Geometric Quantization of real polarizations via sheaves

TL;DR

This paper develops sheaf cohomology tools to compute the geometric quantization of symplectic manifolds with real polarizations, providing new proofs and explicit calculations for Gelfand-Cetlin systems and flows on a 2-torus.

Contribution

It introduces a sheaf cohomology framework for geometric quantization of real polarizations, including new proofs and explicit examples on tori and Gelfand-Cetlin systems.

Findings

Sheaf cohomology satisfies Mayer-Vietoris and K"unneth formulae.

Provides explicit geometric quantization computations for Gelfand-Cetlin systems.

Calculates geometric quantization for irrational flows on a 2-torus.

Abstract

In this article we develop tools to compute the Geometric Quantization of a symplectic manifold with respect to a regular Lagrangian foliation via sheaf cohomology and obtain important new applications in the case of real polarizations. The starting point is the definition of representation spaces due to Kostant. Besides the classical examples of Gelfand-Cetlin systems due to Guillemin and Sternberg very few examples of explicit computations of real polarizations are known. The computation of Geometric Quantization for Gelfand-Cetlin systems is based on a theorem due to \'Sniatycki for fibrations which identifies the representation space with the set of Bohr-Sommerfeld leaves determined by the integral action coordinates. In this article we check that the associated sheaf cohomology apparatus of Geometric Quantization satisfies Mayer-Vietoris and K\"unneth formulae. As a consequence, a new short proof of this classical result for fibrations due to \'Sniatycki is obtained. We also compute Geometric Quantization with respect to any generic regular Lagrangian foliation on a 2-torus and the case of the irrational flow. In the way, we recover some classical results in the computation of foliated cohomology of these polarizations.