On the Classification of Modular Fusion Algebras

Details On the Classification of Modular Fusion Algebras On the Classification of Modular Fusion Algebras

1994-08-29

arxiv.org/abs/hep-th/9408160v2

Commun.Math.Phys. 172 (1995) 623-659

Physics

High Energy Physics

High Energy Physics - Theory

39 pages (AMS TeX) and one postscript figure, one exceptional case added in Main theorem 4, some typos corrected

Scientific paper

10.1007/BF02101810

We introduce the notion of (nondegenerate) strong-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group SL(2,Z) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups SL(2,Z_{p^l}) where p is a prime and l a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24.

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