One-modulus Calabi-Yau fourfold reductions with higher-derivative terms

  • Thomas W. Grimm
  • Kilian Mayer
  • Matthias Weissenbacher
Open Access
Regular Article - Theoretical Physics


In this note we consider M-theory compactified on a warped Calabi-Yau four-fold including the eight-derivative terms in the eleven-dimensional action known in the literature. We dimensionally reduce this theory on geometries with one Kähler modulus and determine the resulting three-dimensional Kähler potential and complex coordinate. The logarithmic form of the corrections suggests that they might admit a physical interpretation in terms of one-loop corrections to the effective action. Including only the known terms the no-scale condition in three dimensions is broken, but we discuss caveats to this conclusion. In particular, we consider additional new eight-derivative terms in eleven dimensions and show that they are strongly constrained by compatibility with the Calabi-Yau threefold reduction. We examine their impact on the Calabi-Yau fourfold reduction and the restoration of the no-scale property.


F-Theory Flux compactifications M-Theory String Duality 


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]
    A.C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity compactified on Calabi-Yau threefolds, Phys. Lett. B 357 (1995) 76 [hep-th/9506144] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    M. Haack and J. Louis, M theory compactified on Calabi-Yau fourfolds with background flux, Phys. Lett. B 507 (2001) 296 [hep-th/0103068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162 [hep-th/9307158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    I. Antoniadis, S. Ferrara, R. Minasian and K.S. Narain, R 4 couplings in M and type-II theories on Calabi-Yau spaces, Nucl. Phys. B 507 (1997) 571 [hep-th/9707013] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    T.W. Grimm, K. Mayer and M. Weissenbacher, Higher derivatives in Type II and M-theory on Calabi-Yau threefolds, JHEP 02 (2018) 127 [arXiv:1702.08404] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    P. Kraus and F. Larsen, Microscopic black hole entropy in theories with higher derivatives, JHEP 09 (2005) 034 [hep-th/0506176] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    A. Castro, J.L. Davis, P. Kraus and F. Larsen, 5D Black Holes and Strings with Higher Derivatives, JHEP 06 (2007) 007 [hep-th/0703087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Castro, J.L. Davis, P. Kraus and F. Larsen, Precision Entropy of Spinning Black Holes, JHEP 09 (2007) 003 [arXiv:0705.1847] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Castro, J.L. Davis, P. Kraus and F. Larsen, String Theory Effects on Five-Dimensional Black Hole Physics, Int. J. Mod. Phys. A 23 (2008) 613 [arXiv:0801.1863] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. Denef, Les Houches Lectures on Constructing String Vacua, in proceedings of String theory and the real world: From particle physics to astrophysics. Summer School in Theoretical Physics, 87th Session, Les Houches, France, 2–27 July 2007, Les Houches 87 (2008) 483 [arXiv:0803.1194] [INSPIRE].
  12. [12]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    T.W. Grimm, R. Savelli and M. Weissenbacher, On αcorrections in N = 1 F-theory compactifications, Phys. Lett. B 725 (2013) 431 [arXiv:1303.3317] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T.W. Grimm, J. Keitel, R. Savelli and M. Weissenbacher, From M-theory higher curvature terms to αcorrections in F-theory, Nucl. Phys. B 903 (2016) 325 [arXiv:1312.1376] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    D. Junghans and G. Shiu, Brane curvature corrections to the \( \mathcal{N}=1 \) type-II/F-theory effective action, JHEP 03 (2015) 107 [arXiv:1407.0019] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  16. [16]
    T.W. Grimm, T.G. Pugh and M. Weissenbacher, On M-theory fourfold vacua with higher curvature terms, Phys. Lett. B 743 (2015) 284 [arXiv:1408.5136] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T.W. Grimm, T.G. Pugh and M. Weissenbacher, The effective action of warped M-theory reductions with higher derivative terms — part I, JHEP 01 (2016) 142 [arXiv:1412.5073] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    R. Minasian, T.G. Pugh and R. Savelli, F-theory at order α ′3, JHEP 10 (2015) 050 [arXiv:1506.06756] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    T.W. Grimm, T.G. Pugh and M. Weissenbacher, The effective action of warped M-theory reductions with higher-derivative terms - Part II, JHEP 12 (2015) 117 [arXiv:1507.00343] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    K. Becker and M. Becker, M theory on eight manifolds, Nucl. Phys. B 477 (1996) 155 [hep-th/9605053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    K. Becker and M. Becker, Supersymmetry breaking, M-theory and fluxes, JHEP 07 (2001) 038 [hep-th/0107044] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. de Boer, K. Hori, H. Ooguri, Y. Oz and Z. Yin, Mirror symmetry in three-dimensional theories, SL(2, ℤ) and D-brane moduli spaces, Nucl. Phys. B 493 (1997) 148 [hep-th/9612131] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    N. Dorey, V.V. Khoze, M.P. Mattis, D. Tong and S. Vandoren, Instantons, three-dimensional gauge theory and the Atiyah-Hitchin manifold, Nucl. Phys. B 502 (1997) 59 [hep-th/9703228] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    N. Dorey, D. Tong and S. Vandoren, Instanton effects in three-dimensional supersymmetric gauge theories with matter, JHEP 04 (1998) 005 [hep-th/9803065] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    D. Tong and C. Turner, Quantum dynamics of supergravity on R 3× S 1, JHEP 12 (2014) 142 [arXiv:1408.3418] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    B. de Wit, H. Nicolai and H. Samtleben, Gauged supergravities in three-dimensions: a panoramic overview, PoS(jhw2003)018 [hep-th/0403014] [INSPIRE].
  29. [29]
    M. Berg, M. Haack and H. Samtleben, Calabi-Yau fourfolds with flux and supersymmetry breaking, JHEP 04 (2003) 046 [hep-th/0212255] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    T.W. Grimm, M. Kerstan, E. Palti and T. Weigand, Massive Abelian Gauge Symmetries and Fluxes in F-theory, JHEP 12 (2011) 004 [arXiv:1107.3842] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    D.J. Gross and E. Witten, Superstring Modifications of Einstein’s Equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string-string duality: A One loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M.B. Green and P. Vanhove, D instantons, strings and M-theory, Phys. Lett. B 408 (1997) 122 [hep-th/9704145] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    J.G. Russo and A.A. Tseytlin, One loop four graviton amplitude in eleven-dimensional supergravity, Nucl. Phys. B 508 (1997) 245 [hep-th/9707134] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A.A. Tseytlin, R 4 terms in 11 dimensions and conformal anomaly of (2,0) theory, Nucl. Phys. B 584 (2000) 233 [hep-th/0005072] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    K. Peeters, J. Plefka and S. Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 08 (2005) 095 [hep-th/0507178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    G. Policastro and D. Tsimpis, R 4 , purified, Class. Quant. Grav. 23 (2006) 4753 [hep-th/0603165] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    Y. Hyakutake, Toward the Determination of R 3 F 2 Terms in M-theory, Prog. Theor. Phys. 118 (2007) 109 [hep-th/0703154] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J.T. Liu and R. Minasian, Higher-derivative couplings in string theory: dualities and the B-field, Nucl. Phys. B 874 (2013) 413 [arXiv:1304.3137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    L. Martucci, Warping the Kähler potential of F-theory/ IIB flux compactifications, JHEP 03 (2015) 067 [arXiv:1411.2623] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477] [hep-th/9906070] [INSPIRE].
  42. [42]
    S. Sethi, Supersymmetry Breaking by Fluxes, arXiv:1709.03554 [INSPIRE].
  43. [43]
    T.W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories, JHEP 03 (2012) 027 [arXiv:1111.1232] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    T.W. Grimm and A. Kapfer, Anomaly Cancelation in Field Theory and F-theory on a Circle, JHEP 05 (2016) 102 [arXiv:1502.05398] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    P. Corvilain, T.W. Grimm and D. Regalado, Chiral anomalies on a circle and their cancellation in F-theory, arXiv:1710.07626 [INSPIRE].
  46. [46]
    F. Bonetti and T.W. Grimm, Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. Bonetti, T.W. Grimm and S. Hohenegger, One-loop Chern-Simons terms in five dimensions, JHEP 07 (2013) 043 [arXiv:1302.2918] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    J.M. Martin-Garcia, R. Portugal and L.R.U. Manssur, The Invar Tensor Package, Comput. Phys. Commun. 177 (2007) 640 [arXiv:0704.1756] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  49. [49]
    T. Nutma, xTras: a field-theory inspired xAct package for mathematica, Comput. Phys. Commun. 185 (2014) 1719 [arXiv:1308.3493] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  50. [50]
    J.M. Martin-Garcia, xPerm: fast index canonicalization for tensor computer algebra, Comput. Phys. Commun. 179 (2008) 597 [arXiv:0803.0862].ADSCrossRefzbMATHGoogle Scholar
  51. [51]
    A. Strominger, Loop corrections to the universal hypermultiplet, Phys. Lett. B 421 (1998) 139 [hep-th/9706195] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Thomas W. Grimm
    • 1
  • Kilian Mayer
    • 1
  • Matthias Weissenbacher
    • 2
  1. 1.Institute for Theoretical PhysicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoTokyoJapan

Personalised recommendations