Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
arXiv:0911.3693 · doi:10.1142/9789814304634_0001
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.
Plenary talk at the XVI International Congress on Mathematical Physics, 3-8 August 2009, Prague, Czech Republic
References in corpus (5)
- Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
- Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
- Quantum geometry of 3-dimensional lattices
- Super-tetrahedra and super-algebras
- Tetrahedron equations, boundary states and hidden structure of U_q(D_n^1)
Cited by in corpus (10)
- An integrable 3D lattice model with positive Boltzmann weights
- Super-tetrahedra and super-algebras
- The pentagon relation and incidence geometry
- Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras
- Stochasticization of Solutions to the Yang-Baxter Equation
- The Yang-Baxter relation and gauge invariance
- New solutions to the tetrahedron equation associated with quantized six-vertex models
- Solution of tetrahedron equation and cluster algebras
- Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras
- Automorphisms of the Cuntz algebras