Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index

TL;DR

This paper proves a conjecture relating discrete $S_n$-quotients of 3D N=4 Coulomb branches to Higgs branch phases in higher dimensions, extending the use of these quotients as a versatile analytical tool.

Contribution

It proves a conjecture on Coulomb branches with unitary nodes and extends it to other classical groups, enhancing the understanding of discrete quotients in gauge theory moduli spaces.

Findings

Proved the conjecture for Coulomb branches with unitary nodes.

Extended the conjecture to Coulomb branches with classical groups.

Demonstrated the utility of $S_n$-quotients in studying Coulomb branches.

Abstract

The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global $S_n$ symmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric $S_n$-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, extend it to Coulomb branches with other classical groups. The results promote discrete $S_n$-quotients to a versatile tool in the study of Coulomb branches.