Coset Construction and Character Sumrules for the Doubly Extended $N=4$ Superconformal Algebras

Character sumrules associated with the realization of the $N=4$ superconformal algebra $\At$ on manifolds corresponding to the group cosets $SU(3)_{\ktp }/U(1)$ are derived and developed as an important tool in obtaining the modular properties of $\At$ characters as well as information on certain extensions of that algebra. Their structure strongly suggests the existence of rational conformal field theories with central charges in the range $1 \le c\le 4$. The corresponding characters appear in the massive sector of the sumrules and are completely specified in terms of the characters for the parafermionic theory $SU(3)/(SU(2)\times U(1))$ and in terms of the branching functions of massless $\At$ characters into $SU(2)_{\ktp }\times SU(2)_1$ characters.