On Diagonalization in Map(M,G)

TL;DR

The paper asks when a smooth map g:M→G can be conjugated into a maximal torus T and proves that regular maps admit local diagonalization on contractible neighborhoods, with global obstructions arising from nontrivial Weyl-group actions and nontrivial T-bundles. It develops a two-step lifting framework that clarifies how local diagonalizations patch together and relates these obstructions to reductions of the structure group to T and regular sections of the adjoint bundle. These results underpin a Weyl integral formula for functional integrals that sums over topological T-sectors, aligning abelianization techniques with gauge-theory path integrals in two dimensions. The work also discusses complications from non-regular maps and nontrivial G-bundles, connecting diagonalization to gauge-field behavior and structure-group restrictions, and provides a foundation for understanding Weyl symmetry in non-Abelian gauge theories.

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.