Temperley-Lieb modules and local operators for critical ADE models

TL;DR

This paper analyzes critical ADE models with boundary conditions, decomposes their state spaces into Temperley-Lieb modules, and constructs local operators that reflect conformal field theory relations.

Contribution

It provides a detailed decomposition of the state space into Temperley-Lieb modules and introduces local operators satisfying lattice analogs of CFT singular-vector relations.

Findings

Decomposition of state space into irreducible Temperley-Lieb modules

Construction of local connectivity operators on the lattice

Operators satisfy linear difference relations mirroring CFT singular vectors

Abstract

We investigate critical restricted solid-on-solid models associated to Dynkin diagrams of type $A$, $D$ and $E$, with fixed, periodic and twisted periodic boundary conditions. These models are endowed with an action of the diagrams of the Temperley-Lieb category. For each model, we obtain the decomposition of the state space as a direct sum of irreducible modules over the Temperley-Lieb algebra $\mathsf{TL}_N(\beta)$ or its periodic incarnation $\mathsf{\mathcal EPTL}_N(\beta)$. This allows us to recover the known conformal partition functions for these models in the continuum scaling limit. For each irreducible factor arising in the decompositions, we define an associated local operator on the lattice, which behaves like a connectivity operator. Using knowledge from the Temperley-Lieb representation theory at roots of unity, we show that these operators satisfy certain linear difference relations, which are lattice counterparts of the singular-vector relations in conformal field theory.