Shifted Poisson Structures and Deformation Quantization

2015/06/30 by D. Calaque, T. Pantev, B. Toen +2 · 2 citations
Mathematics · #math.AG #math.AT

paper · pdf · doi:10.1112/topo.12012

published as Journal of Topology, Volume 10, Issue 2, June 2017, Pages 483-584 · 111 pages. Minor changes, to appear in Journal of Topology

arxiv created 2017/02/22 · arxiv updated 2018/05/10

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.

Cited by

• Shifted Symplectic Fibrations and Derived Thurston Theorem
• A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

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