Quantum geometry of 3-dimensional lattices

Abstract

We study geometric consistency relations between angles on 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 leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). 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. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

27 pages, 10 figures. Minor corrections, references added

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Cited by in corpus (32)

• On the Yang-Baxter equation for the six-vertex model
• Stochastic $R$ matrix for $U_q(A_n^{(1)})$
• Q-operators in the six-vertex model
• Desargues maps and the Hirota-Miwa equation
• Tetrahedron and 3D reflection equations from quantized algebra of functions
• Set-theoretical reflection equation: Classification of reflection maps
• Construction of $R$-matrices for symmetric tensor representations related to $U_q(\widehat{s{l}_n})$
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Non-commutative lattice modified Gel'fand-Dikii systems
• An integrable 3D lattice model with positive Boltzmann weights
• Tetrahedron Equation and Quantum R Matrices for Spin Representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}
• Super-tetrahedra and super-algebras
• The pentagon relation and incidence geometry
• Tetrahedron equation and quantum R matrices for $q$-oscillator representations of $U_q(A_{2n}^{(2)}),U_q(C_n^{(1)})$ and $U_q(D_{n+1}^{(2)})$
• Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
• An Ising-type formulation of the six-vertex model
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Boundary matrices for the higher spin six vertex model
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Tetrahedron equation and quantum R matrices for infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2)
• Classical integrable field theories in discrete 2+1 dimensional space-time
• Tetrahedron Equation and Quantum $R$ Matrices for modular double of $U_q(D_{n+1}^{(2)}),U_q(A_{2n}^{(2)})$ and $U_q(C_n^{(1)})$
• An algebraic approach to discrete time integrability
• An Elliptic Parameterisation of the Zamolodchikov Model
• Quadrangular sets in projective line and in Moebius space, and geometric interpretation of the non-commutative discrete Schwarzian Kadomtsev-Petviashvili equation
• Tetrahedron equation and Schur functions
• Miquel dynamics for circle patterns
• A polynomial formula for the solution of 3D reflection equation
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
• Geometry of quadrilateral nets: second Hamiltonian form
• State integral models and the tetrahedron equation