Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

TL;DR

This paper explores the geometric relations in 3D circular quadrilateral lattices, leading to new solutions of the tetrahedron equation and insights into integrable models in statistical mechanics and quantum field theory.

Contribution

It introduces a geometric framework for 3D lattices that yields new solutions to the tetrahedron equation and connects lattice geometry with quantum integrable models.

Findings

Generated new solutions to the tetrahedron equation.

Reproduced all previously known solutions.

Linked lattice geometry to integrable 3D models.

Abstract

We study geometric consistency relations between angles of 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure allowed us to obtain new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as reproduce all those that were previously known. These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry.