Brane quantization of toric Poisson varieties

TL;DR

This paper introduces a noncommutative deformation framework for toric K"ahler manifolds using holomorphic Poisson structures, leading to the concept of noncommutative toric varieties via generalized complex branes.

Contribution

It develops a novel method to deform classical geometric structures into noncommutative counterparts using generalized complex branes and applies it specifically to toric K"ahler manifolds with R-matrix Poisson structures.

Findings

Constructed noncommutative deformations of homogeneous coordinate rings.

Defined morphisms between generalized complex branes via coisotropic A-branes.

Implemented the framework for all compact toric K"ahler manifolds with R-matrix structures.

Abstract

In this paper we propose a noncommutative generalization of the relationship between compact K\"ahler manifolds and complex projective algebraic varieties. Beginning with a prequantized K\"ahler structure, we use a holomorphic Poisson tensor to deform the underlying complex structure into a generalized complex structure, such that the prequantum line bundle and its tensor powers deform to a sequence of generalized complex branes. Taking homomorphisms between the resulting branes, we obtain a noncommutative deformation of the homogeneous coordinate ring. As a proof of concept, this is implemented for all compact toric K\"ahler manifolds equipped with an R-matrix holomorphic Poisson structure, resulting in what could be called noncommutative toric varieties. To define the homomorphisms between generalized complex branes, we propose a method which involves lifting each pair of generalized complex branes to a single coisotropic A-brane in the real symplectic groupoid of the underlying Poisson structure, and compute morphisms in the A-model between the Lagrangian identity bisection and the lifted coisotropic brane. This is done with the use of a multiplicative holomorphic Lagrangian polarization of the groupoid.