Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture

Abstract

We study the representation theory of the superconformal algebra $\mathcal{W}_k(\mathfrak{g},f_{\theta})$ associated with a minimal gradation of $\frak{g}$. Here, $\frak{g}$ is a simple finite-dimensional Lie superalgebra with a nondegenerate, even supersymmetric invariant bilinear form. Thus, $\mathcal{W}_k(\mathfrak{g},f_{\theta})$ 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 [17, Conjecture 3.1B] for $\mathcal{W}_k(\frak{g},f_{\theta})$. In fact, we show that any irreducible highest-weight character of $\mathcal{W}_k(\frak{g},f_{\theta})$ at any level $k\in mathbb{C}$ is determined by the corresponding irreducible highest-weight character of the Kac-Moody affinization of $\frak{g}$