Skip to main content
SpringerLink
Log in

Six Line Configurations and String Dualities

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen–Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen–Sarti partners is a four-parameter family of K3 surfaces with \({H \oplus E_7(-1) \oplus E_7(-1)}\) lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type IV. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aspinwall, P.S., Morrison, D.R.: Non-simply-connected gauge groups and rational points on elliptic curves, 1029-8479. J. High Energy Phys. 7, Paper 12, 16 pp. (electronic) (1998)

  2. Baily W.L. Jr, Borel A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 84, 442–528 (1966)

    Article  MathSciNet  Google Scholar 

  3. Beshaj, L., Hidalgo, R., Kruk, S., Malmendier, A., Quispe, S., Shaska, T.: Rational points in the moduli space of genus two. In: Higher genus curves in mathematical physics and arithmetic geometry, Contemporary Mathematics, vol. 703, pp. 83–115. American Mathematical Society, Providence (2018)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Clingher A., Doran C.F.: Lattice polarized K3 surfaces and Siegel modular forms. Adv. Math. 231(1), 172–212 (2012)

    Article  MathSciNet  Google Scholar 

  7. Clingher, A., Doran, C.F., Malmendier, A.: Configurations of six lines and Shioda–Inose structures (to appear)

  8. Clingher, A., Malmendier, A.: On Nikulin involutions and the CHL string, arXiV:1805.10242 [math.AG]

  9. Clingher, A., Malmendier, A.: On the geometry of (1,2)-polarized kummer surfaces, [math.AG], arXiV:1704.04884

  10. Clingher A., Morgan J.W.: Mathematics underlying the F-theory/heterotic string duality in eight dimensions. Commun. Math. Phys. 254(3), 513–563 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  11. Comparin P., Garbagnati A.: Van Geemen–Sarti involutions and elliptic fibrations on K3 surfaces double cover of \({{\mathbb{P}}^2}\). J. Math. Soc. Japan 66(2), 479–522 (2014)

    Google Scholar 

  12. Dolgachev, I., Ortland, D.: Point sets in projective spaces and theta functions. Astérisque 165, 210 (1989)

  13. Donagi, R.: Heterotic F-theory duality. In: XIIth International Congress of Mathematical Physics (ICMP ’97) (Brisbane), pp. 206–213. International Press, Cambridge (1999)

  14. Friedman R., Morgan J., Witten E.: Vector bundles and F theory. Commun. Math. Phys. 187(3), 679–743 (1997)

    Article  ADS  Google Scholar 

  15. Gelfand I.M., Kapranov M.M., Zelevinsky A.V.: Generalized Euler integrals and A-hypergeometric functions. Adv. Math. 84(2), 255–271 (1990)

    Article  MathSciNet  Google Scholar 

  16. Gu, J., Jockers, H.: Nongeometric F-theory–heterotic duality. Phys. Rev. D 91(8), 086007, 10 (2015)

  17. Hoyt, W.L., Schwartz, C.F., Yoshida surfaces with Picard number \({\rho\geq 17}\). In: Proceedings on Moonshine and related topics (Montréal, QC, 1999), pp. 71–78 (2001)

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

    Article  MathSciNet  Google Scholar 

  19. Igusa J.-I.: Modular forms and projective invariants. Am. J. Math. 89, 817–855 (1967)

    Article  MathSciNet  Google Scholar 

  20. Igusa J.-I.: On the ring of modular forms of degree two over Z. Am. J. Math. 101(1), 149–183 (1979)

    Article  MathSciNet  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. Kloosterman R.: Classification of all Jacobian elliptic fibrations on certain K3 surfaces. J. Math. Soc. Japan 58(3), 665–680 (2006)

    Article  MathSciNet  Google Scholar 

  23. Kumar, A.: K3 surfaces associated with curves of genus two. Int. Math. Res. Not. IMRN, 6, 26, Art. ID rnm165 (2008)

  24. Kumar A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. 23(4), 599–667 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  27. Malmendier, A., Shaska, T.: A universal genus-two curve from Siegel modular forms. In: SIGMA Symmetry Integrability Geom. Methods Appl. 13, Paper No. 089, 17 (2017)

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

    Article  MathSciNet  Google Scholar 

  29. Matsumoto K., Sasaki T., Takayama N., Yoshida M.: Monodromy of the hypergeometric differential equation of type (3,6). I. Duke Math. J. 71(2), 403–426 (1993)

    Article  MathSciNet  Google Scholar 

  30. Matsumoto, K., Sasaki, T., Takayama, N., Yoshida, M.: Monodromy of the hypergeometric differential equation of type (3,6). II. The unitary reflection group of order \({2^9\cdot 3^7\cdot 5\cdot 7}\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20(4), 617–631 (1993)

  31. Matsumoto K., Sasaki T., Yoshida M.: The monodromy of the period map of a 4-parameter family of K3 surfaces and the hypergeometric function of type (3,6). Intern. J. Math. 3(1), 164 (1992)

    Article  MathSciNet  Google Scholar 

  32. Matsumoto K., Sasaki T., Yoshida M.: Monodromy of the hypergeometric differential equation of type (3,6) III. Kumamoto J. Math. 23, 37–47 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Morrison D.R., Vafa C.: Compactifications of F-theory on Calabi–Yau threefolds. I. Nuclear Phys. B 473(1–2), 74–92 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  35. 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  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. Previato E., Shaska T., Wijesiri G.S.: Thetanulls of cyclic curves of small genus. Albanian J. Math. 1(4), 253–270 (2007)

    MathSciNet  MATH  Google Scholar 

  38. Satake I.: On representations and compactifications of symmetric Riemannian spaces. Ann. Math. (2) 71, 77–110 (1960)

    Article  MathSciNet  Google Scholar 

  39. Schütt, M., Shioda, T.: Elliptic surfaces. In: Algebraic geometry in East Asia—Seoul 2008, pp. 51–160 (2010)

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

    Article  ADS  MathSciNet  Google Scholar 

  41. Vazzana D.R.: Invariants and projections of six lines in projective space. Trans. Am. Math. Soc. 353(7), 2673–2688 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  43. Vinberg, E.B.: On the algebra of Siegel modular forms of genus 2. Trans. Moscow Math. Soc. 74, 1–13 (2013)

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

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    A. Clingher

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

    A. Malmendier

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    T. Shaska

Authors
  1. A. Clingher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Malmendier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Shaska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Malmendier.

Additional information

Communicated by C. Schweigert

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Clingher, A., Malmendier, A. & Shaska, T. Six Line Configurations and String Dualities. Commun. Math. Phys. 371, 159–196 (2019). https://doi.org/10.1007/s00220-019-03372-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03372-0

Search

Navigation