The Coulomb Gauge in Non-associative Gauge Theory
The Coulomb Gauge in Non-associative Gauge Theory
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非结合规范理论中的库仑规范
DOI：
10.1007/s12220-023-01445-0
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Grigorian, Sergey
中科院分区：
文献类型：
--
作者：
Grigorian, Sergey
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, its tangent algebraa finite-dimensional Lie group, that is the pseudoautomorphism group of, a smooth manifoldMwith a principal-bundle, and associated bundlesandwith fibersand, respectively. A configuration in this theory is defined as a pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( s,\omega \right) $$\end{document}, wheresis a section ofandis a connection on. The torsion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\left( s,\omega \right) }$$\end{document} 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-geometry, and the questions of existence of-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.
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, its tangent algebraa finite-dimensional Lie group, that is the pseudoautomorphism group of, a smooth manifoldMwith a principal-bundle, and associated bundlesandwith fibersand, respectively. A configuration in this theory is defined as a pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( s,\omega \right) $$\end{document}, wheresis a section ofandis a connection on. The torsion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\left( s,\omega \right) }$$\end{document} 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-geometry, and the questions of existence of-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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