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

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 $\mathcal{M}$ and the partition function equals the Euler number of $\mathcal{M}$. 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.

50 p. (with a4.sty: 40 p.)

References in corpus (2)

• The Mathai-Quillen Formalism and Topological Field Theory
• Topological gauge theories from supersymmetric quantum mechanics on spaces of connections

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