pre-quantization on cotangent bundle $T^*M$

Question

Let M be a manifold. Is there any pre-quantization on cotangent bundle $T^*M$?. In which case this pre-quantization is unique?

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 $H^1(M; \Bbb R)/H^1(M; \Bbb Z)$.

Note that the canonical symplectic form $\omega_\text{can}$ on $T^\ast M$ is exact, so that $(2\pi)^{-1} \omega_\text{can}$ is automatically an integral class. So a prequantization of $(T^\ast M, \omega_\text{can})$ always exists. $T^\ast M$ is homotopy equivalent to $M$, so different choices of prequantization are parametrized by $H^1(M; \Bbb R)/H^1(M; \Bbb Z)$. In general this is nonzero. If $M$ is simply connected then we can conclude that $(T^\ast M, \omega_\text{can})$ has a unique prequantization, but in general it depends.

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