Canonical expressions for the surfaces obtained from classical sigma models

Abstract:
The construction of real surfaces in N²-1 dimensions, obtained from the sigma model, has been discussed in different papers during the last years. It is based on the properties of the relevant projectors constructed from finite action solutions of the sigma model. Let us remind that one gets three classes of such solutions, namely holomorphic, antiholomorphic and mixed. The mixed solutions can be determined from either the holomorphic or the antiholomorphic nonconstant functions. The procedure is well-known.

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.