Quantum geometry of 3-dimensional lattices and tetrahedron equation

Vladimir V. Bazhanov, Vladimir Mangazeev, Sergey M. Sergeev

Research output: A Conference proceeding or a Chapter in BookChapter

4 Citations (Scopus)

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.

Original languageEnglish
Title of host publicationXVIth International Congress on Mathematical Physics
EditorsPavel Exner
PublisherWorld Scientific Publishing Co.
Pages23-44
Number of pages22
ISBN (Electronic)9789814304634
ISBN (Print)9789814304627
DOIs
Publication statusPublished - 1 Jan 2010

Fingerprint

Triangular pyramid
tetrahedrons
Yang-Baxter Equation
geometry
Canonical Transformation
Quantum Fluctuations
Poisson Bracket
brackets
Nontrivial Solution
Quantum Field Theory
statistical mechanics
Statistical Mechanics
Quantization
algebra
Circle
Face
analogs
Analogue
Angle
Algebra

Cite this

Bazhanov, V. V., Mangazeev, V., & Sergeev, S. M. (2010). Quantum geometry of 3-dimensional lattices and tetrahedron equation. In P. Exner (Ed.), XVIth International Congress on Mathematical Physics (pp. 23-44). World Scientific Publishing Co.. https://doi.org/10.1142/9789814304634_0001
Bazhanov, Vladimir V. ; Mangazeev, Vladimir ; Sergeev, Sergey M. / Quantum geometry of 3-dimensional lattices and tetrahedron equation. XVIth International Congress on Mathematical Physics. editor / Pavel Exner. World Scientific Publishing Co., 2010. pp. 23-44
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Bazhanov, VV, Mangazeev, V & Sergeev, SM 2010, Quantum geometry of 3-dimensional lattices and tetrahedron equation. in P Exner (ed.), XVIth International Congress on Mathematical Physics. World Scientific Publishing Co., pp. 23-44. https://doi.org/10.1142/9789814304634_0001

Quantum geometry of 3-dimensional lattices and tetrahedron equation. / Bazhanov, Vladimir V.; Mangazeev, Vladimir; Sergeev, Sergey M.

XVIth International Congress on Mathematical Physics. ed. / Pavel Exner. World Scientific Publishing Co., 2010. p. 23-44.

Research output: A Conference proceeding or a Chapter in BookChapter

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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.

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Bazhanov VV, Mangazeev V, Sergeev SM. Quantum geometry of 3-dimensional lattices and tetrahedron equation. In Exner P, editor, XVIth International Congress on Mathematical Physics. World Scientific Publishing Co. 2010. p. 23-44 https://doi.org/10.1142/9789814304634_0001