From CFT's to Graphs

TL;DR

This paper explores the deep connections between rational conformal field theories and graph structures, generalizing previous work to include $sl(n)$ theories and providing methods to determine graph eigenvectors from conformal data.

Contribution

It extends the relation between CFT operator algebra constants and graph algebras to $sl(n)$ theories, enabling the determination of graph eigenvectors from conformal data.

Findings

Established linear systems linking conformal data to Pasquier algebra matrices

Provided methods to compute eigenvectors of adjacency matrices from conformal field theory data

Generalized previous $sl(2)$ results to $sl(n)$ theories

Abstract

In this paper, we pursue the discussion of the connections between rational conformal field theories (CFT) and graphs. We generalize our recent work on the relations of operator product algebra (OPA) structure constants of $sl(2)\,$ theories with the Pasquier algebra attached to the graph. We show that in a variety of CFT built on $sl(n)\,$ -- typically conformal embeddings and orbifolds, similar considerations enable one to write a linear system satisfied by the matrix elements of the Pasquier algebra in terms of conformal data -- quantum dimensions and fusion coefficients. In some cases, this provides a sufficient information for the determination of all the eigenvectors of an adjacency matrix, and hence of a graph.