Geometric quantization and mirror symmetry

TL;DR

This paper explores the connections between geometric quantization, spectral curves, and mirror symmetry, motivated by prior work on special Lagrangian geometry and Calabi-Yau manifolds, emphasizing phase geometry and spectral curve prototypes.

Contribution

It introduces constructions linking spectral curve techniques with phase geometry in the context of mirror symmetry and geometric quantization, building on previous related frameworks.

Findings

Spectral curve constructions are closely related to mirror symmetry.

Phase geometry plays a crucial role in the development of the theory.

The work provides a geometric framework connecting quantization and mirror symmetry.

Abstract

After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that the subject is closely related to that of his preprint [V] (Extending mirror conjecture to Calabi-Yau with bundles, hep-th/9804131, 7 pp.). This text started out as an e-mail ``reply'' to his letter. All the constructions we propose have well known ``spectral curve'' prototypes (see for example Friedman and other [FMW], Bershadsky and other [BJPS] and a number of others). Roughly speaking, our constructions are the spectral curve construction plus the phase geometry described in [T3]. So this text should really come before [T3], as motivation for the development of the geometry of the phase map in [T3].