Supersymmetric partition functions on Riemann surfaces

TL;DR

This paper derives a unified compact formula for supersymmetric partition functions of various gauge theories on Riemann surfaces times tori, enabling new tests of dualities and connections to black hole entropy.

Contribution

It provides a general formula for supersymmetric partition functions on Riemann surfaces with partial topological twists across multiple dimensions, including new applications and tests.

Findings

Formula computes partition functions for 2d, 3d, and 4d theories on $ imes$ Riemann surfaces and tori.

Validates non-perturbative dualities using explicit examples.

Reproduces black hole entropy from large N ABJM theory on $ imes$ Riemann surfaces.

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 $\Sigma_g \times T^n$ with partial topological twist on $\Sigma_g$, where $\Sigma_g$ 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 $\Sigma_g \times S^1$ reproduces the Bekenstein-Hawking entropy of BPS black holes in AdS$_4$ whose horizon has $\Sigma_g$ topology.