Publications mathématiques de l'IHÉS

, Volume 117, Issue 1, pp 271–328 | Cite as

Shifted symplectic structures

  • Tony Pantev
  • Bertrand Toën
  • Michel Vaquié
  • Gabriele Vezzosi


This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toën and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toën in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (nd)-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-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 explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π 1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T X), for E an elliptic curve and T X is the total space of the cotangent bundle of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π 1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).


Modulus Space Symplectic Form Symplectic Structure Mixed Complex Chern Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Tony Pantev
    • 1
  • Bertrand Toën
    • 2
  • Michel Vaquié
    • 3
  • Gabriele Vezzosi
    • 4
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Institut de Mathématiques et de Modélisations de MontpellierUniversité de Montpellier2MontpellierFrance
  3. 3.Institut de Math. de ToulouseUniversité Paul SabatierToulouseFrance
  4. 4.Institut de Mathématiques de JussieuUniversité Paris DiderotParisFrance

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