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

TL;DR

This work builds a bridge between N=2 topological gauge theories and the geometry of moduli spaces by recasting partition functions as Euler characteristics via the Mathai–Quillen formalism. It shows that the universal geometry of the space of connections together with moduli-space constraints yields Z = χ(\mathcal{M}) for various moduli (flat connections in 2D/3D and instantons in 4D), and relates these results to Floer cohomology and the Casson invariant through an Atiyah–Jeffrey interpretation. The authors propose the Euler characteristic of moduli spaces, χ(\mathcal{M}), as a natural generalization of the Casson invariant to arbitrary 3-manifolds and discuss the mathematical and physical subtleties, including reducible connections and orbifold/singular structures. They also outline potential extensions, including a topological Penner-type model, and emphasize the deep ties between supersymmetric quantum mechanics, Gauss–Codazzi geometry, and gauge theory in low dimensions.

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.