Advertisement

Towards Kaluza-Klein Dark Matter on nilmanifolds

  • David AndriotEmail author
  • Giacomo Cacciapaglia
  • Aldo Deandrea
  • Nicolas Deutschmann
  • Dimitrios Tsimpis
Open Access
Regular Article - Theoretical Physics

Abstract

We present a first study of the field spectrum on a class of negatively-curved compact spaces: nilmanifolds or twisted tori. This is a case where analytical results can be obtained, allowing to check numerical methods. We focus on the Kaluza-Klein expansion of a scalar field. The results are then applied to a toy model where a natural Dark Matter candidate arises as a stable massive state of the bulk scalar.

Keywords

Beyond Standard Model Compactification and String Models 

Notes

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.

References

  1. [1]
    I. Antoniadis, A Possible new dimension at a few TeV, Phys. Lett. B 246 (1990) 377 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, The Hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263 [hep-ph/9803315] [INSPIRE].
  3. [3]
    I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, New dimensions at a millimeter to a Fermi and superstrings at a TeV, Phys. Lett. B 436 (1998) 257 [hep-ph/9804398] [INSPIRE].
  4. [4]
    N. Arkani-Hamed and M. Schmaltz, Hierarchies without symmetries from extra dimensions, Phys. Rev. D 61 (2000) 033005 [hep-ph/9903417] [INSPIRE].
  5. [5]
    N. Kaloper, J. March-Russell, G.D. Starkman and M. Trodden, Compact hyperbolic extra dimensions: Branes, Kaluza-Klein modes and cosmology, Phys. Rev. Lett. 85 (2000) 928 [hep-ph/0002001] [INSPIRE].
  6. [6]
    A. Kehagias and J.G. Russo, Hyperbolic spaces in string and M-theory, JHEP 07 (2000) 027 [hep-th/0003281] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D. Orlando, String Theory: Exact solutions, marginal deformations and hyperbolic spaces, Fortsch. Phys. 55 (2007) 161 [hep-th/0610284] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. Silverstein, Simple de Sitter Solutions, Phys. Rev. D 77 (2008) 106006 [arXiv:0712.1196] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    M.R. Douglas and R. Kallosh, Compactification on negatively curved manifolds, JHEP 06 (2010) 004 [arXiv:1001.4008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
  11. [11]
    N. Maru, T. Nomura, J. Sato and M. Yamanaka, The Universal Extra Dimensional Model with S 2 /Z 2 extra-space, Nucl. Phys. B 830 (2010) 414 [arXiv:0904.1909] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    H. Dohi and K.-y. Oda, Universal Extra Dimensions on Real Projective Plane, Phys. Lett. B 692 (2010) 114 [arXiv:1004.3722] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    G. Cacciapaglia, A. Deandrea and N. Deutschmann, Dark matter and localised fermions from spherical orbifolds?, JHEP 04 (2016) 083 [arXiv:1601.00081] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D. Orlando and S.C. Park, Compact hyperbolic extra dimensions: a M-theory solution and its implications for the LHC, JHEP 08 (2010) 006 [arXiv:1006.1901] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-ph/9905221] [INSPIRE].
  16. [16]
    G.D. Starkman, D. Stojkovic and M. Trodden, Homogeneity, flatness and ‘large’ extra dimensions, Phys. Rev. Lett. 87 (2001) 231303 [hep-th/0106143] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    C.-M. Chen, P.-M. Ho, I.P. Neupane, N. Ohta and J.E. Wang, Hyperbolic space cosmologies, JHEP 10 (2003) 058 [hep-th/0306291] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    I.P. Neupane, Accelerating cosmologies from exponential potentials, Class. Quant. Grav. 21 (2004) 4383 [hep-th/0311071] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Mostow, Quasi-conformal mapping in n-space and the rigidity of the hyperbolic space forms, Publ. Math. 34 (1968) 53.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S. Nasri, P.J. Silva, G.D. Starkman and M. Trodden, Radion stabilization in compact hyperbolic extra dimensions, Phys. Rev. D 66 (2002) 045029 [hep-th/0201063] [INSPIRE].ADSGoogle Scholar
  21. [21]
    B. Greene, D. Kabat, J. Levin and D. Thurston, A bulk inflaton from large volume extra dimensions, Phys. Lett. B 694 (2011) 485 [arXiv:1001.1423] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    Y. Kim and S.C. Park, Hyperbolic Inflation, Phys. Rev. D 83 (2011) 066009 [arXiv:1010.6021] [INSPIRE].ADSGoogle Scholar
  23. [23]
    N.J. Cornish and N.G. Turok, Ringing the eigenmodes from compact manifolds, Class. Quant. Grav. 15 (1998) 2699 [gr-qc/9802066] [INSPIRE].
  24. [24]
    N.J. Cornish and D.N. Spergel, On the eigenmodes of compact hyperbolic 3-manifolds, arXiv:math/9906017.
  25. [25]
    K.T. Inoue, Computation of eigenmodes on a compact hyperbolic space, Class. Quant. Grav. 16 (1999) 3071 [astro-ph/9810034] [INSPIRE].
  26. [26]
    M. Graña, R. Minasian, M. Petrini and A. Tomasiello, A Scan for new N = 1 vacua on twisted tori, JHEP 05 (2007) 031 [hep-th/0609124] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Andriot, E. Goi, R. Minasian and M. Petrini, Supersymmetry breaking branes on solvmanifolds and de Sitter vacua in string theory, JHEP 05 (2011) 028 [arXiv:1003.3774] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M.-P. Gong, Classification of Nilpotent Lie Algebras of Dimension 7, Ph.D. Thesis, University of Waterloo, Ontario Canada (1998).Google Scholar
  29. [29]
    C. Bock, On Low-Dimensional Solvmanifolds, arXiv:0903.2926.
  30. [30]
    J. Brezin, Harmonic Analysis on Nilmanifolds, Trans. Am. Math. Soc. 150 (1970) 611.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Thangavelu, Harmonic Analysis on Heisenberg Nilmanifolds, Rev. Un. Mat. Argentina 50 (2009) 2.MathSciNetzbMATHGoogle Scholar
  32. [32]
    C.S. Gordon and E.N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986) 253.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    L. Schubert, Spectral properties of the Laplacian on p-forms on the Heisenberg group, Ph.D. Thesis, University of Adelaide, Adelaide Australia (1997).Google Scholar
  34. [34]
    S. Kachru, M.B. Schulz, P.K. Tripathy and S.P. Trivedi, New supersymmetric string compactifications, JHEP 03 (2003) 061 [hep-th/0211182] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    D. Andriot, New supersymmetric vacua on solvmanifolds, JHEP 02 (2016) 112 [arXiv:1507.00014] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    E. Silverstein and A. Westphal, Monodromy in the CMB: Gravity Waves and String Inflation, Phys. Rev. D 78 (2008) 106003 [arXiv:0803.3085] [INSPIRE].ADSGoogle Scholar
  37. [37]
    G. Gur-Ari, Brane Inflation and Moduli Stabilization on Twisted Tori, JHEP 01 (2014) 179 [arXiv:1310.6787] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    D. Andriot, A no-go theorem for monodromy inflation, JCAP 03 (2016) 025 [arXiv:1510.02005] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    D. Andriot, M. Larfors, D. Lüst and P. Patalong, (Non-)commutative closed string on T-dual toroidal backgrounds, JHEP 06 (2013) 021 [arXiv:1211.6437] [INSPIRE].
  40. [40]
    M. Graña, J. Louis and D. Waldram, Hitchin functionals in N = 2 supergravity, JHEP 01 (2006) 008 [hep-th/0505264] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    C. Caviezel, P. Koerber, S. Körs, D. Lüst, D. Tsimpis and M. Zagermann, The Effective theory of type IIA AdS 4 compactifications on nilmanifolds and cosets, Class. Quant. Grav. 26 (2009) 025014 [arXiv:0806.3458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    G. Servant and T.M.P. Tait, Is the lightest Kaluza-Klein particle a viable dark matter candidate?, Nucl. Phys. B 650 (2003) 391 [hep-ph/0206071] [INSPIRE].
  43. [43]
    G. Cacciapaglia, A. Deandrea and J. Llodra-Perez, A Dark Matter candidate from Lorentz Invariance in 6D, JHEP 03 (2010) 083 [arXiv:0907.4993] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  44. [44]
    A. Rezaei-Aghdam, M. Sephid and S. Fallahpour, Automorphism group and ad-invariant metric on all six dimensional solvable real Lie algebras, arXiv:1009.0816.
  45. [45]
    B. Gough, GNU Scientific Library Reference Manual, third edition, Network Theory Ltd (2009).Google Scholar
  46. [46]
    A. Arbey, G. Cacciapaglia, A. Deandrea and B. Kubik, Dark Matter in a twisted bottle, JHEP 01 (2013) 147 [arXiv:1210.0384] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    K. Agashe, A. Falkowski, I. Low and G. Servant, KK Parity in Warped Extra Dimension, JHEP 04 (2008) 027 [arXiv:0712.2455] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A. Ahmed, B. Grzadkowski, J.F. Gunion and Y. Jiang, Higgs dark matter from a warped extra dimension — the truncated-inert-doublet model, JHEP 10 (2015) 033 [arXiv:1504.03706] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    A.I. Malcev, On a class of homogeneous spaces, Trans. Am. Math. Soc. 39 (1951) 1.MathSciNetzbMATHGoogle Scholar
  50. [50]
    A.D. Poularikas, Transforms and Applications Handbook, third edition, CRC Press (2010).Google Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • David Andriot
    • 1
    • 2
    Email author
  • Giacomo Cacciapaglia
    • 3
  • Aldo Deandrea
    • 3
    • 4
  • Nicolas Deutschmann
    • 3
    • 5
  • Dimitrios Tsimpis
    • 3
  1. 1.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdam-GolmGermany
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  3. 3.Univ Lyon, Université Lyon 1, CNRS/IN2P3, IPNLVilleurbanneFrance
  4. 4.Institut Universitaire de FranceParisFrance
  5. 5.Centre for Cosmology, Particle Physics and Phenomenology (CP3)Université catholique de LouvainLouvain-la-NeuveBelgium

Personalised recommendations