Communications in Mathematical Physics

, Volume 255, Issue 2, pp 469–512

Higher-Level Appell Functions, Modular Transformations, and Characters

  • A.M. Semikhatov
  • A. Taorimina
  • I.Yu. Tipunin
Article

DOI: 10.1007/s00220-004-1280-7

Cite this article as:
Semikhatov, A., Taorimina, A. & Tipunin, I. Commun. Math. Phys. (2005) 255: 469. doi:10.1007/s00220-004-1280-7

Abstract

We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functionshttps://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the “period.” Generalizing the well-known interpretation of theta functions as sections of line bundles, the https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif function enters the construction of a section of a rank-(ℓ+1) bundle https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb2.gif. We evaluate modular transformations of the https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif functions and construct the action of an SL(2,ℤ) subgroup that leaves the section of https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb2.gif constructed from https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif invariant.

Modular transformation properties of https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif are applied to the affine Lie superalgebra https://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb3.gif at a rational level k>−1 and to the N=2 super-Virasoro algebra, to derive modular transformations of “admissible” characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • A.M. Semikhatov
    • 1
  • A. Taorimina
    • 2
  • I.Yu. Tipunin
    • 1
  1. 1.Tamm Theory Department, Lebedev Physics InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Department of Mathematical SciencesUniversity of Durham, Science LaboratoriesDurhamUnited Kingdom