On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold PP, there are two approaches to its geometric quantization: one involves a circle bundle QQ over PP endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of PP. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of PP is obtained from the Lie groupoid of QQ via an S1S1 reduction that preserves both the Lie groupoid and the geometric structures.