Higgs branch localization in three dimensions

TL;DR

This work develops Higgs branch localization for three-dimensional $\mathcal{N}=2$ Chern-Simons-matter theories on $S^3_b$ and $S^2\times S^1$, showing that the partition function or index can be written as a finite sum over Higgs vacua multiplied by classical, one-loop, and vortex factors. The authors derive the BPS equations via a $\mathcal{Q}$-exact deformation, classify Coulomb, Higgs, and vortex solutions, and compute the associated determinants with an equivariant index theorem; in a suitable limit the deformed Coulomb branch is suppressed and the full answer factorizes into a vortex/anti-vortex partition function controlled by a 3d vortex counting problem. They demonstrate the equivalence with the conventional Coulomb-branch integral through residues and show that, in a 2d limit, the 3d vortex partition function reduces to the familiar 2d vortex counting, highlighting a unifying structure across geometries. The results connect to holomorphic blocks and suggest broad applicability to other manifolds and even higher dimensions, where vortex membranes may play a similar organizing role. The analysis provides a concrete, computable framework for exact results in 3d supersymmetric gauge theories and clarifies the non-perturbative content encoded in vortex sectors.

Abstract

We show that the supersymmetric partition function of three-dimensional N=2 R-symmetric Chern-Simons-matter theories on the squashed S^3 and on S^2 x S^1 can be computed with the so-called Higgs branch localization method, alternative to the more standard Coulomb branch localization. For theories that could be completely Higgsed by Fayet-Iliopoulos terms, the path integral is dominated by BPS vortex strings sitting at two circles in the geometry. In this way, the partition function directly takes the form of a sum, over a finite number of points on the classical Coulomb branch, of a vortex-string times an antivortex-string partition functions.