[論文レビュー] Temperley-Lieb modules and local operators for critical ADE models

Yacine Ikhlef, Alexi Morin-Duchesne|arXiv (Cornell University)|Feb 17, 2026

Algebraic structures and combinatorial models被引用数 0

ひとこと要約

The paper analyzes critical ADE lattice models with Temperley–Lieb symmetry, decomposes their state spaces into irreducible TL modules, shows the continuum scaling to minimal CFTs, and constructs lattice local operators satisfying discrete singular-vector relations.

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 $\mathsf{TL}_N(β)$ or its periodic incarnation $\mathsf{\mathcal EPTL}_N(β)$. 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.

研究の動機と目的

• Motivate understanding of critical ADE RSOS models and their ADE classifications (A, D, E) through Temperley–Lieb (TL) algebra actions.
• Determine the TL-module structure of the state space under fixed, periodic, and twisted periodic boundary conditions.
• Show that cylinder and torus partition functions reproduce known minimal-CFT results in the scaling limit.
• Construct lattice local operators associated to irreducible TL modules and derive discrete linear relations akin to Virasoro singular-vector equations.

提案手法

• Define critical ADE lattice models with heights on Dynkin diagrams and specify adjacency-based Boltzmann weights.
• Describe diagram spaces and the Temperley–Lieb category and its modules at roots of unity.
• Decompose the transfer-matrix action on state spaces into irreducible TL modules for fixed, periodic, and twisted boundary conditions.
• Compute cylinder and torus partition functions and verify their scaling to minimal-CFT partition functions.
• Construct bulk and boundary local operators corresponding to each irreducible module and establish linear difference equations they satisfy.

実験結果

リサーチクエスチョン

• RQ1 How does the state space of critical ADE lattice models decompose under the action of the Temperley–Lieb category?
• RQ2 What are the scaling limits of these TL-module decompositions to Virasoro minimal models and their characters?
• RQ3 Can one define lattice local operators associated to irreducible TL modules that satisfy discrete singular-vector type relations?
• RQ4 How do boundary, periodic, and twisted boundary conditions affect the TL-module structure and resulting partition functions?

主な発見

• The state space decomposes as a direct sum of irreducible TL modules for each ADE model under various boundary conditions.
• Cylinder and torus partition functions reproduce the expected conformal partition functions in the scaling limit.
• For each irreducible module arising in the periodic/twisted periodic cases, a corresponding local operator is defined on a loop visiting 2k lattice sites.
• Local operators satisfy linear difference relations that serve as lattice counterparts of conformal singular-vector relations.
• The approach uses TL representation theory at roots of unity to organize the decomposition and connect to minimal CFT data.