Twisted traces on abelian quantum Higgs and Coulomb branches

TL;DR

This paper investigates twisted traces on quantum Higgs branches of 3d N=4 gauge theories, revealing their structure, relation to mirror symmetry, and confirming a conjecture about their expansion in abelian cases.

Contribution

It introduces a natural twisted trace from gauge theory correlators, analyzes its properties, and confirms a conjecture about its expansion in abelian gauge theories.

Findings

The twisted trace induces an inner product and star product on the Higgs branch algebra.

In abelian cases, the trace expands in terms of Verma modules' traces, confirming a conjecture.

The expansion relates to 3d mirror symmetry and Atiyah-Bott fixed-point formulas.

Abstract

We study twisted traces on the quantum Higgs branches $A_{\operatorname{Higgs}}$ of $3d, \mathcal{N}=4$ gauge theories, that is, the quantum Hamiltonian reductions of Weyl algebras. In theories which are good, we define a twisted trace that arises naturally from the correlation functions of the gauge theory. We show that this trace induces an inner product and a short star product on $A_{\operatorname{Higgs}}$. We analyze this trace in the case of an abelian gauge group and show that it has a natural expansion in terms of the twisted traces of Verma modules, confirming a conjecture of the first author and Okazaki. This expansion has a natural interpretation in terms of 3-d mirror symmetry, and we predict that it can be interpreted as an Atiyah-Bott fixed-point formula under the quantum Hikita isomorphism.