Suppose we have a principal $G$-bundle $P \rightarrow M$. Then let's consider it's affine space of connections $\mathcal{A}$. I've heard people refer to $\mathcal{A}$ as a smooth infinite dimensional manifold.

My question is: How can we view $\mathcal{A}$ as a smooth manifold?

For context, let me give two examples of where I have seen $\mathcal{A}$ treated as a manifold.

Firstly, in their paper Yang-Mills Equations over Riemann Surfaces, Atiyah and Bott define their well-known symplectic form over $\mathcal{A}$ as $\omega_A : T_A \mathcal{A} \times T_A \mathcal{A} \rightarrow \mathbb{R}$ where: $$ \omega_A \left( \alpha, \beta \right) = \int_M \langle \alpha \wedge \beta \rangle $$ for a connection $A \in \mathcal{A}$

This involves the tangent space of $\mathcal{A}$, which suggests it is being viewed as a manifold.

To give another example in their paper Conditions of Smoothness of Moduli Spaces of Flat Connections Ho, Wilkin and Wu discuss conditions for points in $\mathcal{A}$ to be smooth (before relating this to the smoothness of the flat moduli space $\mathcal{A}/ \text{Gauge} \left( P \right)$).

I'm not very familiar with infinite dimensional smooth manifolds (although I've heard the terms Banach and Frechet Manifolds thrown about).

EDIT: In response to the comment by Kajelad, let me specify the question a little further.

As Kajelad states: "An infinite dimensional manifold is just a topological space locally modelled on some infinite-dimensional topological vector space"

Suppose we fix an origin $A_0 \in \mathcal{A}$ to end up with the vector space $\tilde{\mathcal{A}}$.

My points of confusion are:

  1. What natural/standard topology does $\tilde{\mathcal{A}}$ have which allows it to be viewed as an infinite dimensional manifold?

  2. For $\tilde{\mathcal{A}}$, what is our notion of differentiation in concrete terms?

  • 4
    $\begingroup$ An infinite dimensional manifold is just a topological space locally modelled on some infinite-dimensional topological vector space rather than $\mathbb{R}^n$, together with some prescribed notion of differentiation and an atlas whose transition maps are "sufficiently nice" in some prescribed sense. There's no "standard" choices of these things, which is why one encounters various types. Your case is simpler though: an affine space is just a topological vector space without a preferred choice of origin, so you can choose an origin and just do analysis on a TVS. $\endgroup$
    – Kajelad
    Sep 4 at 21:22
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    $\begingroup$ @Kajelad: I think, OP simply does not realize that its is an affine space. $\endgroup$ Sep 5 at 1:35
  • $\begingroup$ I had realized it was an affine space, but was unsure about some other points. Hopefully Kajelad's comment has helped me state my confusion a little more precisely. I'd edited the question with the details. $\endgroup$
    – leob
    Sep 5 at 14:02
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    $\begingroup$ @leob Choosing an origin gives an identification of $\mathcal{A}$ with the space of smooth sections of the adjoint bundle $\Gamma(\operatorname{adj}P)$, so it suffices to define a topology/differentiation on $\Gamma(\operatorname{adj}P)$. The standard (but by no means only) choice is the Fréchet topology with the standard notion of differention. $\endgroup$
    – Kajelad
    Sep 7 at 16:39

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