### 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 language | English |
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Title of host publication | XVIth International Congress on Mathematical Physics |

Editors | Pavel Exner |

Publisher | World Scientific Publishing Co. |

Pages | 23-44 |

Number of pages | 22 |

ISBN (Electronic) | 9789814304634 |

ISBN (Print) | 9789814304627 |

DOIs | |

Publication status | Published - 1 Jan 2010 |

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### Cite this

*XVIth International Congress on Mathematical Physics*(pp. 23-44). World Scientific Publishing Co.. https://doi.org/10.1142/9789814304634_0001

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*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.

Research output: A Conference proceeding or a Chapter in Book › Chapter

TY - CHAP

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

AU - Bazhanov, Vladimir V.

AU - Mangazeev, Vladimir

AU - Sergeev, Sergey M.

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 - Quantum geometry

KW - discrete differential geometry

KW - integrable quantum systems

KW - Yang-Baxter equation

KW - tetrahedron equation

KW - quadrilateral and circular 3D lattices

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 - 9789814304627

SP - 23

EP - 44

BT - XVIth International Congress on Mathematical Physics

A2 - Exner, Pavel

PB - World Scientific Publishing Co.

ER -