N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant

TL;DR

This paper explores a topological N=2 gauge theory that relates to the Euler characteristic of moduli spaces, connecting mathematical invariants like the Casson invariant with supersymmetric quantum mechanics and proposing new topological models.

Contribution

It introduces a novel topological gauge theory framework that links the Euler characteristic of moduli spaces to invariants like the Casson invariant and develops new supersymmetric quantum mechanics based on geometric equations.

Findings

Partition function equals the Euler number of the moduli space

Proposes the Euler number of flat connection moduli space as a generalization of the Casson invariant

Discusses potential for a topological Penner matrix model

Abstract

We discuss gauge theory with a topological N=2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space $\cal M$ and the partition function equals the Euler number of $\cal M$. We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi equations. We interpret the gauge theory actions from the Atiyah-Jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence of these considerations we propose the Euler number of the moduli space of flat connections as a generalization to arbitrary three-manifolds of the Casson invariant. We also comment on the possibility of constructing a topological version of the Penner matrix model.