Integrability of central extensions of the Poisson Lie algebra via prequantization

We present a geometric construction of central S ¹-extensions of the quantomorphism group of a prequantizable, compact, symplectic manifold, and explicitly describe the corresponding lattice of integrable cocycles on the Poisson Lie algebra. We use this to find nontrivial central S ¹-extensions of the universal cover of the group of Hamiltonian diffeomorphisms. In the process, we obtain central S1-extensions of Lie groups that act by exact strict contact transformations.