Integrability of the modular vector field and quantization

Abstract

We present an approach to the problem of quantization of Poisson manifolds based on Poisson-Nijenhuis (PN) structures of symplectic type. This geometry describes the integrability of the flux of the modular vector fields of all Poisson structures appearing in the PN hierarchy. The integrable models can be lifted to multiplicative integrable models on their symplectic groupoid and are regarded as a singular real polarization. The output of the construction is the convolution algebra of the groupoid of Bohr-Sommerfeld leaves; such algebra takes into accout the topology of the space of symplectic leaves. We sketch here the main ingredients of the construction and discuss as an example the case of a PN structure on the two sphere, the simplest case of a class of PN structures on compact hermitian symmetric spaces.