Letters in Mathematical Physics

, Volume 108, Issue 5, pp 1203–1223 | Cite as

Constant symplectic 2-groupoids

  • Rajan Amit Mehta
  • Xiang Tang


We propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.


Dirac Symplectic 2-Groupoids Courant algebroids 

Mathematics Subject Classification

53D17 58H05 



We would like to thank Ezra Getzler for inspiring discussions and explanations about symplectic structures on differentiable n-stacks. We would like to thank Damien Calaque for a discussion about the relation between integration of Courant algebroids and derived symplectic geometry. The research of the second author is partially supported by NSF Grant DMS 1363250.


  1. 1.
    Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, vol. 87, Univ. Claude-Bernard, Lyon, pp. i–ii, 1–62 (1987)Google Scholar
  2. 2.
    Courant, T., Weinstein, A.: Beyond Poisson structures, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, vol. 27, Hermann, Paris, pp. 39–49 (1988)Google Scholar
  3. 3.
    Courant, T.J.: Dirac manifolds. Trans. Am. Math. Soc. 319(2), 631–661 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dorfman, I.: Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science: Theory and Applications. Wiley, Chichester (1993)Google Scholar
  5. 5.
    Duskin, J.: Higher-dimensional torsors and the cohomology of topoi the abelian theory, Applications of sheaves. In: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis, University Durham, Durham (1977). Lecture Notes in Mathemathics, vol. 753, pp. 255–279. Springer, Berlin (1979)Google Scholar
  6. 6.
    Getzler, E.: Differential forms on stacks, 2014, Slides from minicourse at Winter School in Mathematical Physics, Les Diablerets (2014)Google Scholar
  7. 7.
    Goerss, Paul G., Jardine, John F.: Simplicial Homotopy Theory. Birkhäuser, Basel (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Henriques, A.: Integrating \(L_\infty \)-algebras. Compos. Math. 144(4), 1017–1045 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li-Bland, D., Ševera, P.: Integration of exact Courant algebroids. Electron. Res. Announc. Math. Sci. 19, 58–76 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liu, Z., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids. J. Differ. Geom. 45(3), 547–574 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mehta, R.A., Tang, X.: Symplectic Structures on the Integration of Exact Courant Algebroids. arXiv:1310.6587
  12. 12.
    Mehta, R.A., Tang, X.: From double Lie groupoids to local Lie 2-groupoids. Bull. Braz. Math. Soc. (N.S.) 42(4), 651–681 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pantev, T., Toën, M., Vaquié, B., Vezzosi, G.: Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. In: Quantization, Poisson Brackets and Beyond (Manchester, 2001) Contemporary Mathematics, vol. 315, pp. 169–185. American Mathematical Society, Providence, RI (2002)Google Scholar
  15. 15.
    Ševera, P.: Some title containing the words “homotopy” and “symplectic”, e.g. this one, Travaux mathématiques. Fasc. XVI, Trav. Math., XVI, Univ. Luxemb., Luxembourg, pp. 121–137 (2005)Google Scholar
  16. 16.
    Sheng, Y., Zhu, C.: Higher Extensions of Lie Algebroids and Application to Courant Algebroids (2011). arXiv:1103.5920
  17. 17.
    Ševera, P., Širaň, M.: Integration of Differential Graded Manifolds (2015). arXiv:1506.04898
  18. 18.
    Xu, P.: Momentum maps and Morita equivalence. J. Differ. Geom. 67(2), 289–333 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhu, C.: Kan replacement of simplicial manifolds. Lett. Math. Phys. 90(1–3), 383–405 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSmith CollegeNorthamptonUSA
  2. 2.Department of MathematicsWashington University in Saint LouisSaint LouisUSA

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