Shifted Poisson Structures and Deformation Quantization

TL;DR

This paper develops a comprehensive framework for shifted Poisson structures and their deformation quantization in derived geometry, enabling new approaches to quantizing derived moduli spaces.

Contribution

It introduces formal derived stacks, proves localization and gluing results, and establishes a link between non-degenerate shifted Poisson structures and shifted symplectic forms.

Findings

Defined shifted Poisson structures on derived Artin stacks

Proved correspondence between non-degenerate Poisson and symplectic forms

Outlined shifted deformation quantization for derived stacks

Abstract

This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the way for shifted deformation quantization of many interesting derived moduli spaces, like those studied in [PTVV] and probably many others.