Letters in Mathematical Physics

, Volume 102, Issue 1, pp 31–64 | Cite as

Boundary Coupling of Lie Algebroid Poisson Sigma Models and Representations up to Homotopy

  • Alexander Quintero Vélez


A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin–Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.

Mathematics Subject Classification (2000)

Primary 81T45 Secondary 81T70 81T30 


topological field theories Poisson sigma models Lie algebroids Batalin–Vilkovisky formalism D-branes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexandrov M., Schwarz A., Zaboronsky O., Kontsevich M.: The geometry of the master equation and topological quantum field theory. Int. J. Modern Phys. A 12(7), 1405–1429 (1997)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Arias Abad, C., Crainic, M.: Representations up to homotopy of Lie algebroids.
  3. 3.
    Aspinwall, P.S., Bridgeland, T., Craw, A., Douglas, M.R., Gross, M., Kapustin, A., Moore, G.W., Segal, G., Szendrői, B., Wilson, P.M.H.: Dirichlet Branes and Mirror Symmetry. Clay Mathematics Monographs, vol. 4. American Mathematical Society, Providence (2009)Google Scholar
  4. 4.
    Bergman, A.: Topological D-branes from descent.
  5. 5.
    Block, J.: Duality and equivalence of module categories in noncommutative geometry. In: A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, pp. 311–339. American Mathematical Society, Providence (2010)Google Scholar
  6. 6.
    Bonechi, F., Qiu, J., Zabzine, M.: Wilson lines from representations of NQ-manifolds.
  7. 7.
    Bonechi F., Zabzine M.: Poisson sigma model over group manifolds. J. Geom. Phys. 54(2), 173–196 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Bonechi F., Zabzine M.: Lie algebroids, Lie groupoids and TFT. J. Geom. Phys. 57(3), 731–744 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  9. 9.
    Bonechi F., Zabzine M.: Poisson sigma model on the sphere. Commun. Math. Phys. 285(3), 1033–1063 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Brunner, I., Herbst, M., Lerche, W., Scheuner, B.: Landau–Ginzburg realization of open string TFT. J. High Energy Phys. 11, 043 (2006, electronic)Google Scholar
  11. 11.
    Calvo I., Falceto F.: Poisson reduction and branes in Poisson-sigma models. Lett. Math. Phys. 70(3), 231–247 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Cattaneo A.S., Felder G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212(3), 591–611 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Cattaneo, A.S., Felder, G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56(2), 163–179 (2001, EuroConférence Moshé Flato 2000, Part II (Dijon))Google Scholar
  14. 14.
    Cattaneo A.S., Felder G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157–175 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  15. 15.
    Cattaneo A.S., Qiu J., Zabzine M.: 2D and 3D topological field theories for generalized complex geometry. Adv. Theor. Math. Phys. 14(2), 695–725 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Cavalcanti, G.R.: New aspects of the dd c-lemma. Ph.D. dissertation, Oxford University (2004)Google Scholar
  17. 17.
    Diaconescu, D.-E.: Enhanced D-brane categories from string field theory. J. High Energy Phys. 6, Paper 16, 19 (2001)Google Scholar
  18. 18.
    Fernandes R.L.: Lie algebroids, holonomy and characteristic classes. Adv. Math. 170(1), 119–179 (2002)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gómez-Mont X.: Transversal holomorphic structures. J. Diff. Geom. 15(2), 161–185 (1980)MATHGoogle Scholar
  20. 20.
    Gualtieri, M.: Generalized complex geometry.
  21. 21.
    Gualtieri, M.: Generalized complex geometry. Ph.D. dissertation, Oxford University (2003)Google Scholar
  22. 22.
    Gualtieri, M.: Branes on Poisson varieties. In: The Many Facets of Geometry, pp. 368–394. Oxford University Press, Oxford (2010)Google Scholar
  23. 23.
    Henneaux M., Teitelboim C.: Quantization of gauge systems. Princeton University Press, Princeton (1992)MATHGoogle Scholar
  24. 24.
    Herbst, M.: On higher rank coisotropic A-branes.
  25. 25.
    Herbst, M., Hori, K., Page, D.: Phases of N =  2 theories in 1 +  1 dimensions with boundary.
  26. 26.
    Herbst, M., Lazaroiu, C.-I.: Localization and traces in open-closed topological Landau-Ginzburg models. J. High Energy Phys. 5, 044 (2005, electronic)Google Scholar
  27. 27.
    Hitchin N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Ikeda N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235(2), 435–464 (1994)ADSMATHCrossRefGoogle Scholar
  29. 29.
    Kapustin A.: Topological strings on noncommutative manifolds. Int. J. Geom. Methods Mod. Phys. 1(1–2), 49–81 (2004)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Kapustin, A., Li, Y.: D-branes in Landau–Ginzburg models and algebraic geometry. J. High Energy Phys. 12, 005 (2003, electronic)Google Scholar
  31. 31.
    Kapustin A., Li Y.: Open-string BRST cohomology for generalized complex branes. Adv. Theor. Math. Phys. 9(4), 559–574 (2005)MathSciNetMATHGoogle Scholar
  32. 32.
    Kapustin A., Li Y.: Topological sigma-models with H-flux and twisted generalized complex manifolds. Adv. Theor. Math. Phys. 11(2), 261–290 (2007)MathSciNetMATHGoogle Scholar
  33. 33.
    Katz, S., Sharpe, E.: D-branes, open string vertex operators, and Ext groups. Adv. Theor. Math. Phys. 6(6), 979–1030 (2003)Google Scholar
  34. 34.
    Kraus, P., Larsen, F.: Boundary string field theory of the \({{\rm D \overline D}}\) system. Phys. Rev. D (3), 63(10):106004, 17 (2001)Google Scholar
  35. 35.
    Lazaroiu, C.I.: On the boundary coupling of topological Landau–Ginzburg models. J. High Energy Phys. 5, 037 (2005, electronic)Google Scholar
  36. 36.
    Li, Y.: Anomalies and graded coisotropic branes. J. High Energy Phys. 3, 100 (2006, electronic)Google Scholar
  37. 37.
    Roytenberg D.: AKSZ-BV formalism and Courant algebroid-induced topological field theories. Lett. Math. Phys. 79(2), 143–159 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  38. 38.
    Schaller P., Strobl T.: Poisson structure induced (topological) field theories. Modern Phys. Lett. A 9(33), 3129–3136 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  39. 39.
    Schwarz A.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155(2), 249–260 (1993)ADSMATHCrossRefGoogle Scholar
  40. 40.
    Takayanagi, T., Terashima, S., Uesugi, T.: Brane-antibrane action from boundary string field theory. J. High Energy Phys. 3, Paper 19, 37 (2001)Google Scholar
  41. 41.
    Vaĭntrob A.Y.: Lie algebroids and homological vector fields. Uspekhi Mat. Nauk 52(2(314)), 161–162 (1997)Google Scholar
  42. 42.
    Voronov, T.: Graded manifolds and Drinfeld doubles for Lie bialgebroids. In: Quantization, Poisson Brackets and Beyond (Manchester, 2001), Contemp. Math., vol. 315, pp. 131–168. American Mathematical Society, Providence (2002)Google Scholar
  43. 43.
    Witten E.: A note on the antibracket formalism. Modern Phys. Lett. A 5(7), 487–494 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  44. 44.
    Witten, E.: Chern-Simons gauge theory as a string theory. In: The Floer Memorial Volume, Progr. Math., vol. 133, pp. 637–678. Birkhäuser, Basel (1995)Google Scholar
  45. 45.
    Zucchini, R.: The Lie algebroid Poisson sigma model. J. High Energy Phys. 12, 062, 29 (2008)Google Scholar

Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GlasgowGlasgowUK

Personalised recommendations