On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundles

TL;DR

This paper generalizes optimal regularity and compactness results for affine connections to connections on vector bundles, including applications to Yang-Mills theory on Lorentzian manifolds, by developing new elliptic gauge transformation equations.

Contribution

It introduces vector bundle RT-equations and establishes an existence theory, extending regularity and compactness results to broader geometric and physical contexts.

Findings

Derived vector bundle RT-equations for gauge transformations.

Proved existence of solutions down to curvature in L^p, excluding L^1.

Extended Uhlenbeck compactness to non-Riemannian manifolds and Lorentzian settings.

Abstract

We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in $L^p$. The existence theory handles curvature regularity all the way down to, but not including, $L^1$. Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity.