Geometric Quantization Without Polarizations

TL;DR

This paper introduces a polarization-free geometric quantization method using the Poisson sigma model, unifying various schemes and applying it to the torus to recover the noncommutative torus representation.

Contribution

It provides a novel polarization-free quantization approach via path integrals, connecting different quantization schemes and addressing invariance issues.

Findings

Derived the quantization map for the torus resulting in the noncommutative torus

Unified multiple known quantization schemes through a path integral approach

Addressed the invariance of polarization problem using Schur's lemma

Abstract

We derive the quantization map in geometric quantization of symplectic manifolds via the Poisson sigma model. This gives a polarization-free (path integral) definition of quantization which pieces together most known quantization schemes. We explain how this allows Schur's lemma to address the invariance of polarization problem. We compute this quantization map for the torus and obtain the noncommutative torus and its standard irreducible representation.