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Shifted Poisson structures and deformation quantization

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  • Additional Information
    • Contributors:
      Institut Montpelliérain Alexander Grothendieck (IMAG); Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS); Institut universitaire de France (IUF); Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.); Department of Mathematics Philadelphia; University of Pennsylvania; Institut de Mathématiques de Toulouse UMR5219 (IMT); Université Toulouse Capitole (UT Capitole); Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse); Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J); Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3); Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS); Institut de Mathématiques de Jussieu (IMJ); Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS); ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011); ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011)
    • Publication Information:
      HAL CCSD
      Oxford University Press
    • Publication Date:
      2017
    • Collection:
      Université Toulouse 2 - Jean Jaurès: HAL
    • Abstract:
      International audience ; This paper is a sequel to [PTVV]. We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and poly-vector 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.
    • Relation:
      hal-01253029; https://hal.science/hal-01253029; https://hal.science/hal-01253029v2/document; https://hal.science/hal-01253029v2/file/derpoiss-afterSubmIHES.pdf
    • Accession Number:
      10.1112/topo.12012
    • Online Access:
      https://doi.org/10.1112/topo.12012
      https://hal.science/hal-01253029
      https://hal.science/hal-01253029v2/document
      https://hal.science/hal-01253029v2/file/derpoiss-afterSubmIHES.pdf
    • Rights:
      info:eu-repo/semantics/OpenAccess
    • Accession Number:
      edsbas.F7DDF8A4