Article

Communications in Mathematical Physics

, Volume 255, Issue 2, pp 469-512

Higher-Level Appell Functions, Modular Transformations, and Characters

  • A.M. SemikhatovAffiliated withTamm Theory Department, Lebedev Physics Institute, Russian Academy of Sciences
  • , A. TaoriminaAffiliated withDepartment of Mathematical Sciences, University of Durham, Science Laboratories
  • , I.Yu. TipuninAffiliated withTamm Theory Department, Lebedev Physics Institute, Russian Academy of Sciences

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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 functions http://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 http://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 http://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb2.gif . We evaluate modular transformations of the http://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 http://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb2.gif constructed from http://static-content.springer.com/image/art%3A10.1007%2Fs00220-004-1280-7/MediaObjects/s00220-004-1280-7flb1.gif invariant.

Modular transformation properties of http://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 http://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.