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The duality between F-theory and the heterotic string in \(D=8\) with two Wilson lines

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Abstract

We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank 16. We construct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial Higgsing of the heterotic string gauge algebra using two Wilson line parameters.

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References

  1. Aspinwall, P.S.: Some Relationships Between Dualities in String Theory, 1996, 30–38, \(S\)-duality and mirror symmetry (Trieste 1995)

  2. Aspinwall, P.S.: Point-like instantons and the \({\rm Spin}(32), Z_2\) heterotic string. Nuclear Phys. B 496(1–2), 149–176 (1997)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Aspinwall, P.S., Douglas, M.R.: D-brane stability and monodromy. J. High Energy Phys. 5(31), 35 (2002)

    MathSciNet  Google Scholar 

  4. Aspinwall, P.S., Gross, M.: The \({\rm SO}(32)\) heterotic string on a \(K3\) surface. Phys. Lett. B 387(4), 735–742 (1996)

    ADS  MathSciNet  Google Scholar 

  5. Aspinwall, P.S., Morrison, D.R.: Non-simply-connected gauge groups and rational points on elliptic curves, (1998). J. High Energy Phys. 7, Paper 12, 16

  6. Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact complex surfaces, Second, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer, Berlin (2004)

  7. Bershadsky, M., Pantev, T., Sadov, V.: F-theory with quantized fluxes. Adv. Theor. Math. Phys. 3(3), 727–773 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bianchi, M.: A note on toroidal compactifications of the type I superstring and other superstring vacuum configurations with 16 supercharges. Nuclear Phys. B 528(1–2), 73–94 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Clingher, A., Malmendier, A., Shaska, T.: Six line configurations and string dualities. Commun. Math. Phys. 371(1), 159–196 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Clingher, A., Hill, T., Malmendier, A.: Jacobian elliptic fibrations on a special family of K3 surfaces of picard rank sixteen, arXiv:1908.09578 [math.AG] (2019)

  11. Clingher, A., Doran, C.F.: Modular invariants for lattice polarized K3 surfaces. Michigan Math. J. 55(2), 355–393 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Clingher, A., Doran, C.F.: Note on a geometric isogeny of K3 surfaces. Int. Math. Res. Not. IMRN 16, 3657–3687 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cvetič, M., Klevers, D., Piragua, H.: F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections. J. High Energy Phys. 6, 067 (2013). front matter+53

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Dolgachev, I.V.: Mirror Symmetry for Lattice Polarized K3 Surfaces, pp. 2599–2630. Algebraic geometry, 4 (1996)

  15. Dolgachev, I.: Integral quadratic forms: applications to algebraic geometry (after V. Nikulin), Bourbaki seminar, Vol. 1982/83, pp. 251–278 (1983)

  16. Douglas, M.R.: D-branes, categories and \(\text{N} = 1\) supersymmetry, pp. 2818–2843. Strings, branes, and M-theory (2001)

  17. Gritsenko, V.A., Hulek, K., Sankaran, G.K.: The Kodaira dimension of the moduli of K3 surfaces. Invent. Math. 169(3), 519–567 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Gu, J., Jockers, H.: Nongeometric F-theory-heterotic duality. Phys. Rev. D 91(8), 086007 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  19. Hosono, S., Lian, B.H, Yau, S.-T.: K3 surfaces from configurations of six lines in \({\mathbb{P}} ^{2}\) and mirror symmetry. Int Math Res Notices (2019)

  20. Igusa, J.-I.: On Siegel modular forms of genus two. Am. J. Math. 84, 175–200 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  21. Inose, H: Defining equations of singular K3 surfaces and a notion of isogeny. In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), pp. 495–502 (1978)

  22. Katz, S., Morrison, D.R., Schäfer-Nameki, S., Sully, J.: Tate’s algorithm and F-theory. J. High Energy Phys. 8, 09428 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Kimura, Y.: Nongeometric heterotic strings and dual F-theory with enhanced gauge groups. J. High Energy Phys. 2, 036 (2019). front matter+38

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Kimura, Y.: Unbroken \(e_7\times e_7\) nongeometric heterotic strings, stable degenerations and enhanced gauge groups in f-theory duals, High Energy Physics - Theory (2019)

  25. Kodaira, K.: On compact analytic surfaces. II, III, Ann. Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40

  26. Lawrie, C., Schäfer-Nameki, S.: The Tate form on steroids: resolution and higher codimension fibers. J. High Energy Phys. 4, 061 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Lüst, D., Massai, S., Camell, V.V.: The monodromy of T -folds and T -fects. J. High Energy Phys. 9, 127 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Malmendier, A., Shaska, T.: The Satake sextic in F-theory. J. Geom. Phys. 120, 290–305 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Malmendier, A.: The signature of the Seiberg–Witten surface, Surveys in differential geometry. Volume XV. Perspectives in Mathematics and Physics, pp. 255–277 (2011)

  30. Malmendier, A.: Kummer surfaces associated with Seiberg–Witten curves. J. Geom. Phys. 62(1), 107–123 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Malmendier, A., Morrison, D.R.: K3 surfaces, modular forms, and non-geometric heterotic compactifications. Lett. Math. Phys. 105(8), 1085–1118 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Matsumoto, K.: Theta functions on the bounded symmetric domain of type I2, 2 and the period map of a 4-parameter family of K3 surfaces. Math. Ann. 295(3), 383–409 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mayrhofer, C., Font, A., Garcia-Etxebarria, I., Lüst, D., Massai, S.: Non-geometric heterotic backgrounds and 6D SCFTS. In: Proceedings of Corfu Summer Institute 2016 School and Workshops on Elementary Particle Physics and Gravity—PoS(CORFU2016) (2017)

  34. Mayrhofer, C., Morrison, D.R., Oskar, T., Weigand, T.: Mordell-Weil torsion and the global structure of gauge groups in F-theory. J. High Energy Phys. 10, 016 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. McOrist, J., Morrison, D.R., Sethi, S.: Geometries, non-geometries, and fluxes. Adv. Theor. Math. Phys. 14(5), 1515–1583 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Morrison, D.R., Vafa, C.: Compactifications of F-theory on Calabi–Yau threefolds. II. Nuclear Phys. B 476(3), 437–469 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Namikawa, Y., Ueno, K.: On geometrical classification of fibers in pencils of curves of genus two. Proc. Jpn. Acad. 48, 373–376 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  38. Namikawa, Y., Ueno, K.: The complete classification of fibres in pencils of curves of genus two. Manuscr. Math. 9, 143–186 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  39. Namikawa, Y., Ueno, K.: On fibres in families of curves of genus two. I. Singular fibres of elliptic type, 297–371 (1973)

  40. Narain, K.S.: New heterotic string theories in uncompactified dimensions \(< 10\). Phys. Lett. B 169(1), 41–46 (1986)

    ADS  MathSciNet  Google Scholar 

  41. Narain, K.S., Sarmadi, M.H., Witten, E.: A note on toroidal compactification of heterotic string theory. Nuclear Phys. B 279(3–4), 369–379 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  42. Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ. Math. 21, 128 (1964)

    Article  MATH  Google Scholar 

  43. Nikulin, V.V.: An analogue of the Torelli theorem for Kummer surfaces of Jacobians. Izv. Akad. Nauk SSSR Ser. Mat. 38, 22–41 (1974)

    MathSciNet  MATH  Google Scholar 

  44. Nikulin, V.V.: Kummer surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 39(2), 278–293, 471 (1975)

    MathSciNet  Google Scholar 

  45. Nikulin, V.V.: Finite groups of automorphisms of Kählerian K3 surfaces. Trudy Moskov. Mat. Obshch. 38, 75–137 (1979)

    MathSciNet  MATH  Google Scholar 

  46. Nikulin, V.V.: Integer symmetric bilinear forms and some of their geometric applications. Izv. Akad. Nauk SSSR Ser. Mat. 43(1), 111–177, 238 (1979)

    MathSciNet  MATH  Google Scholar 

  47. Nikulin, V.V.: Quotient-groups of groups of automorphisms of hyperbolic forms by subgroups generated by 2-reflections. Curr. Probl. Math. 18, 3–114 (1981)

    Google Scholar 

  48. Ovrut, B.A., Pantev, T., Park, J.: Small instanton transitions in heterotic Mtheory. J. High Energy Phys. 5, Paper 45, 37 (2000)

  49. Park, J.: Orientifold and F-theory duals of CHL strings. Phys. Lett. B 418(1–2), 91–97 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  50. John, H.: Schwarz, An SL(2, Z) multiplet of type IIB superstrings. Phys. Lett. B 360(1–2), 13–18 (1995)

    MathSciNet  Google Scholar 

  51. Vafa, C.: Evidence for F-theory. Nuclear Phys. B 469(3), 403–415 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Vinberg, E.B.: On automorphic forms on symmetric domains of type IV. Uspekhi Mat. Nauk 65(3(393)), 193–194 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  53. Vinberg, E.B.: On the algebra of Siegel modular forms of genus 2. Trans. Moscow Math. Soc. pp. 1–13 (2013)

  54. Witten, E.: String theory dynamics in various dimensions. Nuclear Phys. B 443(1–2), 85–126 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Witten, E.: Non-perturbative superpotentials in string theory. Nuclear Phys. B 474(2), 343–360 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Witten, E.: Small instantons in string theory. Nuclear Phys. B 460(3), 541–559 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Witten, E.: Toroidal compactification without vector structure. J. High Energy Phys. 2, Paper 6, 43 (1998)

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  1. Department of Mathematics, University of Missouri St. Louis, St. Louis, MO, 63121, USA

    Adrian Clingher

  2. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Thomas Hill & Andreas Malmendier

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A.M. acknowledges support from the Simons Foundation through Grant No. 202367.

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Clingher, A., Hill, T. & Malmendier, A. The duality between F-theory and the heterotic string in \(D=8\) with two Wilson lines. Lett Math Phys 110, 3081–3104 (2020). https://doi.org/10.1007/s11005-020-01323-8

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