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 $\mathfrak{g}$. Here, $\mathfrak{g}$ 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 ${\displaystyle N=2}$ superconformal algebra, the ${\displaystyle N=4}$ superconformal algebra, the ${\displaystyle N=3}$ superconformal algebra, and the big ${\displaystyle N=4}$ superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan, and M. Wakimoto [17, Conjecture 3.1B] for ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$. In fact, we show that any irreducible highest-weight character of ${\mathcal{W}}_k(\mathfrak{g},f_{\theta })$ at any level $k\in ℂ$ is determined by the corresponding irreducible highest-weight character of the Kac-Moody affinization of $\mathfrak{g}$