Quantization of vector bundles on Lagrangian subvarieties

TL;DR

This paper investigates conditions under which vector bundles on Lagrangian subvarieties can be deformed into modules over a given deformation quantization of the ambient variety, providing a classification of such quantizations.

Contribution

It establishes necessary conditions and describes the classification of deformation quantizations of vector bundles on Lagrangian subvarieties within symplectic algebraic varieties.

Findings

Identifies when vector bundles admit deformation quantization as modules.

Provides a classification of equivalence classes of such quantizations.

Clarifies the relationship between the geometry of the subvariety and quantization conditions.

Abstract

We consider a smooth Lagrangian subvariety Y in a smooth algebraic variety X with an algebraic symplectic from. For a vector bundle E on Y and a choice Oh of deformation quantization of the structure sheaf of X, we establish when E admits a deformation quantization to a module over Oh. If the necessary conditions hold, we describe the set of equivalence classes of such quantizations.