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I'm not sure about a reference, but the relationship is quite easy to see if you look at things from a homotopy theoretic point of view. Let $G$ compact connected Lie group and fix a unitary representation $\rho:G\to U$. Let $$\chi:G\overset{\rho}{\to} U \to U(1)$$ be the multiplicative group character of the representation. Then the delooping of $\chi$ ...


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No, because your formula does not make sense: $T\in Hom(E_x,X_x)$ and $\phi\in Hom(E_x,E_y)$ invertible means that $$\phi^{-1}\circ T\circ \phi$$ is not well-defined unless $x=y.$ If you define $$\psi=\phi\circ T\circ \phi^{-1},$$ then $\psi$ is actually the parallel-transport of the induced connection $\nabla^{End}$ on the endomorphism bundle which is ...



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