Geometric Quantization Without Polarizations

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.