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Motivated by some questions in the path integral approach to (topological) gauge theories, we are led to address the following question: given a smooth map from a manifold M to a compact group G, is it possible to smoothly 'diagonalize' it, i.e. conjugate it into a map to a maximal torusWe analyze the local and global obstructions and give a complete solution to the problem for regular maps. We establish that these can always be smoothly diagonalized locally and that the obstructions to doing this globally are non-trivial Weyl group and torus bundles on M . We show how the patching of local diagonalizing maps gives rise to non-trivial T-bundles, explain the relation to winding numbers of maps into G/T and restrictions of the structure group and examine the behaviour of gauge fields under this diagonalization. We also discuss the complications that arise for nonregular maps and in the presence of non-trivial G-bundles. In particular, we establish a relation between the existence of regular sections of a nontrivial adjoint bundle and restrictions of the structure group of a principal G-bundle to T.We use these results to justify a Weyl integral formula for functional integrals which, as a novel feature not seen in the finite-dimensional case, contains a summation over all those topological T-sectors which arise as restrictions of a trivial principal G bundle and which was used previously to solve completely Yang-Mills theory and the G/G model in two dimensions.

We give a derivation of the Verlinde formula for the G k WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function Z Σ×S 1 of Σ × S 1 with an arbitrary number of labelled punctures. By a suitable gauge choice, Z Σ×S 1 is reduced to the partition function of an Abelian topological field theory on Σ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of Σ × S 1 .We derive the G k /G k model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the G k /G k path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift k → k + h is given, based on the index of the twisted Dolbeault complex.

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We comment on various aspects of topological gauge theories possessing N T ≥ 2 topological symmetry:1. We show that the construction of Vafa-Witten and Dijkgraaf-Moore of 'balanced' topological field theories is equivalent to an earlier construction in terms of N T = 2 superfields inspired by supersymmetric quantum mechanics.2. We explain the relation between topological field theories calculating signed and unsigned sums of Euler numbers of moduli spaces.3. We show that the topological twist of N = 4 d = 4 Yang-Mills theory recently constructed by Marcus is formally a deformation of four-dimensional super-BF theory.4. We construct a novel N T = 2 topological twist of N = 4 d = 3 Yang-Mills theory, a 'mirror' of the Casson invariant model, with certain unusual features (e.g. no bosonic scalar field and hence no underlying equivariant cohomology) 5. We give a complete classification of the topological twists of N = 8 d = 3 Yang-Mills theory and show that they are realised as world-volume theories of Dirichlet two-brane instantons wrapping supersymmetric three-cycles of Calabi-Yau three-folds and G 2 -holonomy Joyce manifolds.6. We describe the topological gauge theories associated to D-string instantons on holomorphic curves in K3s and Calabi-Yau 3-folds. 1

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2dimensional surfaces Σ and show that the method of Abelianisation, previously employed for trivial bundles Σ × S 1 , can be adapted to this case. This reduces the non-Abelian theory on M to a 2-dimensional Abelian theory on Σ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and contrast our results with those obtained by Beasley and Witten using the method of non-Abelian localisation, and determine the surgery and framing presecription implicit in this path integral evaluation. We also comment on the extension of these methods to BF theory and other generalisations.

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.

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