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Theses

Structures symplectiques sur les espaces de superlacets

Abstract : The goal of the thesis was to study superloopspaces, the geometric version of superstrings in Physics, by extending the classification results contained in Oleg Mokov’s paper : Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems to the supergeometric setting. In it, lies the classification of local homogeneous symplectic forms of order 0, 1 and 2 on the loopspace LM = C1(S1;M) by means of geometric objects on the manifold M. In this thesis, the manifold M becomes a supermanifold Mpjq, the circle S1 becomes a supercircle S1jn and we consider the superloopspace as the space of morphisms of supermanifolds Mor(S1jn;Mpjq). In the two first chapters, we look at the classical and super geometric structures of the superloopspaces. To do this, we restrict ourselves to the two supercircles S1j1 and using the previous works on LM, we define a Fréchet manifold structure on the superloopspaces SLM = Mor(S1j1;M). Then we bring in what we consider as the most natural superstructure on SLM by means of sheaves. In order to work with coordinates, we adopt another point of view considering SLM as the functor of points SLM. Moreover, rewriting Mokhov results in terms of jets allows us to give a rigorous proof of those calculations and also to extend right away the methods of calculations in coordinates. The third chapter contains the new classification results we obtained. Similarly to the classical case, we first show that the order of the jets in the forms of order 0 and 1 is bounded. Then we give the complete classification of the symplectics forms of order 0 by means of differential forms on the manifold Mpjq and of homogeneous symplectics forms of order 1 and 2 using Riemannian metrics and connections on Mpjq. Finally, the fourth chapter is devoted to extending the cohomology results of an other Mokhov’s article : Complex homogeneous forms on loop spaces of smooth manifolds and their cohomology groups. We first discuss the dependance of the odd variable in the homogeneous forms on SLM, and show that with the exterior derivative, the space of homogeneous forms on SLM of a given order m 2 N is a complex. We then calculate the cohomological spaces, completely for the order m = 0 and 1, partially for the order 2 and 3 and we identify the exact forms amongst those of the third chapter.
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Nicolas Bovetto. Structures symplectiques sur les espaces de superlacets. Autre [cond-mat.other]. Université Claude Bernard - Lyon I, 2011. Français. ⟨NNT : 2011LYO10328⟩. ⟨tel-00739570⟩

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