Shifted Poisson structures and deformation quantization

Abstract

This paper is a sequel to 'Shifted symplectic structures' [T. Pantev, B. Toën, M. Vaquié and G. Vezzosi, Publ. Math. Inst. Hautes E'tudes Sci. 117 (2013) 271-328]. We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and the mixed algebra of 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 to 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 road for shifted deformation quantization of many interesting derived moduli spaces, like those studied in our earlier paper and many others.