Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories

TL;DR

This paper introduces a universal formula for computing the Hilbert series of Coulomb branches in 3d N=4 gauge theories, enabling algebraic and geometric analysis of these moduli spaces beyond previous limitations.

Contribution

The authors present a simple, general formula for the Hilbert series of Coulomb branches, applicable to all good or ugly 3d N=4 gauge theories, advancing the understanding of their chiral rings.

Findings

Derived a universal formula for Coulomb branch Hilbert series

Validated the formula through multiple gauge theory examples

Confirmed mirror symmetry predictions via Hilbert series comparisons

Abstract

This paper addresses a long standing problem - to identify the chiral ring and moduli space (i.e. as an algebraic variety) on the Coulomb branch of an N = 4 superconformal field theory in 2+1 dimensions. Previous techniques involved a computation of the metric on the moduli space and/or mirror symmetry. These methods are limited to sufficiently small moduli spaces, with enough symmetry, or to Higgs branches of sufficiently small gauge theories. We introduce a simple formula for the Hilbert series of the Coulomb branch, which applies to any good or ugly three-dimensional N = 4 gauge theory. The formula counts monopole operators which are dressed by classical operators, the Casimir invariants of the residual gauge group that is left unbroken by the magnetic flux. We apply our formula to several classes of gauge theories. Along the way we make various tests of mirror symmetry, successfully comparing the Hilbert series of the Coulomb branch with the Hilbert series of the Higgs branch of the mirror theory.