Higher-Level Appell Functions, Modular Transformations, and Characters

We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The $\mathit{\text{level-}}\mathrm{\ell }\mathit{\text{Appell}}\mathit{\text{functions}}$ $K_{\mathrm{\ell }}$ 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.