仿射代数的Kac-Peterson模矩阵的对称性
Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras
摘要：
非扭仿射代数在固定水平上的特征标$χ_μ$以一种自然的方式定义了模群$SL_2(Z)$的表示$R$。图像$R(SL_2(Z))$中的矩阵被称为Kac-Peterson模化矩阵，描述了字符的模化行为。在本文中，我们考虑所有级别的$(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$，并为这些找到所有的排列的最高权与相应的Kac-Peterson矩阵交换。这个问题等价于共形场论的自同构不变量的分类，它的解决，特别是考虑到它的简单性，是朝着所有Wess-Zumino-Witten共形场论的分类迈出的重要一步。
英文摘要：
The characters $χ_μ$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $SL_2(Z)$. The matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular matrices, and describe the modular behaviour of the characters. In this paper we consider all levels of $(A_{r_1}\oplus\cdots\oplus A_{r_s})^{(1)}$, and for each of these find all permutations of the highest weights which commute with the corresponding Kac-Peterson matrices. This problem is equivalent to the classification of automorphism invariants of conformal field theories, and its solution, especially considering its simplicity, is a major step toward the classification of all Wess-Zumino-Witten conformal field theories.
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