Extensions supersymétriques des équations structurelles des supervariétés plongées dans des superespaces

The goal of this thesis consisting of articles is to study certain geometric aspects of supermanifolds associated with integrable suspersymmetric systems. This work is contained in four published articles and one currently submitted article in international peer-reviewed journals. In the first article, two supersymmetric extensions of the Gauss–Weingarten and Gauss–Codazzi equations for surfaces immersed in Euclidean superspaces were constructed. This allowed us to provide a geometric characterization of such surfaces with linearly independent tangent vectors oriented in the directions of the infinitesimal displacement of the fermionic covariant derivatives. In addition, a study of the symmetries of the supersymmetric versions of the Gauss–Codazzi equations led to the construction of invariant solutions, involving bosonic and fermionic variables, through the symmetry reduction method, which led to nontrivial surfaces, e.g. vanishing Gauss curvature surfaces. In the second article, a conjecture stating the necessary conditions for a system to be integrable in the sense of soliton theory was extended to the supersymmetric cases. This was accomplished by introducing a new projection operator and by comparing the symmetries of the original system to those of the associated linear problem. This conjecture was applied to some examples and a fermionic “spectral” parameter was introduced in one of the systems. In the third article, two supersymmetric versions of the Fokas–Gel’fand formula for the immersion of soliton surfaces in Lie superalgebras were constructed. The geometric characterization of the immersion function presented in this article allowed us to investigate the behavior of the associated surfaces. These theoretical considerations were applied to the supersymmetric sine-Gordon equation, for which constant Gaussian curvature surfaces and nonlinear-type surfaces were obtained. The fourth article was devoted to integrability properties of the supersymmetric sine-Gordon equation and to the construction of explicit multisoliton solutions. Two types of linear spectral problems, a set of coupled super-Riccati equations and the auto-Bäcklund transformation, all equivalent to the supersymmetric sine-Gordon equation, were studied. In addition, a detailed analysis of the nth Darboux transformations allowed us to find nontrivial multisoliton solutions of the supersymmetric sine-Gordon equation. These solutions were used to investigate the bosonic supersymmetric version of the Sym–Tafel immersion formula. In the fifth article, a new geometric characterization of the Fokas–Gel’fand immersion formula was presented. In order to do this, three different types of linear spectral problems were studied, one involving the covariant fermionic derivatives, one involving the bosonic variable derivatives and one involving the fermionic variable derivatives. This geometric characterization involves eight linearly independent coefficients for both the first and second fundamental forms, in constrast with three such coefficients in the third article, which leads to a richer geometry in the sense that curve-like supermanifolds in the third article are of higher dimensions in the fifth article.

Keyphrases: Integrable systems, Supersymmetric systems, Supermanifolds, Conformally parametrized surfaces, Lie superalgebras, Gauss–Weingarten and Gauss–Codazzi equations, Sym–Tafel and Fokas–Gel’fand immersion formulas, Supersymmetric sine-Gordon equation, Supersymetric Bäcklund and Darboux transformations, Symmetry reduction