Cluster algebras from dualities of 2d N=(2,2) quiver gauge theories

TL;DR

This work exposes a deep link between two-dimensional $\mathcal{N}=(2,2)$ quiver gauge theories and cluster algebras by showing that Seiberg-like dualities enact cluster mutations on quivers and map FI parameters to dual cluster coordinates. It provides a detailed mapping of chiral and twisted chiral rings, and demonstrates that the $S^2$ partition function is invariant under dualities up to calculable contact terms, linking geometry of Kahler moduli spaces to cluster transformations. The results imply a cluster-algebra structure on the extended Kahler moduli spaces of Calabi–Yau geometries engineered by these GLSMs and offer a framework to relate dualities to dual coordinates, with connections to integrable systems via $N=(2,2)^*$ theories. These insights pave the way for exploring cluster structures in broader QFT contexts and their geometric and combinatorial consequences.

Abstract

We interpret certain Seiberg-like dualities of two-dimensional N=(2,2) quiver gauge theories with unitary groups as cluster mutations in cluster algebras, originally formulated by Fomin and Zelevinsky. In particular, we show how the complexified Fayet-Iliopoulos parameters of the gauge group factors transform under those dualities and observe that they are in fact related to the dual cluster variables of cluster algebras. This implies that there is an underlying cluster algebra structure in the quantum Kahler moduli space of manifolds constructed from the corresponding Kahler quotients. We study the S^2 partition function of the gauge theories, showing that it is invariant under dualities/mutations, up to an overall normalization factor whose physical origin and consequences we spell out in detail. We also present similar dualities in N=(2,2)* quiver gauge theories, which are related to dualities of quantum integrable spin chains.

Figures (8)

• Figure 1: Seiberg-like duality of SQCD-like theories in quiver notation. The circles represent unitary gauge groups, and the Kähler coordinates associated to the gauge groups are indicated above them. The squares represent unitary flavor groups, and arrows represent chiral multiplets transforming in the fundamental representation of the group at the tail and in the antifundamental of the group at the head. Since all matter fields are bifundamental, the flavor symmetry is actually given by $S[ U(N_f) \times U(N_a) ]$. On the left is theory $\mathfrak{A}$: a $U(N)$ theory with $N_f$ fundamentals and $N_a$ antifundamentals. On the right is theory $\mathfrak{B}$: a $U(N')$ theory, where $N' = \max(N_f,N_a) - N$, with $N_a$ fundamentals, $N_f$ antifundamentals and $N_fN_a$ extra gauge singlets, as well as a superpotential $W = q'M\tilde{q}'$. The two theories are IR dual.
• Figure 2: Gauging the flavor symmetry. We promote a $U(N_1) \times \cdots \times U(N_g)$ subgroup of the $U(N_f)$ flavor symmetry to be a gauge symmetry, in such a way that $U(N_i)$ has embedding index $a_i$, with $\sum_i a_i N_i + M = N_f$. This leaves an unbroken $U(M)$ flavor symmetry, as well as a $\prod_i SU(a_i)$ symmetry, the latter of which is not represented by any node. These additional flavor symmetries can be broken by superpotential terms. The resulting quiver has $a_i$ arrows pointing to $U(N_i)$, and one arrow pointing to $U(M)$.
• Figure 3: The "UV completion" of a non-conformal quiver where we extend the theory by a $U(1)$ flavor node. The twisted mass with respect to this flavor symmetry sets the UV renormalization scale $u$. The new node is connected to the $i$-th node by $|\beta_i|$ arrows, in the direction such that the extended quiver is conformal.
• Figure 4: Local picture of cluster duality. From the point of view of the dualized node $U(N)$, the Coulomb branch parameters of adjacent nodes play the role of twisted masses. The duality properties of $Z_{S^2}$ can be inferred by: 1) isolating the terms of the partition function participating in the duality and treating the Coulomb branch parameters as twisted masses (upper arrow); 2) dualizing (right arrow); 3) regrouping the twisted masses into Coulomb parameters of neighboring gauge groups (lower arrow). In this example all embedding indices are 1.
• Figure 5: Cluster mutations on the quiver diagram for the Gulliksen-Negård Calabi-Yau threefold. The numbers next to the arrows denote the multiplicity. Figure (a) is the GLSM model proposed in Jockers:2012zr, while the other ones are obtained via mutation on the left ($\mu_1$) or the right ($\mu_2$) node.

Theorems & Definitions (10)

• Proposition B.1
• proof
• Corollary B.2
• proof
• Corollary B.3
• proof
• Lemma D.1
• proof
• Proposition D.2
• proof