Gauged supergravities from M-theory reductions

  • Stefanos Katmadas
  • Alessandro Tomasiello
Open Access
Regular Article - Theoretical Physics


In supergravity compactifications, there is in general no clear prescription on how to select a finite-dimensional family of metrics on the internal space, and a family of forms on which to expand the various potentials, such that the lower-dimensional effective theory is supersymmetric. We propose a finite-dimensional family of deformations for regular Sasaki-Einstein seven-manifolds M7, relevant for M-theory compactifications down to four dimensions. It consists of integrable Cauchy-Riemann structures, corresponding to complex deformations of the Calabi-Yau cone M8 over M7. The non-harmonic forms we propose are the ones contained in one of the Kohn-Rossi cohomology groups, which is finite-dimensional and naturally controls the deformations of Cauchy-Riemann structures. The same family of deformations can be also described in terms of twisted cohomology of the base M6, or in terms of Milnor cycles arising in deformations of M8. Using existing results on SU(3) structure compactifications, we briefly discuss the reduction of M-theory on our class of deformed Sasaki-Einstein manifolds to four-dimensional gauged supergravity.


Differential and Algebraic Geometry Flux compactifications Supergravity Models Superstring Vacua 


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]
    S. Gurrieri, J. Louis, A. Micu and D. Waldram, Mirror symmetry in generalized Calabi-Yau compactifications, Nucl. Phys. B 654 (2003) 61 [hep-th/0211102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Graña, J. Louis and D. Waldram, Hitchin functionals in N = 2 supergravity, JHEP 01 (2006) 008 [hep-th/0505264] [INSPIRE].
  3. [3]
    A.-K. Kashani-Poor and R. Minasian, Towards reduction of type-II theories on SU(3) structure manifolds, JHEP 03 (2007) 109 [hep-th/0611106] [INSPIRE].
  4. [4]
    A.-K. Kashani-Poor, Nearly Kähler reduction, JHEP 11 (2007) 026 [arXiv:0709.4482] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    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].ADSCrossRefGoogle Scholar
  6. [6]
    D. Cassani and A.-K. Kashani-Poor, Exploiting N = 2 in consistent coset reductions of type IIA, Nucl. Phys. B 817 (2009) 25 [arXiv:0901.4251] [INSPIRE].
  7. [7]
    D. Cassani, P. Koerber and O. Varela, All homogeneous N = 2 M-theory truncations with supersymmetric AdS 4 vacua, JHEP 11 (2012) 173 [arXiv:1208.1262] [INSPIRE].
  8. [8]
    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].
  9. [9]
    D. Cassani and A.F. Faedo, A supersymmetric consistent truncation for conifold solutions, Nucl. Phys. B 843 (2011) 455 [arXiv:1008.0883] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    R. D’Auria, S. Ferrara, M. Trigiante and S. Vaula, Gauging the Heisenberg algebra of special quaternionic manifolds, Phys. Lett. B 610 (2005) 147 [hep-th/0410290] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    T. House and E. Palti, Effective action of (massive) IIA on manifolds with SU(3) structure, Phys. Rev. D 72 (2005) 026004 [hep-th/0505177] [INSPIRE].
  12. [12]
    A. Micu, E. Palti and P.M. Saffin, M-theory on seven-dimensional manifolds with SU(3) structure, JHEP 05 (2006) 048 [hep-th/0602163] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    R. Eager, J. Schmude and Y. Tachikawa, Superconformal indices, Sasaki-Einstein manifolds and cyclic homologies, Adv. Theor. Math. Phys. 18 (2014) 129 [arXiv:1207.0573] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  14. [14]
    R. Eager and J. Schmude, Superconformal indices and M 2-branes, JHEP 12 (2015) 062 [arXiv:1305.3547] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    G. Székelyhidi, The Kähler-Ricci flow and K-stability, Amer. J. Math. 132 (2010) 1077 [arXiv:0803.1613].MathSciNetCrossRefGoogle Scholar
  16. [16]
    P. Candelas and X. de la Ossa, Moduli space of Calabi-Yau manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    B. Dubrovin, Integrable systems in topological field theory, Nucl. Phys. B 379 (1992) 62.Google Scholar
  18. [18]
    G. Dall’Agata and N. Prezas, N = 1 geometries for M-theory and type IIA strings with fluxes, Phys. Rev. D 69 (2004) 066004 [hep-th/0311146] [INSPIRE].
  19. [19]
    K. Behrndt, M. Cvetič and T. Liu, Classification of supersymmetric flux vacua in M-theory, Nucl. Phys. B 749 (2006) 25 [hep-th/0512032] [INSPIRE].
  20. [20]
    X.-X. Chen, S. Donaldson and S. Sun, Kähler-einstein metrics and stability, arXiv:1210.7494.
  21. [21]
    P. Candelas and X.C. de la Ossa, Comments on conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
  22. [22]
    M.B. Stenzel, Ricci-flat metrics on the complexification of a compact rank one symmetric space., Manuscr. Math. 80 (1993) 151.MathSciNetCrossRefGoogle Scholar
  23. [23]
    M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Ricci flat metrics, harmonic forms and brane resolutions, Commun. Math. Phys. 232 (2003) 457 [hep-th/0012011] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1982) 775.Google Scholar
  25. [25]
    A.M. Nadel, Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Natl. Acad. Sci. 86 (1989) 7299.ADSCrossRefGoogle Scholar
  26. [26]
    P. Candelas, Yukawa couplings between (2, 1) forms, Nucl. Phys. B 298 (1988) 458 [INSPIRE].
  27. [27]
    V.A. Iskovskikh et al., Algebraic geometry: Fano varieties. V, Springer, Germany (1999).Google Scholar
  28. [28]
    S. Blesneag, E.I. Buchbinder, P. Candelas and A. Lukas, Holomorphic Yukawa couplings in heterotic string theory, JHEP 01 (2016) 152 [arXiv:1512.05322] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    T. Akahori and P.M. Garfield, Hamiltonian flow over deformations of ordinary double points, J. Math. Anal. Appl. 333 (2007) 24.MathSciNetCrossRefGoogle Scholar
  30. [30]
    J. Schmude, Laplace operators on Sasaki-Einstein manifolds, JHEP 04 (2014) 008 [arXiv:1308.1027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  32. [32]
    G.W. Gibbons and S.W. Hawking, Gravitational multi-instantons, Phys. Lett. 78B (1978) 430 [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    A. Dimca, Topics on real and complex singularities: an introduction, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Germany (1987).CrossRefGoogle Scholar
  34. [34]
    R.J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J. 162 (2013) 2855 [arXiv:1205.6347] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  35. [35]
    A.S. Gusein-Zade and A. Varchenko, Singularities of differentiable maps, volume II, Birkhäuser, Geramny (2012).Google Scholar
  36. [36]
    C. Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge University Press, Cambridge U.K. (2002).Google Scholar
  37. [37]
    K. Saito, The higher residue pairings K F( k) for a family of hypersurface singular points, Singularities. Part 2 40 (1981) 441.Google Scholar
  38. [38]
    K. Aleshkin and A. Belavin, Special geometry on the 101 dimesional moduli space of the quintic threefold, arXiv:1710.11609 [INSPIRE].
  39. [39]
    F. Forstnerič, Stein manifolds and holomorphic mappings: the homotopy principle in complex analysis, Springer, Germany (2011).CrossRefGoogle Scholar
  40. [40]
    W. Ebeling, Functions of several complex variables and their singularities, American Mathematical Society, U.S.A. (2007).Google Scholar
  41. [41]
    P. Albin, Analysis on non-compact manifolds, (2008).
  42. [42]
    S. Cappell et al., Cohomology of harmonic forms on Riemannian manifolds with boundary, Forum Math. 18 (2006) 923.MathSciNetCrossRefGoogle Scholar
  43. [43]
    C. Shonkwiler, Poincaré duality angles for Riemannian manifolds with boundary, arXiv:0909.1967.
  44. [44]
    N.J. Hitchin, The geometry of three-forms in six dimensions, J. Diff. Geom. 55 (2000) 547 [math/0010054] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  45. [45]
    S. Dragomir and G. Tomassini, Differential geometry and analysis on CR manifolds, Springer, Germany (2007).Google Scholar
  46. [46]
    S.S.T. Yau, Kohn-Rossi cohomology and its application to the complex Plateau problem. I, Ann. Math. 113 (1981) 67.MathSciNetCrossRefGoogle Scholar
  47. [47]
    H.R.J.J. Kohn, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965) 451.MathSciNetCrossRefGoogle Scholar
  48. [48]
    H.S. Luk et al., Holomorphic de Rham cohomology of strongly pseudoconvex cr manifolds with s1-actions, J. Diff. Geom. 63 (2003) 155.CrossRefGoogle Scholar
  49. [49]
    I. Naruki, On Hodge structure of isolated singularity of complex hypersurface, Proc. Jpn. Acad. 50 (1974) 334.ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Kinokuniya, Japan (1975).Google Scholar
  51. [51]
    X. Huang, H.S. Luk and S.S.T. Yau, Punctured local holomorphic de Rham cohomology, J. Math. Soc. Jpn. 55 (2003) 633.MathSciNetCrossRefGoogle Scholar
  52. [52]
    M. Kuranishi, Application of \( {\overline{\partial}}_b \) to deformation of isolated singularities, Several complex Variables, Proc. Symp. Pure Math. 30 (1977) 97.Google Scholar
  53. [53]
    T. Akahori, Intrinsic Formula for Kuranishis \( \overline{\partial}\phi \), Publ. Res. Inst. Math. Sci. 14 (1978) 615.MathSciNetCrossRefGoogle Scholar
  54. [54]
    T. Akahori and K. Miyajima, Complex analytic construction of the Kuranishi family on a normal strongly pseudo-convex manifold, II, Publ. Res. Inst. Math. Sci. 16 (1980) 811.MathSciNetCrossRefGoogle Scholar
  55. [55]
    T. Akahori, The canonical Kaehler potential on the parameter space of the versal family of CR structures, J. Math. Anal. Appl. 300 (2004) 43.MathSciNetCrossRefGoogle Scholar
  56. [56]
    T. Akahori, P.M. Garfield and J.M. Lee, Deformation theory of five-dimensional CR structures and the Rumin complex, math/0104056.
  57. [57]
    T. Akahori, Homogeneous polynomial hypersurface isolated singularities, J. Korean Math. Soc. 40 (1980) 667.MathSciNetCrossRefGoogle Scholar
  58. [58]
    J. Cao and S.-C. Chang, Pseudo-Einstein and Q-flat metrics with eigenvalue estimates on CR-hypersurfaces, math/0609312.
  59. [59]
    J. Cao and S.-C. Chang, The modified Calabi-Yau problems for CR-manifolds and applications, arXiv:0801.3431.
  60. [60]
    D.C. Chang, S.C. Chang and J. Tie, Calabi-Yau theorem and Hodge-Laplacian heat equation in a closed strictly pseudoconvex CR manifold, J. Diff. Geom. 97 (2014) 395.MathSciNetCrossRefGoogle Scholar
  61. [61]
    R.J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, II, arXiv:1301.5312 [INSPIRE].
  62. [62]
    R. J. Conlon and H.-J. Hein, Asymptotically conical Calabi-Yau manifolds, III, arXiv:1405.7140.
  63. [63]
    G. Tian and S. T. Yau, Complete Kähler manifolds with zero Ricci curvature II, Inv. Math. 106 (1991) 27.ADSCrossRefGoogle Scholar
  64. [64]
    J. Lee, Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988) 157.MathSciNetCrossRefGoogle Scholar
  65. [65]
    E. Cremmer, B. Julia and J. Scherk, Supergravity theory in eleven-dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    J. Louis and A. Micu, Type 2 theories compactified on Calabi-Yau threefolds in the presence of background fluxes, Nucl. Phys. B 635 (2002) 395 [hep-th/0202168] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    B. de Wit, H. Samtleben and M. Trigiante, Magnetic charges in local field theory, JHEP 09 (2005) 016 [hep-th/0507289] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  68. [68]
    B. de Wit and M. van Zalk, Electric and magnetic charges in N = 2 conformal supergravity theories, JHEP 10 (2011) 050 [arXiv:1107.3305] [INSPIRE].
  69. [69]
    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
  70. [70]
    K. Hristov, H. Looyestijn and S. Vandoren, Maximally supersymmetric solutions of D = 4 N = 2 gauged supergravity, JHEP 11 (2009) 115 [arXiv:0909.1743] [INSPIRE].
  71. [71]
    J. Louis, P. Smyth and H. Triendl, Supersymmetric vacua in N = 2 supergravity, JHEP 08 (2012) 039 [arXiv:1204.3893] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  72. [72]
    S. de Alwis, J. Louis, L. McAllister, H. Triendl and A. Westphal, Moduli spaces in AdS 4 supergravity, JHEP 05 (2014) 102 [arXiv:1312.5659] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    H. Erbin and N. Halmagyi, Abelian hypermultiplet gaugings and BPS vacua in \( \mathcal{N}=2 \) supergravity, JHEP 05 (2015) 122 [arXiv:1409.6310] [INSPIRE].
  74. [74]
    M. Shmakova, Calabi-Yau black holes, Phys. Rev. D 56 (1997) 540 [hep-th/9612076] [INSPIRE].ADSMathSciNetGoogle Scholar
  75. [75]
    O. Aharony, M. Berkooz, J. Louis and A. Micu, Non-Abelian structures in compactifications of M-theory on seven-manifolds with SU(3) structure, JHEP 09 (2008) 108 [arXiv:0806.1051] [INSPIRE].
  76. [76]
    H. Looyestijn, E. Plauschinn and S. Vandoren, New potentials from Scherk-Schwarz reductions, JHEP 12 (2010) 016 [arXiv:1008.4286] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  77. [77]
    W. Ebeling, Monodromy, math/0507171.
  78. [78]
    G.G.M. Hamm and A. Helmut., Invarianten quasihomogener vollständiger durchschnitte., Inv. Math. 49 (1978) 67.Google Scholar
  79. [79]
    R. Randell, The Milnor number of some isolated complete intersection singularities with C -action, Proc. Amer. Math. Soc. 72 (1978) 375.MathSciNetzbMATHGoogle Scholar
  80. [80]
    C. Arezzo, A. Ghigi and G. P. Pirola, Symmetries, quotients and Kähler-Einstein metrics, J. Reine Agew. Math. 2006 (2006) 177 [math/0402316] .
  81. [81]
    R. Dervan, On K-stability of finite covers, Bull. London Math. Soc. 48 (2016) 717 [arXiv:1505.07754].MathSciNetCrossRefGoogle Scholar
  82. [82]
    H. Süß, Fano threefolds with 2-torus action — A picture book, arXiv:1308.2379.
  83. [83]
    G. Franchetti, Harmonic forms on ALF gravitational instantons, JHEP 12 (2014) 075 [arXiv:1410.2864] [INSPIRE].ADSCrossRefGoogle Scholar

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© The Author(s) 2018

Authors and Affiliations

  1. 1.Institut de Physique Théorique, Université Paris Saclay, CEA, CNRSGif sur YvetteFrance
  2. 2.Instituut voor Theoretische FysicaKatholieke Universiteit LeuvenLeuvenBelgium
  3. 3.Dipartimento di FisicaUniversità di Milano-BicoccaMilanoItaly
  4. 4.INFN — Sezione di Milano-BicoccaMilanoItaly

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