Fusion in conformal field theory as the tensor product of the symmetry algebra

TL;DR

This paper defines fusion in conformal field theory as a tensor product of symmetry algebra modules, proves its algebraic properties, and applies it to recover known fusion rules in specific models, highlighting quantum group structures.

Contribution

It introduces a rigorous algebraic framework for fusion as a quotient of tensor products, establishing associativity, symmetry, and explicit algebra actions, with applications to known models.

Findings

Fusion is a well-defined algebraic operation as a quotient of tensor products.

The tensor product is associative and symmetric up to equivalence.

The R-matrix of the comultiplication is triangular, indicating quantum group relevance.

Abstract

Following a recent proposal of Richard Borcherds to regard fusion as the ring-like tensor product of modules of a {\em quantum ring}, a generalization of rings and vertex operators, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra ${\cal A}$. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of ${\cal A}$ on it, under which the central extension is preserved. \\ Having given a precise meaning to fusion, determining the fusion rules is now a well-posed algebraic problem, namely to decompose the tensor product into irreducible representations. We demonstrate how to solve it for the case of the WZW- and the minimal models and recover thereby the well-known fusion rules. \\ The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the $R$-matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible r\^{o}le of the quantum group in conformal field theory.