Advertisement

Communications in Mathematical Physics

, Volume 291, Issue 1, pp 1–30 | Cite as

Generalized Kähler Potentials from Supergravity

  • Nick Halmagyi
  • Alessandro TomasielloEmail author
Article

Abstract

We consider supersymmetric \({\mathcal{N} = 2}\) solutions with non–vanishing NS three–form. Building on worldsheet results, we reduce the problem to a single generalized Monge–Ampère equation on the generalized Kähler potential K recently interpreted geometrically by Lindström, Roček, Von Unge and Zabzine. One input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a D3 brane probe. The procedure is hence likely to be useful for finding gravity duals to field theories with non–vanishing abelian superpotential, such as Leigh–Strassler theories. We indeed show that a purely NS precursor of the Lunin–Maldacena dual to the β–deformed \({\mathcal{N} = 4}\) super–Yang–Mills falls in our class.

Keywords

Manifold Sigma Model Pure Spinor Chiral Multiplet Marginal Deformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hitchin N.: Generalized Calabi–Yau manifolds. Quart. J. Math. Oxford Ser. 54, 281–308 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Gualtieri, M.: Generalized complex geometry. Oxford Univ. DPhil. thesis available at http://arxiv.org/abs/math/0401221v1[Math.DG], 2004
  3. 3.
    Graña M., Minasian R., Petrini M., Tomasiello A.: Generalized structures of \({\mathcal{N} = 1}\) vacua. JHEP 11, 020 (2005)CrossRefADSGoogle Scholar
  4. 4.
    Jeschek C., Witt F.: Generalised G2 structures and type IIB superstrings. JHEP 03, 053 (2005)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Lindström U., Roček M., von Unge R., Zabzine M.: Generalized Kaehler manifolds and off–shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Linch W.D., Vallilo B.C.: Hybrid formalism, supersymmetry reduction, and Ramond-Ramond fluxes. JHEP 01, 099 (2007)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Berkovits N.: Covariant quantization of the Green-Schwarz superstring in a Calabi-Yau background. Nucl. Phys. B 431, 258 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Grisaru M.T., Massar M., Sevrin A., Troost J.: The quantum geometry of \({\mathcal{N} = (2, 2)}\) non-linear sigma-models. Phys. Lett. B 412, 53–58 (1997)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Minasian R., Petrini M., Zaffaroni A.: Gravity duals to deformed SYM theories and generalized complex geometry. JHEP 12, 055 (2006)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Apostolov V., Gauduchon P., Grantcharov G.: Bihermitian structures on complex surfaces. Proc. London Math. Soc. 79, 414–428 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Martucci L.: D–branes on general \({\mathcal{N} = 1}\) backgrounds: Superpotentials and D–terms. JHEP 06, 033 (2006)CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Lunin O., Maldacena J.M.: Deforming field theories with U(1) × U(1) global symmetry and their gravity duals. JHEP 05, 033 (2005)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Graña M., Minasian R., Petrini M., Tomasiello A.: A scan for new \({\mathcal{N} = 1}\) vacua on twisted tori. JHEP 0705, 031 (2007)CrossRefADSGoogle Scholar
  14. 14.
    Tomasiello A.: Reformulating supersymmetry with a generalized Dolbeault operator. JHEP 0802, 010 (2008)CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Gates, J., S.J., Hull, C.M., Rčcek, M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nucl. Phys. B248, 157 (1984)Google Scholar
  16. 16.
    Lust D., Tsimpis D.: Supersymmetric AdS4 compactifications of iia supergravity. JHEP 02, 027 (2005)CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Gauntlett J.P., Martelli D., Sparks J., Waldram D.: Supersymmetric AdS5 solutions of type IIB supergravity. Class. Quant. Grav. 23, 4693–4718 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Koerber P., Tsimpis D.: Supersymmetric sources, integrability and generalized- structure compactifications. JHEP 0708, 082 (2007)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Pilch K., Warner N.P.: Generalizing the \({\mathcal{N} = 2}\) supersymmetric rg flow solution of IIB supergravity. Nucl. Phys. B675, 99–121 (2003)CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Pilch K., Warner N.P.: \({\mathcal{N} = 1}\) supersymmetric renormalization group flows from IIB supergravity. Adv. Theor. Math. Phys. 4, 627–677 (2002)MathSciNetGoogle Scholar
  21. 21.
    Myers R.C.: Dielectric-branes. JHEP 12, 022 (1999)CrossRefADSGoogle Scholar
  22. 22.
    Koerber P.: Stable D–branes, calibrations and generalized Calabi–Yau geometry. JHEP 08, 099 (2005)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Martucci L., Smyth P.: Supersymmetric D–branes and calibrations on general n = 1 backgrounds. JHEP 11, 048 (2005)CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Becker K., Becker M., Strominger A.: Five-branes, membranes and nonperturbative string theory. Nucl. Phys. B456, 130–152 (1995)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Mariño M., Minasian R., Moore G.W., Strominger A.: Nonlinear instantons from supersymmetric p-branes. JHEP 01, 005 (2000)CrossRefADSGoogle Scholar
  26. 26.
    Mariotti A.: Supersymmetric D–branes on SU(2) structure manifolds. JHEP 0709, 123 (2007)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Hitchin, N.J.: Bihermitian metrics on Del Pezzo surfaces. http://arxiv.org/abs/math/0608213v1[math.DG], 2006
  28. 28.
    Lindström U., Minasian R., Tomasiello A., Zabzine M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235–256 (2005)zbMATHCrossRefADSGoogle Scholar
  29. 29.
    Lyakhovich S., Zabzine M.: Poisson geometry of sigma models with extended supersymmetry. Phys. Lett. B548, 243–251 (2002)ADSMathSciNetGoogle Scholar
  30. 30.
    Hitchin N.: Instantons, Poisson structures and generalized kaehler geometry. Commun. Math. Phys. 265, 131–164 (2006)zbMATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Wijnholt, M.: Parameter space of quiver gauge theories. http://arxiv.org/abs/hep-th/0512122v2, 2005
  32. 32.
    Seiberg N., Witten E.: String theory and noncommutative geometry. JHEP 09, 032 (1999)CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Schomerus V.: D–branes and deformation quantization. JHEP 06, 030 (1999)CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Kapustin, A., Witten, E.: Electric–magnetic duality and the geometric langlands program. http://arxiv.org/abs/hep-th/0604151v3, 2006
  35. 35.
    Kapustin A.: Topological strings on noncommutative manifolds. Int. J. Geom. Meth. Mod. Phys. 1, 49–81 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Gauntlett J.P., Martelli D., Waldram D.: Superstrings with intrinsic torsion. Phys. Rev. D 69, 086002 (2004)CrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Witten, E.: Mirror manifolds and topological field theory. http://arxiv.org/abs/hep-th/9112056v1, 1991
  38. 38.
    Zucchini R.: The bihermitian topological sigma model. JHEP 12, 039 (2006)CrossRefADSMathSciNetGoogle Scholar
  39. 39.
    Chuang W.-y.: Topological twisted sigma model with H–flux revisited. J. Phys. A 41, 115402 (2008)CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Roček, M.: Modified Calabi–Yau manifolds with torsion. In: Essays on Mirror Manifolds, S.-T. Yau (ed.), Hong Kong: International Press, 1992Google Scholar
  41. 41.
    Bogaerts J., Sevrin A., van der Loo S., Van Gils S.: Properties of semi-chiral superfields. Nucl. Phys. B562, 277–290 (1999)CrossRefADSGoogle Scholar
  42. 42.
    Leigh R.G., Strassler M.J.: Exactly marginal operators and duality in four-dimensional \({\mathcal{N} = 1}\) supersymmetric gauge theory. Nucl. Phys. B447, 95–136 (1995)CrossRefADSMathSciNetGoogle Scholar
  43. 43.
    Kulaxizi, M.: Marginal deformations of N = 4 SYM and open vs. closed string parameters. http://arxiv.org/abs/hep-th/0612160v2, 2006
  44. 44.
    Graña M., Polchinski J.: Supersymmetric three–form flux perturbations on AdS5. Phys. Rev. D 63, 026001 (2001)CrossRefADSMathSciNetGoogle Scholar
  45. 45.
    Aharony O., Kol B., Yankielowicz S.: On exactly marginal deformations of \({\mathcal{N} = 1}\) SYM and type IIB supergravity on AdS5 × S5. JHEP 06, 039 (2002)CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Maldacena J.M., Sheikh-Jabbari M.M., Van Raamsdonk M.: Transverse fivebranes in matrix theory. JHEP 01, 038 (2003)CrossRefADSGoogle Scholar
  47. 47.
    Corrado R., Halmagyi N.: N = 1 field theories and fluxes in iib string theory. Phys. Rev. D71, 046001 (2005)ADSMathSciNetGoogle Scholar
  48. 48.
    Bergman A.: Deformations and D–branes. Adv. Theor. Math. Phys. 12, 781–815 (2008)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Halmagyi N., Pilch K., Romelsberger C., Warner N.P.: Holographic duals of a family of n = 1 fixed points. JHEP 0608, 083 (2006)CrossRefADSMathSciNetGoogle Scholar
  50. 50.
    Benvenuti S., Hanany A.: Conformal manifolds for the conifold and other toric field theories. JHEP 08, 024 (2005)CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    Kol B.: On conformal deformations. JHEP 09, 046 (2002)CrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Lin, H., Tolman, S.: Symmetries in generalized Kähler geometry. http://arxiv.org/abs/math/0509069v1[math.DG], 2005
  53. 53.
    Bursztyn H., Cavalcanti G., Gualtieri M.: Reduction of Courant algebroids and generalized complex structures. Adv. Math. 211, 726–765 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Fidanza S., Minasian R., Tomasiello A.: Mirror symmetric SU(3)–structure manifolds with NS fluxes. Commun. Math. Phys. 254, 401–423 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Enrico Fermi InstituteUniversity of ChicagoChicagoUSA
  2. 2.ITPStanford UniversityStanfordUSA

Personalised recommendations