Representation Theory of Superconformal Algebras and the Kac-Roan-Wakimoto Conjecture
Representation Theory of Superconformal Algebras and the Kac-Roan-Wakimoto Conjecture
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DOI：
10.1215/s0012-7094-05-13032-0
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发表时间：
2004-05
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影响因子：
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通讯作者：
T. Arakawa
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文献类型：
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作者：
T. Arakawa
我们研究了极小阶度为$g$的超共形代数$W_k(g，f_{\theta})$的表示理论。这里，$g$是一个简单的有限维李超代数，具有非退化的，甚至超对称不变的双线性形式。因此，$W_k(g，f_{\theta})$可以是著名的超共形代数之一，包括Virasoro代数、Bershadsky-Polyakov代数、N-Schwarz代数、Bershadsky-Knizhnik代数、N=2超共形代数、N=4超共形代数、N=3超共形代数和大N=4超共形代数。我们证明了V.G.Kac，S.-S.Roan和M.Wakimoto对$W_k(g，f_{\theta})$的猜想。事实上，我们证明了在C$中，$W_k(g，f_{\theta})$的任何不可约最高权特征标都是由$g$的Kac-Moody仿射的对应的不可约最高权特征标决定的。
We study the representation theory of the superconformal algebra $W_k(g,f_{\theta})$ associated with a minimal gradation of $g$. Here, $g$ is a simple finite-dimensional Lie superalgebra with a non-degenerate, even supersymmetric invariant bilinear form. Thus, $W_k(g,f_{\theta})$ 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 for $W_k(g,f_{\theta})$. In fact, we show that any irreducible highest weight character of $W_k(g,f_{\theta})$ at any level $k\in C$ is determined by the corresponding irreducible highest weight character of the Kac-Moody affinization of $g$.