SMRI Seminar: Bazhanov -- Quantum geometry of 3-dimensional lattices

SMRI Seminar 

`Quantum geometry of 3-dimensional lattices’ 

Vladimir Bazhanov (Australian National University) 

October 26, 2021 2:00pm - 3:00pm via Zoom.  

REGISTER HERE:
https://uni-sydney.zoom.us/meeting/register/tZMvceuqrDMsGdCAEu6zpUf-6XeU1nuSR_SJ 

Abstract: 

In this lecture I will explain a relationship between incidence theorems in elementary
geometry and the theory of integrable systems, both classical and quantum.  We will
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.