Twisted Indices of 3d ${\mathcal N} = 4$ Gauge Theories and Enumerative Geometry of Quasi-Maps

TL;DR

This work identifies the twisted index of 3d ${\mathcal N}=4$ theories on $S^1\times\Sigma$ with the virtual Euler characteristics of moduli spaces of twisted quasi-maps to the Higgs branch, providing a geometric interpretation of the Jeffrey-Kirwan residue contour integrals. Localization to generalized vortex equations and the subsequent fixed-point analysis yield a precise correspondence between contour data and enumerative invariants on moduli spaces, including a clean $t\to1$ limit that connects to Rozansky-Witten invariants. The authors also demonstrate mirror symmetry as a duality exchanging the H- and C-twists, thereby equating generating functions of quasi-map invariants for mirror Higgs branches and exchanging degree-counting with equivariant parameters. The results unify physical localization with sophisticated algebraic geometry and open doors to further extensions involving stacks, non-abelian gauge groups, and a deeper categorical understanding of the supersymmetric ground states. These insights have potential implications for computations of invariants in string theory compactifications and for a broader class of supersymmetric gauge theories.

Abstract

We explore the geometric interpretation of the twisted index of 3d ${\mathcal N} =4$ gauge theories on $S^1\times Σ$ where $Σ$ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of generalised vortex equations on $Σ$, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d ${\mathcal N} = 4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters.

Figures (2)

• Figure 1: Quiver for $U(N_c)$ supersymmetric QCD with $N_f$ fundamental hypermultiplets
• Figure 2: $T[SU(N)]$ theory