Towards a classification of modular invariant partition functions for theories based on $N=4$ superconformal algebras

The representation theory of the doubly extended $N=4$ superconformal algebra is reviewed. The modular properties of the corresponding characters can be derived, using characters sumrules for coset realizations of these $N=4$ algebras. Some particular combinations of massless characters are shown to transform as affine $SU(2)$ characters under $S$ and $T$, a fact used to completely classify the massless sector of the partition function.