Communications in Mathematical Physics

, Volume 255, Issue 2, pp 469–512

Higher-Level Appell Functions, Modular Transformations, and Characters


  • A.M. Semikhatov
    • Tamm Theory Department, Lebedev Physics InstituteRussian Academy of Sciences
  • A. Taorimina
    • Department of Mathematical SciencesUniversity of Durham, Science Laboratories
  • I.Yu. Tipunin
    • Tamm Theory Department, Lebedev Physics InstituteRussian Academy of Sciences

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


We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions 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 function enters the construction of a section of a rank-(ℓ+1) bundle We evaluate modular transformations of the functions and construct the action of an SL(2,ℤ) subgroup that leaves the section of constructed from invariant.

Modular transformation properties of are applied to the affine Lie superalgebra 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.

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© Springer-Verlag Berlin Heidelberg 2005