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Journal of High Energy Physics

, 2016:114 | Cite as

P fluxes and exotic branes

  • Davide M. Lombardo
  • Fabio RiccioniEmail author
  • Stefano Risoli
Open Access
Regular Article - Theoretical Physics

Abstract

We consider the \( \mathcal{N} \) = 1 superpotential generated in type-II orientifold models by non-geometric fluxes. In particular, we focus on the family of P fluxes, that are related by T-duality transformations to the S-dual of the Q flux. We determine the general rule that transforms a given flux in this family under a single T-duality transformation. This rule allows to derive a complete expression for the superpotential for both the IIA and the IIB theory for the particular case of a \( {T}^6/\left[{\mathbb{Z}}_2\times {\mathbb{Z}}_2\right] \) orientifold. We then consider how these fluxes modify the generalised Bianchi identities. In particular, we derive a fully consistent set of quadratic constraints coming from the NS-NS Bianchi identities. On the other hand, the P flux Bianchi identities induce tadpoles, and we determine a set of exotic branes that can be consistently included in order to cancel them. This is achieved by determining a universal transformation rule under T-duality satisfied by all the branes in string theory.

Keywords

Flux compactifications Supersymmetry and Duality p-branes Supergravity 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]
    J. Polchinski and A. Strominger, New vacua for type-II string theory, Phys. Lett. B 388 (1996) 736 [hep-th/9510227] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Michelson, Compactifications of type IIB strings to four-dimensions with nontrivial classical potential, Nucl. Phys. B 495 (1997) 127 [hep-th/9610151] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  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].ADSMathSciNetGoogle Scholar
  4. [4]
    S. Kachru, M.B. Schulz and S. Trivedi, Moduli stabilization from fluxes in a simple IIB orientifold, JHEP 10 (2003) 007 [hep-th/0201028] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional string compactifications with D-branes, orientifolds and fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Graña, Flux compactifications in string theory: a comprehensive review, Phys. Rept. 423 (2006) 91 [hep-th/0509003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Denef, Les Houches lectures on constructing string vacua, arXiv:0803.1194 [INSPIRE].
  9. [9]
    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].
  10. [10]
    J. Shelton, W. Taylor and B. Wecht, Nongeometric flux compactifications, JHEP 10 (2005) 085 [hep-th/0508133] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Shelton, W. Taylor and B. Wecht, Generalized flux vacua, JHEP 02 (2007) 095 [hep-th/0607015] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    B. Wecht, Lectures on nongeometric flux compactifications, Class. Quant. Grav. 24 (2007) S773 [arXiv:0708.3984] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Aldazabal, P.G. Camara, A. Font and L.E. Ibáñez, More dual fluxes and moduli fixing, JHEP 05 (2006) 070 [hep-th/0602089] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    J.-P. Derendinger, C. Kounnas, P.M. Petropoulos and F. Zwirner, Superpotentials in IIA compactifications with general fluxes, Nucl. Phys. B 715 (2005) 211 [hep-th/0411276] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    G. Villadoro and F. Zwirner, N=1 effective potential from dual type-IIA D6/O6 orientifolds with general fluxes, JHEP 06 (2005) 047 [hep-th/0503169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    A. Guarino and G.J. Weatherill, Non-geometric flux vacua, S-duality and algebraic geometry, JHEP 02 (2009) 042 [arXiv:0811.2190] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    A. Font, A. Guarino and J.M. Moreno, Algebras and non-geometric flux vacua, JHEP 12 (2008) 050 [arXiv:0809.3748] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    G. Aldazabal, P.G. Camara and J.A. Rosabal, Flux algebra, Bianchi identities and Freed-Witten anomalies in F-theory compactifications, Nucl. Phys. B 814 (2009) 21 [arXiv:0811.2900] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    C. Kounnas and M. Porrati, Spontaneous supersymmetry breaking in string theory, Nucl. Phys. B 310 (1988) 355 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Ferrara, C. Kounnas, M. Porrati and F. Zwirner, Superstrings with spontaneously broken supersymmetry and their effective theories, Nucl. Phys. B 318 (1989) 75 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    C. Kounnas and B. Rostand, Coordinate dependent compactifications and discrete symmetries, Nucl. Phys. B 341 (1990) 641 [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    I. Antoniadis, A possible new dimension at a few TeV, Phys. Lett. B 246 (1990) 377 [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    I. Antoniadis and C. Kounnas, Superstring phase transition at high temperature, Phys. Lett. B 261 (1991) 369 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    C. Vafa and E. Witten, Dual string pairs with N = 1 and N = 2 supersymmetry in four-dimensions, Nucl. Phys. Proc. Suppl. 46 (1996) 225 [hep-th/9507050] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    I. Antoniadis, E. Dudas and A. Sagnotti, Supersymmetry breaking, open strings and M-theory, Nucl. Phys. B 544 (1999) 469 [hep-th/9807011] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    I. Antoniadis, G. D’Appollonio, E. Dudas and A. Sagnotti, Partial breaking of supersymmetry, open strings and M-theory, Nucl. Phys. B 553 (1999) 133 [hep-th/9812118] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    I. Antoniadis, G. D’Appollonio, E. Dudas and A. Sagnotti, Open descendants of Z 2 × Z 2 freely acting orbifolds, Nucl. Phys. B 565 (2000) 123 [hep-th/9907184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    P.G. Camara, E. Dudas, T. Maillard and G. Pradisi, String instantons, fluxes and moduli stabilization, Nucl. Phys. B 795 (2008) 453 [arXiv:0710.3080] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    C. Angelantonj and A. Sagnotti, Open strings, Phys. Rept. 371 (2002) 1 [hep-th/0204089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    G. Aldazabal, E. Andres, P.G. Camara and M. Graña, U-dual fluxes and generalized geometry, JHEP 11 (2010) 083 [arXiv:1007.5509] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    M. Graña, J. Louis and D. Waldram, SU(3) × SU(3) compactification and mirror duals of magnetic fluxes, JHEP 04 (2007) 101 [hep-th/0612237] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    E.A. Bergshoeff, V.A. Penas, F. Riccioni and S. Risoli, Non-geometric fluxes and mixed-symmetry potentials, JHEP 11 (2015) 020 [arXiv:1508.00780] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    M. Ihl, D. Robbins and T. Wrase, Toroidal orientifolds in IIA with general NS-NS fluxes, JHEP 08 (2007) 043 [arXiv:0705.3410] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    D. Geissbuhler, D. Marques, C. Núñez and V. Penas, Exploring double field theory, JHEP 06 (2013) 101 [arXiv:1304.1472] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    R. Blumenhagen, X. Gao, D. Herschmann and P. Shukla, Dimensional oxidation of non-geometric fluxes in type II orientifolds, JHEP 10 (2013) 201 [arXiv:1306.2761] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    D. Andriot and A. Betz, NS-branes, source corrected Bianchi identities and more on backgrounds with non-geometric fluxes, JHEP 07 (2014) 059 [arXiv:1402.5972] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    P. Shukla, Revisiting the two formulations of Bianchi identities and their implications on moduli stabilization, JHEP 08 (2016) 146 [arXiv:1603.08545] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    E.A. Bergshoeff and F. Riccioni, D-brane Wess-Zumino terms and U-duality, JHEP 11 (2010) 139 [arXiv:1009.4657] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    E.A. Bergshoeff and F. Riccioni, The D-brane U-scan, Proc. Symp. Pure Math. 85 (2012) 313 [arXiv:1109.1725] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  41. [41]
    E.A. Bergshoeff and F. Riccioni, String solitons and T-duality, JHEP 05 (2011) 131 [arXiv:1102.0934] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A. Kleinschmidt, Counting supersymmetric branes, JHEP 10 (2011) 144 [arXiv:1109.2025] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    E.A. Bergshoeff, F. Riccioni and L. Romano, Branes, weights and central charges, JHEP 06 (2013) 019 [arXiv:1303.0221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    E.A. Bergshoeff, A. Marrani and F. Riccioni, Brane orbits, Nucl. Phys. B 861 (2012) 104 [arXiv:1201.5819] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Elitzur, A. Giveon, D. Kutasov and E. Rabinovici, Algebraic aspects of matrix theory on T d, Nucl. Phys. B 509 (1998) 122 [hep-th/9707217] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    N.A. Obers and B. Pioline, U duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    E. Lozano-Tellechea and T. Ortín, 7-branes and higher Kaluza-Klein branes, Nucl. Phys. B 607 (2001) 213 [hep-th/0012051] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    J. de Boer and M. Shigemori, Exotic branes and non-geometric backgrounds, Phys. Rev. Lett. 104 (2010) 251603 [arXiv:1004.2521] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Chatzistavrakidis, F.F. Gautason, G. Moutsopoulos and M. Zagermann, Effective actions of nongeometric five-branes, Phys. Rev. D 89 (2014) 066004 [arXiv:1309.2653] [INSPIRE].ADSGoogle Scholar
  51. [51]
    Y. Sakatani, Exotic branes and non-geometric fluxes, JHEP 03 (2015) 135 [arXiv:1412.8769] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  52. [52]
    G. Dibitetto, A. Guarino and D. Roest, Charting the landscape of N = 4 flux compactifications, JHEP 03 (2011) 137 [arXiv:1102.0239] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    J. Blabäck, U. Danielsson and G. Dibitetto, Fully stable dS vacua from generalised fluxes, JHEP 08 (2013) 054 [arXiv:1301.7073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    R. Blumenhagen et al., A flux-scaling scenario for high-scale moduli stabilization in string theory, Nucl. Phys. B 897 (2015) 500 [arXiv:1503.07634] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  55. [55]
    U. Danielsson and G. Dibitetto, Type IIB on S 3 × S 3 through Q & P fluxes, JHEP 01 (2016) 057 [arXiv:1507.04476] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    H. Nicolai and H. Samtleben, Maximal gauged supergravity in three-dimensions, Phys. Rev. Lett. 86 (2001) 1686 [hep-th/0010076] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    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
  58. [58]
    B. de Wit, H. Samtleben and M. Trigiante, The maximal D = 5 supergravities, Nucl. Phys. B 716 (2005) 215 [hep-th/0412173] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    B. de Wit, H. Samtleben and M. Trigiante, Magnetic charges in local field theory, JHEP 09 (2005) 016 [hep-th/0507289] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  60. [60]
    H. Samtleben and M. Weidner, The maximal D = 7 supergravities, Nucl. Phys. B 725 (2005) 383 [hep-th/0506237] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    P. Meessen and T. Ortín, An \( \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \) multiplet of nine-dimensional type-II supergravity theories, Nucl. Phys. B 541 (1999) 195 [hep-th/9806120] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    G. Dall’Agata, K. Lechner and M. Tonin, D = 10, N = IIB supergravity: Lorentz invariant actions and duality, JHEP 07 (1998) 017 [hep-th/9806140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    E. Eyras and Y. Lozano, Exotic branes and nonperturbative seven-branes, Nucl. Phys. B 573 (2000) 735 [hep-th/9908094] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    E.A. Bergshoeff and F. Riccioni, Heterotic wrapping rules, JHEP 01 (2013) 005 [arXiv:1210.1422] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    E.A. Bergshoeff, C. Condeescu, G. Pradisi and F. Riccioni, Heterotic-Type II duality and wrapping rules, JHEP 12 (2013) 057 [arXiv:1311.3578] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    E.A. Bergshoeff, F. Riccioni and L. Romano, Towards a classification of branes in theories with eight supercharges, JHEP 05 (2014) 070 [arXiv:1402.2557] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  67. [67]
    G. Pradisi and F. Riccioni, Non-geometric orbifolds and wrapping rules, JHEP 09 (2014) 170 [arXiv:1407.5576] [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  69. [69]
    P.C. West, E 11 and M-theory, Class. Quant. Grav. 18 (2001) 4443 [hep-th/0104081] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    F. Riccioni and P.C. West, The E 11 origin of all maximal supergravities, JHEP 07 (2007) 063 [arXiv:0705.0752] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  71. [71]
    E.A. Bergshoeff, T. Ortín and F. Riccioni, Defect branes, Nucl. Phys. B 856 (2012) 210 [arXiv:1109.4484] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    E.A. Bergshoeff and F. Riccioni, Dual doubled geometry, Phys. Lett. B 702 (2011) 281 [arXiv:1106.0212] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    E.A. Bergshoeff and F. Riccioni, Branes and wrapping rules, Phys. Lett. B 704 (2011) 367 [arXiv:1108.5067] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    G. Villadoro and F. Zwirner, On general flux backgrounds with localized sources, JHEP 11 (2007) 082 [arXiv:0710.2551] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    A. Kleinschmidt, I. Schnakenburg and P.C. West, Very extended Kac-Moody algebras and their interpretation at low levels, Class. Quant. Grav. 21 (2004) 2493 [hep-th/0309198] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Davide M. Lombardo
    • 1
  • Fabio Riccioni
    • 2
    Email author
  • Stefano Risoli
    • 1
    • 2
  1. 1.Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.INFN — Sezione di Roma, Dipartimento di FisicaUniversità di Roma “La Sapienza”RomaItaly

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