Fusion Algebras and Characters of Rational Conformal Field Theories

TL;DR

This paper classifies strongly-modular fusion algebras of small dimensions and shows how to determine conformal characters of rational models from minimal data, advancing understanding of rational conformal field theories.

Contribution

It introduces the concept of strongly-modular fusion algebras, classifies low-dimensional cases, and provides methods to construct conformal characters from basic invariants.

Findings

Classified all strongly-modular fusion algebras of dimension 2, 3, 4.

Classified all nondegenerate strongly-modular fusion algebras of dimension less than 24.

Developed tools for studying modular group representations in rational conformal field theories.

Abstract

We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group 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 first 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. Secondly, we show that the conformal characters of various rational models of W-algebras can be determined from the mere knowledge of the central charge and the set of conformal dimensions. We also describe how to actually construct conformal characters by using theta series associated to certain lattices. On our way we develop several tools for studying representations of the modular group on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, are in general useful for the analysis rational models.