TY - CHAP

T1 - Quantum geometry of 3-dimensional lattices and tetrahedron equation

AU - Bazhanov, Vladimir V.

AU - Mangazeev, Vladimir V.

AU - Sergeev, Sergey M.

N1 - Publisher Copyright:
© 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

KW - Discrete differential geometry

KW - Integrable quantum systems

KW - Quadrilateral and circular 3D lattices

KW - Quantum geometry

KW - Tetrahedron equation

KW - Yang-baxter equation

UR - http://www.scopus.com/inward/record.url?scp=84969704612&partnerID=8YFLogxK

U2 - 10.1142/9789814304634_0001

DO - 10.1142/9789814304634_0001

M3 - Chapter

SN - 981430462X

SN - 9789814304627

SP - 23

EP - 44

BT - XVIth International Congress on Mathematical Physics

PB - World Scientific Publishing Co

ER -