Representation Theory of Superconformal Algebras and the Kac-Roan-Wakimoto Conjecture

We study the representation theory of the superconformal algebra $W_k(g,f_{\theta})$ associated with a minimal gradation of $g$. Here, $g$ is a simple finite-dimensional Lie superalgebra with a non-degenerate, even supersymmetric invariant bilinear form. Thus, $W_k(g,f_{\theta})$ can be one of the well-known superconformal algebras including the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnik algebras, the N=2 superconformal algebra, the N=4 superconformal algebra, the N=3 superconformal algebra and the big N=4 superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan and M. Wakimoto for $W_k(g,f_{\theta})$. In fact, we show that any irreducible highest weight character of $W_k(g,f_{\theta})$ at any level $k\in C$ is determined by the corresponding irreducible highest weight character of the Kac-Moody affinization of $g$.