The spectra of type IIB flux compactifications at large complex structure

Open Access
Regular Article - Theoretical Physics

Abstract

We compute the spectra of the Hessian matrix, \( \mathrm{\mathscr{H}} \), and the matrix \( \mathrm{\mathcal{M}} \) that governs the critical point equation of the low-energy effective supergravity, as a function of the complex structure and axio-dilaton moduli space in type IIB flux compactifications at large complex structure. We find both spectra analytically in an h1,2 + 3 real-dimensional subspace of the moduli space, and show that they exhibit a universal structure with highly degenerate eigenvalues, independently of the choice of flux, the details of the compactification geometry, and the number of complex structure moduli. In this subspace, the spectrum of the Hessian matrix contains no tachyons, but there are also no critical points. We show numerically that the spectra of \( \mathrm{\mathscr{H}} \) and \( \mathrm{\mathcal{M}} \) remain highly peaked over a large fraction of the sampled moduli space of explicit Calabi-Yau compactifications with 2 to 5 complex structure moduli. In these models, the scale of the supersymmetric contribution to the scalar masses is strongly linearly correlated with the value of the superpotential over almost the entire moduli space, with particularly strong correlations arising for gs< 1. We contrast these results with the expectations from the much-used continuous flux approximation, and comment on the applicability of Random Matrix Theory to the statistical modelling of the string theory landscape.

Keywords

Flux compactifications Supergravity Models 

References

  1. [1]
    M. Graña and J. Polchinski, Supersymmetric three form flux perturbations on AdS 5, Phys. Rev. D 63 (2001) 026001 [hep-th/0009211] [INSPIRE].ADSGoogle Scholar
  2. [2]
    S.S. Gubser, Supersymmetry and F-theory realization of the deformed conifold with three form flux, hep-th/0010010 [INSPIRE].
  3. [3]
    S.B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D 66 (2002) 106006 [hep-th/0105097] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    O. DeWolfe and S.B. Giddings, Scales and hierarchies in warped compactifications and brane worlds, Phys. Rev. D 67 (2003) 066008 [hep-th/0208123] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, de Sitter vacua in string theory, Phys. Rev. D 68 (2003) 046005 [hep-th/0301240] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    S. Kachru, R. Kallosh, A.D. Linde, J.M. Maldacena, L.P. McAllister and S.P. Trivedi, Towards inflation in string theory, JCAP 10 (2003) 013 [hep-th/0308055] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  7. [7]
    M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].MATHMathSciNetCrossRefADSGoogle Scholar
  8. [8]
    F. Denef, M.R. Douglas and S. Kachru, Physics of String Flux Compactifications, Ann. Rev. Nucl. Part. Sci. 57 (2007) 119 [hep-th/0701050] [INSPIRE].CrossRefADSGoogle Scholar
  9. [9]
    F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [INSPIRE].
  10. [10]
    A. Maharana and E. Palti, Models of Particle Physics from Type IIB String Theory and F-theory: A Review, Int. J. Mod. Phys. A 28 (2013) 1330005 [arXiv:1212.0555] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  11. [11]
    A.N. Schellekens, Life at the Interface of Particle Physics and String Theory, Rev. Mod. Phys. 85 (2013) 1491 [arXiv:1306.5083] [INSPIRE].CrossRefADSGoogle Scholar
  12. [12]
    P.K. Tripathy and S.P. Trivedi, Compactification with flux on K3 and tori, JHEP 03 (2003) 028 [hep-th/0301139] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  13. [13]
    D. Martinez-Pedrera, D. Mehta, M. Rummel and A. Westphal, Finding all flux vacua in an explicit example, JHEP 06 (2013) 110 [arXiv:1212.4530] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  14. [14]
    S. Ashok and M.R. Douglas, Counting flux vacua, JHEP 01 (2004) 060 [hep-th/0307049] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  15. [15]
    F. Denef and M.R. Douglas, Distributions of flux vacua, JHEP 05 (2004) 072 [hep-th/0404116] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  16. [16]
    F. Denef and M.R. Douglas, Distributions of nonsupersymmetric flux vacua, JHEP 03 (2005) 061 [hep-th/0411183] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  17. [17]
    T. Eguchi and Y. Tachikawa, Distribution of flux vacua around singular points in Calabi-Yau moduli space, JHEP 01 (2006) 100 [hep-th/0510061] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  18. [18]
    E.P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Math. Proc. Cambridge Philos. Soc. 47 (1951) 548.CrossRefGoogle Scholar
  19. [19]
    M. Mehta, Random Matrices, Academic Press, Boston U.S.A. (1991).MATHGoogle Scholar
  20. [20]
    P. Deift, Universality for mathematical and physical systems, math-ph/0603038.
  21. [21]
    A.B.J. Kuijlaars, Universality, arXiv:1103.5922.
  22. [22]
    E.P. Wigner, On the distribution of the roots of certain symmetric matrices, Ann. Math. 67 (1958) 325.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    J.H. Schenker and H. Schulz-Baldes, Semicircle law and freeness for random matrices with symmetries or correlations, math-ph/0505003.
  24. [24]
    K. Hofmann-Credner and M. Stolz, Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices, arXiv:0707.2333.
  25. [25]
    D. Marsh, L. McAllister and T. Wrase, The Wasteland of Random Supergravities, JHEP 03 (2012) 102 [arXiv:1112.3034] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  26. [26]
    X. Chen, G. Shiu, Y. Sumitomo and S.H.H. Tye, A Global View on The Search for de-Sitter Vacua in (type IIA) String Theory, JHEP 04 (2012) 026 [arXiv:1112.3338] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  27. [27]
    T.C. Bachlechner, D. Marsh, L. McAllister and T. Wrase, Supersymmetric Vacua in Random Supergravity, JHEP 01 (2013) 136 [arXiv:1207.2763] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  28. [28]
    K. Sousa and P. Ortiz, Perturbative Stability along the Supersymmetric Directions of the Landscape, JCAP 02 (2015) 017 [arXiv:1408.6521] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  29. [29]
    A. Kobakhidze and L. Mersini-Houghton, Birth of the universe from the landscape of string theory, Eur. Phys. J. C 49 (2007) 869 [hep-th/0410213] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  30. [30]
    A. Aazami and R. Easther, Cosmology from random multifield potentials, JCAP 03 (2006) 013 [hep-th/0512050] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  31. [31]
    R. Easther and L. McAllister, Random matrices and the spectrum of N-flation, JCAP 05 (2006) 018 [hep-th/0512102] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  32. [32]
    F.G. Pedro and A. Westphal, The Scale of Inflation in the Landscape, Phys. Lett. B 739 (2014) 439 [arXiv:1303.3224] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  33. [33]
    C. Long, L. McAllister and P. McGuirk, Heavy Tails in Calabi-Yau Moduli Spaces, JHEP 10 (2014) 187 [arXiv:1407.0709] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  34. [34]
    M.C.D. Marsh, L. McAllister, E. Pajer and T. Wrase, Charting an Inflationary Landscape with Random Matrix Theory, JCAP 11 (2013) 040 [arXiv:1307.3559] [INSPIRE].CrossRefADSGoogle Scholar
  35. [35]
    T.C. Bachlechner, C. Long and L. McAllister, Planckian Axions in String Theory, arXiv:1412.1093 [INSPIRE].
  36. [36]
    T.C. Bachlechner, M. Dias, J. Frazer and L. McAllister, Chaotic inflation with kinetic alignment of axion fields, Phys. Rev. D 91 (2015) 023520 [arXiv:1404.7496] [INSPIRE].ADSGoogle Scholar
  37. [37]
    T. Battefeld and C. Modi, Local random potentials of high differentiability to model the Landscape, JCAP 03 (2015) 010 [arXiv:1409.5135] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  38. [38]
    A. Altland and M.R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [INSPIRE].CrossRefADSGoogle Scholar
  39. [39]
    T.W. Grimm and J. Louis, The effective action of N = 1 Calabi-Yau orientifolds, Nucl. Phys. B 699 (2004) 387 [hep-th/0403067] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  40. [40]
    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].
  41. [41]
    M. Cicoli, J.P. Conlon, A. Maharana and F. Quevedo, A Note on the Magnitude of the Flux Superpotential, JHEP 01 (2014) 027 [arXiv:1310.6694] [INSPIRE].CrossRefADSGoogle Scholar
  42. [42]
    V. Balasubramanian, P. Berglund, J.P. Conlon and F. Quevedo, Systematics of moduli stabilisation in Calabi-Yau flux compactifications, JHEP 03 (2005) 007 [hep-th/0502058] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  43. [43]
    A. Westphal, de Sitter string vacua from Kähler uplifting, JHEP 03 (2007) 102 [hep-th/0611332] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  44. [44]
    L. Covi, M. Gomez-Reino, C. Gross, J. Louis, G.A. Palma and C.A. Scrucca, de Sitter vacua in no-scale supergravities and Calabi-Yau string models, JHEP 06 (2008) 057 [arXiv:0804.1073] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  45. [45]
    R. Kallosh, A. Linde, B. Vercnocke and T. Wrase, Analytic Classes of Metastable de Sitter Vacua, JHEP 10 (2014) 011 [arXiv:1406.4866] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  46. [46]
    M.C.D. Marsh, B. Vercnocke and T. Wrase, Decoupling and de Sitter Vacua in Approximate No-Scale Supergravities, JHEP 05 (2015) 081 [arXiv:1411.6625] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  47. [47]
    M. Cicoli, D. Klevers, S. Krippendorf, C. Mayrhofer, F. Quevedo and R. Valandro, Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter, JHEP 05 (2014) 001 [arXiv:1312.0014] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  48. [48]
    P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2., Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  49. [49]
    F. Denef, M.R. Douglas and B. Florea, Building a better racetrack, JHEP 06 (2004) 034 [hep-th/0404257] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  50. [50]
    P. Candelas and X. de la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B 355 (1991) 455 [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    A. Hebecker, S.C. Kraus and L.T. Witkowski, D7-Brane Chaotic Inflation, Phys. Lett. B 737 (2014) 16 [arXiv:1404.3711] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  52. [52]
    A. Hebecker, P. Mangat, F. Rompineve and L.T. Witkowski, Tuning and Backreaction in F-term Axion Monodromy Inflation, Nucl. Phys. B 894 (2015) 456 [arXiv:1411.2032] [INSPIRE].CrossRefADSGoogle Scholar
  53. [53]
    H. Hayashi, R. Matsuda and T. Watari, Issues in Complex Structure Moduli Inflation, arXiv:1410.7522 [INSPIRE].
  54. [54]
    A.P. Braun, N. Johansson, M. Larfors and N.-O. Walliser, Restrictions on infinite sequences of type IIB vacua, JHEP 10 (2011) 091 [arXiv:1108.1394] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordU.K.
  2. 2.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeU.K.

Personalised recommendations