References on the moduli space of flat connections as a symplectic reduction

Question

In their Yang Mills equations over Riemann surfaces paper, Atiyah & Bott famously remark that the moduli space of flat connections on a principal bundle over a compact orientable surface may be obtained as a symplectic reduction on the space of connections (with the curvature as the moment map). In that paper, however, it is only a passing remark. One must be careful about this construction since the spaces involved are infinite-dimensional (e.g. the space of connections). I've only found references which ignore this technical detail and present the ideas in analogy to the finite-dimensional case. While they are helpful to give a simple understanding of what's going on, I was wondering if there is a reference that actually goes through the infinite-dimensional analysis to formalize this process. Any recommendation is appreciated.

Answer 1

Some remarks on infinite-dimensional manifolds. There are two approaches which to me still feel very manifold-ish (there are others yet: Frolicher spaces and diffeological spaces, which feel a bit less so); the approach of Banach manifolds and the much more general approach of Frechet manifolds.

The former is almost precisely the same theory as the theory of finite-dimensional smooth manifolds and is a harmless technical addition. The latter is much more complicated, and I would be hesitant to make any actually technical claims about it, as I am not expert in Kriegl-Michor (the lovely book you cited). The advantages of Frechet manifold theory are that the most obvious and naturally occuring manifolds are Frechet manifolds (spaces of smooth mappings; diffeomorphism groups), and that if you honestly want a theory of infinite-dimensional Lie groups, Banach manifolds will not do (it is a theorem that no Banach Lie group can act transitively and effectively on a finite-dimensional smooth manifold; in particular no large group of diffeomorphisms can be given a Banach Lie group structure).

On the other hand, you lose the technical simplicity of Banach manifolds and a bit of the power of the inverse function theorem. If one wants to do analysis, the inverse function theorem is indispensible, which is why one sees lots of Sobolev completions flying around. There's Nash-Moser but I don't understand it very well and do not think it has the wide applicability of the straight-up inverse function theorem.

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Let $\Sigma$ be a surface equipped with a Riemannian metric $g$. Fix a $U(n)$-bundle $E \to \Sigma$, and importantly, fix a smooth connection $A_0$ on $E$. (Whenever I say a connection on $E$, I expect it to preserve the unitary structure; it should be valued in $\mathfrak u(n)$, not just $\mathfrak{gl}(n,\Bbb C)$.) Define the space of $L^2_k$ connections, $\mathcal A_k(E)$, to be the set of connections $A$ on $E$ such that $A-A_0 \in \Omega^1(\text{End}(E))$ is an $L^2_k$ 1-form. (Note that, to make sense of $L^2_k$, I need a connection on $\text{End}(E) \otimes \Lambda^j T^*M$ - but this is provided by my base connection $A_0$ and my metric $g$.) This is topologized precisely so that it is affine over the space of $L^2_k$ sections of $\text{End}(E) \otimes T^*M$ with the $L^2_k$ topology. Henceforth I will suppress the $L^2_k$ in notation. We also have a group $\mathcal G_{k+1}$, of $L^2_{k+1}$ automorphisms of $E$. (These are sections of a principal bundle instead of sections of a bundle, so one needs to be careful about what we mean about $L^2_{k+1}$ section: by Sobolev embedding, $L^2_k$ sections are automatically continuous for $k \geq 0$, though one needs larger $k$ for larger-dimensional base, after which it is easy to make sense of the derivatives being $L^2_k$ and whatnot). This acts on $\mathcal A_k$ in the standard way. We need to increase regularity for the gauge group because its action involves taking a derivative.

Because $\mathcal A_k(E)$ is an affine space over $\Omega^1(\text{End}(E))$, its tangent space at every point $A$ is $\Omega^1(\text{End}(E))$. Slightly more subtle is the tangent space of $\mathcal G_{k+1}(E)$, which is $\Omega^0(\text{End}(E))$. Now as in Atiyah-Bott, we can put a symplectic structure on $\mathcal A_k(E)$, preserved by the action of $\mathcal G_{k+1}(E)$: $\omega(v,w) = \int [v \wedge w]$, where by the inside of the integral I mean the composition of the maps $\wedge: \Omega^1(\text{End}(E)) \to \Omega^2(\text{End}(E) \otimes \text{End}(E))$, and $\Omega^2(\text{End}(E) \otimes \text{End}(E)) \to \Omega^2(B)$ (this is just a fiberwise inner product).

Here is the first infinite-dimensional subtlety. This is nondegenerate in the sense that the induced map $\omega^*: T_x \mathcal A_k \to T_x^* \mathcal A_k$ is injective, but it is not an isomorphism! This is essentially because we're taking the $L^2$ inner product of $L^2_k$ forms, and the dual under the $L^2$ inner product is not $L^2_k$ - it's $L^2_{-k}$. But for the sake of having a symplectic form, this is fine.

Atiyah-Bott gives an argument that the curvature map $F: \mathcal A_k \to \Omega^2_{k-1}(\text{End}(E))$ is the moment map. This argument is perfectly valid. Now in the setting of Hilbert manifolds, the regular value theorem (or transverse intersection theorems etc) is still valid; since you're interested in flat connections, you're interested in particular in $F^{-1}(0)$. Why is $0$ a regular value? Because the derivative of $F$ at $A$ is just the derivative map $d_A$, we're asking why the differential is surjective at any flat connection. Well... it's not. At a flat connection the cokernel of $d_A$ is canonically isomorphic to the $H^2(\text{End}(E);d_A)$, the de Rham cohomology with differential from flat connection $A$. Poincare duality (and the self-duality of $\text{End}(E)$) implies that this is the same as $H^0(\text{End}(E);d_A)$: the dimension of parallel sections of $\text{End}(E)$. How big this dimension is says how reducible the connection $A$ is; as an example, for $G = SU(2)$, there are three possibilities: you could be fully reducible, where $H^0$ is 3-dimensional; you could be a $U(1)$-reducible, where $H^0$ is 1-dimensional; you could be irreducible, where $H^0$ is zero-dimensional. Our notion of transversality needs to have reducibility baked into it.

If $H$ is the centralizer of some subgroup of $G$, $H$ is a possible group to reduce to - in the sense that the holonomy group of a connection can be $H$. We can stratify the space (not linearly, if the structure of these subgroups is complicated) $\mathcal A$ into subsets $\mathcal A_H$ of connections reducible to $H$. On each of these, $F$ is constant rank. So $F^{-1}(0)$ is a stratified space, of which each stratum a manifold.

Unfortunately, we're now in trouble. The quotient by the action of $\mathcal G$ - canonically identified with $\text{Hom}(\pi_1 X, G)/G$ - is not usually a manifold. Consider, for instance, $X = T^2$ and $G = SU(2)$; then this quotient is the "pillowcase", a sphere with four singular points. On the irreducible part of the moduli space, you get a smooth structure and a symplectic structure by the above method (the same way you usually would; we've more or less passed all the technicalities). I think the people who spend a lot of time thinking about this representation variety have an appropriate sort of structure on it even at the singular points, but I'm not really sure what it is. For further reading, you could look at this thesis, which I haven't but looks nice.