Abstract
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 manifoldM to a compact groupG, is it possible to smoothly “diagonalize” it, i.e. conjugate it into a map to a maximal torusT ofG?
We 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 onM. We explain the relation of the obstructions to winding numbers of maps intoG/T and restrictions of the structure group of a principalG bundle toT and examine the behaviour of gauge fields under this diagonalization. We also discuss the complications that arise in the presence of non-trivialG-bundles and for non-regular maps.
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 topologicalT-sectors which arise as restrictions of a trivial principalG bundle and which was used previously to solve completely Yang-Mills theory and theG/G model in two dimensions.
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Communicated by R.H. Dijkgraaf
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Blau, M., Thompson, G. On diagonalization in map(M,G). Commun.Math. Phys. 171, 639–660 (1995). https://doi.org/10.1007/BF02104681
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DOI: https://doi.org/10.1007/BF02104681