Abstract
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\({\mathbb {L}}\), its tangent algebra \({\mathfrak {l}},\) a finite-dimensional Lie group \(\Psi \), that is the pseudoautomorphism group of \({\mathbb {L}}\), a smooth manifold M with a principal \(\Psi \)-bundle \({\mathcal {P}}\), and associated bundles \({\mathcal {Q}}\) and\({\mathcal {A}}\) with fibers \({\mathbb {L}}\) and \({\mathfrak {l}}\), respectively. A configuration in this theory is defined as a pair \(\left( s,\omega \right) \), where s is a section of \({\mathbb {Q}}\) and \(\omega \) is a connection on \({\mathcal {P}}\). The torsion \(T^{\left( s,\omega \right) }\) 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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Appendix
Appendix
Lemma A.1
Let k, \(k^{\prime },n\) be positive integers and \(kp>n\), for a positive real number p, and let \(A_{1},\ldots ,A_{k^{\prime }}\) be real-valued functions on a compact n-dimensional Riemannian manifold M. Also, suppose \(m_{1},\ldots ,m_{k^{\prime }}\) are non-negative integers and \( q_{1},\ldots ,q_{k^{\prime }}\) are positive integers such that \( \sum _{j=1}^{k^{\prime \prime }}q_{j}m_{j}\le k,\) then
Proof
Let \(k^{\prime \prime }=\sum _{j=1}^{k^{\prime \prime }}q_{j}m_{j}\le k.\) Then suppose \(p_{j}=\frac{pk^{\prime \prime }}{q_{j}m_{j}}\) for all j for which \(m_{j}>0,\) so that \(\frac{1}{p_{j}}=\frac{q_{j}m_{j}}{pk^{\prime \prime }},\) and hence \(\sum _{j=1}^{k^{\prime }}\frac{1}{p_{j}}=\frac{1}{p}.\) Thus, from Hölder’s inequality, we have
Now note that using the definition of \(p_{j}\), \(\frac{q_{j}}{k^{\prime \prime }}=\frac{p}{p_{j}m_{j}}\le 1,\) and hence
Since by assumption, \(\frac{k}{n}>\frac{1}{p}\), we obtain
Using a version of the Sobolev Embedding Theorem, this shows that indeed,
and (A.1) follows. \(\square \)
Theorem A.2
(Banach space quantitative implicit function theorem [17, Theorem F.1]) Let \(k\ge 1\) be an integer or \(\infty ,\) and let X, Y, Z be real Banach spaces. Suppose \(U\subset X\) and \(V\subset Y\) are open neighborhoods of points \(x_{0}\in X\) and \(y_{0}\in Y\) and \(f:U\times V\longrightarrow Z\) is a \(C^{k}\) map such that \(f\left( x_{0},y_{0}\right) =0 \) and the partial derivative of f at \(\left( x_{0},y_{0}\right) \) with respect to the second variable, \(\left. \partial _{2}f\right| _{\left( x_{0},y_{0}\right) }\in {Hom}\left( Y,Z\right) \) is an isomorphism of Banach spaces. Define
Let \(\zeta \in (0,1]\) be small enough such that the open ball \(B_{\zeta }\left( x_{0}\right) \subset U\) and \(B_{\zeta }\left( y_{0}\right) \subset V, \) and assume
Then there exist a constant \(\delta \in \left( 0,\min \left\{ \zeta ,\frac{ \zeta }{2\beta N}\right\} \right] \) and unique \(C^{k}\) map \(g:B_{\delta }\left( x_{0}\right) \longrightarrow B_{\zeta }\left( y_{0}\right) \) such that \(y_{0}=g\left( x_{0}\right) \) and
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Grigorian, S. The Coulomb Gauge in Non-associative Gauge Theory. J Geom Anal 34, 7 (2024). https://doi.org/10.1007/s12220-023-01445-0
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DOI: https://doi.org/10.1007/s12220-023-01445-0