D = 3 unification of curious supergravities

Open Access
Regular Article - Theoretical Physics


We consider the dimensional reduction to D = 3 of four maximal-rank super-gravities which preserve minimal supersymmetry in D = 11, 7, 5 and 4. Such “curious” theories were investigated some time ago, and the four-dimensional one corresponds to an \( \mathcal{N}=1 \) supergravity with 7 chiral multiplets spanning the seven-disk manifold. Recently, this latter theory provided cosmological models for α-attractors, which are based on the disk geometry with possible restrictions on the parameter α. A unified picture emerges in D = 3, where the Ehlers group of General Relativity merges with the S-, T- and U-dualities of the D = 4 parent theories.


Supergravity Models Extended Supersymmetry M-Theory Differential and Algebraic Geometry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    M.J. Duff and S. Ferrara, Four curious supergravities, Phys. Rev. D 83 (2011) 046007 [arXiv:1010.3173] [INSPIRE].ADSGoogle Scholar
  2. [2]
    D. Joyce, Compact Riemannian 7-manifolds with holonomy G 2 . I, J. Diff. Geom. 43 (1996) 291.MATHGoogle Scholar
  3. [3]
    D. Joyce, Compact Riemannian 7-manifolds with holonomy G 2 . II, J. Diff. Geom. 43 (1996) 329.MATHGoogle Scholar
  4. [4]
    M.J. Duff, J.T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality, Nucl. Phys. B 459 (1996) 125 [hep-th/9508094] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W.K. Wong, STU black holes and string triality, Phys. Rev. D 54 (1996) 6293 [hep-th/9608059] [INSPIRE].ADSGoogle Scholar
  6. [6]
    Y.S. Tung, Essays on mirror manifolds, Int. Press, Hong Kong (1992).Google Scholar
  7. [7]
    S. Ferrara and R. Kallosh, Seven-disk manifold, α-attractors and B-modes, arXiv:1610.04163 [INSPIRE].
  8. [8]
    B. Kostant, The three-dimensional sub-group and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959) 973.CrossRefMATHGoogle Scholar
  9. [9]
    B. Julia, Group disintegrations, in Superspace and supergravity, S.W. Hawking and M. Rocek eds., Cambridge Univ. Press, Cambridge U.K. (1981) [INSPIRE].
  10. [10]
    E. Cremmer, Supergravities in 5 dimensions, in Superspace and supergravity, S.W. Hawking and M. Rocek eds., Cambridge Univ. Press, Cambridge U.K. (1981) [INSPIRE].
  11. [11]
    B. Julia, Kac-Moody symmetry of gravitation and supergravity theories, invited talk at AMS-SIAM summer seminar on applications of group theory in physics and mathematical physics, Chicago U.S.A. July 6–16 1982 [INSPIRE].
  12. [12]
    B. de Wit, A.K. Tollsten and H. Nicolai, Locally supersymmetric D = 3 nonlinear σ-models, Nucl. Phys. B 392 (1993) 3 [hep-th/9208074] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York U.S.A. (1978).MATHGoogle Scholar
  14. [14]
    L. Borsten, D. Dahanayake, M.J. Duff, H. Ebrahim and W. Rubens, Black holes, qubits and octonions, Phys. Rept. 471 (2009) 113 [arXiv:0809.4685] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    N. Marcus and J.H. Schwarz, Three-dimensional supergravity theories, Nucl. Phys. B 228 (1983) 145 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    L. Borsten, M.J. Duff and P. Lévay, The black-hole/qubit correspondence: an up-to-date review, Class. Quant. Grav. 29 (2012) 224008 [arXiv:1206.3166] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. Ferrara, A. Marrani, M. Trigiante, A. Marrani and M. Trigiante, Super-Ehlers in any dimension, JHEP 11 (2012) 068 [arXiv:1206.1255] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    M.J. Duff and S. Ferrara, E 7 and the tripartite entanglement of seven qubits, Phys. Rev. D 76 (2007) 025018 [quant-ph/0609227] [INSPIRE].
  19. [19]
    P. Breitenlohner, D. Maison and G.W. Gibbons, Four-dimensional black holes from Kaluza-Klein theories, Commun. Math. Phys. 120 (1988) 295 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  20. [20]
    S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    S. Ferrara and A. Marrani, N = 8 non-BPS attractors, fixed scalars and magic supergravities, Nucl. Phys. B 788 (2008) 63 [arXiv:0705.3866] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    L. Andrianopoli, R. D’Auria and S. Ferrara, Supersymmetry reduction of N extended supergravities in four-dimensions, JHEP 03 (2002) 025 [hep-th/0110277] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    S. Ferrara, R. Kallosh and A. Marrani, Degeneration of groups of type E 7 and minimal coupling in supergravity, JHEP 06 (2012) 074 [arXiv:1202.1290] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    P. West, A brief review of E theory, Int. J. Mod. Phys. A 31 (2016) 1630043 [arXiv:1609.06863] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Theoretical Physics, Blackett LaboratoryImperial College LondonLondonU.K.
  2. 2.Mathematical Institute University of OxfordOxfordU.K.
  3. 3.Theoretical Physics DepartmentCERNGenevaSwitzerland
  4. 4.INFN — Laboratori Nazionali di FrascatiFrascatiItaly
  5. 5.Department of Physics and Astronomy and Mani L. Bhaumik Institute for Theoretical PhysicsUCLALos AngelesU.S.A.
  6. 6.Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”RomaItaly
  7. 7.Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova and INFNPadovaItaly

Personalised recommendations