2023/03/01 by Sergey Grigorian, Grigorian, Sergey
Mathematics · Physics and Astronomy · #20N05 #53C07 #53C10 #57R57 #Advanced Differential Geometry Research #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2303.00664
openalex publication_date 2023/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbL, its tangent algebra \mathfrakl, a finite-dimensional Lie group Ψ, that is the pseudoautomorphism group of \mathbbL, a smooth manifold M with a principal Ψ-bundle P, and associated bundles Q and A with fibers \mathbbL and \mathfrakl, 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 G2-geometry, and the questions of existence of G2-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.