The Coulomb gauge in non-associative gauge theory

Abstract

The aim of this paper is to extend existence results for the Coulomb gauge from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogs of Lie groups. The main components of the theory include a finite-dimensional smooth loop $\mathbb{L}$, its tangent algebra $\mathfrak{l},$ a finite-dimensional Lie group $\Psi $, that is the pseudoautomorphism group of $\mathbb{L}$, a smooth manifold $M$ with a principal $\Psi $-bundle $\mathcal{P}$, and associated bundles $\mathcal{Q}$ and $\mathcal{A}$ with fibers $\mathbb{L}$ and $\mathfrak{l}$, respectively. A configuration in this theory is defined as a pair $\left( s,\omega \right) $, where $s$ is a section of $\mathbb{Q}$ and $\omega $ is a connection on $\mathcal{P}$. The torsion $T^{\left( s,\omega \right) }$ is the key object in the theory, with a role similar to that of a connection in standard gauge theory. The original motivation for this study comes from $G_{2}$-geometry, and the questions of existence of $G_{2}$-structures with particular torsion types. In particular, given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm.