Topological gauge theories from supersymmetric quantum mechanics on spaces of connections

Abstract

We rederive the recently introduced $N=2$ topological gauge theories, representing the Euler characteristic of moduli spaces $\mathcal{M}$ of connections, from supersymmetric quantum mechanics on the infinite dimensional spaces $\mathcal{A}/\mathcal{G}$ 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 $\mathcal{A}\to \mathcal{A}/\mathcal{G}$ and inclusions $\mathcal{M}\subset \mathcal{A}/\mathcal{G}$ 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.

~15 pp

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