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Title:
Quantum Geometry of 3-Dimensional Lattices: Existence as Integrability
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Abstract:
We will review various connections between the theory of integrable quantum systems and discrete differential geometry. First, we consider a problem of quantization of 2D circle patterns and the associated discrete conformal transformations connected with the Thurston’s discrete analogue of the Riemann mappings theorem. Next, we study geometric consistency relations between angles on three-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. Quantisation of this structure allowed us to obtain new solutions of the tetrahedron equation, which play a role of an integrability condition for 3D system (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 talk is intended for a general math-phys audience. No special knowledge is assumed.
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