Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

Details Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation

2009-11-19

arxiv.org/abs/0911.3693v1

Physics

Mathematical Physics

Plenary talk at the XVI International Congress on Mathematical Physics, 3-8 August 2009, Prague, Czech Republic

Scientific paper

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.

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