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

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.