Transversally Elliptic Complex and Cohomological Field Theory

TL;DR

This paper explores the mathematical structure of a supersymmetric gauge theory on 4D manifolds, focusing on its cohomological aspects and the role of transversally elliptic complexes in defining and computing the theory.

Contribution

It introduces the concept of flipping instantons derived from 5D contact instantons and demonstrates their deformation theory via transversally elliptic complexes, extending the Atiyah-Jeffrey construction.

Findings

Transversally elliptic complex controls deformation theory of flipping instantons.

Equivariant Atiyah-Jeffrey construction yields the Lagrangian as an equivariant Euler class.

Transversal ellipticity ensures non-degeneracy and computability of the theory.

Abstract

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson-Witten theory is a twisted version of ${\cal N}=2$ SYM and can also be constructed using the Atiyah-Jeffrey construction. This theory is concerned with the moduli space of anti-self-dual gauge connections, with a deformation theory controlled by an elliptic complex. More generally, supersymmetry requires considering configurations that look like either instantons or anti-instantons around fixed points, which we call flipping instantons. The flipping instantons of our 4D ${\cal N}=2$ theory are derived from the 5D contact instantons. The novelty is that their deformation theory is controlled by a transversally elliptic complex, which we demonstrate here. We repeat the Atiyah-Jeffrey construction in the equivariant setting and arrive at the Lagrangian (an equivariant Euler class in the relevant field space) that was also obtained from our previous work arXiv:1812.06473. We show that the transversal ellipticity of the deformation complex is crucial for the non-degeneracy of the Lagrangian and the calculability of the theory. Our construction is valid on a large class of quasi toric 4 manifolds.