The moduli spaces of $3d$ ${\cal N} \ge 2$ Chern-Simons gauge theories and their Hilbert series
Stefano Cremonesi, Noppadol Mekareeya, Alberto Zaffaroni
TL;DR
The paper presents a comprehensive monopole-based Hilbert-series framework for counting gauge-invariant chiral operators in 3d ${\cal N}\ge 2$ YM-CS theories, including abelian and nonabelian cases such as ABJM/ABJ(M). It systematically incorporates background magnetic charges, real masses, FI parameters, and superpotentials, and demonstrates exact results through a suite of explicit examples and dualities (notably Dorey–Tong mirror symmetry and ABJM/ABJM). It then connects these field-theory counts to the geometric moduli spaces of M2-brane worldvolumes, yielding detailed Hilbert-series descriptions of CY4 and hyperKähler moduli spaces, both with and without quantum corrections. The framework further yields powerful insights into the structure of moduli spaces under various CS level configurations, fractional branes, and toric geometries, enabling precise predictions for the spectrum and operator relations in these theories.
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 hyperKähler twofold singularities with ${\cal N} = 2$ and ${\cal N} = 3$ supersymmetry.