Skip to main content
SpringerLink
Log in

An explicit formula of the normalized Mumford form

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

Abstract

We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary.

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. Beilinson, A.A., Manin, Y.I.: The Mumford form and the Polyakov measure in string theory. Commun. Math. Phys. 107, 359–376 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  2. Belavin, A.A., Knizhnik, V.G.: Algebraic geometry and the geometry of quantum strings. Phys. Lett. B 168, 201–206 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  3. Deligne, P.: Le detérminant de la cohomologie, current trends in arithmetical algebraic geometry. Contemp. Math. 67, 93–177 (1987)

    Article  Google Scholar 

  4. Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math. IHES 36, 75–109 (1969)

    Article  MathSciNet  Google Scholar 

  5. Fay, J.: Kernel functions, analytic torsion and moduli spaces. Mem. Am. Math. Soc. 96, 123 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Freixas i Montplet, G.: An arithmetic Riemann-Roch theorem for pointed stable curves. Ann. Scient. Éc. Norm. Sup. 42, 335–369 (2009)

    Article  MathSciNet  Google Scholar 

  7. Gillet, H., Soulé, C.: Arithmetic intersection theory. Publ. Math. IHES 72, 94–174 (1990)

    Article  Google Scholar 

  8. Ichikawa, I.: \(P\)-adic theta functions and solutions of the KP hierarchy. Commun. Math. Phys. 176, 383–399 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  9. Ichikawa, I.: Generalized Tate curve and integral Teichmüller modular forms. Am. J. Math. 122, 1139–1174 (2000)

    Article  Google Scholar 

  10. Ichikawa, I.: Chern-Simons invariant and Deligne-Riemann-Roch isomorphism, Preprint

  11. Manin, Y., Drinfeld, V.: Periods of \(p\)-adic Schottky groups. J. Reine Angew. Math. 262/263, 239–247 (1973)

  12. Matone, M.: Extending the Belavin-Knizhnik “wonderful formula” by the characterization of the Jacobian. JHEP 1210, 175 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  13. Matone, M., Volpato, R.: Vector-valued modular forms from the Mumford forms, Schottky-Igusa form, product of thetanullwerte and the amazing Klein formula. Proc. Am. Math. Soc. 141, 2575–2587 (2013)

    Article  MathSciNet  Google Scholar 

  14. McIntyre, A., Takhtajan, L.: Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula. GAFA 16, 1291–1323 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Mumford, D.: An analytic construction of degenerating curves over complete local rings. Compositio Math. 24, 129–174 (1972)

    MathSciNet  MATH  Google Scholar 

  16. Mumford, D.: Stability of projective varieties. L’Ens. Math. 23, 39–110 (1977)

    MathSciNet  MATH  Google Scholar 

  17. Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1986)

    Article  Google Scholar 

  18. Verlinde, E.P., Verlinde, H.L.: Chiral bosonization, determinants and the string partition function. Nucl. Phys. B 288, 357–396 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  19. Weng, L.: \(\Omega \)-admissible theory II, Deligne pairings over moduli spaces of punctured Riemann surfaces. Math. Ann. 320, 239–283 (2001)

    Article  MathSciNet  Google Scholar 

  20. Zograf, P.G.: Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces, Algebra, i Analiz 1(1989), 136–160 (Russian). English translation in Leningrad Math. J. 1, 941–965 (1990)

  21. Zograf, P.G.: Determinants of Laplacians, Liouville action, and an analogue of the Dedekind \(\eta \)-function on Teichmüller space, Unpublished manuscript (1997)

  22. Zograf, P.G., Takhtajan, L.A.: On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces. Math. USSR-Sb. 60, 297–313 (1988)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

    Takashi Ichikawa

Authors
  1. Takashi Ichikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takashi Ichikawa.

Additional information

Publisher's Note

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

This work is partially supported by the JSPS Grant-in-Aid for Scientific Research No. 17K05179.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ichikawa, T. An explicit formula of the normalized Mumford form. Lett Math Phys 111, 2 (2021). https://doi.org/10.1007/s11005-020-01339-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-020-01339-0

Keywords

Mathematics Subject Classification

Search

Navigation