On the Classification of Modular Fusion Algebras

Abstract

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.

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

References in corpus (4)

• Meromorphic c=24 Conformal Field Theories
• Fusion rules in conformal field theory
• Fusion Algebras Induced by Representations of the Modular Group
• Towards a Classification of Fusion Rule Algebras in Rational Conformal Field Theories

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