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In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369).

Let $P\rightarrow M$ be a principal bundle, $M$ compact (or closed if you require it). Define $g_{t}:[0,T)\rightarrow Gau(P)$, where $Gau(P)$ is the gauge group of $P$. Then the author is able to write $\frac{\partial}{\partial t}g_{t}$, i.e. there is some kind of differentiable structure on $Gau(P)$.

My questions:

  1. What is the differentiable structure on $Gau(P)$?

  2. What is the tangent space at an element of $Gau(P)$?

My first inclination would be to turn $Gau(P)$ into a Fr├ęchet space, but the only references I could find for this were from 2005 onwards. Since the paper is from a few years earlier, the author might have something else in mind.

Thank you very much.

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If I remember correctly, gauge transformations may be seen as sections of some principal bundle associated to a principal $G$ bundle $P\to X$. Can't you simply differentiate the family of sections with respect to time over some open subset where the bundle is trivialized, and show that the result is independent of the trivialization chosen? –  Olivier Bégassat Jun 15 at 13:17
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You are correct that gauge transformations of a G-bundle are also global sections of $G(P)=P\times_{Ad} G$. I believe that $C^{\infty}(M,G(P))$ can be made into a Fréchet space, as $M$ is compact? Then maybe differentiating $g_{t}$ is the Fréchet differential after all. But I do not think we can trivialise $G(P)$ and work it out locally, because the 'operator' we are working with will goes from an interval to a function space. So we would have to localise that function space, which would be redundant if we model our manifolds on Fréchet spaces. –  David Hornshaw Jun 15 at 14:25

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