Topological gauge theories of antisymmetric tensor fields

Abstract

We introduce a new class of topological gauge field theories in any dimension, based on anti-symmetric tensor fields, and discuss the BRST-quantization of these reducible systems as well as the equivalence of BRST-quantization and Schwarz's method of resolvents in detail. As a consequence we can use path-integral techniques and BRST-symmetry to prove metric independence and other properties of the Ray-Singer torsion. We pay particular attention to the presence of zero modes and discuss various methods of treating them in these models and other topological field theories. Non-Abelian models in two dimensions provide us with a complete Nicolai map for Yang-Mills theory on an arbitrary two-surface, as well as with a theory of topological gravity which is closely related to Hitchin's self-duality equation on a Riemann surface. Candidate observables for Abelian models in any dimension are linking and intersection numbers of manifolds for which we give explicit path-integral representations.