Deformation surfaces, integrable systems and Chern - Simons theory

Details Deformation surfaces, integrable systems and Chern - Simons theory Deformation surfaces, integrable systems and Chern - Simons theory

2000-06-23

arxiv.org/abs/nlin/0006039v1

J.Math.Phys. 42 (2001) 1397-1417

Nonlinear Sciences

Exactly Solvable and Integrable Systems

24 pages, 1 figure, submitted to J. Math. Phys

Scientific paper

10.1063/1.1339831

A few years ago, some of us devised a method to obtain integrable systems in (2+1)-dimensions from the classical non-Abelian pure Chern-Simons action via reduction of the gauge connection in Hermitian symmetric spaces. In this paper we show that the methods developed in studying classical non-Abelian pure Chern-Simons actions, can be naturally implemented by means of a geometrical interpretation of such systems. The Chern-Simons equation of motion turns out to be related to time evolving 2-dimensional surfaces in such a way that these deformations are both locally compatible with the Gauss-Mainardi-Codazzi equations and completely integrable. The properties of these relationships are investigated together with the most relevant consequences. Explicit examples of integrable surface deformations are displayed and discussed.

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