School Seminars and Colloquia

Quantum Geometry of 3-Dimensional Lattices: Existence as Integrability

Miscellaneous Seminar

by Prof. Vladimir Bazhanov

Institution: Australian National University
Date: Wed 23rd November 2011
Time: 11:00 AM
Location: Room 213, Richard Berry Building

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.

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