Minimizers of higher order gauge invariant functionals

TL;DR

This paper introduces higher order gauge invariant functionals extending Yang-Mills theory, proving existence and regularity of minimizers in various dimensions, and generalizing a removable singularity theorem for connections.

Contribution

It develops higher order variants of the Yang-Mills functional, establishes existence of smooth minimizers, and generalizes a singularity removal result for connections.

Findings

Existence of smooth minimizers in subcritical dimensions.

Construction of minimizers in critical dimension with matching Chern classes.

Removable singularity theorem for $W^{n-1,2}$-connections.

Abstract

We introduce higher order variants of the Yang-Mills functional that involve $(n-2)$th order derivatives of the curvature. We prove coercivity and smoothness of critical points in Uhlenbeck gauge in dimensions $\mathrm{dim}M\le 2n$. These results are then used to establish the existence of smooth minimizers on a given principal bundle $P\to M$ for subcritical dimensions $\mathrm{dim}M<2n$. In the case of critical dimension $\mathrm{dim}M=2n$ we construct a minimizer on a bundle which might differ from the prescribed one, but has the same Chern classes $c_1,\ldots,c_{n-1}$. A key result is a removable singularity theorem for bundles carrying a $W^{n-1,2}$-connection. This generalizes a recent result by Petrache and Rivi\`ere.