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.