Smoothness in Banach space

Question

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 $A$?

Answer 1

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.

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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 $G = \mathrm{SU}(2)$.