Algebraic properties of the monopole formula

TL;DR

This paper develops an algebraic framework for computing the Hilbert series of Coulomb branches in 3D N=4 gauge theories, providing explicit formulas, pole order relations, and monopole operator classifications.

Contribution

It introduces a fan-based algebraic method to explicitly compute Hilbert series and classify monopole operators for arbitrary gauge groups, advancing the understanding of Coulomb branch structures.

Findings

Explicit Hilbert series formulas for any gauge group

Pole order at t=1 and t=infinity matches moduli space dimensions

Classification of generating monopole operators

Abstract

The monopole formula provides the Hilbert series of the Coulomb branch for a 3-dimensional N=4 gauge theory. Employing the concept of a fan defined by the matter content, and summing over the corresponding collection of monoids, allows the following: firstly, we provide explicit expressions for the Hilbert series for any gauge group. Secondly, we prove that the order of the pole at t=1 and t=infinity equals the complex or quaternionic dimension of the moduli space, respectively. Thirdly, we determine all bare and dressed BPS monopole operators that are sufficient to generate the entire chiral ring. As an application, we demonstrate the implementation of our approach to computer algebra programs and the applicability to higher rank gauge theories.