Quantization of Abelian Varieties: distributional sections and the transition from K\"ahler to real polarizations

TL;DR

This paper investigates how geometric quantization of symplectic tori varies with different polarizations, unifying the description for all cases and analyzing the properties of the BKS pairing maps.

Contribution

It provides a unified analytical framework for quantization spaces across all polarizations and demonstrates the unitarity and transitivity of BKS pairing maps.

Findings

Quantization spaces are described uniformly for all polarizations.

BKS pairing maps are transitive and unitary.

The action of Sp(2g,R) relates different polarization quantizations.

Abstract

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributional sections, we give a unified analytical description of the quantization spaces for all nonnegative polarizations. The Blattner-Kostant-Sternberg (BKS) pairing maps between half-form corrected quantization spaces for different polarizations are shown to be transitive and related to an action of $Sp(2g,\R)$. Moreover, these maps are shown to be unitary.