Skip to main content
SpringerLink
Log in

Tri-Sasakian consistent reduction

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We establish a universal consistent Kaluza-Klein truncation of M-theory based on seven-dimensional tri-Sasakian structure. The four-dimensional truncated theory is an \( \mathcal{N} \) = 4 gauged supergravity with three vector multiplets and a non-abelian gauge group, containing the compact factor SO(3). Consistency follows from the fact that our truncation takes exactly the same form as a left-invariant reduction on a specific coset manifold, and we show that the same holds for the various universal consistent truncations recently put forward in the literature. We describe how the global symmetry group SL(2, \( \mathbb{R} \)) × SO(6, 3) is embedded in the symmetry group E7(7) of maximally supersymmetric reductions, and make the connection with the approach of Exceptional Generalized Geometry. Vacuum AdS4 solutions spontaneously break the amount of supersymmetry from \( \mathcal{N} \) = 4 to \( \mathcal{N} \) = 3, 1 or 0, and the spectrum contains massive modes. We find a subtruncation to minimal \( \mathcal{N} \) = 3 gauged supergravity as well as an \( \mathcal{N} \) = 1 subtruncation to the SO(3)-invariant sector. We also show that a reduction on the homogeneous space N 010 enhances the universal tri-Sasakian truncation with a Betti vector multiplet.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. de Wit and H. Nicolai, The consistency of the S7 truncation in D = 11 supergravity, Nucl. Phys. B 281 (1987) 211 [INSPIRE].

    Article  ADS  Google Scholar 

  2. H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11d supergravity on AdS 7 × S 4 and selfduality in odd dimensions, Phys. Lett. B 469 (1999) 96 [hep-th/9905075] [INSPIRE].

    ADS  Google Scholar 

  3. H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS 7 × S 4 reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B 581 (2000) 179 [hep-th/9911238] [INSPIRE].

    Article  ADS  Google Scholar 

  4. A. Khavaev, K. Pilch and N.P. Warner, New vacua of gauged \( \mathcal{N} \) = 8 supergravity in five-dimensions, Phys. Lett. B 487 (2000) 14 [hep-th/9812035] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  5. M. Cvetič, H. Lü, C. Pope, A. Sadrzadeh and T.A. Tran, Consistent SO(6) reduction of type IIB supergravity on S 5, Nucl. Phys. B 586 (2000) 275 [hep-th/0003103] [INSPIRE].

    Article  ADS  Google Scholar 

  6. D. Cassani and A.-K. Kashani-Poor, Exploiting \( \mathcal{N} \) = 2 in consistent coset reductions of type IIA, Nucl. Phys. B 817 (2009) 25 [arXiv:0901.4251] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  8. J.P. Gauntlett, S. Kim, O. Varela and D. Waldram, Consistent supersymmetric Kaluza-Klein truncations with massive modes, JHEP 04 (2009) 102 [arXiv:0901.0676] [INSPIRE].

    Article  ADS  Google Scholar 

  9. J. Maldacena, D. Martelli and Y. Tachikawa, Comments on string theory backgrounds with non-relativistic conformal symmetry, JHEP 10 (2008) 072 [arXiv:0807.1100] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. D. Cassani, G. Dall’Agata and A.F. Faedo, Type IIB supergravity on squashed Sasaki-Einstein manifolds, JHEP 05 (2010) 094 [arXiv:1003.4283] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. J.P. Gauntlett and O. Varela, Universal Kaluza-Klein reductions of type IIB to \( \mathcal{N} \) = 4 supergravity in five dimensions, JHEP 06 (2010) 081 [arXiv:1003.5642] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  12. J.T. Liu, P. Szepietowski and Z. Zhao, Consistent massive truncations of IIB supergravity on Sasaki-Einstein manifolds, Phys. Rev. D 81 (2010) 124028 [arXiv:1003.5374] [INSPIRE].

    ADS  Google Scholar 

  13. K. Skenderis, M. Taylor and D. Tsimpis, A Consistent truncation of IIB supergravity on manifolds admitting a Sasaki-Einstein structure, JHEP 06 (2010) 025 [arXiv:1003.5657] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. D. Cassani and A.F. Faedo, A Supersymmetric consistent truncation for conifold solutions, Nucl. Phys. B 843 (2011) 455 [arXiv:1008.0883] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. I. Bena, G. Giecold, M. Graña, N. Halmagyi and F. Orsi, Supersymmetric consistent truncations of IIB on T 1,1, JHEP 04 (2011) 021 [arXiv:1008.0983] [INSPIRE].

    Article  ADS  Google Scholar 

  16. I. Bah, A. Faraggi, J.I. Jottar, R.G. Leigh and L.A. Pando Zayas, Fermions and D = 11 Supergravity on squashed Sasaki-Einstein manifolds, JHEP 02 (2011) 068 [arXiv:1008.1423] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. I. Bah, A. Faraggi, J.I. Jottar and R.G. Leigh, Fermions and type IIB supergravity on squashed Sasaki-Einstein manifolds, JHEP 01 (2011) 100 [arXiv:1009.1615] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. J.T. Liu, P. Szepietowski and Z. Zhao, Supersymmetric massive truncations of IIB supergravity on Sasaki-Einstein manifolds, Phys. Rev. D 82 (2010) 124022 [arXiv:1009.4210] [INSPIRE].

    ADS  Google Scholar 

  19. J.T. Liu and P. Szepietowski, Supersymmetry of consistent massive truncations of IIB supergravity, arXiv:1103.0029 [INSPIRE].

  20. S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon, Phys. Rev. D 78 (2008) 065034 [arXiv:0801.2977] [INSPIRE].

    ADS  Google Scholar 

  21. S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].

    Article  ADS  Google Scholar 

  22. D. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  23. K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  25. F. Denef and S.A. Hartnoll, Landscape of superconducting membranes, Phys. Rev. D 79 (2009) 126008 [arXiv:0901.1160] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  26. S.S. Gubser, C.P. Herzog, S.S. Pufu and T. Tesileanu, Superconductors from superstrings, Phys. Rev. Lett. 103 (2009) 141601 [arXiv:0907.3510] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. J.P. Gauntlett, J. Sonner and T. Wiseman, Holographic superconductivity in M-theory, Phys. Rev. Lett. 103 (2009) 151601 [arXiv:0907.3796] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. S.S. Gubser, S.S. Pufu and F.D. Rocha, Quantum critical superconductors in string theory and M-theory, Phys. Lett. B 683 (2010) 201 [arXiv:0908.0011] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  29. J.P. Gauntlett, J. Sonner and T. Wiseman, Quantum criticality and holographic superconductors in M-theory, JHEP 02 (2010) 060 [arXiv:0912.0512] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. F. Aprile, D. Roest and J.G. Russo, Holographic superconductors from gauged supergravity, JHEP 06 (2011) 040 [arXiv:1104.4473] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  31. A. Donos and J.P. Gauntlett, Superfluid black branes in AdS 4 × S 7, JHEP 06 (2011) 053 [arXiv:1104.4478] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. N. Bobev, A. Kundu, K. Pilch and N.P. Warner, Minimal holographic superconductors from maximal supergravity, arXiv:1110.3454 [INSPIRE].

  33. A. Donos and J.P. Gauntlett, Lifshitz Solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. A. Donos, J.P. Gauntlett, N. Kim and O. Varela, Wrapped M5-branes, consistent truncations and AdS/CMT, JHEP 12 (2010) 003 [arXiv:1009.3805] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. D. Cassani and A.F. Faedo, Constructing Lifshitz solutions from AdS, JHEP 05 (2011) 013 [arXiv:1102.5344] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  37. E. Ó Colgáin and H. Samtleben, 3D gauged supergravity from wrapped M5-branes with AdS/CMT applications, JHEP 02 (2011) 031 [arXiv:1012.2145] [INSPIRE].

    Google Scholar 

  38. A. Tomasiello, New string vacua from twistor spaces, Phys. Rev. D 78 (2008) 046007 [arXiv:0712.1396] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  39. P. Koerber, D. Lüst and D. Tsimpis, Type IIA AdS 4 compactifications on cosets, interpolations and domain walls, JHEP 07 (2008) 017 [arXiv:0804.0614] [INSPIRE].

    Article  ADS  Google Scholar 

  40. C.P. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys Diff. Geom. 7 (1999) 123 [hep-th/9810250] [INSPIRE].

    MathSciNet  Google Scholar 

  41. B.S. Acharya, J. Figueroa-O’Farrill, C. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].

    MathSciNet  Google Scholar 

  42. M. Billó, D. Fabbri, P. Fré, P. Merlatti and A. Zaffaroni, Shadow multiplets in AdS 4 /CFT 3 and the superHiggs mechanism: Hints of new shadow supergravities, Nucl. Phys. B 591 (2000) 139 [hep-th/0005220] [INSPIRE].

    Article  ADS  Google Scholar 

  43. J. Schön and M. Weidner, Gauged \( \mathcal{N} \) = 4 supergravities, JHEP 05 (2006) 034 [hep-th/0602024] [INSPIRE].

    Article  ADS  Google Scholar 

  44. B. de Wit, H. Samtleben and M. Trigiante, Magnetic charges in local field theory, JHEP 09 (2005) 016 [hep-th/0507289] [INSPIRE].

    Article  Google Scholar 

  45. P.P. Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  46. C. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].

    Article  ADS  Google Scholar 

  47. H. Samtleben, Lectures on gauged supergravity and flux compactifications, Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. C. Pope and N. Warner, An SU(4) invariant compactification of D = 11 supergravity on a stretched seven-sphere, Phys. Lett. B 150 (1985) 352 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  49. N. Bobev, N. Halmagyi, K. Pilch and N.P. Warner, Supergravity instabilities of non-supersymmetric quantum critical points, Class. Quant. Grav. 27 (2010) 235013 [arXiv:1006.2546] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N} \) = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. D.L. Jafferis and A. Tomasiello, A Simple class of \( \mathcal{N} \) = 3 gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  52. C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-matrix models and tri-Sasaki Einstein spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].

    ADS  Google Scholar 

  53. M. Billó, D. Fabbri, P. Fré, P. Merlatti and A. Zaffaroni, Rings of short \( \mathcal{N} \) = 3 superfields in three-dimensions and M-theory on AdS 4 × N 0,1,0, Class. Quant. Grav. 18 (2001) 1269 [hep-th/0005219] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  54. S.S. Gubser, Colorful horizons with charge in anti-de Sitter space, Phys. Rev. Lett. 101 (2008) 191601 [arXiv:0803.3483] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  55. S.S. Gubser and S.S. Pufu, The gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. C.P. Herzog and S.S. Pufu, The second sound of SU(2), JHEP 04 (2009) 126 [arXiv:0902.0409] [INSPIRE].

    Article  ADS  Google Scholar 

  57. M. Duff, B. Nilsson and C. Pope, Kaluza-Klein supergravity, Phys. Rept. 130 (1986) 1 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  58. J. Louis and A. Micu, Type II theories compactified on Calabi-Yau threefolds in the presence of background fluxes, Nucl. Phys. B 635 (2002) 395 [hep-th/0202168] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. T. Danckaert, J. Louis, D. Martínez-Pedrera, B. Spanjaard and H. Triendl, The N = 4 effective action of type IIA supergravity compactified on SU(2)-structure manifolds, JHEP 08 (2011) 024 [arXiv:1104.5174] [INSPIRE].

    Article  ADS  Google Scholar 

  60. P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  61. L. Magnin, Sur les algèbres de Lie nilpotentes de dimension ≤ 7 (in French), J. Geom. Phys. 3 (1986) 119.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. M. Graña, R. Minasian, M. Petrini and A. Tomasiello, A scan for new \( \mathcal{N} \) = 1 vacua on twisted tori, JHEP 05 (2007) 031 [hep-th/0609124] [INSPIRE].

    Article  ADS  Google Scholar 

  63. M. Duff, B. Nilsson and C. Pope, The criterion for vacuum stability in Kaluza-Klein supergravity, Phys. Lett. B 139 (1984) 154 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  64. F. Englert, Spontaneous Compactification of Eleven-Dimensional Supergravity, Phys. Lett. B 119 (1982) 339 [INSPIRE].

    ADS  Google Scholar 

  65. D.N. Page and C. Pope, New squashed solutions of D = 11 supergravity, Phys. Lett. B 147 (1984) 55 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  66. P. Fré, L. Gualtieri and P. Termonia, The structure of \( \mathcal{N} \) = 3 multiplets in AdS 4 and the complete Osp(3|4) × SU(3) spectrum of M-theory on AdS 4 × N 010, Phys. Lett. B 471 (1999) 27 [hep-th/9909188] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  67. W. Heidenreich, All linear unitary irreducible representations of de Sitter supersymmetry with positive energy, Phys. Lett. B 110 (1982) 461 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  68. D.N. Page and C. Pope, Instabilities in englert type supergravity solutions, Phys. Lett. B 145 (1984) 333 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  69. K. Ito, Instability of englert solution on squashed seven-sphere in eleven-dimensional supergravity, Phys. Lett. B 147 (1984) 52 [INSPIRE].

    ADS  Google Scholar 

  70. F. Müller-Hoissen and R. Stückl, Coset spaces and ten-dimensional unified theories, Class. Quant. Grav. 5 (1988) 27 [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  71. L. Castellani, L. Romans and N. Warner, Symmetries of coset spaces and Kaluza-Klein supergravity, Annals Phys. 157 (1984) 394 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  72. O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, \( \mathcal{N} \) = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  73. A.-K. Kashani-Poor, Nearly Kähler reduction, JHEP 11 (2007) 026 [arXiv:0709.4482] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  74. L. Castellani and L. Romans, \( \mathcal{N} \) = 3 and N = 1 supersymmetry in a new class of solutions for d = 11 supergravity, Nucl. Phys. B 238 (1984) 683 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  75. R. D’Auria and P. Fré, Universal Bose-Fermi mass relations in Kaluza-Klein supergravity and harmonic analysis on coset manifolds with Killing spinors, Annals Phys. 162 (1985) 372 [INSPIRE].

    Article  ADS  Google Scholar 

  76. P. Termonia, The Complete \( \mathcal{N} \) = 3 Kaluza-Klein spectrum of 11D supergravity on AdS 4 × N 010, Nucl. Phys. B 577 (2000) 341 [hep-th/9909137] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  77. D.Z. Freedman and A.K. Das, Gauge internal symmetry in extended supergravity, Nucl. Phys. B 120 (1977) 221 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  78. J.M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  79. T. House and A. Micu, M-Theory compactifications on manifolds with G2 structure, Class. Quant. Grav. 22 (2005) 1709 [hep-th/0412006] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  80. D.N. Page, classical stability of round and squashed seven-spheres in eleven-dimensional supergravity, Phys. Rev. D 28 (1983) 2976 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  81. H. Ooguri and C.-S. Park, Superconformal Chern-Simons theories and the squashed seven-sphere, JHEP 11 (2008) 082 [arXiv:0808.0500] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, King’s College London, The Strand, London, WC2R 2LS, United Kingdom

    Davide Cassani

  2. Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001, Leuven, Belgium

    Paul Koerber

Authors
  1. Davide Cassani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Koerber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davide Cassani.

Additional information

ArXiv ePrint: 1110.5327

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cassani, D., Koerber, P. Tri-Sasakian consistent reduction. J. High Energ. Phys. 2012, 86 (2012). https://doi.org/10.1007/JHEP01(2012)086

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2012)086

Keywords

Search

Navigation