A search for AdS 5 × S 2 IIB supergravity solutions dual to \( \mathcal{N} = 2 \) SCFTs

Article

Abstract

We present a systematic search for Type IIB supergravity solutions whose spacetimes include AdS 5 and S 2 factors, which would be candidate duals to \( \mathcal{N} = 2 \) four-dimensional Superconformal field theories. The candidate solutions encode the SU(2) R-symmetry geometrically on the S 2 and an additional Killing vector generates the U(1) R-symmetry. By analysing the Killing spinor equations we show that no such solutions exist. This suggests that if Type IIB backgrounds dual to \( \mathcal{N} = 2 \) SCFTs exist, the SU(2) R-symmetry is realised non-geometrically. Finally, we also show that, in the context of both \( \mathcal{N} = 1 \) and \( \mathcal{N} = 2 \) Type IIB backgrounds with an AdS 5 factor, the only candidate U(1) R-symmetry Killing vector directions are the ones that appear for generic values of the Killing spinors; no further Killing vectors exist for special values of the Killing spinors.

Keywords

AdS-CFT Correspondence Supergravity Models 

References

  1. [1]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].
  2. [2]
    J. Gomis, T. Okuda and V. Pestun, Exact results for ’t Hooft loops in gauge theories on S 4, arXiv:1105.2568 [SPIRES].
  3. [3]
    F. Passerini and K. Zarembo, Wilson loops in N = 2 super-Yang-Mills from matrix model, JHEP 09 (2011) 102 [arXiv:1106.5763] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  5. [5]
    N. Wyllard, A (N − 1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, arXiv:1103.5832 [SPIRES].
  10. [10]
    D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES]
  11. [11]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [SPIRES].MathSciNetCrossRefGoogle Scholar
  12. [12]
    N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [SPIRES].ADSMATHCrossRefGoogle Scholar
  13. [13]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [SPIRES].
  14. [14]
    N. Nekrasov and E. Witten, The Ω deformation, branes, integrability and Liouville theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    N. Dorey, S. Lee and T.J. Hollowood, Quantization of integrable systems and a 2D/4D duality, arXiv:1103.5726 [SPIRES].
  16. [16]
    H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A new 2D/4D duality via integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [SPIRES].ADSCrossRefGoogle Scholar
  17. [17]
    D. Gaiotto and J. Maldacena, The gravity duals of N = 2 superconformal field theories, arXiv:0904.4466 [SPIRES].
  18. [18]
    R. Reid-Edwards and B. Stefański Jr., On type IIA geometries dual to N = 2 SCFTs, Nucl. Phys. B 849 (2011) 549 [arXiv:1011.0216] [SPIRES].ADSCrossRefGoogle Scholar
  19. [19]
    X.C. de la Ossa and F. Quevedo, Duality symmetries from nonAbelian isometries in string theory, Nucl. Phys. B403 (1993) 377 [hep-th/9210021] [SPIRES].ADSCrossRefGoogle Scholar
  20. [20]
    K. Sfetsos and D.C. Thompson, On non-abelian T -dual geometries with Ramond fluxes, Nucl. Phys. B 846 (2011) 21 [arXiv:1012.1320] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    B. Chen, E. O Colgain, J.-B. Wu and H. Yavartanoo, N = 2 SCFTs: an M 5-brane perspective, JHEP 04 (2010) 078 [arXiv:1001.0906] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    E. Ocolgain, J.-B. Wu and H. Yavartanoo, On the generality of the LLM geometries in M-theory, JHEP 04 (2011) 002 [arXiv:1010.5982] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    E. D’Hoker and Y. Guo, Rigidity of SU(2, 2|2)-symmetric solutions in type IIB, JHEP 05 (2010) 088 [arXiv:1001.4808] [SPIRES].MathSciNetCrossRefGoogle Scholar
  26. [26]
    G. Compere, S. Detournay and M. Romo, Supersymmetric Gödel and warped black holes in string theory, Phys. Rev. D 78 (2008) 104030 [arXiv:0808.1912] [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    D. Orlando and L.I. Uruchurtu, Warped Anti-de Sitter spaces from brane intersections in type-II string theory, JHEP 06 (2010) 049 [arXiv:1003.0712] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    J.P. Gauntlett, N. Kim, D. Martelli and D. Waldram, W rapped five-branes and N = 2 super Yang-Mills theory, Phys. Rev. D 64 (2001) 106008 [hep-th/0106117] [SPIRES].MathSciNetADSGoogle Scholar
  29. [29]
    F. Bigazzi, A. Cotrone and A. Zaffaroni, N = 2 gauge theories from wrapped five-branes, Phys. Lett. B 519 (2001) 269 [hep-th/0106160] [SPIRES].MathSciNetADSGoogle Scholar
  30. [30]
    J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS 5 solutions of type IIB supergravity, Class. Quant. Grav. 23 (2006) 4693 [hep-th/0510125] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  31. [31]
    J.H. Schwarz, Covariant field equations of chiral N = 2 D = 10 supergravity, Nucl. Phys. B 226 (1983) 269 [SPIRES].ADSCrossRefGoogle Scholar
  32. [32]
    P.S. Howe and P.C. West, The complete N = 2, D = 10 supergravity, Nucl. Phys. B 238 (1984) 181 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  33. [33]
    S. Hassan, T duality, space-time spinors and RR fields in curved backgrounds, Nucl. Phys. B 568 (2000) 145 [hep-th/9907152] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    E.O Colgain, J.-B. Wu and H. Yavartanoo, Supersymmetric AdS 3 × S 2 M-theory geometries with fluxes, JHEP 08 (2010) 114 [arXiv:1005.4527] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  35. [35]
    H. Lü, C. Pope and J. Rahmfeld, A Construction of Killing spinors on S n, J. Math. Phys. 40 (1999) 4518 [hep-th/9805151] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  36. [36]
    S. Kobayashi and K. Nomizu, Foundations of differential geometry: volume 1, John Wiley, U.S.A. (1996).Google Scholar
  37. [37]
    J.P. Gauntlett, O.A. MacConamhna, T. Mateos and D. Waldram, AdS spacetimes from wrapped M5 branes, JHEP 11 (2006) 053 [hep-th/0605146] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    H. Kim, K.K. Kim and N. Kim, 1/4-BPS M-theory bubbles with SO(3) × SO(4) symmetry, JHEP 08 (2007) 050 [arXiv:0706.2042] [SPIRES].ADSCrossRefGoogle Scholar
  39. [39]
    J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Sasaki-Einstein metrics on S 2 × S 3, Adv. Theor. Math. Phys. 8 (2004) 711 [hep-th/0403002] [SPIRES].MathSciNetMATHGoogle Scholar
  40. [40]
    J.P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS 5 solutions of M-theory, Class. Quant. Grav. 21 (2004) 4335 [hep-th/0402153] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  41. [41]
    J.P. Gauntlett, E.O Colgain and O. Varela, Properties of some conformal field theories with M-theory duals, JHEP 02 (2007) 049 [hep-th/0611219] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    O. DeWolfe, D.Z. Freedman, S.S. Gubser, G.T. Horowitz and I. Mitra, Stability of AdS (p) × M (q) compactifications without supersymmetry, Phys. Rev. D 65 (2002) 064033 [hep-th/0105047] [SPIRES].MathSciNetADSGoogle Scholar
  43. [43]
    M. Sohnius, Introducing supersymmetry, Phys. Rept. 128 (1985) 39 [SPIRES].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Korea Institute for Advanced StudySeoulKorea
  2. 2.Centre for Mathematical ScienceCity University LondonLondonU.K.

Personalised recommendations