I‑AIMS Institute for Artificial Intelligence
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The Satake sextic in F-theory

Andreas Malmendier, Tanush Shaska

I-AIMS paper 2016-4 · Published in J. Geom. Phys. 120 (2017), 290–305 · 2017

We describe explicit formulas relevant to the F-theory/heterotic string duality that reconstruct from a specific Jacobian elliptic fibration on the Shioda-Inose surface covering a generic Kummer surface the corresponding genus-two curve using the level-two Satake coordinate functions. We derive explicitly the rational map on the moduli space of genus-two curves realizing the algebraic correspondence between a sextic curve and its corresponding Satake sextic. We will prove that it is not the original sextic defining the genus-two curve, but its corresponding Satake sextic which is manifest in the F-theory model, dual to the 𝔰𝔬(32) heterotic string with an unbroken 𝔰𝔬(28) ⊕ 𝔰𝔲(2) gauge algebra.

MSC classes: 11F03, 14J28, 14J81

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References

  1. Paul S. Aspinwall and Mark Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B 387 (1996), no. 4, 735–742.
  2. Paul S. Aspinwall and David R. Morrison, String theory on K3 surfaces, Mirror symmetry, II, 1997, pp. 703–716.
  3. ———, Non-simply-connected gauge groups and rational points on elliptic curves, J. High Energy Phys. 7 (1998), Paper 12, 16 pp. (electronic).
  4. Oskar Bolza, On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1887), no. 1, 47–70.
  5. Adrian Clingher and Charles F. Doran, Modular invariants for lattice polarized K3 surfaces, Michigan Math. J. 55 (2007), no. 2, 355–393.
  6. ———, Note on a geometric isogeny of K3 surfaces, Int. Math. Res. Not. IMRN 16 (2011), 3657–3687.
  7. ———, Lattice polarized K3 surfaces and Siegel modular forms, Adv. Math. 231 (2012), no. 1, 172–212.
  8. Anamaría Font, In̂aki García-Etxebarria, Dieter Lüst, Stefano Massai, and Christoph Mayrhofer, Heterotic T-fects, 6d SCFTs, and F-theory, arXiv:1603.09361 [hep-th], available at 1603.09361.
  9. Robert Friedman, John W. Morgan, and Edward Witten, Vector bundles over elliptic fibrations, J. Algebraic Geom. 8 (1999), no. 2, 279–401.
  10. Federica Galluzzi and Giuseppe Lombardo, Correspondences between K3 surfaces, Michigan Math. J. 52 (2004), no. 2, 267–277. With an appendix by Igor Dolgachev.
  11. V. A. Gritsenko and V. V. Nikulin, Igusa modular forms and “the simplest” Lorentzian Kac-Moody algebras, Mat. Sb. 187 (1996), no. 11, 27–66.
  12. David Gruenewald, Computing Humbert surfaces and applications, Arithmetic, geometry, cryptography and coding theory 2009, 2010, pp. 59–69.
  13. R. W. H. T. Hudson, Kummer’s quartic surface, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. With a foreword by W. Barth, Revised reprint of the 1905 original.
  14. Jun-Ichi Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175–200.
  15. ———, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817–855.
  16. Hiroshi Inose, Defining equations of singular K3 surfaces and a notion of isogeny, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), 1978, pp. 495–502.
  17. K. Kodaira, On compact analytic surfaces. II, III, Ann. of Math. (2) 77 (1963), 563–626; ibid. 78 (1963), 1–40.
  18. Abhinav Kumar, K3 surfaces associated with curves of genus two, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm165, 26.
  19. ———, Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geom. 23 (2014), no. 4, 599–667.
  20. Andreas Malmendier and David R. Morrison, K3 surfaces, modular forms, and non-geometric heterotic compactifications, Lett. Math. Phys. 105 (2015), no. 8, 1085–1118.
  21. Andreas Malmendier and Tony Shaska, A universal pair of genus-2 curves, arXiv:1607.08294
  22. [math.AG], available at 1607.08294.
  23. Afsaneh Mehran, Double covers of Kummer surfaces, Manuscripta Math. 123 (2007), no. 2, 205–235.
  24. David Mumford, Tata lectures on theta., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. With the collaboration of C. Musili, M. Nori, E. Previato and M. Stillman, Reprint of the 1983 edition.
  25. Yukihiko Namikawa and Kenji Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math. 9 (1973), 143–186.
  26. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238.
  27. E. Previato, T. Shaska, and G. S. Wijesiri, Thetanulls of cyclic curves of small genus, Albanian J. Math. 1 (2007), no. 4, 253–270.
  28. ———, Thetanulls of cyclic curves of small genus, Albanian J. Math. 1 (2007), no. 4, 253–270.
  29. T. Shaska and G. S. Wijesiri, Theta functions and algebraic curves with automorphisms, Algebraic aspects of digital communications, 2009, pp. 193–237.
  30. Tanush Shaska and Helmut Voelklein, Elliptic subfields and automorphisms of genus 2 function fields, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000), 2004, pp. 703–723.
  31. Tetsuji Shioda, Kummer sandwich theorem of certain elliptic K3 surfaces, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 137–140.
  32. E. B. Vinberg, On the algebra of Siegel modular forms of genus 2, Trans. Moscow Math. Soc. (2013), 1–13.

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