Positive Traces on Certain ${\rm SL}(2)$ Coulomb Branches

TL;DR

This paper classifies positive traces on Coulomb branch algebras associated with certain gauge theories, enhancing understanding of their mathematical structure and physical implications.

Contribution

It provides a classification of positive traces on Coulomb branch algebras for specific cases related to ${ m SL}(2)$ gauge theories and Kleinian singularities.

Findings

Classified positive traces for quantizations of Kleinian singularities of type D.

Classified positive traces for algebras involving $K$-theoretic Coulomb branches of ${ m SL}(2)$ gauge theories.

Abstract

For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $\rho$ of $\mathcal{A}$, there is a notion of a positive trace. When we have a three-dimensional $\mathcal{N}=4$ gauge theory or four-dimensional $\mathcal{N}=2$ gauge theory compactified on a circle, classification of positive traces on its Coulomb branch $\mathcal{A}$ can give a better understanding of this theory. We classify positive traces on $\mathcal{A}$ in two cases. The first case is when $\mathcal{A}$ is a quantization of a Kleinian singularity of type $D$, with certain restriction on the quantization parameter. The second case is when $\mathcal{A}=K^{{\rm SL}(2,\mathbb{C}[[t]])\rtimes \mathbb{C}_q^{\times}}({\rm Gr}_{{\rm PGL}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure ${\rm SL}(2)$ and ${\rm PGL}(2)$ gauge theories.