Derived Algebraic Geometry and Deformation Quantization

TL;DR

This paper reviews recent advances in derived algebraic geometry and deformation quantization, focusing on shifted structures and their applications to quantizing moduli spaces of G-bundles.

Contribution

It introduces derived algebraic stacks, shifted symplectic and Poisson structures, and constructs deformation quantization for shifted Poisson structures, applying these to moduli spaces.

Findings

Introduction of derived algebraic stacks and shifted structures

Construction of deformation quantization for shifted Poisson structures

Application to quantization of moduli spaces of G-bundles

Abstract

This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the construction of deformation quantization of shifted Poisson structures. As an application we propose a general construction of the quantization of the moduli space of $G$-bundles on an oriented space of arbitrary dimension.