Speaker: Sergey Grigorian, University of Texas Rio Grande Valley

Abstract: In this talk, we generalize some results from standard gauge theory to a non-associative setting. Non-associative gauge theory is based on smooth loops, which are the non-associative analogues of Lie groups. The main components of this theory include a finite-dimensional smooth loop $\mathbb{L}$, together with its tangent algebra $\mathfrak{l}$ and pseudoautomorphism group $\Psi $, and a smooth manifold $M$ with a principal $\Psi $-bundle $\mathcal{P}$. A configuration in this theory is defined as a pair $\left( s,\omega \right) $, where $s$ is an $\mathbb{L}$-valued section and $\omega $ is a connection on $\mathcal{P}$. Each such pair determines the torsion $T^{\left( s,\omega \right) }$, which is a key object in theory. Given a fixed connection, we prove existence of configurations with divergence-free torsion, given a sufficiently small torsion in a Sobolev norm. We will also show how these results apply to $G_2$-geometry on $7$-dimensional manifolds.