On the classification of modular fusion algebras

1994/08/31 by Wolfgang Eholzer · 3 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #hep-th

paper · pdf · doi:10.1007/bf02101810

published as Commun.Math.Phys. 172 (1995) 623-659 · 39 pages (AMS TeX) and one postscript figure, one exceptional case added in Main theorem 4, some typos corrected

arxiv created 1995/05/08 · openalex publication_date 1995/09/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

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.

Citations

• Infinite conformal symmetry in two-dimensional quantum field theory
• Meromorphicc=24 conformal field theories
• Towards a Classification of Fusion Rule Algebras in Rational Conformal Field Theories

Cited by

• Finite modular groups and lepton mixing
• Galois currents and the projective kernel in rational conformal field theory
• Holomorphic modular bootstrap revisited