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 loopL, its tangent algebra l, a finite-dimensional Lie group Psi, that is the pseudoautomorphism group of L, a smooth manifold M with a principal Psi-bundle P, and associated bundles Q andA with fibers L and l, respectively. A configuration in this theory is defined as a pair (s,omega), where s is a section of Q and omega is a connection on P. The torsion T-(s,T-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 G2-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.