Skip to main content

Heterotic/type II duality and non-geometric compactifications

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

A preprint version of the article is available at arXiv.


We present a new class of dualities relating non-geometric Calabi-Yau com- pactifications of type II string theory to T-fold compactifications of the heterotic string, both preserving four-dimensional \( \mathcal{N} \) = 2 supersymmetry. The non-geometric Calabi-Yau space is a K 3 fibration over T2 with non-geometric monodromies in the duality group O4,20); this is dual to a heterotic reduction on a T4 fibration over T2 with the O4,20) monodromies now viewed as heterotic T-dualities. At a point in moduli space which is a minimum of the scalar potential, the type II compactification becomes an asymmetric Gepner model and the monodromies become automorphisms involving mirror symmetries, while the heterotic dual is an asymmetric toroidal orbifold. We generalise previous constructions to ones in which the automorphisms are not of prime order. The type II construction is perturbatively consistent, but the naive heterotic dual is not modular invariant. Modular invariance on the heterotic side is achieved by including twists in the circles dual to the winding numbers round the T2, and this in turn introduces non-perturbative phases depending on NS5-brane charge in the type II construction.


  1. [1]

    A. Sen and C. Vafa, Dual pairs of type-II string compactification, Nucl. Phys.B 455 (1995) 165 [hep-th/9508064] [INSPIRE].

  2. [2]

    D. Israël and V. Thiéry, Asymmetric Gepner models in type-II, JHEP02 (2014) 011 [arXiv:1310.4116] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. [3]

    D. Israël, Nongeometric Calabi-Yau compactifications and fractional mirror symmetry, Phys. Rev.D 91 (2015) 066005 [Erratum ibid.D 91 (2015) 129902] [arXiv:1503.01552] [INSPIRE].

  4. [4]

    R. Blumenhagen, M. Fuchs and E. Plauschinn, Partial SUSY Breaking for Asymmetric Gepner Models and Non-geometric Flux Vacua, JHEP01 (2017) 105 [arXiv:1608.00595] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    K.A. Intriligator and C. Vafa, Landau-Ginzburg orbifolds, Nucl. Phys.B 339 (1990) 95 [INSPIRE].

  6. [6]

    A.N. Schellekens and S. Yankielowicz, New Modular Invariants for N = 2 Tensor Products and Four-dimensional Strings, Nucl. Phys.B 330 (1990) 103 [INSPIRE].

  7. [7]

    C. Hull, D. Israël and A. Sarti, Non-geometric Calabi-Yau Backgrounds and K 3 automorphisms, JHEP11 (2017) 084 [arXiv:1710.00853] [INSPIRE].

  8. [8]

    C.M. Hull, A geometry for non-geometric string backgrounds, JHEP10 (2005) 065 [hep-th/0406102] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. [9]

    P. Comparin and N. Priddis, BHK mirror symmetry for K3 surfaces with non-symplectic automorphism, arXiv:1704.00354.

  10. [10]

    C.J. Bott, P. Comparin and N. Priddis, Mirror symmetry for K 3 surfaces, arXiv:1901.09373 [INSPIRE].

  11. [11]

    C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys.B 438 (1995) 109 [hep-th/9410167] [INSPIRE].

  12. [12]

    S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett.B 361 (1995) 59 [hep-th/9505162] [INSPIRE].

  13. [13]

    P.S. Aspinwall, K 3 surfaces and string duality, in Differential geometry inspired by string theory, pp. 421–540, 1996, hep-th/9611137 [INSPIRE].

  14. [14]

    A. Dabholkar and C. Hull, Duality twists, orbifolds and fluxes, JHEP09 (2003) 054 [hep-th/0210209] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP05 (2006) 009 [hep-th/0512005] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  16. [16]

    J.A. Harvey and G.W. Moore, Conway Subgroup Symmetric Compactifications of Heterotic String, J. Phys.A 51 (2018) 354001 [arXiv:1712.07986] [INSPIRE].

  17. [17]

    J. McOrist, D.R. Morrison and S. Sethi, Geometries, Non-Geometries and Fluxes, Adv. Theor. Math. Phys.14 (2010) 1515 [arXiv:1004.5447] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  18. [18]

    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].

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    Y. Gautier, C. Hull and D. Isräel, Moduli spaces of non-geometric type II/heterotic dual pairs, to appear.

  20. [20]

    N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys.B 303 (1988) 286 [INSPIRE].

  21. [21]

    P.S. Aspinwall and D.R. Morrison, String theory on K 3 surfaces, hep-th/9404151 [INSPIRE].

  22. [22]

    W. Nahm and K. Wendland, Mirror symmetry on Kummer type K 3 surfaces, Commun. Math. Phys.243 (2003) 557 [hep-th/0106104] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  23. [23]

    R.A. Reid-Edwards and B. Spanjaard, N = 4 Gauged Supergravity from Duality-Twist Compactifications of String Theory, JHEP12 (2008) 052 [arXiv:0810.4699] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    B.R. Greene and M.R. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys.B 338 (1990) 15 [INSPIRE].

  25. [25]

    M. Artebani, S. Boissière and A. Sarti, The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces, J. Math. Pures Appl.102 (2014) 758.

    MathSciNet  Article  Google Scholar 

  26. [26]

    P. Comparin, C. Lyons, N. Priddis and R. Suggs, The mirror symmetry of K 3 surfaces with non-symplectic automorphisms of prime order, Adv. Theor. Math. Phys.18 (2014) 1335 [arXiv:1211.2172] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  27. [27]

    D. Gepner, Space-Time Supersymmetry in Compactified String Theory and Superconformal Models, Nucl. Phys.B 296 (1988) 757 [INSPIRE].

  28. [28]

    E. Witten, String theory dynamics in various dimensions, Nucl. Phys.B 443 (1995) 85 [hep-th/9503124] [INSPIRE].

  29. [29]

    P.S. Aspinwall and J. Louis, On the ubiquity of K 3 fibrations in string duality, Phys. Lett.B 369 (1996) 233 [hep-th/9510234] [INSPIRE].

  30. [30]

    K.S. Narain, New Heterotic String Theories in Uncompactified Dimensions < 10, Phys. Lett.169B (1986) 41 [INSPIRE].

  31. [31]

    K.S. Narain, M.H. Sarmadi and E. Witten, A Note on Toroidal Compactification of Heterotic String Theory, Nucl. Phys.B 279 (1987) 369 [INSPIRE].

  32. [32]

    J.A. Harvey and A. Strominger, The heterotic string is a soliton, Nucl. Phys.B 449 (1995) 535 [Erratum ibid.B 458 (1996) 456] [hep-th/9504047] [INSPIRE].

  33. [33]

    S. Groot Nibbelink and P.K.S. Vaudrevange, T-duality orbifolds of heterotic Narain compactifications, JHEP04 (2017) 030 [arXiv:1703.05323] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  34. [34]

    M.A. Walton, The Heterotic String on the Simplest Calabi-Yau Manifold and Its Orbifold Limits, Phys. Rev.D 37 (1988) 377 [INSPIRE].

  35. [35]

    M.R. Gaberdiel and R. Volpato, Mathieu Moonshine and Orbifold K 3s, Contrib. Math. Comput. Sci.8 (2014) 109 [arXiv:1206.5143] [INSPIRE].

    Article  Google Scholar 

  36. [36]

    R. Vaidyanathaswamy, Integer-roots of the unit matrix, J. Lond. Math. Soc.1 (1928) 121.

  37. [37]

    K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric Orbifolds, Nucl. Phys.B 288 (1987) 551 [INSPIRE].

  38. [38]

    K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds: Path integral and operator formulations, Nucl. Phys.B 356 (1991) 163 [INSPIRE].

  39. [39]

    A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP10 (2005) 096 [hep-th/0507014] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. [40]

    C. Hull and R.J. Szabo, Noncommutative gauge theories on D-branes in non-geometric backgrounds, JHEP09 (2019) 051 [arXiv:1903.04947] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  41. [41]

    E. Kiritsis, Introduction to superstring theory, vol. B9 of Leuven notes in mathematical and theoretical physics, Leuven University Press, Leuven, Belgium (1998).

  42. [42]

    A. Dabholkar and J.A. Harvey, Nonrenormalization of the Superstring Tension, Phys. Rev. Lett.63 (1989) 478 [INSPIRE].

    ADS  Article  Google Scholar 

  43. [43]

    P.S. Aspinwall, An N = 2 dual pair and a phase transition, Nucl. Phys.B 460 (1996) 57 [hep-th/9510142] [INSPIRE].

  44. [44]

    R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Theoretical and Mathematical Physics, Springer, Heidelberg, Germany, (2013).

  45. [45]

    T.M. Apostol, Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York, U.S.A. (1976).

Download references

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

Author information



Corresponding author

Correspondence to D. Israël.

Additional information

ArXiv ePrint: 1906.02165

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gautier, Y., Hull, C.M. & Israël, D. Heterotic/type II duality and non-geometric compactifications. J. High Energ. Phys. 2019, 214 (2019).

Download citation


  • Conformal Field Models in String Theory
  • String Duality
  • Superstring Vacua
  • Superstrings and Heterotic Strings