Duke Mathematical Journal

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

Tomoyuki Arakawa

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 $\frak{g}$

Article information

Source
Duke Math. J., Volume 130, Number 3 (2005), 435-478.

Dates
First available in Project Euclid: 1 December 2005

https://projecteuclid.org/euclid.dmj/1133447439

Digital Object Identifier
doi:10.1215/S0012-7094-05-13032-0

Mathematical Reviews number (MathSciNet)
MR2184567

Zentralblatt MATH identifier
1112.17026

Citation

Arakawa, Tomoyuki. Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture. Duke Math. J. 130 (2005), no. 3, 435--478. doi:10.1215/S0012-7094-05-13032-0. https://projecteuclid.org/euclid.dmj/1133447439

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