Fusion rules in conformal field theory

TL;DR

This paper explores the mathematical structure of fusion rules in conformal field theory, including their algebraic properties, graphical representations, and significance in the conformal bootstrap approach.

Contribution

It provides a comprehensive analysis of fusion rings, their algebraic presentations, and their applications in classifying conformal field theories.

Findings

Fusion rings can be diagonalized and linked to modular invariance.

Fusion graphs offer visual insights into algebraic structures.

Strategies for partial classification of fusion rules are discussed.

Abstract

Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme.