  In this paper, we establish automorphic properties and asymptotic behaviors
of characters due to Kac-Wakimoto pertaining to $s\ell(m|n)^\wedge$ highest
weight modules in the case $m=n$, extending work of the first author and Ono
\cite{BOKac} and the first two authors \cite{BF} which pertains to the case
$m>n$.
