Shifted Poisson Structures and Deformation Quantization

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.

111 pages. Minor changes, to appear in Journal of Topology

References in corpus (3)

• Completions and derived de Rham cohomology
• Shifted Poisson and symplectic structures on derived N-stacks
• Weyl n-algebras

Cited by in corpus (39)

• On dualizability of braided tensor categories
• Shifted cotangent stacks are shifted symplectic
• Derived coisotropic structures II: stacks and quantization
• Homotopy theory of algebraic quantum field theories
• Derived coisotropic structures I: affine case
• Braces and Poisson additivity
• Topological twists of supersymmetric algebras of observables
• Shifted Coisotropic Correspondences
• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• Linear Yang-Mills theory as a homotopy AQFT
• Homological perspective on edge modes in linear Yang-Mills and Chern-Simons theory
• Higher Structures in Algebraic Quantum Field Theory
• Quantisation of derived Lagrangians
• Categorification of algebraic quantum field theories
• BV equivalence with boundary
• The topological chiral homology of the spherical category
• A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes
• Moduli problems for operadic algebras
• Quantisation of derived Poisson structures
• Batalin--Vilkovisky quantization and supersymmetric twists
• On Quantum Obstruction Spaces and Higher Codimension Gauge Theories
• Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism
• Infinitesimal 2-braidings from 2-shifted Poisson structures
• Frobenius objects in the category of spans
• Derived gluing construction of chiral algebras
• Secondary products in supersymmetric field theory
• Shifted coisotropic structures for differentiable stacks
• 4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism
• Classical BV formalism for group actions
• Derived symplectic geometry
• Deformation Quantization via Categorical Factorization Homology
• Notes on Derived Geometric Formulations in Physics
• Calabi-Yau structures on (quasi-)bisymplectic algebras
• A $t$-structure on the $\infty$-category of mixed graded modules
• Holomorphic field theories and higher algebra
• Mixed graded structure on Chevalley-Eilenberg functors
• Shifted Contact Structures and Their Local Theory
• Algebras over not too little discs
• Shifted Poisson structures on higher Chevalley-Eilenberg algebras