Supersymmetric Field Theories on Three-Manifolds

TL;DR

This paper develops a framework for constructing and analyzing N=2 supersymmetric field theories on three-manifolds, revealing geometric conditions for supersymmetry and enabling calculations like energy-momentum tensor correlators via localization.

Contribution

It introduces a method to define supersymmetric theories on arbitrary three-manifolds using new minimal supergravity, identifying geometric structures needed for supersymmetry and providing explicit Lagrangians.

Findings

Supersymmetry requires an almost contact metric structure on the manifold.

Two supercharges of opposite R-charge exist on every Seifert manifold.

Localization enables computation of energy-momentum tensor correlators in superconformal theories.

Abstract

We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N=2 theories with a U(1)_R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on M possesses a single supercharge, if and only if M admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on M, even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on M and discuss their flat-space limit, which can be analyzed using the R-current supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in N=2 superconformal theories can be calculated using localization on a squashed sphere.