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I am currently reading this paper in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most common" differentiable structure on $Gauge(P)$ would be.

Thank you very much!

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  • $\begingroup$ The differentiable structure is just part of the structure of a Lie group. So when talking about G-principal bundles one means that G is a Lie group (with a given differentiable structure). $\endgroup$
    – ThiKu
    Jun 17, 2014 at 9:09
  • $\begingroup$ Then the gauge group can be identified with $C^\infty(M,G)$ and differentiation of paths in this space is discussed in the second-to-last example of en.wikipedia.org/wiki/Fréchet_space together with en.wikipedia.org/wiki/Differentiation_in_Fréchet_spaces $\endgroup$
    – ThiKu
    Jun 17, 2014 at 9:18
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    $\begingroup$ Maybe I'm wrong, but I thought the gauge group would be identified with $C^{\infty}(P,G)$ which are $G$-equivariant, or global sections of $P\times_{Ad}G$. Would you have references how to get a Frechet space structure on either? $\endgroup$
    – David Hornshaw
    Jun 17, 2014 at 9:32

2 Answers 2

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For compact base manifold and reasonably well behaved structure groups $G$ (like a finite-dimensinoal or a Banach-Lie group) there is the structure of a Frechet-Lie group on the group $Gauge(P)$. This is worked out in Chapter 1 of Lie Group Structures on Symmetry Groups of Principal Bundles (what I have called "property SUB" there is satisfied under the above conditions on $G$). It is not stated explicitly that $Gauge(P)$ is modelled on a Frechet space, but the modelling space is a closed subspace of a finite product of Frechet spaces (Proposition 1.4) and thus a Frechet space.
The same is also true for the group $Aut(P)$ of all bundle automotphisms of $P$, see Theorem 2.14. The unifying theme here is that these groups can all be seen as groups of bisections in locally trivial Lie groupoids (work in progress).

If you are interested in smooth maps into $Gauge(P)$, then you can take the following criterion: let $f\colon I\to Gauge(P)$ be a map and let $f_i\colon I\to C^\infty(\overline{V_i},G)$ be the corresponding maps with respect to a chosen local trivialsation $\Phi_i\colon P|_{U_i}\to U_i\times G$ (such that $\overline{V_i}$ is a manifold with corners and $P|_{\overline{V_i}}$ is also trivial to be precise). Then $f$ is smooth if and only if all $f_i$ are smooth. This is due to the fact that $Gauge(P)$ is diffeomorphic to a submanifold of $\prod_{i=1}^n C^\infty(\overline{V_i},G)$ (which in turn is is implicitly contained in the proof of Theorem 1.11). For the smoothness of maps into mapping spaces there exist many useful criteria, like smoothness of pull-back, push-forward and composition maps.

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  • $\begingroup$ Thank you! I am indeed in the compact case, so this helps a lot. I will make sure to have a look at your paper. $\endgroup$
    – David Hornshaw
    Jun 18, 2014 at 8:43
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I believe the following paper may be helpful to you:

Abbati, M. C., Cirelli, R., Mania', A. & Michor, P.
The lie group of automorphisms of a principle bundle
Journal of Geometry and Physics 6, 215-235 (1989)

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    $\begingroup$ Thank you! I have also looked into Michor's book "The convenient setting of global analysis" where the lie group structure is explained as well. $\endgroup$
    – David Hornshaw
    Jun 18, 2014 at 8:42

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