The moduli spaces of $3d$ ${\cal N} \ge 2$ Chern-Simons gauge theories and their Hilbert series

TL;DR

This paper derives a formula for the Hilbert series in 3d ${ m N} \\ge 2$ Chern-Simons theories, enabling counting of gauge-invariant operators, including monopoles, with applications to various supersymmetric gauge theories.

Contribution

It provides a general Hilbert series formula for 3d ${ m N} \\ge 2$ theories, including abelian and nonabelian cases, with detailed analysis of ABJ(M) and M2-brane worldvolume theories.

Findings

Derived a Hilbert series formula for abelian theories without nonperturbative corrections.

Applied the formula to nonabelian theories, including ABJ(M) models.

Analyzed the operator spectrum of M2-brane theories probing Calabi-Yau and hyperKähler singularities.

Abstract

We present a formula for the Hilbert series that counts gauge invariant chiral operators in a large class of 3d ${\cal N} \ge 2$ Yang-Mills-Chern-Simons theories. The formula counts 't Hooft monopole operators dressed by gauge invariants of a residual gauge theory of massless fields in the monopole background. We provide a general formula for the case of abelian theories, where nonperturbative corrections are absent, and consider a few examples of nonabelian theories where nonperturbative corrections are well understood. We also analyze in detail nonabelian ABJ(M) theories as well as worldvolume theories of M2-branes probing Calabi-Yau fourfold and hyperK\"ahler twofold singularities with ${\cal N} = 2$ and ${\cal N} = 3$ supersymmetry.