Symmetry in Nonlinear Mathematical Physics - 2009
Véronique Hussin (Université de Montréal, Canada)
Canonical expressions for the surfaces obtained from classical sigma models
It has been shown recently that some of these surfaces are related to a fundamental projector P0, constructed out of holomorphic solutions. New surfaces were also obtained by constructing new projectors out of mixed solutions. In this paper, we want to give a description of such surfaces through a canonical formalism. Indeed, starting from an orthogonal projector P constructed from solutions of the sigma model, we will describe a procedure to get the coordinates of the radius real vector X in N2-1 dimensions which give rise to a canonical expression of the corresponding surface. This surface is characterised by a quadratic equation on the components of this vector.
The question of independence of the coordinates is also considered. We give a complete proof of the fact that we have only 2(N-1) real independent quantities for projectors of rank 1 and N-1. For the other cases, partial results are exhibited for relevant special solutions constructed form the holomorphic Veronese sequence.