Existence of prequantization on a simply connected manifold

Question

Let $M$ be a simply connected manifold. Then when, $M$ has a unique pre-quantization and when there is no pre-quantization on $M$.

Answer 1

The following holds for prequantizations of arbitrary symplectic manifolds:

Existence: A prequantization of $(M, \omega)$ exists if and only if $(2\pi)^{-1} \omega \in H^2(M; \Bbb R)$ is an integral class.

Uniqueness: The different choices of prequantization of $(M, \omega)$ are parametrized by $H^1(M; \Bbb R)/H^1(M; \Bbb Z)$.

In particular, if $M$ is simply connected then $H^1(M; \Bbb R) = 0$ and hence if a prequantization exists, it is unique. For a prequantization to exist, the symplectic form must satisfy the given integrality condition.

See Proposition 8.3.1 of Woodhouse's Geometric Quantization for more details.