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Solving the tetrahedron equation by Teichmüller TQFT

Myungbo Shim, Xiaoyue Sun, Hao Ellery Wang, Junya Yagi

Abstract

We propose an approach to construct three-dimensional lattice models using line defects in state integral models on shaped triangulations of 3-manifolds. The Boltzmann weights for these models satisfy a variant of the tetrahedron equation, which implies integrability under suitable assumptions on R-matrices and transfer matrices. As an explicit example, we present a solution produced by Teichmüller TQFT.

Solving the tetrahedron equation by Teichmüller TQFT

Abstract

We propose an approach to construct three-dimensional lattice models using line defects in state integral models on shaped triangulations of 3-manifolds. The Boltzmann weights for these models satisfy a variant of the tetrahedron equation, which implies integrability under suitable assumptions on R-matrices and transfer matrices. As an explicit example, we present a solution produced by Teichmüller TQFT.
Paper Structure (11 sections, 5 theorems, 40 equations, 9 figures)

This paper contains 11 sections, 5 theorems, 40 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Three-dimensional bipartite lattice models
  3. Spin models on bipartite cubic lattices
  4. Bicolored tetrahedron equations and integrability
  5. State integrable models and the BTEs
  6. State integral models on shaped triangulations
  7. BTEs from state integral models
  8. Solution by Teichmüller TQFT
  9. Teichmüller TQFT
  10. Teichmüller TQFT solves the BTEs
  11. Conclusions

Key Result

Lemma 2.1

If $\dddot{R}$ is invertible, then $\mathbf{R}$, $\dot\mathbf{R}$, $\ddot\mathbf{R}$ satisfy the bicolored Yang--Baxter equation

Figures (9)

  • Figure 1: The tetrahedron equation for IRC models.
  • Figure 2: (a) A neighborhood of vertex $(1_l 2_m 3_n)$ with $l+m+n$ even. (b) A neighborhood of plane $3_n$ with $n$ even.
  • Figure 3: Spins around vertex $(\alpha\beta\gamma)$ in (a) a vertex model and (b) an IRC model.
  • Figure 4: The tetrahedral weights for a positive tetrahedron (left) and a negative tetrahedron (right).
  • Figure 5: An example of a shaped $2$--$3$ move.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof