Higher Coxeter graphs associated to affine su(3) modular invariants

TL;DR

This paper explores the structure of affine su(3) modular invariants in 2D RCFT, linking them to generalized Coxeter graphs and introducing a modular operator to classify partition functions.

Contribution

It introduces a modular operator and a method to generate all Type II partition functions via vertex twists on subgroup graphs.

Findings

Type I partition functions derived from the modular operator.

Type II functions obtained through vertex twists that preserve the modular operator.

Establishment of a graph-theoretic framework for classifying affine su(3) invariants.

Abstract

The affine $su(3)$ modular invariant partition functions in 2d RCFT are associated with a set of generalized Coxeter graphs. These partition functions fall into two classes, the block-diagonal (Type I) and the non block-diagonal (Type II) cases, associated, from spectral properties, to the subsets of subgroup and module graphs respectively. We introduce a modular operator $\hat{T}$ taking values on the set of vertices of the subgroup graphs. It allows us to obtain easily the associated Type I partition functions. We also show that all Type II partition functions are obtained by the action of suitable twists $\vartheta$ on the set of vertices of the subgroup graphs. These twists have to preserve the values of the modular operator $\hat{T}$.