Coulomb branches for rank 2 gauge groups in 3d N=4 gauge theories

TL;DR

This paper develops a new method to compute the Coulomb branch's chiral ring in 3d N=4 gauge theories with rank 2 groups, using semi-group and Hilbert basis techniques for efficient Hilbert series calculation.

Contribution

It introduces a novel geometric approach involving fans and semi-groups to identify monopole operators generating the Coulomb branch, enabling explicit Hilbert series computation for rank 2 groups.

Findings

Finite set of monopole operators generate the chiral ring.

Explicit Hilbert series can be computed from minimal generators.

Method applies efficiently to all rank 2 gauge groups.

Abstract

The Coulomb branch of 3-dimensional N=4 gauge theories is the space of bare and dressed BPS monopole operators. We utilise the conformal dimension to define a fan which, upon intersection with the weight lattice of a GNO-dual group, gives rise to a collection of semi-groups. It turns out that the unique Hilbert bases of these semi-groups are a sufficient, finite set of monopole operators which generate the entire chiral ring. Moreover, the knowledge of the properties of the minimal generators is enough to compute the Hilbert series explicitly. The techniques of this paper allow an efficient evaluation of the Hilbert series for general rank gauge groups. As an application, we provide various examples for all rank two gauge groups to demonstrate the novel interpretation.