Kac-Moody and Virasoro algebras
Kac-Moody and Virasoro algebras
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发表时间：
2010-04
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通讯作者：
A. Wassermann
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作者：
A. Wassermann
这些是1998年秋季在剑桥大学讲授的第三部分课程的笔记。它们包含一个阐述的代表性理论的李代数紧凑矩阵群，仿射卡茨-穆迪代数和Virasoro代数从一个单一的观点。治疗使用了许多方法的共形场理论，特别是戈达德肯特橄榄建设和风见铃木supercharge运营商，一个概括的狄拉克运营商。外尔特征标公式的证明取自彼得戈达德未发表的笔记。近似在同一时间，格雷格·兰德韦伯（Greg Landweber）在他的哈佛博士学位中也独立地发现了仿射Kac-Moody代数的Kac特征公式的超对称证明。论文这种方法的主要新颖之一是一个非常快速的证明费金-富克斯字符公式的离散系列表示的Virasoro代数。它只依赖于Friedan-Qiu-Shenker么正性定理的第一部分，对参数h施加限制。他们的结果只使用了平面曲线的基本性质，沿着在这些注释中详细介绍了c的较难的结果。如果最终的目标只是Feigin-Fuchs公式，则可以进行更短的处理，因为在这种情况下只需要仿射sl（2）的表示理论（参见MSRI 2000年夏季课程，此http URL）。S. Palcoux推广到Neveu-Schwarz代数，并且也推广到N=2超共形代数.
These are the notes for a Part III course given in the University of Cambridge in autumn 1998. They contain an exposition of the representation theory of the Lie algebras of compact matrix groups, affine Kac-Moody algebras and the Virasoro algebra from a unitary point of view. The treatment uses many of the methods of conformal field theory, in particular the Goddard-Kent-Olive construction and the Kazami-Suzuki supercharge operator, a generalisation of the Dirac operator. The proof of the Weyl character formula is taken from unpublished notes of Peter Goddard. The supersymmetric proof of the Kac character formula for affine Kac-Moody algebras was also found independently at roughly the same time by Greg Landweber in his Harvard Ph.D. dissertation. One of the main novelties of this approach is a very rapid proof of the Feigin-Fuchs character formula for the discrete series representations of the Virasoro algebra. It relies only on the first part of the Friedan-Qiu-Shenker unitarity theorem placing restrictions on the parameter h. Their result, which only uses elementary properties of plane curves, is presented in detail in these notes along with the somewhat harder result for c. A substantially shorter treatment is possible if the final goal is just the Feigin-Fuchs formula, since in that case only the representation theory of affine sl(2) is needed (cf the MSRI summer course given in 2000, this http URL). The same method has been applied by S. Palcoux to the Neveu-Schwarz algebra and also works for the N=2 superconformal algebra.