Localization on Three-Manifolds
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.