Duke Mathematical Journal

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

Tomoyuki Arakawa

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We study the representation theory of the superconformal algebra Wk(g,fθ) associated with a minimal gradation of g. Here, g is a simple finite-dimensional Lie superalgebra with a nondegenerate, even supersymmetric invariant bilinear form. Thus, Wk(g,fθ) 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 Wk(g,fθ). In fact, we show that any irreducible highest-weight character of Wk(g,fθ) at any level k is determined by the corresponding irreducible highest-weight character of the Kac-Moody affinization of g

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

First available in Project Euclid: 1 December 2005

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Zentralblatt MATH identifier

Primary: 17B68: Virasoro and related algebras
Secondary: 17B10: Representations, algebraic theory (weights) 17B55: Homological methods in Lie (super)algebras 17B69: Vertex operators; vertex operator algebras and related structures


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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