Topology of weak $G$-bundles via Coulomb gauges in critical dimensions

TL;DR

This paper develops a framework to define and analyze the topology of Sobolev G-bundles with connections in critical dimensions, establishing invariance under gauge changes and criteria for topological flatness via Yang-Mills energy.

Contribution

It introduces a method to associate a topological class to Sobolev G-bundles with connections in critical dimensions, invariant under gauge transformations, and provides approximation and stability results.

Findings

Topological isomorphism class can be assigned to Sobolev bundles with connections in critical dimensions.

Any such bundle with a Coulomb gauge Sobolev connection is actually Hölder continuous.

Topological stability occurs under bounded Yang-Mills energy with equiintegrable curvature norms.

Abstract

The transition maps for a Sobolev $G$-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle $P$ is equipped with a Sobolev connection $A$, then one can associate a topological isomorphism class to the pair $\left( P, A\right),$ which is invariant under Sobolev gauge changes and coincides with the usual notions for regular bundles and connections. This is based on a regularity result which says any bundle in the critical dimension in which a Sobolev connection is in Coulomb gauges are actually $C^{0,\alpha}$ for any $\alpha < 1.$ We also show any such pair can be strongly approximated by smooth connections on smooth bundles. Finally, we prove that for sequences $(P^{\nu},A^{\nu})$ with uniformly bounded $n/2$-Yang-Mills energy, the topology stabilizes if the $n/2$ norm of the curvatures are equiintegrable. This implies a criterion to detect topological flatness in Sobolev bundles in critical dimensions via $n/2$-Yang-Mills energy.