Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I need a reference about a definition. Let $n$ be an integer and $G$ be a group of $H^n$(Sobolev) automorphisms of a vector bundle $E$ on some manifold $M$ and $C$ be the space of connections of class $H^{n-1}$, what is the definition that $G$ acts "smoothly" on $A$?

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

To talk about smoothness of an action, we need to be sure that the group is a Lie group, and in the case of Sobolev gauge transformations (bundle automorphisms) the smoothness relies on the use of some standard Sobolev theory.

Here's an outline of a special case. Let $G \subset \mathrm{GL}(n, \Bbb R)$ be a Lie group. If $\mathscr{G}_{p,k}$ is the group of $W^{p,k}$ gauge transformations of a principal $G$-bundle $P \longrightarrow M^n$, then it is known that $\mathscr{G}_{p,k}$ is a Banach Lie group for $k - 1 > n/p$ (in your question you have $p = 2$, and in that case $\mathscr{G}_{2,k}$ is a Hilbert Lie group for $k - 1 > n/2$). Hence when $k - 1 > n/p$, $\mathscr{G}_{p,k}$ is a smooth (infinite-dimensional) manifold. Of course, the space $\mathscr{A}_{p,k-1}$ of $W^{p,k-1}$ connections is a smooth Banach manifold (it is an affine space).

Hence when $G \subset \mathrm{GL}(n, \Bbb R)$ and $k - 1 > n/p$, the phrase "$\mathscr{G}_{p,k}$ acts smoothly on $\mathscr{A}_{p,k-1}$" means exactly what you think it should mean: the map $$\mathscr{A}_{p,k-1} \times \mathscr{G}_{p,k} \longrightarrow \mathscr{A}_{p,k-1},$$ $$(A, g) \mapsto g^\ast A$$ is a smooth map of Banach manifolds.


Here are some references that might be worth checking out for more details:

An Introduction to Gauge Theory by John Morgan, in the book Gauge Theory and the Topology of $4$-Manifolds.

The first few lectures of Tom Mrowka's Master Class at Aarhus (videos included!).

Appendix A of Instantons and Four-Manifolds by Freed and Uhlenbeck. The ideas are similar to the general case, but since this is a book on instantons they assume $M$ is a closed $4$-manifold and $G = \mathrm{SU}(2)$.

share|improve this answer
    
(+1) I do not understand anything but looks very smart –  Norbert May 12 '13 at 1:14
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.