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State integral models and the tetrahedron equation

Junya Yagi

Abstract

It is shown that for a class of state integral models on shaped pseudo 3-manifolds, including the edge formulation of Teichmüller TQFT, the Boltzmann weight assigned to a tetrahedron solves the tetrahedron equation. The dihedral angles of the tetrahedron play the role of spectral parameters.

State integral models and the tetrahedron equation

Abstract

It is shown that for a class of state integral models on shaped pseudo 3-manifolds, including the edge formulation of Teichmüller TQFT, the Boltzmann weight assigned to a tetrahedron solves the tetrahedron equation. The dihedral angles of the tetrahedron play the role of spectral parameters.
Paper Structure (16 sections, 1 theorem, 48 equations, 5 figures)

This paper contains 16 sections, 1 theorem, 48 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Integrable IRC models
  3. IRC models
  4. Integrability and the tetrahedron equation
  5. Six-parameter tetrahedron equation
  6. State integral models on shaped pseudo $3$-manifolds
  7. Shaped pseudo $3$-manifolds
  8. Tetrahedral weights
  9. Shaped pentagon identity
  10. Tetrahedral symmetry
  11. Transpose of the tetrahedral weight
  12. Examples
  13. Meromorphic 3D index
  14. Kashaev--Luo--Vartanov model
  15. Teichmüller TQFT
  16. ...and 1 more sections

Key Result

Proposition 1

The six-parameter tetrahedron equation eq:TE6 is satisfied in $\mathcal{D} \times S^8$ by

Figures (5)

  • Figure 1: The tetrahedron equation in the IRC form.
  • Figure 2: The local Boltzmann weight for an IRC model.
  • Figure 3: The wiring diagram presentation of the local Boltzmann weight \ref{['eq:W']} and the tetrahedron equation \ref{['eq:TE']}.
  • Figure 4: The tetrahedral weights for a positive tetrahedron (left) and a negative tetrahedron (right).
  • Figure 5: A shaped $2$--$3$ move for positive tetrahedra. The vertices of the bipyramid in the middle are labeled by the number of incoming edges, including an invisible edge from vertex 1 to vertex 3. In the shaped pentagon identity \ref{['eq:P']}, the tetrahedron on the left or right that does not contain vertex $v$ is labeled $(v)$.

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Remark 2