Prequantum line bundle ($F_{\nabla }=i\omega /\hslash$)

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.