The Coulomb Gauge in Non-associative Gauge Theory

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.

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

$$\begin{aligned} \left\| \prod _{j=1}^{k^{\prime }}A_{j}^{m_{j}}\right\| _{L^{p}}\lesssim \prod _{j=1}^{k^{\prime }}\left\| A_{j}\right\| _{W^{k-q_{j},p}}^{m_{j}}. \end{aligned}$$

(A.1)

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

$$\begin{aligned} \left\| \prod _{j=1}^{k^{\prime }}A_{j}^{m_{j}}\right\| _{L^{p}}\lesssim \prod _{j=1}^{k^{\prime }}\left\| A_{j}\right\| _{L^{m_{j}p_{j}}}^{m_{j}}. \end{aligned}$$

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

$$\begin{aligned} \frac{k-q_{j}}{n}= & {} \frac{k}{n}\left( 1-\frac{q_{j}}{k}\right) \\\ge & {} \frac{k}{n}\left( 1-\frac{q_{j}}{k^{\prime \prime }}\right) \\= & {} \frac{k}{n}\left( 1-\frac{p}{p_{j}m_{j}}\right) . \end{aligned}$$

Since by assumption, $\frac{k}{n}>\frac{1}{p}$, we obtain

$$\begin{aligned} \frac{k-q_{j}}{n}>\frac{1}{p}-\frac{1}{p_{j}m_{j}}. \end{aligned}$$

Using a version of the Sobolev Embedding Theorem, this shows that indeed,

$$\begin{aligned} \left\| A_{j}\right\| _{L^{m_{j}p_{j}}}\lesssim \left\| A_{j}\right\| _{W^{k-q_{j},p}}, \end{aligned}$$

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

$$\begin{aligned} N=\left\| \left( \left. \partial _{2}f\right| _{\left( x_{0},y_{0}\right) }\right) ^{-1}\right\| _{\mathop {\textrm{Hom}} \nolimits \left( Z,Y\right) }. \end{aligned}$$

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

$$\begin{aligned} \sup _{\left( x,y\right) \in B_{\zeta }\left( x_{0}\right) \times B_{\zeta }\left( y_{0}\right) }\left\| \left. \partial _{2}f\right| _{\left( x,y\right) }-\left. \partial _{2}f\right| _{\left( x_{0},y_{0}\right) }\right\| _{_{\mathop {\textrm{Hom}}\nolimits \left( Y,Z\right) }}\le & {} \frac{1 }{2N} \\ \beta =\sup _{\left( x,y\right) \in B_{\zeta }\left( x_{0}\right) \times B_{\zeta }\left( y_{0}\right) }\left\| \left. \partial _{1}f\right| _{\left( x,y\right) }\right\| _{\mathop {\textrm{Hom}}\nolimits \left( X,Z\right) }< & {} \infty . \end{aligned}$$

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

$$\begin{aligned} f\left( g\left( x\right) ,x\right)= & {} 0,\ \forall x\in B_{\delta }\left( x_{0}\right) \\ \left. Dg\right| _{x}= & {} -\left( \left. \partial _{2}f\right| _{\left( x,g\left( x\right) \right) }\right) ^{-1}\left. \partial _{1}f\right| _{\left( x,g\left( x\right) \right) }\in \mathop {\textrm{Hom}} \nolimits \left( X,Y\right) ,\ \forall x\in B_{\delta }\left( x_{0}\right) \\ \left\| g\left( x_{1}\right) -g\left( x_{2}\right) \right\| _{Y}\le & {} 2\beta N\left\| x_{1}-x_{2}\right\| _{X,}\ \forall x_{1},x_{2}\in B_{\delta }\left( x_{0}\right) . \end{aligned}$$