  We resolve a question of Kac, and explain the automorphic properties of
characters due to Kac-Wakimoto pertaining to sl(m|n)^ highest weight modules,
for n \geq 1. We prove that the Kac-Wakimoto characters are essentially
holomorphic parts of certain generalizations of harmonic weak Maass forms which
we call "almost harmonic Maass forms". Using a new approach, this generalizes
prior work of the first author and Ono, and the authors, both of which treat
only the case n = 1. We also provide an explicit asymptotic expansion for the
characters.
