= Infinitesimal gauge transformation with an anti-Hermitian connection
{title2=$\delta\mathcal A_\mu=-\partial_\mu\epsilon+[\epsilon,\mathcal A_\mu]$}
With $D_\mu=\partial_\mu+\rho(\mathcal A_\mu)$ and $\Phi\mapsto R(U)\Phi$, a local $U=1+\epsilon$ gives $\delta\mathcal A_\mu=-\partial_\mu\epsilon+[\epsilon,\mathcal A_\mu]$, $\delta\Phi=\rho(\epsilon)\Phi$ and $\delta\mathcal F_{\mu\nu}=[\epsilon,\mathcal F_{\mu\nu}]$. Derivative terms cancel in $\delta(D_\mu\Phi)=\rho(\epsilon)D_\mu\Phi$. Here the <gauge coupling> is absorbed into the connection.
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