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.
Self-duality of the 6j-symbol and Fisher zeros for the Tetrahedron
LIPN, CNRS UMR 7030, Institut Galilée, Université Paris 13（巴黎第十三大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；Université de Lyon, ENS de Lyon, Laboratoire de Physique, CNRS UMR 5672（里昂大学）
On spectral equations for an evolution operator of a $q$-oscillator lattice
Faculty of Science and Technology, University of Canberra（堪培拉大学科学技术学院）；Department of Fundamental and Theoretical Physics, Research School of Physics and Engineering, Australian National University（澳大利亚国立大学物理与工程学院基础与理论物理系）
