Advertisement

Geometry of ℝ+ × E3(3) exceptional field theory and F-theory

  • Lilian ChabrolEmail author
Open Access
Regular Article - Theoretical Physics
  • 9 Downloads

Abstract

We consider a non trivial solution to the section condition in the context of ℝ+ ×E3(3) exceptional field theory and show that allowing fields to depend on the additional stringy coordinates of the extended internal space permits to describe the monodromies of (p, q) 7-branes in the context of F-theory. General expressions of non trivial fluxes with associated linear and quadratic constraints are obtained via a comparison to the embedding tensor of eight dimensional gauged maximal supergravity with gauged trombone symmetry. We write an explicit generalised Christoffel symbol for E3(3) EFT and show that the equations of motion of F-theory, namely the vanishing of a 4 dimensional Ricci tensor with two of its dimensions fibered, can be obtained from a generalised Ricci tensor and an appropriate type IIB ansatz for the metric.

Keywords

D-branes F-Theory p-branes String Duality 

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]
    C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
  3. [3]
    C. Hull and B. Zwiebach, The gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J. Math. 54 (2003) 281 [math/0209099].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Gualtieri, Generalized complex geometry, math/0401221.
  6. [6]
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    C.M. Hull, Doubled geometry and T-folds, JHEP 07 (2007) 080 [hep-th/0605149] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006) 009 [hep-th/0512005] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    C.M. Hull and R.A. Reid-Edwards, Gauge symmetry, T-duality and doubled geometry, JHEP 08 (2008) 043 [arXiv:0711.4818] [INSPIRE].
  10. [10]
    C.M. Hull and R.A. Reid-Edwards, Flux compactifications of M-theory on twisted Tori, JHEP 10 (2006) 086 [hep-th/0603094] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    D.S. Berman, H. Godazgar, M.J. Perry and P. West, Duality invariant actions and generalised geometry, JHEP 02 (2012) 108 [arXiv:1111.0459] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D.S. Berman, H. Godazgar, M. Godazgar and M.J. Perry, The Local symmetries of M-theory and their formulation in generalised geometry, JHEP 01 (2012) 012 [arXiv:1110.3930] [INSPIRE].
  13. [13]
    D.S. Berman and M.J. Perry, Generalized geometry and M-theory, JHEP 06 (2011) 074 [arXiv:1008.1763] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    D.C. Thompson, Duality invariance: from M-theory to double field theory, JHEP 08 (2011) 125 [arXiv:1106.4036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    O. Hohm and H. Samtleben, Exceptional field theory. I: E 6(6) covariant Form of M-theory and Type IIB, Phys. Rev. D 89 (2014) 066016 [arXiv:1312.0614] [INSPIRE].
  18. [18]
    O. Hohm and H. Samtleben, Exceptional field theory. II: E 7(7), Phys. Rev. D 89 (2014) 066017 [arXiv:1312.4542] [INSPIRE].
  19. [19]
    O. Hohm and H. Samtleben, Exceptional field theory. III. E 8(8), Phys. Rev. D 90 (2014) 066002 [arXiv:1406.3348] [INSPIRE].
  20. [20]
    C.M. Hull, Generalised geometry for M-theory, JHEP 07 (2007) 079 [hep-th/0701203] [INSPIRE].
  21. [21]
    P. Pires Pacheco and D. Waldram, M-theory, exceptional generalised geometry and superpotentials, JHEP 09 (2008) 123 [arXiv:0804.1362] [INSPIRE].
  22. [22]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry II: E d(d) ×+ and M-theory, JHEP 03 (2014) 019 [arXiv:1212.1586] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    O. Hohm, S.K. Kwak and B. Zwiebach, Unification of Type II strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    O. Hohm, S.K. Kwak and B. Zwiebach, Double field theory of Type II strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as generalised geometry I: Type II theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    D.S. Berman, M. Cederwall, A. Kleinschmidt and D.C. Thompson, The gauge structure of generalised diffeomorphisms, JHEP 01 (2013) 064 [arXiv:1208.5884] [INSPIRE].
  27. [27]
    O. Hohm and Y.-N. Wang, Tensor hierarchy and generalized Cartan calculus in SL(3) × SL(2) exceptional field theory, JHEP 04 (2015) 050 [arXiv:1501.01600] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    A. Abzalov, I. Bakhmatov and E.T. Musaev, Exceptional field theory: SO(5, 5), JHEP 06 (2015) 088 [arXiv:1504.01523] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    E.T. Musaev, Exceptional field theory: SL(5), JHEP 02 (2016) 012 [arXiv:1512.02163] [INSPIRE].
  30. [30]
    D.S. Berman, C.D.A. Blair, E. Malek and F.J. Rudolph, An action for F-theory: SL(2)ℝ+ exceptional field theory, Class. Quant. Grav. 33 (2016) 195009 [arXiv:1512.06115] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    O. Hohm and S.K. Kwak, Massive Type II in double field theory, JHEP 11 (2011) 086 [arXiv:1108.4937] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    F. Ciceri, A. Guarino and G. Inverso, The exceptional story of massive IIA supergravity, JHEP 08 (2016) 154 [arXiv:1604.08602] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. Cassani et al., Exceptional generalised geometry for massive IIA and consistent reductions, JHEP 08 (2016) 074 [arXiv:1605.00563] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].
  35. [35]
    D. Tong, N S5-branes, T duality and world sheet instantons, JHEP 07 (2002) 013 [hep-th/0204186] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    J.A. Harvey and S. Jensen, Worldsheet instanton corrections to the Kaluza-Klein monopole, JHEP 10 (2005) 028 [hep-th/0507204] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    S. Jensen, The KK-Monopole/N S5-brane in doubled geometry, JHEP 07 (2011) 088 [arXiv:1106.1174] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    I. Bakhmatov, A. Kleinschmidt and E.T. Musaev, Non-geometric branes are DFT monopoles, JHEP 10 (2016) 076 [arXiv:1607.05450] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J.J. Fernández-Melgarejo, T. Kimura and Y. Sakatani, Weaving the exotic web, JHEP 09 (2018) 072 [arXiv:1805.12117] [INSPIRE].
  40. [40]
    I. Bakhmatov et al., Exotic branes in exceptional field theory: the SL(5) duality group, JHEP 08 (2018) 021 [arXiv:1710.09740] [INSPIRE].
  41. [41]
    D.S. Berman, E.T. Musaev and R. Otsuki, Exotic branes in exceptional field theory: E 7(7) and beyond, JHEP 12 (2018) 053 [arXiv:1806.00430] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    E. Bergshoeff, J. Hartong and D. Sorokin, Q7-branes and their coupling to IIB supergravity, JHEP 12 (2007) 079 [arXiv:0708.2287] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    R. Blumenhagen, Basics of F-theory from the Type IIB perspective, Fortsch. Phys. 58 (2010) 820 [arXiv:1002.2836] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  45. [45]
    T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    T. Weigand, F-theory, PoS(TASI2017) 016 [arXiv:1806.01854] [INSPIRE].
  47. [47]
    E. Cremmer, H. Lü, C.N. Pope and K.S. Stelle, Spectrum generating symmetries for BPS solitons, Nucl. Phys. B 520 (1998) 132 [hep-th/9707207] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A. Le Diffon and H. Samtleben, Supergravities without an action: gauging the trombone, Nucl. Phys. B 811 (2009) 1 [arXiv:0809.5180] [INSPIRE].
  49. [49]
    G. Aldazabal, M. Graña, D. Marqués and J.A. Rosabal, Extended geometry and gauged maximal supergravity, JHEP 06 (2013) 046 [arXiv:1302.5419] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Coimbra, C. Strickland-Constable and D. Waldram, E d(d) ×+ generalised geometry, connections and M-theory, JHEP 02 (2014) 054 [arXiv:1112.3989] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  51. [51]
    H. Samtleben, Lectures on gauged supergravity and flux compactifications, Class. Quant. Grav. 25 (2008) 214002 [arXiv:0808.4076] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    B. de Wit, H. Samtleben and M. Trigiante, On Lagrangians and gaugings of maximal supergravities, Nucl. Phys. B 655 (2003) 93 [hep-th/0212239] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    M. de Roo, G. Dibitetto and Y. Yin, Critical points of maximal D = 8 gauged supergravities, JHEP 01 (2012) 029 [arXiv:1110.2886] [INSPIRE].
  54. [54]
    E.A. Bergshoeff, J. Hartong, T. Ortín and D. Roest, Seven-branes and Supersymmetry, JHEP 02 (2007) 003 [hep-th/0612072] [INSPIRE].
  55. [55]
    K. Peeters, A field-theory motivated approach to symbolic computer algebra, Comput. Phys. Commun. 176 (2007) 550 [cs/0608005].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    K. Peeters, Symbolic field theory with Cadabra, https://cadabra.science/car.pdf.
  57. [57]
    D. Andriot and A. Betz, β-supergravity: a ten-dimensional theory with non-geometric fluxes and its geometric framework, JHEP 12 (2013) 083 [arXiv:1306.4381] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institut de Physique ThéoriqueUniversité Paris SaclayGif-sur-Yvette CedexFrance

Personalised recommendations