Symmetries of the Kac-Peterson Modular Matrices of Affine Algebras

Abstract

The characters ${\chi }_{\mu }$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $S{L}_2(Z)$. The matrices in the image $R(S{L}_2(Z))$ 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.

16 pp, plain tex

References in corpus (6)

• The Classification of Affine SU(3) Modular Invariant Partition Functions
• WZW Commutants, Lattices, and Level 1 Partition Functions
• Automorphism Modular Invariants of Current Algebras
• Implications of an arithmetical symmetry of the commutant for modular invariants
• Towards a Classification of su(2)$⨁\cdots ⨁$su(2) Modular Invariant Partition Functions
• Automorphisms of the affine SU(3) fusion rules

Cited by in corpus (4)

• Automorphism Modular Invariants of Current Algebras
• $U(1{)}^m$ modular invariants, N=2 minimal models, and the quantum Hall effect
• Weight-Lattice Discretization of Weyl-Orbit Functions
• The W_N minimal model classification