N=2 TOPOLOGICAL GAUGE-THEORY, THE EULER CHARACTERISTIC OF MODULI SPACES, AND THE CASSON INVARIANT

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 M and the partition function equals the Euler number chi(M)) of 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 chi(M) 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.