Higher-Level Appell Functions, Modular Transformations, and Characters

Details Higher-Level Appell Functions, Modular Transformations, and Characters Higher-Level Appell Functions, Modular Transformations, and Characters

2003-11-18

arxiv.org/abs/math/0311314v3

Commun. Math. Phys. 255 (2005) 469--512

Mathematics

Quantum Algebra

amsart++, xy, times. 46pp. References added, minor changes

Scientific paper

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 \textit{level-$\ell$ Appell functions} $K_\ell$ 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 $K_\ell$ function enters the construction of a section of a rank-$(\ell+1)$ bundle $V(\ell,\tau)$. We evaluate modular transformations of the $K_\ell$ functions and construct the action of an SL(2,Z) subgroup that leaves the section of $V(\ell,\tau)$ constructed from $K_\ell$ invariant. Modular transformation properties of $K_\ell$ are applied to the affine Lie superalgebra ^sl(2|1) at 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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