Darboux, Moser and Weinstein theorems for prequantum systems and applications to geometric quantization

Abstract

We establish analogs of the Darboux, Moser and Weinstein theorems for prequantum systems. We show that two prequantum systems on a manifold with vanishing first cohomology, with symplectic forms defining the same cohomology class and homotopic to each other within that class, differ only by a symplectomorphism and a gauge transformation. As an application, we show that the Bohr-Sommerfeld quantization of a prequantum system on a manifold with trivial first cohomology is independent of the choice of the connection.