Propriétés géométriques des surfaces associées aux solutions des modèles sigma grassmanniens en deux dimensions

In this Ph. D. thesis, we analyze the geometric properties of surfaces obtained from the classical solutions of the two-dimensional bosonic and supersymmetric sigma models which has Grassmann manifolds G(m,n) as target space. In particular, we consider the metric, the fundamental forms and the gaussian curvature induced by these surfaces which naturally live in the su(n) Lie algebra.   The first chapter presents some preliminary tools to understand the elements of the following chapters. We present non-abelian gauge theories and bosonic grassmannian sigma models as well as its supersymmetric counterpart. Another section presents a construction of surfaces in the Lie algebra su(n) from the solutions of the bosonic sigma models.  The three last chapters contained in this thesis presents the constraints that have to be imposed on the solutions of the models in order to generate constant gaussian curvature surfaces. From these constraints, we can give a classification of the solutions depending on the possible values of the curvature. The first two papers presents an investigation of these surfaces and of their associated solutions for the bosonic grassmannian sigma models. In the third paper, we generalize our approach to a supersymmetric extension of the bosonic CP^(n-1)= G(1,n) sigma model including some results for the general Grassmann manifold G(m,n).  In chapter 2, we study the geometric properties of surfaces associated to holomorphic solutions of the grassmannian sigma models. We give a complete classification of these constant curvature solutions for the particular models G(2,n) with n=3,4,5. Furthermore, we establish two conjectures on the possible values of the gaussian curvature. We also give some elements of proof for these conjectures in terms of Plücker coordinates and immersions as well as Veronese curves. These results are published in the Journal of Geometry and Physics.  The third chapter presents a similar analysis as in the second chapter in the case of non-holomorphic solutions of the bosonic grassmannian sigma models. This work generalizes the results obtained in the first paper and give a systematic algorithm to obtain such surfaces from the known solutions of the models. These results are published in the Journal of Geometry and Physics.  In the last chapter of this thesis, we consider a N=2 supersymmetric extension of the bosonic sigma model which has the CP^(n-1)=G(1,n) manifold as target space. This chapter presents a geometric description of the surfaces obtained from the solutions of the model and shows, in the holomorphic case, that they have constant gaussian curvature if and only if the solutions consists of a generalization of the Veronese curve. Furthermore, using a gauge invariant formulation of the model in terms of orthogonal projectors, we obtain explicit non-holomorphic solutions and study the geometry of their associated surfaces. These results are submitted to Communications in Mathematical Physics.

Keyphrases: Grassmannian sigma models, Veronese curve, Supersymmetry, Holomorphic and non-holomorphic solutions, Geometry of surfaces, Orthogonal projectors