In her Lectures on Symplectic Geometry on page 159, Ana Cannas da Silva writes "It turns out that $\mathcal{M}$ is a finite-dimensional symplectic orbifold."
Can somebody give me a reference for this result, preferably including a detailed definition of symplectic orbifold?
I'm trying to become familiar with orbifolds, and I am familiar with symplectic manifolds, but I imagine a symplectic orbifold could have a more stringent definition than just an orbifold whose "smooth part" is equipped with a symplectic form (I expect maybe there is some condition on how the symplectic structure behaves near the singular part).
Best Answer
The original reference for this fact is the paper Yang-Mills Equations over Riemann Surfaces by Atiyah and Bott.
The idea is as follows. Let $P \longrightarrow \Sigma$ be a principal $G$-bundle over a Riemann surface and let $\mathcal{A}$ denote the space of connections on $P$. For a connection $A \in \mathcal{A}$, we define $$\omega_A : T_A \mathcal{A} \times T_A \mathcal{A} \longrightarrow \Bbb R,$$ $$\omega_A(\alpha, \beta) = \int_\Sigma \langle \alpha \wedge \beta \rangle.$$ Here we have identified $T_A \mathcal{A}$ with $\Omega_\Sigma^1(\mathrm{Ad}(P))$ since $\mathcal{A}$ is an affine space modeled on $\Omega_\Sigma^1(\mathrm{Ad}(P))$. $\langle \alpha \wedge \beta \rangle$ is the composition $$\Omega_\Sigma^1(\mathrm{Ad}(P)) \times \Omega_\Sigma^1(\mathrm{Ad}(P)) \xrightarrow{~\wedge~} \Omega_\Sigma^2(\mathrm{Ad}(P)) \xrightarrow{~\langle \cdot, \cdot \rangle ~} \Omega_\Sigma^2,$$ where $\langle \cdot, \cdot \rangle$ is an $\mathrm{Ad}$-invariant inner product on the Lie algebra $\mathfrak{g}$. Now we have the following.
Here we consider the curvature map $$F: \mathcal{A} \longrightarrow \Omega_\Sigma^2(\mathrm{Ad}(P))$$ as a map $$F: \mathcal{A} \longrightarrow \mathrm{Lie}(\mathcal{G})^\ast$$ via the identification $$\Omega_\Sigma^2(\mathrm{Ad}(P)) = \Omega_\Sigma^0(\mathrm{Ad}(P))^0 \cong \mathrm{Lie}(\mathcal{G})^\ast.$$
When we have a moment map, we can form the Marsden-Weinstein quotient $$\mathcal{A} /\!\!/ \mathcal{G} = \mu^{-1}(0)/\mathcal{G},$$ which is a stratified symplectic space (a symplectic orbifold if $0$ is a regular value of $\mu$, and a symplectic manifold if $0$ is a regular value of $\mu$ and the action of $\mathcal{G}$ is free). Since $\mu$ is minus the curvature, we see that $\mu^{-1}(0) = \mathcal{A}^\flat$, the space of flat connections on $P$. Therefore $$\mathcal{A} /\!\!/ \mathcal{G} = \mathcal{M}^\flat,$$ where $\mathcal{M}^\flat$ denotes the moduli space of flat connections on $P$.
There are some caveats here. The usual Marsden-Weinstein quotient is defined for finite-dimensional symplectic manifolds with a Hamiltonian action; here $\mathcal{A}$ is infinite-dimensional. Nevertheless, one can show that the formal process above works and $\mathcal{M}^\flat$ is a finite-dimensional.
As for the definition of symplectic orbifold, recall that an orbifold $\mathcal{O}$ has an orbifold atlas $\{(U_i, \tilde{U}_i, \phi_i, \Gamma_i)\}$, where $U_i \subset \mathcal{O}$ is open, $\tilde{U}_i \subset \Bbb R^n$ is open and connected, $\phi_i: U_i \longrightarrow \tilde{U}_i$ is a continuous map, and $\Gamma_i$ is a finite group of diffeomorphisms of $\tilde{U}_i$. A symplectic form on $\mathcal{O}$ is specified in terms of the orbifold atlas by a family of symplectic forms $\{\omega_i\}$, where $\omega_i$ is a symplectic form on $\tilde{U}_i \subset \Bbb R^n$ that is invariant under the action of $\Gamma_i$. We require the following compatibility condition between the $\omega_i$. Recall that the overlap condition for an orbifold goes as follows: if $x \in \tilde{U}_i$ and $y \in \tilde{U}_j$ are such that $\phi_i(x) = \phi_j(y)$, then there is a neighborhood $V_x$ of $x$ and $V_y$ of $y$ and a diffeomorphism $$\psi: V_x \longrightarrow V_y$$ such that $$\phi_i(z) = \phi_j(\psi(z)) \text{ for all } z \in V_x.$$ Then our compatibility condition is that $$\omega_i = \psi^\ast \omega_j.$$