The Coulomb gauge in non-associative gauge theory

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 $\mathrm{\Psi }$, that is the pseudoautomorphism group of $\mathbb{L}$, a smooth manifold $M$ with a principal $\mathrm{\Psi }$-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 $(s,\omega )$, where $s$ is a section of $\mathbb{Q}$ and $\omega$ is a connection on $\mathcal{P}$. The torsion $T^{(s,\omega )}$ 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.