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

TL;DR

This paper reveals that certain dualities in 2d N=(2,2) quiver gauge theories can be understood as cluster mutations, uncovering an underlying cluster algebra structure in their quantum Kähler moduli space and analyzing their invariance properties.

Contribution

It establishes a connection between Seiberg-like dualities in 2d N=(2,2) quiver gauge theories and cluster algebra mutations, providing new insights into their mathematical structure.

Findings

Dualities correspond to cluster mutations in cluster algebras.

S^2 partition function remains invariant under dualities, up to normalization.

Dualities in N=(2,2)* theories relate to quantum integrable spin chain dualities.

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.