Topological gauge theories from supersymmetric quantum mechanics on spaces of connections

TL;DR

This paper derives topological gauge theories from supersymmetric quantum mechanics on spaces of connections, linking geometric structures to physical models and illustrating their relations to Donaldson theory and other examples.

Contribution

It introduces supersymmetric quantum mechanics models for infinite-dimensional spaces of connections, providing a new derivation of topological gauge theories and exploring their geometric and physical relations.

Findings

Reformulation of topological gauge theories via supersymmetric quantum mechanics.

Extension of supersymmetric models to infinite-dimensional target spaces.

Connections established between Donaldson theory and gauge theories of flat connections.

Abstract

We rederive the recently introduced $N=2$ topological gauge theories, representing the Euler characteristic of moduli spaces ${\cal M}$ of connections, from supersymmetric quantum mechanics on the infinite dimensional spaces ${\cal A}/{\cal G}$ of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces and introduce supersymmetric quantum mechanics actions modelling the Riemannian geometry of submersions and embeddings, relevant to the projections ${\cal A}\rightarrow {\cal A}/{\cal G}$ and inclusions ${\cal M}\subset{\cal A}/{\cal G}$ respectively. We explain the relation between Donaldson theory and the gauge theory of flat connections in $3d$ and illustrate the general construction by other $2d$ and $4d$ examples.