Three conjectures on lagrangian tori in the projective plane

TL;DR

This paper explores the relationship between Lagrangian tori in the projective plane and homological mirror symmetry, extending known results to more general symplectic manifolds and introducing the Bohr-Sommerfeld condition.

Contribution

It extends the Homological Mirror Symmetry framework to broader classes of symplectic manifolds and highlights the role of Bohr-Sommerfeld conditions in this context.

Findings

Bridge between Geometric Quantization and Homological Mirror Symmetry

Extension of HMS predictions beyond toric Fano varieties

Identification of Bohr-Sommerfeld conditions as key in Lagrangian geometry

Abstract

In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa prediction, proven by Cho and Oh for toric Fano varieties, can be checked in much more wider context. Then the notion of Bohr - Sommerfeld with respect to the canonical class lagrangian submanifold appears and plays an important role. The discussion presents a bridge between Geometric Quantization and Homological Mirror Symmetry programmes both applied to the projective plane in terms of its lagrangian geometry. Due to this relation one could exploit some standard facts known in GQ to produce results in HMS.