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On the solutions of the local Zamolodchikov tetrahedron equation

M. Chirkov, S. Konstantinou-Rizos

Abstract

We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by $3\times 3$ matrices. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation. Moreover, we demonstrate that, from some of our solutions, we can recover the 4-simplex extensions of Kashaev--Korepanov--Sergeev and Hirota type tetrahedron maps. Finally, we construct several novel 4-simplex maps.

On the solutions of the local Zamolodchikov tetrahedron equation

Abstract

We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by matrices. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation. Moreover, we demonstrate that, from some of our solutions, we can recover the 4-simplex extensions of Kashaev--Korepanov--Sergeev and Hirota type tetrahedron maps. Finally, we construct several novel 4-simplex maps.
Paper Structure (9 sections, 6 theorems, 43 equations, 1 figure)

This paper contains 9 sections, 6 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Local Zamolodchikov tetrahedron equation and 4-simplex maps
  5. Solutions of the local tetrahedron equation
  6. Set-theoretical solutions to the 4-simplex equation
  7. Recovery of known 4-simplex maps
  8. Construction of novel 4-simplex maps
  9. Conclusions

Key Result

Theorem 3.1

Let $x_i\in \mathbb{C}$, $i=1,\ldots, 9$, be free variables. Then, the essentially different solutions of the local Zamolodchikov tetrahedron equation local-tetrahedron are given in the form of correspondences generated by matrices of the form: where ${\rm A}= {\rm A} (x_1,x_2,x_3,x_4,x_5)$, ${\rm B}= {\rm B} (x_1,x_2,x_3,x_4,x_5)$, ${\rm C}= {\rm C}(x_1,x_2,x_3,x_4)$ and ${\rm D}= {\r

Figures (1)

  • Figure 1: Reflection symmetry example.

Theorems & Definitions (13)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Example 4.1
  • Example 4.2
  • Proposition 4.3
  • proof
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 3 more