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Temperley-Lieb modules and local operators for critical ADE models

Yacine Ikhlef, Alexi Morin-Duchesne

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.

Temperley-Lieb modules and local operators for critical ADE models

Abstract

We investigate critical restricted solid-on-solid models associated to Dynkin diagrams of type , and , 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 or its periodic incarnation . 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.
Paper Structure (86 sections, 27 theorems, 352 equations, 2 figures, 2 tables)

This paper contains 86 sections, 27 theorems, 352 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Definition of the ADE lattice models
  3. Dynkin diagrams and automorphisms
  4. Interaction round-a-face models and transfer matrices
  5. Diagram spaces and families of modules
  6. Diagram spaces
  7. Temperley--Lieb algebras and families of modules
  8. The algebra $\boldsymbol{\mathsf{\mathcal{E}PTL}_N(\beta)}$.
  9. The algebra $\boldsymbol{\mathsf{TL}_N(\beta)}$.
  10. The uncoiled algebras $\boldsymbol{\mathsf{\mathcal{E}PTL}_N(\beta,\gamma)}$, $\boldsymbol{\mathsf{\mathcal{E}PTL}^{\textrm{\tiny$(1)$}}_N(\beta,\alpha)}$ and $\boldsymbol{\mathsf{\mathcal{E}PTL}^{\textrm{\tiny$(2)$}}_N(\beta,\gamma)}$.
  11. Transfer matrices.
  12. Families of modules.
  13. Jones--Wenzl projectors
  14. Link modules over $\boldsymbol{\mathsf{TL}_N(\beta)}$
  15. Link modules over $\boldsymbol{\mathsf{\mathcal{E}PTL}_N(\beta)}$
  16. ...and 71 more sections

Key Result

Proposition 3.1

Let $q \in \mathbb C^\times$ and $k \in \frac{1}{2} \mathbb Z_{\geqslant 0}$. Let also $\mathsf{M}$ be a family of modules over $\mathsf{TL}_N(\beta)$, and $\xi$ be a nonzero element in $\mathsf{M}(2k)$ satisfying The linear map $\phi_\xi: \mathsf{V}_k(N) \to \mathsf{M}(N)$ defined as is well-defined for all admissible $N$, and defines a family of homomorphisms from $\mathsf{V}_k$ to $\mathsf{M}

Figures (2)

  • Figure 1: The Dynkin diagrams for the ADE series.
  • Figure 2: The lattices corresponding to the $8\times 10$ torus and cylinder. The red and green segments indicate periodic boundary conditions.

Theorems & Definitions (27)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Theorem 1
  • ...and 17 more