Graph fusion algebras of WLM(p,p')

TL;DR

This paper analyzes the fusion algebra of W-extended logarithmic minimal models, revealing a detailed structure of fusion matrices, Jordan webs, and their eigenvalues, with applications to critical percolation models.

Contribution

It introduces a detailed description of the fusion algebra's graph structure and Jordan form, extending understanding of WLM(p,p') models and their boundary conditions.

Findings

Fusion matrices can be simultaneously brought to block upper-triangular form.

The Jordan web structure organizes generalized eigenvectors into connected subwebs.

Eigenvalues are given by 2cos(jπ/n) for specific p and p' values.

Abstract

We consider the W-extended logarithmic minimal model WLM(p,p'). As in the rational minimal models, the so-called fundamental fusion algebra of WLM(p,p') is described by a simple graph fusion algebra. The fusion matrices in the regular representation thereof are mutually commuting, but in general not diagonalizable. Nevertheless, we show that they can be brought simultaneously to block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 3, 5 or 9. The directed graphs associated with the two fundamental modules are described in detail. The corresponding adjacency matrices share a complete set of common generalized eigenvectors organized as a web constructed by interlacing the Jordan chains of the two matrices. This web is here called a Jordan web and it consists of connected subwebs with 1, 3, 5 or 9 generalized eigenvectors. The similarity matrix, formed by concatenating these vectors, simultaneously brings the two fundamental adjacency matrices to Jordan canonical form modulo permutation similarity. The ranks of the participating Jordan blocks are 1 or 3, and the corresponding eigenvalues are given by 2cos(j\pi/n) where j=0,...,n and n=p,p'. For p>1, only some of the modules in the fundamental fusion algebra of WLM(p,p') are associated with boundary conditions within our lattice approach. The regular representation of the corresponding fusion subalgebra has features similar to the ones in the regular representation of the fundamental fusion algebra, but with dimensions of the upper-triangular blocks and connected Jordan-web components given by 1, 2, 3 or 8. Some of the key results are illustrated for W-extended critical percolation WLM(2,3).