Localization on Three-Manifolds

TL;DR

This paper derives a formula for the partition function of supersymmetric gauge theories on three-spheres, showing it depends solely on the Reeb vector field and matches supergravity predictions in the large N limit.

Contribution

It provides an explicit expression for the partition function on three-manifolds with an almost contact structure, linking field theory results to supergravity solutions.

Findings

Partition function depends only on the Reeb vector field.

Explicit formula involves the double sine function.

Large N limit matches supergravity duals.

Abstract

We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with an almost contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.