Twisted supersymmetric 5D Yang-Mills theory and contact geometry

TL;DR

This paper extends 3D Chern-Simons localization techniques to 5D supersymmetric Yang-Mills theory on contact manifolds, deriving a matrix model for the partition function on a 5-sphere.

Contribution

It introduces a twisted 5D supersymmetric Yang-Mills theory on contact manifolds and computes its partition function, generalizing previous 3D results and contact geometry applications.

Findings

Partition function expressed as a matrix model

Construction of 5D contact instanton equations

Generalization to higher-dimensional contact manifolds

Abstract

We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90's, and in a way it is covariantization of their ideas for a contact manifold.