Quantization of Poisson manifolds from the integrability of the modular function

TL;DR

This paper presents a novel framework for quantizing Poisson manifolds using symplectic groupoids and Renault's C*-algebra theory, with applications to Poisson structures on complex projective spaces.

Contribution

It introduces the concept of multiplicative integrability of the modular function for quantization and applies it to Poisson structures on CP_n, extending existing theories.

Findings

Defined quantum algebra as convolution algebra of Bohr-Sommerfeld subgroupoid.

Applied the framework to Poisson structures on CP_n, showing multiplicative integrability.

Extended Sheu's description of quantum homogeneous spaces as groupoid C*-algebras.

Abstract

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on CP_n, seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU(n+1). We show that a bihamiltoniam system on CP_n defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C*-algebras.