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Transformation Groups

, Volume 22, Issue 4, pp 979–1004 | Cite as

A CHARACTERIZATION OF MODIFIED MOCK THETA FUNCTIONS

  • VICTOR G. KACEmail author
  • MINORU WAKIMOTO
Article
  • 99 Downloads

Abstract

We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic transformation properties, and by being annihilated by certain second order differential operators.

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References

  1. [Ap]
    P. Appell, Sur le fonctions doublement periodique de troisieme espece, Annals Sci. l'Ecole norm. Sup., 3e Sér. 1 (1884), 135–164; 2 (1885), 9–36; 3 (1886), 9–42.Google Scholar
  2. [G]
    M. Gorelik, Weyl denominator identity for affine Lie superalgebras with non-zero dual Coxeter number, Jpn. J. Algebra 337 (2011), 50–62.Google Scholar
  3. [GK]
    M. Gorelik, V. G. Kac, Characters of (relatively) integrable modules over affine Lie superalgebras, Jpn. J. Math 10 (2015), no. 2, 135–235.Google Scholar
  4. [K1]
    V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), no. 1, 8–96.Google Scholar
  5. [K2]
    V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. Russian transl.: Кaц, Бecкoнeчнoмepныe aлгeбpыЛи, Mиp, M., 1993.Google Scholar
  6. [KP]
    V. G. Kac, D. H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125–264.Google Scholar
  7. [KW1]
    V. G. Kac, M. Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, in: Lie Theory and Geometry, Progress in Math., Vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415–456.Google Scholar
  8. [KW2]
    V. G. Kac, M. Wakimoto, Integrable highest weight modules over affine superalgebras and Appell's function, Commun. Math. Phys. 215 (2001), 631–682.Google Scholar
  9. [KW3]
    V. G. Kac, M. Wakimoto, Representations of affine superalgebras and mock theta functions, Transform. Groups 19 (2014), 387–455.Google Scholar
  10. [KW4]
    V. G. Kac, M. Wakimoto, Representations of affine superalgebras and mock theta functions II, Adv. Math. 300 (2016), DOI: 10.1016/j.aim.2016.03.015.
  11. [KW5]
    V. G. Kac, M. Wakimoto, Representations of affine superalgebras and mock theta functions III, Izv. Math 80 (2016), no. 4, DOI:  10.1070/IM8408.
  12. [M]
    D. Mumford, Tata Lectures on Theta I, Progress in Math., Vol. 28, Birkhäuser Boston, Boston, MA, 1983. Russian transl.: Д. Maмфopд, Лeкции o mзmaфункцияx, Mиp, M., 1988.Google Scholar
  13. [Za]
    D. Zagier, Ramanujan's mock theta functions and their applications (after Zwegers and Ono–Bringmann), Seminaire Bourbaki, Vol. 2007/2008, Asterisque 326 (2009), Exp. no. 986 (2010), vii-viii, 143–164.Google Scholar
  14. [Z]
    S. P. Zwegers, Mock theta functions, arXiv:0807.4834 (2008).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.KobeJapan

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