Quantization of symplectic tori in a real polarization

TL;DR

This paper explores the geometric quantization of symplectic tori using real polarizations, establishing a canonical projective Hilbert space and analyzing the role of the Maslov-Kashiwara index and metaplectic structures.

Contribution

It introduces a canonical quantization framework for symplectic tori with real polarizations, connecting projective factors to the Maslov-Kashiwara index and constructing unitary representations of the metaplectic group.

Findings

Projective Hilbert space associated with symplectic torus is characterized.

The projective factor relates to the Maslov-Kashiwara index.

Hilbert space realizes a unitary representation of the integer metaplectic group.

Abstract

We apply the geometric quantization method with real polarizations to the quantization of a symplectic torus. By quantizing with half-densities we canonically associate to the symplectic torus a projective Hilbert space and prove that the projective factor is expressible in terms of the Maslov-Kashiwara index. As in the quantization of a linear symplectic space, we have two ways of resolving the projective ambiguity: (i) by introducing a metaplectic structure and using half-forms in the definition of the Hilbert space; (ii) by choosing a 4-fold cover of the Lagrangian Grassmannian of the linear symplectic space covering the torus. We show that the Hilbert space constructed through either of these approaches realizes a unitary representation of the integer metaplectic group.