On Diagonalization in Map(M,G)

TL;DR

This paper investigates the possibility of smoothly diagonalizing maps from manifolds to compact groups, analyzing local and global obstructions, and applying these insights to gauge theories and functional integrals.

Contribution

It provides a complete analysis of diagonalization obstructions for regular maps and relates these to topological bundles, extending the Weyl integral formula in gauge theory contexts.

Findings

Local diagonalization is always possible for regular maps.

Global obstructions are characterized by non-trivial Weyl group and torus bundles.

The results justify a Weyl integral formula involving topological T-sectors.

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 manifold $M$ to a compact group $G$, is it possible to smoothly `diagonalize' it, i.e.~conjugate it into a map to a maximal torus $T$ of $G$? 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 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 non-regular maps and in the presence of non-trivial $G$-bundles. In particular, we establish a relation between the existence of regular sections of a non-trivial 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.