Representation Theory of Superconformal Algebras and the Kac-roan-wakimoto Conjecture

We study the representation theory of the superconformal algebra Wk(g, fθ) associated to a minimal gradation of g. Here, g is a simple finite-dimensional Lie superalgebra with a non-degenerate even supersymmetric invariant bilinear form. Thus, Wk(g, fθ) can be the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the BershadskyKnizhnik algebras, the N = 2 superconformal algebra, the N = 4 superconformal algebra, the N = 3 superconformal algebra, the big N = 4 superconformal algebra, and so on. The conjecture of V. Kac, S.-S. Roan and M. Wakimoto for Wk(g, fθ) is proved. In fact, we show that any irreducible highest weight character of Wk(g, fθ) at any level k ∈ C is determined by the corresponding irreducible highest weight character of the Kac-Moody affinization of g.