Higher-Level Appell Functions, Modular Transformations, and Characters

Abstract

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.

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

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