Dualities of $K$-theoretic Coulomb branches from a once-punctured torus

TL;DR

This paper explores the structure of the quantized SL(2)-character variety of a once-punctured torus, revealing three symmetric subalgebras linked to K-theoretic Coulomb branches, and confirms physics predictions about dualities in 4d theories.

Contribution

It demonstrates the existence of three $bZ_2$-invariant subalgebras within the quantized character variety, corresponding to Coulomb branches, permuted by the mapping class group, aligning mathematical structures with physical dualities.

Findings

Identifies three $bZ_2$-invariant subalgebras isomorphic to Coulomb branches.

Shows these subalgebras are permuted by the $ m SL_2(bZ)$ action.

Confirms predictions from physics literature about dualities in 4d $ m N=2^*$ theories.

Abstract

We consider the quantized $\mathrm{SL}_2$-character variety of a once-punctured torus. We show that this quantized algebra has three $\mathbb{Z}_2$-invariant subalgebras that are isomorphic to quantized $K$-theoretic Coulomb branches in the sense of Braverman, Finkelberg, and Nakajima. These subalgebras are permuted by the $\mathrm{SL}_2(\mathbb{Z})$ mapping class group action. Our results confirm various predictions from the physics literature about 4d $\mathcal{N}=2^*$ theories and their dualities.