# Supersymmetric partition functions and the three-dimensional A-twist

## TL;DR

This paper computes supersymmetric partition functions and loop operator correlations for 3D $ ext{N}=2$ gauge theories on circle bundles over Riemann surfaces, revealing relations between observables on different topologies and connecting 3D results to 2D A-twisted theories.

## Contribution

It introduces a method to derive 3D supersymmetric partition functions on $ ext{M}_{g,p}$ from 2D A-twisted theories, providing new insights into topological relations and dualities.

## Key findings

- Partition functions on $ ext{M}_{g,p}$ are determined by 2D A-twisted theories.
- The $S^3$ partition function is an expectation value of a fibering operator on $S^2 imes S^1$.
- Results have applications to F-maximization and supersymmetric dualities.

## Abstract

We study three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $\mathcal{M}_{g,p}$, an oriented circle bundle of degree $p$ over a closed Riemann surface, $\Sigma_g$. We compute the $\mathcal{M}_{g,p}$ supersymmetric partition function and correlation functions of supersymmetric loop operators. This uncovers interesting relations between observables on manifolds of different topologies. In particular, the familiar supersymmetric partition function on the round $S^3$ can be understood as the expectation value of a so-called "fibering operator" on $S^2 \times S^1$ with a topological twist. More generally, we show that the 3d $\mathcal{N}=2$ supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on $\mathcal{M}_{g,p}$ are fully determined by the two-dimensional A-twisted topological field theory obtained by compactifying the 3d theory on a circle. We give two complementary derivations of the result. We also discuss applications to F-maximization and to three-dimensional supersymmetric dualities.