Cluster Algebras from Dualities of 2d $${\mathcal{N}}$$ N = (2, 2) Quiver Gauge Theories

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 Kähler moduli space of manifolds constructed from the corresponding Kähler 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.

Defects and quantum Seiberg-Witten geometry

“…(4.67) up to the factor 68) which is independent of the mass and FI parameters. 10 We have confirmed this relation up to u 2 (Q/u) 2 order.…”

“…Further connections to integrability in N = (2, 2) * theories were studied in [68], in particular, Q Q-type relations, similar to (2.7) were examined using cluster algebra methods.…”

Agt/ℤ2

“…Each ZZ brane boundary of Σ corresponds to a vector multiplet on S 4 b constrained by a Z 2 identification (3.8) at opposite points around the equator, in other 17 A pants decomposition is a collection of curves in Σ whose complement consists of spheres with three punctures.…”

“…In the S 2 case these were understood as regularization ambiguities and background FI parameters [17].…”

Gauge and matter superfield theories on S 2

“…In particular, quantum partition functions of two-dimensional N = (2, 2) supergauge models on S 2 were computed in [4,5] and used for studying Seiberg-like dualities of these models [6]. Some mathematical aspects of such dualities were investigated in a recent paper [7]. In [8][9][10] it was shown that partition functions of N = (2, 2) gauge theories on S 2 compute exact Kähler potentials for Calabi-Yau target spaces of N = (2, 2) non-linear sigma-models.…”