Supersymmetric partition functions on Riemann surfaces

TL;DR

This work delivers a universal localization-based recipe for computing supersymmetric partition functions on $Σ_g imes T^n$ across 2d, 3d, and 4d gauge theories, expressing the higher-genus twisted index as a Jeffrey–Kirwan residue sum over Bethe Ansatz equations. It carefully treats non-Abelian gauge dynamics via a regularization of W-boson singularities and accommodates flavor fluxes, Wilson lines, and operator insertions, unifying the computation with known $g=0$ results. The authors validate the framework through Abelian and non-Abelian examples, show non-perturbative dualities (Aharony, Seiberg, Giveon–Kutasov) hold at higher genus, and demonstrate that the large-$N$ ABJM index reproduces the entropy of AdS$_4$ black holes with horizons $Σ_g$. The approach thus links detailed field-theoretic index calculations to holographic entropy and provides a robust toolkit for exploring dualities and non-perturbative data in supersymmetric gauge theories.

Abstract

We present a compact formula for the supersymmetric partition function of 2d N=(2,2), 3d N=2 and 4d N=1 gauge theories on $Σ_g \times T^n$ with partial topological twist on $Σ_g$, where $Σ_g$ is a Riemann surface of arbitrary genus and $T^n$ is a torus with n=0,1,2, respectively. In 2d we also include certain local operator insertions, and in 3d we include Wilson line operator insertions along $S^1$. For genus g=1, the formula computes the Witten index. We present a few simple Abelian and non-Abelian examples, including new tests of non-perturbative dualities. We also show that the large N partition function of ABJM theory on $Σ_g \times S^1$ reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS$_4$ whose horizon has $Σ_g$ topology.

Figures (1)

• Figure 1: Analytic structure of $e^{iB}$ in the complex $z$ plane, for $k=4$ and $y=0.7$. The black dots are the points $\{0, y, y^{-1}\}$ from left to right. The red dots are the solutions to $e^{iB} = 1$. The blue delimiter is the line $|e^{iB}|=1$, while the shaded and white regions have $|e^{iB}| >1$ and $<1$, respectively. The dashed contour inside the shaded region is $\mathcal{C}_a$, the one inside the white region is $\mathcal{C}_b$.