Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)

TL;DR

This paper explores the fusion algebra of logarithmic minimal models WLM(1,p), deriving generalized Verlinde formulas and partition functions through spectral and Jordan form analysis of fusion matrices, based on modular data.

Contribution

It introduces a method to diagonalize fusion matrices via Jordan forms and derives a generalized Verlinde formula linked to modular data and partition functions.

Findings

Fusion matrices can be simultaneously brought to Jordan form.

A generalized Verlinde formula relates fusion matrices to modular data.

Partition functions are connected to the spectral properties of fusion matrices.

Abstract

The infinite series of logarithmic minimal models LM(1,p) is considered in the W-extended picture where they are denoted by WLM(1,p). As in the rational models, the fusion algebra of WLM(1,p) is described by a simple graph fusion algebra. The corresponding fusion matrices are mutually commuting, but in general not diagonalizable. Nevertheless, they can be simultaneously brought to Jordan form by a similarity transformation. The spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into Jordan canonical form consisting of Jordan blocks of rank 1, 2 or 3. The various similarity transformations and Jordan forms are determined from the modular data. This gives rise to a generalized Verlinde formula for the fusion matrices. Its relation to the partition functions in the model is discussed in a general framework. By application of a particular structure matrix and its Moore-Penrose inverse, this Verlinde formula reduces to the generalized Verlinde formula for the associated Grothendieck ring.