Journal of High Energy Physics

, 2018:20 | Cite as

A note on T-folds and T3 fibrations

  • Ismail Achmed-Zade
  • Mark J. D. Hamilton
  • Dieter Lüst
  • Stefano MassaiEmail author
Open Access
Regular Article - Theoretical Physics


We study stringy modifications of T3-fibered manifolds, where the fiber undergoes a monodromy in the T-duality group. We determine the fibration data defining such T-folds from a geometric model, by using a map between the duality group and the group of large diffeomorphisms of a four-torus. We describe the monodromies induced around duality defects where such fibrations degenerate and we argue that local solutions receive corrections from the winding sector, dual to the symmetry-breaking modes that correct semi-flat metrics.


D-branes String Duality Superstring Vacua 


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. Strominger, S.-T. Yau and E. Zaslow, Mirror symmetry is T duality, Nucl. Phys. B 479 (1996) 243 [hep-th/9606040] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    S. Hellerman, J. McGreevy and B. Williams, Geometric constructions of nongeometric string theories, JHEP 01 (2004) 024 [hep-th/0208174] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  4. [4]
    D. Vegh and J. McGreevy, Semi-flatland, JHEP 10 (2008) 068 [arXiv:0808.1569] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Kumar and C. Vafa, U manifolds, Phys. Lett. B 396 (1997) 85 [hep-th/9611007] [INSPIRE].CrossRefGoogle Scholar
  6. [6]
    J.T. Liu and R. Minasian, U-branes and T 3 fibrations, Nucl. Phys. B 510 (1998) 538 [hep-th/9707125] [INSPIRE].CrossRefzbMATHGoogle Scholar
  7. [7]
    L. Martucci, J.F. Morales and D. Ricci Pacifici, Branes, U-folds and hyperelliptic fibrations, JHEP 01 (2013) 145 [arXiv:1207.6120] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. de Boer and M. Shigemori, Exotic branes in string theory, Phys. Rept. 532 (2013) 65 [arXiv:1209.6056] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Lüst, S. Massai and V. Vall Camell, The monodromy of T-folds and T-fects, JHEP 09 (2016) 127 [arXiv:1508.01193] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    J. McOrist, D.R. Morrison and S. Sethi, Geometries, non-geometries and fluxes, Adv. Theor. Math. Phys. 14 (2010) 1515 [arXiv:1004.5447] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Malmendier and D.R. Morrison, K3 surfaces, modular forms and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015) 1085 [arXiv:1406.4873] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I. García-Etxebarria, D. Lüst, S. Massai and C. Mayrhofer, Ubiquity of non-geometry in heterotic compactifications, JHEP 03 (2017) 046 [arXiv:1611.10291] [INSPIRE].
  14. [14]
    A. Font et al., Heterotic T-fects, 6D SCFTs and F-theory, JHEP 08 (2016) 175 [arXiv:1603.09361] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  15. [15]
    A. Font and C. Mayrhofer, Non-geometric vacua of the Spin(32)/2 heterotic string and little string theories, JHEP 11 (2017) 064 [arXiv:1708.05428] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  17. [17]
    C. Bock, On low-dimensional solvmanifolds, arXiv:0903.2926.
  18. [18]
    R. Donagi, P. Gao and M.B. Schulz, Abelian fibrations, string junctions and flux/geometry duality, JHEP 04 (2009) 119 [arXiv:0810.5195] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  19. [19]
    M.B. Schulz, Calabi-Yau duals of torus orientifolds, JHEP 05 (2006) 023 [hep-th/0412270] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973) 143.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    A.B. Altman and S.L. Kleiman, The presentation functor and the compactified Jacobian, in The Grothendieck Festschrift , P. Cartier et al. eds., Birkhäuser Boston, Boston, U.S.A. (2007).Google Scholar
  22. [22]
    J. Kass, Notes on compactified Jacobian, lecture notes (2008).Google Scholar
  23. [23]
    R. Gompf and A. Stipsicz, 4-manifolds and Kirby Calculus, Graduate studies in mathematics. American Mathematical Society, U.S.A. (1999).Google Scholar
  24. [24]
    B. Farb and D. Margalit, A primer on mapping class groups, Princeton University Press, Princeton U.S.A. (2011).CrossRefzbMATHGoogle Scholar
  25. [25]
    B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau, Stringy cosmic strings and noncompact Calabi-Yau manifolds, Nucl. Phys. B 337 (1990) 1 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    N.A. Obers and B. Pioline, U duality and M-theory, Phys. Rept. 318 (1999) 113 [hep-th/9809039] [INSPIRE].CrossRefzbMATHGoogle Scholar
  28. [28]
    F. Hassler and D. Lüst, Non-commutative/non-associative IIA (IIB) Q- and R-branes and their intersections, JHEP 07 (2013) 048 [arXiv:1303.1413] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    H. Ooguri and C. Vafa, Summing up D instantons, Phys. Rev. Lett. 77 (1996) 3296 [hep-th/9608079] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    K. Becker and S. Sethi, Torsional heterotic geometries, Nucl. Phys. B 820 (2009) 1 [arXiv:0903.3769] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R. Gregory, J.A. Harvey and G.W. Moore, Unwinding strings and t duality of Kaluza-Klein and h monopoles, Adv. Theor. Math. Phys. 1 (1997) 283 [hep-th/9708086] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    J.A. Harvey and S. Jensen, Worldsheet instanton corrections to the Kaluza-Klein monopole, JHEP 10 (2005) 028 [hep-th/0507204] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  33. [33]
    T. Kimura and S. Sasaki, Worldsheet instanton corrections to 522 -brane geometry, JHEP 08 (2013) 126 [arXiv:1305.4439] [INSPIRE].CrossRefzbMATHGoogle Scholar
  34. [34]
    D. Lüst, E. Plauschinn and V. Vall Camell, Unwinding strings in semi-flatland, JHEP 07 (2017) 027 [arXiv:1706.00835] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    A. Giveon and D. Kutasov, Little string theory in a double scaling limit, JHEP 10 (1999) 034 [hep-th/9909110] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    D.R. Morrison, On the structure of supersymmetric T 3 fibrations, arXiv:1002.4921 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Arnold Sommerfeld Center for Theoretical PhysicsMünchenGermany
  2. 2.Institute for Geometry and TopologyUniversity of StuttgartStuttgartGermany
  3. 3.Max-Planck-Institut für PhysikMünchenGermany
  4. 4.Enrico Fermi InstituteUniversity of ChicagoChicagoU.S.A.

Personalised recommendations