A Supersymmetric Quantum Field Theory Formulation of the Donaldson Polynomial Invariants

TL;DR

This paper develops a mathematical framework for a supersymmetric topological quantum field theory on 4-manifolds, linking physical theories to Donaldson invariants and setting the stage for proving Witten's Conjecture.

Contribution

It constructs a rigorous mathematical formulation of twisted N=2 supersymmetric theories related to Donaldson invariants, highlighting geometric structures and connections to gauge theory.

Findings

Established a supermanifold structure over 4-manifolds.

Derived the supersymmetric action as an infinite-dimensional Euler class.

Connected the theory to anti-self-dual connections and gauge group actions.

Abstract

We construct a mathematical framework for twisted N=2 supersymmetric topological quantum field theory on a 4-manifold. Supersymmetry in flat space is defined and the twist homomorphism is constructed, giving us a supermanifold that is the total space of an odd vector bundle over the even 4-manifold. A special category of connections on this space is defined and a decomposition into so-called component fields is proved. The twisted supersymmetric action is computed, and the structure of the action, the decomposition, and the action of a special odd vector field are all shown to have a rich geometrical structure that was partially interpred by Atiyah and Jeffrey. In short, the action is an infinite-dimensional analogue of the Euler class of the vector bundle of self-dual 2-forms over the space of connections mod gauge. This geometrical insight serves two purposes: first, it motivates the study of anti-self-dual connections, intersection theory, and the action of the group of gauge transformations, all of which appear by themselves after the twist. Secondly, it sets the stage for an eventual proof of Witten's Conjecture, relating the Donaldson and Seiberg-Witten invariants. What we build here amounts to a mathematical treatment of a physical treatment of a mathematical construction of Donaldson.