Twisted Deformation Quantization of Algebraic Varieties

TL;DR

This paper introduces a new framework for twisted deformation quantization of algebraic varieties, extending classical results by establishing a canonical bijection between twisted Poisson and associative deformations.

Contribution

It develops the concept of twisted deformations as stack-like structures and proves a canonical bijection between twisted Poisson and associative deformations, extending Kontsevich's work.

Findings

Established a twisted quantization operation that is canonical.

Proved the bijectivity of the quantization map on gauge equivalence classes.

Extended deformation quantization theory to stack-like twisted structures.

Abstract

Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.