indices. For a monomial $B_{i,j}^{(m-2-c;m+1-c)}\Phi^{(c;c+1)}_{j,i}$ in $\{W,W\}$ we then have: \begin{equation} B_{i,j}^{(m-2-c;m+1-c)}\Phi^{(c;c+1)}_{j,i} = (-1)^{m+1}\Phi^{(c;c+1)}_{j,i}B_{i,j}^{(m-2-c;m+1-c)} ~ . \end{equation} Using this rule, it is straightforward to verify that $\{W,W\}=0$. ]]>

j) \\ \hline s_{i} \,\,\,(1 \le i \le m+1) & \Phi_{i-1,i}^{(0;1)} & \Phi_{j,k}^{(k-j-1;k-j)} \,\,\, (j < i \mbox{ and } j < k) \\ & & \bar{\Phi}^{(m+1-k+j;m+2-k+j)}_{k,j} \,\,\, (k > j \ge i)\\ \hline \end{array} \end{equation} We have indicated the chiral field content separately, since it is what matters for the moduli space. From the expression of the superpotential~\eqref{potential_Cn_Zn}, $s_0$ is evidently a perfect matching. All the $s_{i}$ can be determined by the following simple rule. Given an unbarred field $\Phi_{j,k}^{(k-j-1;k-j)}$, it is in the perfect matching iff $j < i$; otherwise, its conjugate is in the perfect matching. It is straightforward to verify that this results in a collection of fields which covers each term in the superpotential exactly once. \paragraph{Corners.} The $\SU(m+2)$ symmetry permutes the corners $v_\mu$, $\mu=1,\ldots,m+2$, of the toric diagram. Thus, the perfect matching associated to any corner breaks the $\SU(m+2)$ down to $\SU(m+1)\times \U(1)$. In order to find the perfect matching corresponding to a corner it is sufficient to consider how a given representation of $\SU(m+2)$ decomposes under $\SU(m+1)$. Since this breaking corresponds to picking a particular $\SU(m+2)$ fundamental index $\mu$, this behavior is very simple: $\Phi^{(k-1;k)}_{i,i+k}$ decomposes into two representation, $\Phi^{(k-1;k;\mu)}_{i,i+k}$ and $\Phi^{(k-1;k;\cancel{\mu})}$, of $\SU(m+1)$. They are in the $(k-1)-$ and $k$-index antisymmetric representations of $\SU(m+1)$, respectively. Explicitly: \beq \begin{aligned} (\Phi^{(k-1;k;\mu)}_{i,i+k})_{\nu_{1}\cdots\nu_{k-1}} &= (\Phi^{(c;k)}_{i,i+k})_{\mu\nu_{1}\cdots\nu_{k-1}} \\[1mm] (\Phi^{(k-1;k;\cancel{\mu})}_{i,i+k})_{\nu_{1}\cdots\nu_{k}} &= (\Phi^{(c;k)}_{i+k,k})_{\nu_{1}\cdots\nu_{k}} \qquad \nu_{j} \ne \mu \end {aligned} \eeq Similarly, $\bar{\Phi}_{i+k,i}^{(m+1-k;m+2-k)}$ decomposes into two representations and, in keeping with our convention of making all quantum numbers explicit, the conjugate of $\Phi^{(k-1;k;\mu)}_{i,i+k}$ is $\bar{\Phi}^{(m+1-k;m+2-k;\cancel{\mu})}$. Under this breaking, the terms in the superpotential decompose as \beq \begin{array}{cr} & \Phi_{i,i+j}^{(j-1;j;\mu)}\Phi_{i+j,i+j+k}^{(k-1;k;\cancel{\mu})}\bar{\Phi}_{i+j+k,i}^{(m+1-j-k;m+2-j-k;\cancel{\mu})} \\[1.5mm] \Phi_{i,i+j}^{(j-1;j)}\Phi_{i+j,i+j+k}^{(k-1;k)}\bar{\Phi}_{i+j+k,i}^{(m+1-j-k;m+2-j-k)} \to & + \, \Phi_{i,i+j}^{(j-1;j;\cancel{\mu})}\Phi_{i+j,i+j+k}^{(k-1;k;\mu)}\bar{\Phi}_{i+j+k,i}^{(m+1-j-k;m+2-j-k;\cancel{\mu})} \\[1.5mm] & + \, \Phi_{i,i+j}^{(j-1;j;\cancel{\mu})}\Phi_{i+j,i+j+k}^{(k-1;k;\cancel{\mu})}\bar{\Phi}_{i+j+k,i}^{(m+1-j-k;m+2-j-k;\mu)} \end{array} \eeq Hence we see that, for every $\mu$, we get a perfect matching $p_{\mu}$ containing the following fields: \begin{equation} \renewcommand{\arraystretch}{1.3} \begin{array}{|c|c|c|} \hline \ \mbox{ Perfect matching } \ \ & \mbox{ Chirals } & \ \mbox{ Additional fields } \ \ \\ \hline p_{\mu} & \Phi_{i,i+1}^{(0;1;\mu)} & \Phi_{i,i+k}^{(k-1;k;\mu)} \\ & \bar{\Phi}_{m+1,0}^{(0;1;\mu)} & \bar{\Phi}_{i+k,i}^{(m+1-k;m+2-k;\mu)} \\ \hline \end{array} \end{equation} In summary, the perfect matchings give rise to the toric diagrams in~\eqref{eq_toric_Cn_Zn}, confirming that the moduli spaces of these quiver theories are indeed the desired $\mathbb{C}^{m+2}/\mathbb{Z}_{m+2}$ orbifolds. ]]>

0$, $0\leq q

\floor{m\ov 2}$, we consider their conjugates. Nodes 0 and $m$ are identical, up to conjugation of all the fields in the quiver. The rest of the nodes, 1 to $m-1$, are all equivalent. Let us consider the behavior of these quivers under mutations, which are reviewed in appendix~\ref{app: mutations}. Interestingly, node 0 is the only \emph{toric node} of the quiver for $m>1$. By this, we mean that it is the only node with two incoming chiral arrows, which results in a toric phase when mutated. Similarly, node 1 is an \emph{inverse toric node}, i.e.\ we obtain a toric phase when acting on it with the inverse mutation. We plan to carry out a more detailed investigation of the mutations of these quivers in future work. For $m=1$ we have the conifold quiver. In this case, the naive $\SU(2)$ global symmetry is enhanced to $\SU(2)\times \SU(2)$, with the two chiral fields that go from node 1 to node 0 combining to form a doublet of the new $\SU(2)$. The $m=2$ quiver (with its superpotential) first appeared in the mathematical literature in~\cite{lam2014calabi}; see also~\cite{Eager-ml-2018oww}. \begin{figure} \captionsetup[subfigure]{labelformat=empty} \begin{subfigure}[b]{0.13\textwidth} \newcommand{\posnew}{0.35} \newcommand{\arrowHeadPosition}{0.65} \includegraphics{figure20.pdf} \vspace{5.5em} \subcaption{$m=1$. \ \ } \end{subfigure} \hspace{.2cm} \begin{subfigure}[b]{0.18\textwidth} \newcommand{\posnew}{0.4} \newcommand{\arrowHeadPosition}{0.6} \includegraphics{figure21.pdf} \vspace{3.7em} \subcaption{$m=2$. \ \ \ \ \ } \end{subfigure} \begin{subfigure}[b]{0.29\textwidth} \newcommand{\posnew}{0.4} \newcommand{\arrowHeadPosition}{0.6} \includegraphics{figure22.pdf} \vspace{2.85em} \subcaption{$m=3$. \ \ \ \ \ \ \ \ \ \ } \end{subfigure} \hspace{-.5cm} \begin{subfigure}[b]{0.35\textwidth} \newcommand{\posnew}{0.42} \newcommand{\arrowHeadPosition}{0.58} \includegraphics{figure23.pdf} \subcaption{$m=4$. \ \ } \end{subfigure} \begin{subfigure}[b]{0.46\textwidth} \newcommand{\posnew}{0.42} \newcommand{\arrowHeadPosition}{0.58} \includegraphics{figure24.pdf} \renewcommand{\arrowHeadPosition}{0.68} \renewcommand{\arrowHeadPosition}{0.58} \vspace{0.5em} \subcaption{$m=5$. \ \ \ \ \ \ \ } \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \newcommand{\posnew}{0.42} \newcommand{\arrowHeadPosition}{0.58} \includegraphics[width=\textwidth]{figure25.pdf} \renewcommand{\arrowHeadPosition}{0.61} \renewcommand{\arrowHeadPosition}{0.58} \subcaption{$m=6$.} \end{subfigure} ]]>

0$ they give rise to multiple toric phases related by the corresponding order $m+1$ dualities. The $m=1$~\cite{Feng-ml-2002zw} and $2$~\cite{Franco-ml-2016nwv,Franco-ml-2016qxh,Franco-ml-2018qsc} cases have been extensively studied in the literature. In particular, $\mathbb{F}_0^{(1)}$ has 2 toric phases and $\mathbb{F}_0^{(2)}$ has 14 toric phases. ]]>

0$, i.e.\ iff $\beta_i\geq \alpha_i$ for all $1\leq i \leq m+1$. \item The degree of the arrow is \be c=d_{\alpha\beta}-1 ~. \label{degrees_F0^m} \eeq \item The multiplicity of the arrow is $2^{c+1}$. More specifically, the arrow represents $2^{c+1}$ fields that transform in the \be {\bf 2}_1^{\beta_1-\alpha_1}\times {\bf 2}_2^{\beta_2-\alpha_2}\times \ldots \times {\bf 2}_{m+1}^{\beta_{m+1}-\alpha_{m+1}} \eeq representation of the $\SU(2)^{m+1}$ global symmetry, where the subindices run over the different $\SU(2)$ factors. \end{itemize} As usual, we can restrict to fields with $c\leq {m\ov 2}$ by conjugating the arrows with $c>{m\ov 2}$. \paragraph{Superpotential.} As for the $\mathbb{C}^{m+2}/\mathbb{Z}_{m+2}$ family, it is possible to show the construction of these models via iterative orbifold reduction implies that all the terms in the superpotential are cubic. The superpotential terms are given by cubic terms of degree $m-1$ combined into $\SU(2)^{m+1}$ invariants. Once again, it is possible to show that terms for all possible integer partitions of $m-1$ into three integers are present. In fact we can regard the purely cubic superpotential as the characteristic property of the specific toric phases of $\mathbb{F}_0^{(m)}$ that we construct. Let us be more explicit about the superpotential for the $\mathbb{F}_0^{(m)}$ family. From our previous discussion of the field content, there is an arrow connecting nodes $i$ and $j$ whenever $d_{ij}\neq 0$. We will consider the arrow $X_{ij}$ which has $d_{ij}>0$ as the field while we will write $\overline{X}_{ji}$ for its conjugate.\footnote{Note the convention we use for this argument is not the usual one in which we restrict to degrees $c\leq {m\ov 2}$. For example the arrow directed from $(1,1,\cdots,1)$ to $(0,0,\cdots,0)$ is a chiral but in this notation it will be written as the conjugate of $X_{(0,0,\cdots,0),(1,1,\cdots,1)}$.} It is also useful to define a partial ordering relation $\succ$ between two nodes by $j \succ i$ iff $d_{ij}>0$. The superpotential can then be written as \beq W = \sum_{i}\sum_{j \succ i}\sum_{k \succ j}s(i,j,k)X_{ij}X_{jk}\bar{X}_{ki} ~ , \label{W_F0m_general} \eeq where we omit $\SU(2)^{m+1}$ indices and their contractions, and the $s(i,j,k)$ are signs that are necessary for the vanishing of $\{W,W\}$. According to~\eqref{degrees_F0^m}, $X_{ij}$ has degree $d_{ij}-1$, $X_{jk}$ has degree $d_{jk}-1$ and $\bar{X}_{ki}$ has degree $m+1-d_{ik}$. Gauge invariance implies that $d_{ik} = d_{ij} + d_{jk}$, which in turn implies that the degree of any such term is equal to $m-1$ and it is hence present in the superpotential. ]]>

3$. Naively, it may seem that it is physically impossible to go beyond $m=3$, since it would require the gauge theory to live below 0d and the CY$_{m+2}$ to go beyond the critical dimension of Type IIB string theory. In this work we have shown that $m$-graded quivers describe the open string sector of the topological B-model on CY $(m+2)$-folds, for any $m$. To illustrate this correspondence, we constructed toric quivers associated to three infinite families of toric singularities indexed by $m$.\footnote{As usual, due to the dualities discussed in appendix~\ref{app: mutations}, the map between geometry and quivers is not one-to-one. For a given CY singularity, it is possible to start from the quiver we presented and construct all the corresponding duals, by quiver mutations.} We first derived these families using a variety of powerful tools that are available in the toric case, which include: algebraic dimensional reduction (sometimes combined with orbifolding), orbifold reduction, 3d printing and partial resolution. We independently derived all these quiver theories via B-model computations. Our results provide the first explicit examples of $m$-graded quivers with superpotentials for CY $(m+2)$-folds with $m > 4$. Previously, only a few orbifold examples had been presented for $m=4$~\cite{Franco-ml-2017lpa} and $m=3$~\cite{Diaconescu-ml-2000ec,Douglas-ml-2002fr,Franco-ml-2016tcm,Closset-ml-2017yte}. Quivers for more general geometries were studied only up to $m=2$, both in physics and mathematics. In this work, we considerably expanded the exploration of quiver theories associated to CY $(m+2)$-folds. Until now, quiver gauge theories were typically studied at fixed $m$. For each $m$ (and only for $m \leq 2$, so far), one could then consider various infinite families of geometries and construct their dual quiver gauge theories. In the toric case, this approach was significantly accelerated by the study of brane tilings ($m=1$) and brane brick models ($m=2$). In this work, we have included a new ``theory space'' direction to the problem, considering all possible CY dimensions at once. New tools for studying toric quivers, for any $m$, will be discussed in~\cite{toappear1}. \looseness=-1 Various interesting aspects of SUSY gauge theories extend to the more general context of $m$-graded quivers. For instance, we have shown that some of these theories admit periodic duality cascades. Generalizing the well-known behavior of the conifold, we presented explicit examples based on the $C(Y^{1,0}(\mathbb{P}^m))$ family, in which the number of fractional branes remains constant while the number of regular branes depends linearly on the step of the cascade. It would be interesting to investigate the significance of such formal cascades for arbitrary $m$. Interestingly, gravity duals with a running number of regular branes exist for systems of branes at CY 4-folds, namely for $m=2$~\cite{Herzog-ml-2000rz}. It would be interesting to elucidate whether those solutions have a field theoretic interpretation in terms of cascades of trialities. It is natural to expect that order $m+1$ dualities correspond to mutations of exceptional collections of B-branes. This expectation is supported by the known $m=1$~\cite{Herzog-ml-2003zc,Herzog-ml-2004qw,Aspinwall-ml-2004vm} and $m=2$~\cite{Closset-ml-2017yte} cases, mirror symmetry~\cite{Franco-ml-2016qxh,Franco-ml-2016tcm} and the general discussion in~\cite{Franco-ml-2017lpa}. We plan to elaborate on this correspondence in the near future. ]]>

0$. We will follow the method proposed in~\cite{Aspinwall-ml-2004bs} to compute the composition maps $m_k$ of $D^b(\t{\bf X}_{m+2})$. Any object in $D^b(\t{\bf X}_{m+2})$ can be represented by a cochain complex $\mathcal{E}^\bullet$ of locally-free sheaves over ${\t{\bf X}_{m+2}}$. For any pair of complexes, the $\Ext$ groups $\mathrm{Ext}^d_{\t{\bf X}_{m+2}}(\mathcal{E}^\bullet,\mathcal{F}^\bullet)$ can be viewed as the cohomology of the single complex associated with the double complex $(K^{\bullet,\bullet},d,\delta)$ with: \beq K^{p,q}(\mathcal{E}^\bullet,\mathcal{F}^\bullet) = \check{C}^p (\mathcal{U}, \mathcal{H}om^q(\mathcal{E}^\bullet,\mathcal{F}^\bullet))\,, \eeq where $\check{C}^p(\mathcal{U},\cdot)$ denotes the \u{C}ech cochains of degree $p$ associated with some acyclic covering $\mathcal{U}$, and $\mathcal{H}om^q$ denotes the maps of degree $q$ between complexes, i.e.: \beq \mathcal{H}om^q(\mathcal{E}^\bullet,\mathcal{F}^\bullet) = \bigoplus_i \mathcal{H}om(\mathcal{E}^i,\mathcal{F}^{i+q}) ~ . \eeq In the double complex $(K^{\bullet,\bullet},d,\delta)$, $d$ is the differential of \u{C}ech cochains and $\delta$ is defined as follows. Let $\partial_j$ and $\partial'_k$ be differentials of $\mathcal{E}^j$ and $\mathcal{F}^k$ respectively, then for any $\sum_i \phi_{q,i} \in \mathcal{H}om^q(\mathcal{E}^\bullet,\mathcal{F}^\bullet)$ with $\phi_{q,i} \in \mathcal{H}om(\mathcal{E}^i,\mathcal{F}^{i+q})$, we have: \beq \delta_q \phi_{q,i} = \partial'_{q+i} \circ \phi_{q,i} - (-1)^q \phi_{i+1,q} \circ \partial_i~. \eeq For any $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$, we associate to every $a \in \mathrm{Ext}^d_{\t{\bf X}_{m+2}}(\mathcal{E}^\bullet,\mathcal{F}^\bullet)$ an element $\iota(a) \in \oplus_{p+q=d}K^{p,q}(\mathcal{E}^\bullet,\mathcal{F}^\bullet)$, such that the cohomology class of $\iota(a)$ is $a$. Then, there exist maps: \beq f_k: \mathrm{Ext}^\bullet_{\t{\bf X}_{m+2}}(\mathcal{E}_{k-1}^\bullet,\mathcal{E}_k^\bullet) \otimes \cdots \otimes \mathrm{Ext}^\bullet_{\t{\bf X}_{m+2}}(\mathcal{E}_0^\bullet,\mathcal{E}_1^\bullet) \rightarrow \oplus_{p,q}K^{p,q}(\mathcal{E}_0^\bullet,\mathcal{E}_k^\bullet)\,, \eeq of degree $1-k$ for any $k\geq 1$, such that: \beq f_1 = \iota\,, \eeq and \begin{equation}\label{fk} \sum_{r+s+t=k} (-1)^{r+st} f_{n+1-s}(\mathrm{id}^{\otimes r} \otimes m_s \otimes \mathrm{id}^{\otimes t}) = \sum_{2 \leq r \leq n \atop i_1+\cdots +i_r = k} (-1)^w f_{i_1} \circ f_{i_2} \circ \cdots \circ f_{i_r} + d f_k\,, \end{equation} where $w=(r-1)(i_1-1)+(r-2)(i_2-1)+\cdots +(i_{r-1}-1)$ and $\circ$ denotes the composition of maps in $\oplus_{p,q} K^{p,q}(\bullet,\bullet)$. For example, we have: \begin{equation}\label{f2} \iota m_2 = \iota \circ \iota + d f_2\,, \end{equation} and \begin{equation}\label{f3} \iota m_3 = f_2(\mathrm{id} \otimes m_2) - f_2(m_2 \otimes \mathrm{id}) + (\iota \circ f_2) - (f_2 \circ \iota) + d f_3~. \end{equation} To compute the $A_\infty$ structure, the first step is to find representatives for a basis of the $\Ext$ groups, which in turn defines $\iota$. Then, we can employ~\eqref{fk} to compute the composition maps $m_k$. Specifically, we can use~\eqref{f2} to determine $m_2$ and $f_2$, then use~\eqref{f3} to determine $m_3$ and $f_3$ and so forth. In the theories we consider, the B-branes of interest are of the form: \be i_* \mathcal{E}\,, \ee with $i$ the embedding of a complex submanifold $S$ in $\t{\bf X}_{m+2}$, and $\mathcal{E}$ a holomorphic vector bundle over $S$. Suppose that $\mathcal{E}^\bullet_l$ is the Koszul resolution of $i_* \mathcal{E}_l$: \beq \cdots \rightarrow \mathcal{E}^{-i}_l \rightarrow \mathcal{E}^{-i+1}_l \rightarrow \cdots \rightarrow \mathcal{E}^{0}_l \rightarrow i_* \mathcal{E}_l \rightarrow 0 ~ . \eeq Then, $\mathrm{Ext}^d_{\t{\bf X}_{m+2}} (i_*\mathcal{E}_1,i_*\mathcal{E}_2)$ is the same as $\mathrm{Ext}^d_{\t{\bf X}_{m+2}} (\mathcal{E}_1^\bullet,\mathcal{E}_2^\bullet)$, so that we can use the method discussed above to compute the composition maps $m_k$. ]]>

0$, is the sheaf which locally has as its basis the $p^{th}$ tensor power $e_{\mu}^{p}$ of $e_{\mu}$. The sheaf $\Ol(p)$, for $p>0$, is defined to be the dual sheaf of $\Ol(-p)$. In particular let $e^{*}_{\mu}$ be the basis of $\Ol(1)$ in the chart $\mu$ then the transition functions for it are determined by: \begin{equation} e^{*}_{i} = w_{0,i}e_{0}^{*} ~ . \end{equation} $(e^{*}_{\mu})^{p}$ form a basis of $\Ol(p)$ in $U_{\mu}$. Finally, $\Ol(0)$, which is often denoted as $\Ol$, is the trivial sheaf. ]]>

0$. The same is true for $\om{p}$. However their dual bundles do have global sections. For $\Ol(p)$ with $p \ge 0$, a local section is determined by a homogeneous polynomial of degree $p$ in the homogeneous coordinates $z_{\mu}$. These obviously transform in the symmetric $(p,0)$-index tensor representation of $\SU(n+1)$, which has dimension $\binom{n+1+p}{p}$. The tangent bundle $\Omega^{*}$ has $(n+1)^{2}-1$ global sections. In the homogeneous coordinates, these are given by: \begin{equation} z_{\mu}\pdv{}{z_{\nu}}\,, \end{equation} with the linear relation $\sum_{\mu}z_{\mu}\pdv{}{z_{\mu}} = 0$. They transform in the adjoint representation of $\SU(n+1)$. More relevant for us will be the sheaf $\Omega^{*}(-1)$.\footnote{For any sheaf $F$ we define $F(p)$ to be $F$ tensored with $\Ol(p)$.} It has $(n+1)$ of global sections transforming in the $(0,1)$-index representation of $\SU(n+1)$. Locally in $U_{0}$, they can be written as: \begin{align} \varphi^{0} &= -\sum_{i}w_{0,i}\pdv{}{w_{0,i}}\otimes e_{0}\,, \nonumber \nonumber \\ \varphi^{i} &= \pdv{}{w_{0,i}}\otimes e_{0}~. \label{global_sections} \end{align} The maps between two sheaves $E$ and $F$ form a sheaf denoted by $\Hom(E,F)$. The sections~\eqref{global_sections} can also be regarded as the global sections of $\Hom(\Omega,\Ol(-1))$. More generally, they can be regarded as global sections of $\Hom(\Omega^{i+1}(j+1),\Omega^{i}(j))$. \looseness=-1 We can also easily compute the global sections of $\Hom(\Omega^{i+k}(j+k),\Omega^{i}(j))$. These are given by antisymmetric compositions of $\lambda^{i}$ defined above and they transform in the antisymmetric $k$-index\footnote{More formally $(0,k)$, but throughout the paper all the representations we mention are of this form and we will just write $k$ to simplify the notation.} representation of $\SU(n+1)$. Concretely, a basis of them is given by: \begin{equation} \varphi^{\mu_{1}\cdots \mu_{k}} = \frac{1}{k!}\varphi^{[\mu_{1}}\circ \varphi^{\mu_{2}} \circ \cdots \circ \varphi^{\mu_{k}]}~. \label{section_composition} \end{equation} The square brackets represent the antisymmetrization of the indices they enclose. ]]>

3$. In this appendix, we summarize the effect of a mutation on a node, which we denote by $\star$~\cite{Franco-ml-2017lpa}. \paragraph{1. Flavors.} As it is standard, we refer to the arrows connected to the mutated node as \emph{flavors}. It is possible to take all flavors as incoming into the mutated note, simply by trading any arrow that is oriented outward for its conjugate. Once this is done, there is a natural cyclic order for flavors around the node, in which the degree of incoming arrows increases clockwise, as shown on the left of figure~\ref{mutation_flavors}. There can be multiple or no arrows of a given degree. \begin{figure} \centering \includegraphics[width=12.5cm]{mutation_flavors.pdf} ]]>

}[r]^{(c)} & \star} with the arrow \xymatrix{ i \ar@{->}[r]^{(c-1)} & \star}. In terms of the cyclic ordering of flavors previously introduced, this transformation is elegantly implemented as a clockwise rotation of the degrees of the flavors while keeping the spectator nodes fixed, as shown in figure~\ref{mutation_flavors}. \paragraph{2. Mesons.} \looseness=-1 The second step in the transformation of the quiver involves the addition of composite arrows, to which we refer as \emph{mesons}. For every $2$-path \xymatrix{ i \ar@{->}[r]^{(0)} & \star \ar@{->}[r]^{(c)} & j} in $\overline{Q}$, where $c\neq m$, {add a new arrow} \xymatrix{ i \ar@/^0.5pc/[rr]^{(c)} & \star & j}. In other words, we generate all possible mesons involving incoming chiral fields. Sometimes, we might chose to represent the field to be composed with a chiral field as an arrow that goes into the mutated node. The orientation of both arrows, both incoming, naively seems incompatible for composition. The general rule above is equivalent to saying that, in such cases, we use the conjugate of the incoming chiral field for the composition. This phenomenon, dubbed \emph{anticomposition}, was first discussed in the physics literature in the context of quadrality of 0d $\mathcal{N}=1$ theories~\cite{Franco-ml-2016tcm}. \begin{figure} \centering \includegraphics[width=12cm]{composition_anticomposition.pdf} ]]>

}[r]^{(0)} & \star \ar@{->}[r]^{(c)} & j} in $\overline{Q}$, with $c\neq m$, add the new arrow \xymatrix{ i \ar@{->}[r]^{(c)} & j} in $\overline{Q}$ and the new cubic term $\Phi_{ij}^{(c)}\Phi_{\star j}^{(c+1)}\Phi_{i\star}^{(m)} = \Phi_{ij}^{(c)}\Phi_{j\star}^{(m-c-1)}\Phi_{\star i}^{(0)}$ to $W$. Figure~\ref{mutation_cubic_couplings} shows the general form of these terms, which are in one-to-one correspondence with the mesons. \begin{figure} \centering \includegraphics[width=11cm]{mutation_cubic_couplings.pdf} ]]>