Abstract

Given a Poisson (or more generally Dirac) manifold P, there are two approaches to its geometric quantization: one involves a circle bundle Q over P 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 P. We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P is obtained from the Lie groupoid of Q via an S1 reduction that preserves both the Lie groupoid and the geometric structures.