School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The Coulomb Gauge in Non-associative Gauge Theory

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


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J Geom Anal