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Generalized Complex Manifolds and Supersymmetry

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We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two–dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in the context of deformation quantization.

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References

  1. Hitchin, N.: Generalized Calabi-Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)

    Article  Google Scholar 

  2. Gualtieri, M.: Generalized complex geometry. Oxford University DPhil thesis. http://xxx.lanl.gov/abs/math.DG/0401221, 2004

  3. Huybrechts, D.: Generalized Calabi-Yau structures, K3 surfaces, and B-fields. http://arxiv.org/abs/math.AG/0306162, 2003

  4. Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: The case of complex tori. Commun. Math. Phys. 233, 79 (2003)

    Article  Google Scholar 

  5. Fidanza, S., Minasian, R., Tomasiello, A.: Mirror symmetric SU(3)-structure manifolds with NS fluxes. http://arxiv.org/abs/hep-th/0311122, 2003

  6. Vafa, C.: Superstrings and topological strings at large N. J. Math. Phys. 42, 2798 (2001)

    Article  Google Scholar 

  7. Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror symmetry in generalized Calabi-Yau compactifications. Nucl. Phys. B 654, 61 (2003)

    Article  Google Scholar 

  8. Berkovits, N.: Super-Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000)

    Article  Google Scholar 

  9. Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton, NJ: Princeton Univ. Press, 1989

  10. Hellerman, S., McGreevy, J., Williams, B.: Geometric constructions of nongeometric string theories. JHEP 0401, 024 (2004)

    Article  Google Scholar 

  11. Flournoy, A., Wecht, B., Williams, B.: Constructing nongeometric vacua in string theory. http://arxiv.org/abs/hep-th/0404217, 2004

  12. Lindstrom, U.: Generalized N = (2,2) supersymmetric non-linear sigma models. Phys. Lett. B 587, 216–224 (2004)

    Article  CAS  MathSciNet  Google Scholar 

  13. Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591 (2000)

    Article  Google Scholar 

  14. Gates, S.J., Hull, C.M., Rocek, M.: Twisted Multiplets And New Supersymmetric Nonlinear Sigma Models. Nucl. Phys. B 248, 157 (1984)

    Google Scholar 

  15. Lyakhovich, S., Zabzine, M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B 548, 243 (2002)

    CAS  Google Scholar 

  16. Hassan, S.F.: O(D,D:R) Deformations of Complex Structures And Extended World Sheet Supersymmetry. Nucl. Phys. B 454, 86 (1995)

    Google Scholar 

  17. Lindstrom, U., Zabzine, M.: N = 2 boundary conditions for non-linear sigma models and Landau-Ginzburg models. JHEP 0302, 006 (2003)

    Google Scholar 

  18. Lindstrom, U., Zabzine, M.: D-branes in N = 2 WZW models. Phys. Lett. B 560, 108 (2003)

    CAS  Google Scholar 

  19. Kapustin, A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49–81 (2004)

    Google Scholar 

  20. Grange, P.: Branes as stable holomorphic line bundles on the non-commutative torus. JHEP 0410, 002 (2004)

    Google Scholar 

  21. Baulieu, L., Losev, A.S., Nekrasov, N.A.: Target space symmetries in topological theories. I. JHEP 0202, 021 (2002)

    Google Scholar 

  22. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999)

    Article  Google Scholar 

  23. Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235, 435 (1994)

    CAS  Google Scholar 

  24. Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A 9, 3129 (1994)

    Google Scholar 

  25. Klimcik, C., Strobl, T.: WZW-Poisson manifolds. J. Geom. Phys. 43, 341 (2002)

    Article  Google Scholar 

  26. Courant, T.: Dirac manifolds. Trans. Amer. Math. Soc. 319(2), 631–661 (1990)

    Google Scholar 

  27. Courant, T., Weinstein, A. Beyond Poisson structures. In: Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, 27, Paris: Hermann, 1988, pp. 39–49

  28. Cavalcanti, G., Gualtieri, M.: Generalized complex structures on nilmanifolds. http://arxiv.org/abs/math.DG/0404451, 2004

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Communicated by M.R. Douglas

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Lindström, U., Minasian, R., Tomasiello, A. et al. Generalized Complex Manifolds and Supersymmetry. Commun. Math. Phys. 257, 235–256 (2005). https://doi.org/10.1007/s00220-004-1265-6

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  • DOI: https://doi.org/10.1007/s00220-004-1265-6

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