Shifted Symplectic Structures

Abstract

This is the first of a series of papers about \emph{quantization} in the context of \emph{derived algebraic geometry}. In this first part, we introduce the notion of \emph{$n$-shifted symplectic structures}, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks. We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures. Our main existence theorem states that for any derived Artin stack $F$ equipped with an $n$-shifted symplectic structure, the derived mapping stack $\text{Map}(X,F)$ is equipped with a canonical $(n-d)$-shifted symplectic structure as soon a $X$ satisfies a Calabi-Yau condition in dimension $d$. These two results imply the existence of many examples of derived moduli stacks equipped with $n$-shifted symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We also show that Lagrangian intersections carry canonical (-1)-shifted symplectic structures.

52 pages. To appear in Publ. Math. IHES

References in corpus (5)

• Higher Topos Theory
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Lectures on deformations of complex manifolds
• Derived critical loci I - Basics
• Derived equivalence for quantum symplectic resolutions

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• The Derived Marsden-Weinstein Quotient is Symplectic
• The Thom-Sebastiani theorem for the Euler characteristic of cyclic L-infinity algebras
• Higher dimensional formal loop spaces
• The moduli space of stable coherent sheaves via non-archimedean geometry
• The Tanaka-Thomas's Vafa-Witten invariants via surface Deligne-Mumford stacks
• Note on the motivic DT/PT correspondence and the motivic Flop formula