Letters in Mathematical Physics

, Volume 104, Issue 3, pp 243–270 | Cite as

Coisotropic Submanifolds and Dual Pairs

Article

Abstract

The Poisson sigma model is a widely studied two-dimensional topological field theory. This note shows that boundary conditions for the Poisson sigma model are related to coisotropic submanifolds (a result announced in [math.QA/0309180]) and that the corresponding reduced phase space is a (possibly singular) dual pair between the reduced spaces of the given two coisotropic submanifolds. In addition the generalization to a more general tensor field is considered and it is shown that the theory produces Lagrangian evolution relations if and only if the tensor field is Poisson.

Mathematics Subject Classification (1991)

Primary 53D17 Secondary 81T45 53D20 58H05 

Keywords

coisotropic submanifolds dual pairs Poisson sigma model Lagrangian field theories with boundary 

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References

  1. 1.
    Calvo I., Falceto F.: Poisson reduction and branes in Poisson sigma models. Lett. Math. Phys. 70, 231–247 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Cattaneo A.S.: On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds. Lett. Math. Phys 67, 33–48 (2004)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cattaneo, A.S., Contreras, I.: Groupoids and Poisson sigma models with boundary. http://arxiv.org/abs/1206.4330 (arXiv:1206.4330). In: Geometric, Algebraic and Topological Methods for Quantum Field Theory, Proceedings of the 7th Villa de Leyva Summer School, World Scientific (2013, to appear)
  4. 4.
    Cattaneo, A.S., Mnëv P., Reshetikhin, N.: Classical BV theories on manifolds with boundaries. http://arxiv.org/abs/1201.0290 (math-ph/1201.0290)
  5. 5.
    Cattaneo, A.S., Mnëv, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. http://pos.sissa.it/archive/conferences/155/044/CORFU2011_044.pdf (PoS(CORFU2011)044), http://arxiv.org/abs/1207.0239 (arXiv:1207.0239)
  6. 6.
    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, vol. 198. Birkhauser, Basel, pp. 61–93 (2001)Google Scholar
  7. 7.
    Cattaneo A.S., Felder G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157–175 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cattaneo A.S., Zambon M.: Pre-Poisson submanifolds. Travaux mathématiques 17, 61–74 (2007)MathSciNetGoogle Scholar
  9. 9.
    Contreras, I.: Relational Symplectic Groupoids and Poisson Sigma Models with Boundary, Ph. D. thesis (Zurich, 2013), http://arxiv.org/abs/1306.4119
  10. 10.
    Crainic M., Fernandes R.L.: Integrability of Lie brackets. Ann. Math. 157, 575–620 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ikeda N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235, 435–464 (1994)ADSCrossRefMATHGoogle Scholar
  12. 12.
    Landsman, N.P.: Quantized reduction as a tensor product. In: Landsman, N.P., Pflaum, M., Schlichenmeier, M. (eds.) Quantization of Singular Symplectic Quotients, Progress in Mathematics, vol.~198. Birkhäuser, Basel, pp. 137–180 (2001)Google Scholar
  13. 13.
    Schaller P., Strobl T.: Poisson structure induced (topological) field theories. Modern Phys. Lett. A 9, 3129–3136 (1994)ADSCrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Weinstein A.: The local structure of Poisson manifolds. J. Differ. Geom 18, 523–557 (1983)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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