When studying Hamiltonian group actions, a very nice set up might be to take the following:
Let $M$ be a compact Kähler manifold with (integral) Kähler form $\omega$, endowed with a Hamiltonian $G$ action with moment map $\mu: M \to \mathfrak g^*$ such that $\iota_{X^\sharp}\omega = d\mu^X$ for $X \in \mathfrak g$, where $X^\sharp$ is the corresponding fundamental vector field, and $\mu^X(p) = \mu(p)(X)$.
However, several recent papers (for example, [1], [2], and [3]) recast this into the language of pre-quantum line bundles. For example,
Let $M$ be a compact Kähler manifold, $L \to M$ a holomorphic, Hermitian line bundle with connection $\nabla$ such that the curvature is $F_\nabla = \omega$. Define the comoment map $\mu^*: \mathfrak g \to C^\infty(M,\mathbb R)$ by $$ \mu^*(X) = \frac1{2\pi i} \left[ \mathcal L(X) - \nabla(X^\sharp) \right]$$ where $\mathcal L$ is Lie differentiation of sections $C^\infty(M,L)$ along $X$. In particular, one again has $d\mu^*(X) = \iota_{X^\sharp}\omega$.
The two are used seemingly interchangeably, and I would hazard a guess is that the relationship is related to the notion of quantization. Unfortunately, in my exploration of the literature I rarely see discussions of quantization that also include Hamiltonian group actions, and when it is mentioned the reader is often referred to the tome that is Kostant's Quantization and Unitary Representations, which has neither an index, nor a table of contents, nor is searchable electronically. So my questions are as follows:
- Does there exist a good reference for the relationship between these two frameworks, specifically in the context of Hamiltonian group actions (and even better would be convexity theorems).
- Is it clear that the moment map in the first framework is equivalent to the comoment map in the second framework? If so, a few remarks commenting on why this is the case would be greatly appreciated.
- Precisely when does the datum of a Hamiltonian group action on an integral Kähler manifold have a one-to-one correspondence with the prequantum line bundle set up? Is being integral, compact, and Kähler sufficient?
Edit: It has been pointed out that $\mu^*(X)$ is not a map on $M$. I agree that I cannot see why it should be a function on $M$ rather than a function on sections, so perhaps this is a typo in [2,page 5] and [3,page 3]. All the more reason why a reference would be greatly appreciated.