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Multivariable Appell functions and nonholomorphic Jacobi forms

  • Sander ZwegersEmail author
Research
  • 23 Downloads

Abstract

Multivariable Appell functions show up in the work of Kac and Wakimoto in the computation of character formulas for certain \(s \ell (m,1)^\wedge \) modules. Bringmann and Ono showed that the character formulas for the \(s \ell (m,1)^\wedge \) modules \(L(\varLambda _{(s)})\), where \(L(\varLambda _{(s)})\) is the irreducible \(s \ell (m,1)^\wedge \) module with the highest weight \(\varLambda _{(s)}\), can be seen as the “holomorphic parts” of certain nonholomorphic modular functions. Here, we consider more general multivariable Appell functions and relate them to nonholomorphic Jacobi forms.

Keywords

Appell functions Jacobi forms Mock modular forms Nonholomorphic modular forms 

Mathematics Subject Classification

11F27 11F30 11F50 

Notes

Acknowledgements

The author wishes to thank Kathrin Bringmann and the referees for their comments.

References

  1. 1.
    Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, Colloquium Publications, No. 64. American Mathematical Society, Providence (2017)Google Scholar
  2. 2.
    Bringmann, K., Ono, K.: Some characters of Kac and Wakimoto and nonholomorphic modular functions. Math. Ann. 345, 547–558 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bringmann, K., Ono, K.: Dyson’s ranks and Maass forms. Ann. Math. 171(1), 419–449 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Eichler, M., Zagier, D.B.: The Theory of Jacobi Forms, Progress in Mathematics, No. 55. Birkhäuser, Boston (1985)Google Scholar
  5. 5.
    Hickerson, D.R.: A proof of the mock theta conjectures. Invent. Math. 94, 639–660 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kac, V.G., Peterson, D.H.: Infinite-dimensional Lie algebra, theta functions, and modular forms. Adv. Math. 53, 125–264 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affince superalgebras and number theory. In: Lie Theory and Geometry, Progress in Mathematics, No. 123, pp. 415–456. Birkhäuser, Boston (1994)Google Scholar
  8. 8.
    Kac, V.G., Wakimoto, M.: Integrable highest weight modules over affine superalgebras and Appell’s function. Commun. Math. Phys. 215(3), 631–682 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mellit, A., Okada, S.: Joyce invariants for K3 surfaces and mock theta functions. Commun. Number Theory Phys. 3(4), 655–676 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mumford, D.: Tata Lectures on Theta I, Progress in Mathematics, No. 28. Birkhäuser, Boston (1983)Google Scholar
  11. 11.
    Polishchuk, A.M.P.: Appell’s function and vector bundles of rank 2 on elliptic curves. Ramanujan J. 5, 111–128 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Semikhatov, A.M., Taormina, A., Tipunin, IYu.: Higher-level Appell functions, modular transformations, and characters. Commun. Math. Phys. 255(2), 469–512 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Weil, A.: Elliptic Functions According to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 88. Springer, Berlin (1976)CrossRefGoogle Scholar
  14. 14.
    Zagier, D.B.: Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann–Ono], Séminaire Bourbaki, 2007–2008, No. 986, Astérisque 326, Soc. Math. de France, pp. 143–164 (2009)Google Scholar
  15. 15.
    Zwegers, S.P.: Mock theta functions. Ph.D. Thesis, Universiteit Utrecht (2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of CologneCologneGermany

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