Discrete Integrable Geometry and Physics

by Bobenko, Alexander I.; Seiler, Ruedi

• ISBN13: 9780198501602
• ISBN10: 0198501609
• Format: Hardcover
• Copyright: 1999-08-26
• Publisher: Oxford University Press

Summary

Recent interactions between the fields of geometry, classical and quantum dynamical systems, and the visualization of geometric objects have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discrete analogues. For these analogues the corresponding difference equations are often integrable, which has in turn led to important results in such areas as condensed matter physics and quantum field theory. This book combines the efforts of a distinguished team of authors from various fields in mathematics and physics to provide an accessible overview of the subject. The book begins with the mathematical concepts of discrete geometry and discrete integrable systems, which are interesting on their own, and then proceeds to develop the many connections with classical and quantum dynamics.

Table of Contents

List of contributors

Introduction

Geometry

Discretization of Surfaces and Integrable Systems

A Discrete Version of the Darboux Transform for Isothermic Surfaces

Discrete Amsler Surfaces and a Discrete Painleve III Equation

Discrete cmc Surfaces and Discrete Holomorphic Maps

Discrete Indefinite Affine Spheres

Geometry of Discrete Curves and Lattices and Integrable Difference Equations

Classical Systems

R-matrices and Integrable Discretizations

Discrete Painleve Equations and Symmetry Reduction on the Lattice

Lagrangian Description of Doubly Discrete Sine-Gordon Type Models

Quantum Systems

Spectra of Quantum Integrals

Algebraic Quantization of Integrable Models in Discrete Space-Time

Affine Toda Field Theory as a Three-dimensional Integrable System

Quantum Hyperbolic Invariants of Knots

Charge Transport in the Discretized Landau Model

Index

Table of Contents provided by Publisher. All Rights Reserved.