Do metric independent classical actions lead to topological field theories?

Abstract We investigate the quantum theory of non-abelian BF -systems (gauge theories with the classical metric independent action ∫ tr BF A ). The fact that due to a complicated (on-shell reducible) gauge structure the quantum action of these theories does not differ only by a BRST commutator from the classical action, and that moreover the BRST operator turns out to be metric dependent, renders the standard arguments for metric independence inapplicable. We establish the topological nature of these models and argue that in gauge theories the information on gauge invariance is contained entirely in the metric independent part of the BRST operator. We make some general remarks on the relati…

Topological field theory

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 $\cal M$ 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…

QUANTUM YANG-MILLS THEORY ON ARBITRARY SURFACES

We study quantum Maxwell and Yang-Mills theory on orientable two-dimensional surfaces with an arbitrary number of handles and boundaries. Using path integral methods we derive general and explicit expressions for the partition function and expectation values of contractible and noncontractible Wilson loops on closed surfaces of any genus, as well as for the kernels on manifolds with handles and boundaries. In the Abelian case we also compute correlation functions of intersecting and self-intersecting loops on closed surfaces, and discuss the role of large gauge transformations and topologically nontrivial bundles.

TOPOLOGICAL GAUGE THEORIES FROM SUPERSYMMETRIC QUANTUM MECHANICS ON SPACES OF CONNECTIONS

We rederive the recently introduced $N=2$ topological gauge theories, representing the Euler characteristic of moduli spaces ${\cal M}$ of connections, from supersymmetric quantum mechanics on the infinite dimensional spaces ${\cal A}/{\cal G}$ of gauge orbits. To that end we discuss variants of ordinary supersymmetric quantum mechanics which have meaningful extensions to infinite-dimensional target spaces and introduce supersymmetric quantum mechanics actions modelling the Riemannian geometry of submersions and embeddings, relevant to the projections ${\cal A}\rightarrow {\cal A}/{\cal G}$ and inclusions ${\cal M}\subset{\cal A}/{\cal G}$ respectively. We explain the relation between Donal…