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Suppose we consider $\operatorname{ad}P_G \to T^k$ as the associated adjoint bundle (maybe this is not the correct name, but I just mean with the associated vector bundle ${\rm Lie}G$ as standard fiber in the adjoint rep.) over $T^k$, which is parameterized by standard coordinates $(\varphi_1, ..., \varphi_k)$.

Let vector field $R$ be $\partial_{\varphi_1}$ (or any vector field that generated cycles on $T^k$). Finally denote $A$ as connection on $\operatorname{ad} P_G$.

My questions are:

  1. Can I always choose a gauge such that $\iota_R A = 0$ or $\mathcal{L}_R A = 0$?

  2. If not, are there known gauge choices that are optimized in the presence of a vector field $R$ (I am guessing the space of connection $\mathcal{A}$ is decomposed into subspaces, and in each subspace has its own "optimal" gauge.)?

I think the answer to (1) is probably NO, since counter examples ($A$ with non-zero holonomy around the cycle of $R$, which cannot be made into $\iota_R A = 0$ via gauge transformations) are there; but I am not sure if $\mathcal{L}_R A = 0$ is also ruled out somehow.

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Non-zero holonomy implies non-zero connection form only on a simply connected domain. A non-zero holonomy around a cycle can perfectly happen even if $A$ is zero, if said cycle is not a boundary (and if $R=\partial_{\phi_1}$, that is indeed the case). Anyway, it ultimately depends on the bundle. – geodude Mar 19 '14 at 21:26

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