Quasimodular forms and sl(m|m)^ characters

In this paper, we establish automorphic properties and asymptotic behaviors of characters due to Kac-Wakimoto pertaining to $s\mathrm{\ell }(m\mathrm{\mid }n{)}^{\wedge }$ 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 $m>n$.