  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.
