Geometric quantization: not understanding the curvature form and Weil's theorem

Question

I am reading a bit about geometric quantization. The texts that I am following are N.M.J. Woodhouse's "Geometric Quantization" and an article from arXiv. I am having troubles understanding the prequantization step.

First, everybody cites 1958 Weil's theorem about the existence of Hermitian line bundles with prescribed curvature, but nobody provides its exact statement. Could anyone help me with it? In particular, do I prescribe just the curvature form, or a connection and its associated curvature form?

Second, one tries to produce Hermitian line bundles $L$ over a symplectic manifold $(M, \omega)$ having $\omega$ as the curvature form $R$ of $L$. I'm lost: how can we relate these two forms? $R$ takes values in $\operatorname{End} _{\Bbb C} (L)$, which $\omega$ doesn't, so how can I identify them? Had $L$ been trivial, it would have been elementary, but how to do it in general? Or is this an abuse of terminology?

Answer 1

First, $\text{End}(L)$ is trivial by the canonical isomorphisms $L \otimes L^* \to \text{End}(L)$ and $L \otimes L^* \to \Bbb C$. So we may canonically identify $\Omega^2(\text{End}(L)) \cong \Omega^2$ (where here I'm using complex-valued forms).

Then based on looking at your first reference, it seems to me the desired theorem is the following.

Let $L$ be a complex line bundle over some smooth manifold $M$, and $R \in \Omega^2(M) \otimes \Bbb C$. Then if $R$ is closed and satisfies $[R] = c_1(L) \in H^2_{dR}(M)$, then $R$ is the curvature form of some connection on $L$.

You could make everything here Hermitian but I'm not going to bother.

1) First, note that if $A$ is a connection on $L$, the induced connection on $\text{End}(L) \cong \Bbb C$ is the trivial connection.

2) The Bianchi identity is that $d_A F(A) = 0$ if $F(A)$ is the curvature of $A$. By 1), this says that $F(A)$ is closed.

3) You may as well define $c_1(L)$ to be the homology class of the curvature of any connection on $L$. This is well-defined because of the formula $F(A+a) = F(A) + d_Aa + a \wedge a$; in the case of line bundles this simply degenerates to $F(A+a) = F(A) + da$. Weil proved that the set of possible $c_1(L)$ is precisely $2\pi H^2(M;\Bbb Z) \subset H^2_{dR}(M)$, one for each line bundle (modulo torsion classes; these correspond to flat complex line bundles).

4) Now to write down a proof of the theorem. Let $R$ be your desired curvature form, and $F(A)$ the curvature of some random connection. We want to solve $R = F(A+a) = F(A)+da$. This simplifies to $R-F(A) = da$; since we assumed $[R]=[F(A)]$ in cohomology, this is solvable, as desired.