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

TL;DR

This work defines fusion in conformal field theory as a ring-like tensor product of symmetry-algebra modules, realized by quotienting the ordinary tensor product to enforce a consistent holomorphic action. It derives explicit comultiplication maps for Kac-Moody and Virasoro algebras, proves associativity up to equivalence, and finds a triangular R-matrix, clarifying the (non-)braiding structure. Using this framework, it re-derives the fusion rules for WZW and Virasoro minimal models in purely algebraic terms and connects braiding to the process of decomposing tensor products into irreducibles. The results illuminate the interplay between conformal symmetry, quantum groups, and the algebraic structure underlying fusion.

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 rôle of the quantum group in conformal field theory.