Letters in Mathematical Physics

, Volume 105, Issue 5, pp 723–767 | Cite as

Relational Symplectic Groupoids

  • Alberto S. Cattaneo
  • Ivan ContrerasEmail author


This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed.

Mathematics Subject Classification

Primary 53D17 Secondary 70S05 53D20 70G45 


symplectic groupoids canonical relations Poisson sigma models Poisson structures 


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  1. 1.
    Andersen, J.E., Kashaev, R.: A TQFT from quantum Teichmüller theory. arXiv:1109.6295 [math.QA]
  2. 2.
    Aof M., Brown R.: The holonomy groupoid of a locally topological groupoid. Top. Appl. 47, 97–113 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bates, S., Weinstein, A.: Lectures on the geometry of quantization, Berkeley mathematical lecture Notes, vol. 8 (1997)Google Scholar
  4. 4.
    Brown R., Mucuk O.: Foliations, locally Lie groupoids and holonomy. Cah. Top. Geom. Diff. Cat. 37(1), 61–71 (1996)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bursztyn H., Crainic M., Weinstein A., Zhu C.: Integration of twisted Dirac brackets. Duke Math. J. 123(3), 549–607 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cattaneo A.S.: Deformation quantization and reduction. Contemp. Math. 450, 79–101 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cattaneo A.S.: Coisotropic submanifolds and dual pairs. Lett. Math. Phys. 104, 243–270 (2014)CrossRefADSzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cattaneo, A.S., Contreras, I.: Groupoids and Poisson sigma models with boundary. arXiv:1206.4330. In: Proceedings of the Summer School: Geometric and Topological Methods for Quantum Field Theory (2014)
  9. 9.
    Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids, In: Landsman, N.P., Pflaum, M., Schlichenmeier, M. (eds.) Quantization of Singular Symplectic Quotients, Progress in Mathematics 198 (Birkhäuser), 61–93 (2001)Google Scholar
  10. 10.
    Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical BV theories on manifolds with boundaries. arXiv:1201.0290 [math-ph], to appear in Comm. Math. Phys. (2012)
  11. 11.
    Cattaneo, A.S., Mn̈ev, P., Reshetikin, N.: Classical and quantum Lagrangian field theories with boundary. In: Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity,Corfu, Greece (2011)Google Scholar
  12. 12.
    Contreras, I.: Relational symplectic groupoids and Poisson sigma models with boundary. arXiv:1306.3943 [math.SG]. Ph.D. Thesis, Zürich University
  13. 13.
    Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques, Publ. Dept. Math. Univ. Claude-Bernard Lyon I (1987)Google Scholar
  14. 14.
    Crainic M., Fernandes R.L.: Integrability of Lie brackets. Ann. Math. (2) 157, 575–620 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Crainic M., Fernandes R.L.: Integrability of Poisson brackets. J. Diff. Geom. 66, 71–137 (2004)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Dazord P.: Groupödes symplectiques et troisième théorème de Lie non linéaire. Lect. Notes Math. 1416, 39–74 (1990)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Hawkins E.: A groupoid approach to quantization. J. Symplectic Geom. 6(1), 61–125 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kontsevich, M.: Deformation quantization of Poisson manifolds. q-alg/9709040. Lett. Math. Phys. 66(3) (2003)
  19. 19.
    Lang S.: Differentiable manifolds. Addison Wesley, USA (1972)Google Scholar
  20. 20.
    Libermann, P.: Sur les automorphismes infinitesimaux des structures symplectiques et des structure de contact, Louveim, Colloque de Geometrie differentielle globale, Bruxelles (1959)Google Scholar
  21. 21.
    Moerdjik, I., Mrcun, J.: Intoduction to foliations and Lie groupoids. Cambridge studies in advanced mathematics. 91 (2003)Google Scholar
  22. 22.
    Severa, P.: Some title containing the words homotopy and symplectic, e.g. this one. Preprint math.SG/0105080
  23. 23.
    Tseng H.H., Zhu C.: Integrating Poisson manifolds via stacks. Trav. Math. 15, 285–297 (2006)Google Scholar
  24. 24.
    Weinstein A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc. 16, 101–104 (1987)CrossRefzbMATHGoogle Scholar
  25. 25.
    Xu, P.: Symplectic groupoids of reduced Poisson spaces. C. R. Acad. Sci. Paris, Serie I Math (1992)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institut Für MathematikUniversität Zürich IrchelZürichSwitzerland
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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