$K$-theoretic Coulomb branches of quiver gauge theories and cluster varieties

TL;DR

This paper establishes an isomorphism between the quantized K-theoretic Coulomb branch algebra of a quiver gauge theory and a cluster algebra, revealing deep connections between gauge theory, algebra, and cluster transformations.

Contribution

It explicitly constructs the initial seed for the cluster algebra associated with the Coulomb branch and proves the existence of a cluster Donaldson--Thomas transformation.

Findings

Isomorphism between Coulomb branch algebra and cluster algebra.

Explicit initial seed for the cluster algebra.

Existence of a cluster Donaldson--Thomas transformation.

Abstract

For $\Gamma$ a quiver without 1-cycles, we show that the Braverman--Finkelberg--Najakima quantized $K$-theoretic Coulomb branch algebra $\mathscr{A}_\Gamma$ of the corresponding quiver gauge theory is isomorphic to the quantized universally Laurent algebra (upper cluster $X$-algebra) associated to an explicit initial seed $\Pi_\Gamma$. We also show that $\Pi_\Gamma$ admits a cluster Donaldson--Thomas transformation as defined by Keller.