Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review

TL;DR

This review explains how to compute exact supersymmetric partition functions of 3D $ ext{N}=2$ gauge theories on various three-manifolds, highlighting localization techniques, dualities, and recent advances in the field.

Contribution

It provides a comprehensive overview of methods for calculating partition functions of 3D $ ext{N}=2$ theories on Seifert manifolds, including recent developments and duality tests.

Findings

Exact partition functions can be computed using localization techniques.

Supersymmetric partition functions serve as precise tests for infrared dualities.

The review covers a range of three-manifolds including $S^3$, $S^2 imes S^1$, and Seifert manifolds.

Abstract

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D $\mathcal{N}{=}2$ supersymmetric Chern-Simons-matter theories (with an $R$-symmetry) on half-BPS closed three-manifolds---including $S^3$, $S^2 \times S^1$, and any Seifert three-manifold. Three-dimensional gauge theories can flow to non-trivial fixed points in the infrared. In the presence of 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.