The Hilbert series of 3d N=2 Yang-Mills theories with vectorlike matter

TL;DR

This work generalizes the monopole Hilbert-series framework to 3d $\mathcal{N}=2$ Yang-Mills theories with vectorlike matter and no Chern-Simons terms by counting gauge-invariant, dressed monopole operators in a restricted magnetic-charge sum that accounts for instanton lifting. The central tool is a monopole formula $H(t,z,\mathbf{x})=\sum_{m\in \Gamma_q} [ t^{R(m)} z^{J(m)} \prod_i x_i^{F_i(m)} ] \cdot H_{T_m}(t,\mathbf{x})$, where dressing comes from the residual massless theory $T_m$ and the sum runs over the quantum sublattice $\Gamma_q$ determined by nonperturbative effects. The authors compute explicit Hilbert series for $U(1)$, $SU(2)$, $U(N)$, and $USp(2N)$ with fundamental matter, reproducing known moduli spaces and chiral rings without reliance on singular effective superpotentials; the quantum relations arise from the Coulomb–Higgs fibration rather than superpotential constraints. This framework provides a robust, testable method to study moduli spaces and dualities in 3d gauge theories and can be extended to additional gauge groups and matter content.

Abstract

This paper presents a formula for the Hilbert series that counts gauge invariant chiral operators in 3d N=2 Yang-Mills theories with vectorlike matter and no Chern-Simons interactions. The formula counts 't Hooft monopole operators dressed by gauge invariants of a residual gauge theory of massless fields in the monopole background, which is determined by the Higgs mechanism. The sum over magnetic charges is restricted due to instanton effects that partially lift the classical Coulomb branch. The formalism is applied to unitary and symplectic gauge theories with fundamental matter, reproducing old results for the moduli space of vacua and the chiral ring, without resorting to any further effective superpotential on the moduli space.