Twisted supersymmetric 5D Yang-Mills theory and contact geometry

TL;DR

This work constructs a twisted $N=1$ supersymmetric Yang–Mills theory in five dimensions on circle bundles over 4D symplectic manifolds, with contact geometry providing the geometric backbone and a five-dimensional analogue of instantons guiding localization. By defining appropriate BRST-like transformations and a BRST-exact action, the authors localize the theory to contact-instanton equations and compute the full perturbative partition function on $S^5$, obtaining a matrix-model representation that depends on a pair of Chern–Simons couplings. They also lift 3D Chern–Simons observables to 5D, and discuss pure 5D Chern–Simons terms and their twisted extensions, highlighting a key distinction: some observables depend on the contact form while others depend only on the contact planes. The results provide a concrete perturbative window into 5D twisted SYM and its relation to higher-dimensional contact geometry, with potential extensions to nonperturbative sectors and other odd-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.