Graph IRF Models and Fusion Rings

TL;DR

This paper explores the relationship between IRF lattice models and RCFT, introducing a graph algebra framework that links certain IRF models to fusion rings, and investigates which models originate from RCFT.

Contribution

It introduces an algebraic construction connecting IRF models to fusion rings and analyzes specific examples, including A-D-E Dynkin diagram models, to determine their RCFT origins.

Findings

A graph algebra construction links IRF models to fusion rings.

A-D-E models are studied; A-type originates from RCFT, D-E do not.

Open question on whether all IRF models derive from RCFT.

Abstract

Recently, a class of interaction round the face (IRF) solvable lattice models were introduced, based on any rational conformal field theory (RCFT). We investigate here the connection between the general solvable IRF models and the fusion ones. To this end, we introduce an associative algebra associated to any graph, as the algebra of products of the eigenvalues of the incidence matrix. If a model is based on an RCFT, its associated graph algebra is the fusion ring of the RCFT. A number of examples are studied. The Gordon--generalized IRF models are studied, and are shown to come from RCFT, by the graph algebra construction. The IRF models based on the Dynkin diagrams of A-D-E are studied. While the $A$ case stems from an RCFT, it is shown that the $D-E$ cases do not. The graph algebras are constructed, and it is speculated that a natural isomorphism relating these to RCFT exists. The question whether all solvable IRF models stems from an RCFT remains open, though the $D-E$ cases shows that a mixing of the primary fields is needed.