\(N=2\) topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant. (English) Zbl 0769.53017
Summary: 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 \(\chi({\mathcal M})\) 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 \(\chi({\mathcal M})\) 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.
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
Keywords:
Mathai-Quillen formalism; supersymmetric quantum mechanics; Casson invariant; Penner matrix modelReferences:
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