# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1904.12782