Towards an explicit expression of the Seiberg-Witten map at all orders

The Seiberg-Witten map links noncommutative gauge theories to ordinary gauge theories, and allows to express the noncommutative variables in terms of the commutative ones. Its explicit form can be found order by order in the noncommutative parameter theta and the gauge potential A by the requirement that gauge orbits are mapped on gauge orbits. This of course leaves ambiguities, corresponding to gauge transformations, and there is an infinity of solutions. Is there one better, clearer than the others ? In the abelian case, we were able to find a solution, linked by a gauge transformation to already known formulas, which has the property of admitting a recursive formulation, uncovering some pattern in the map. In the special case of a pure gauge, both abelian and non-abelian, these expressions can be summed up, and the transformation is expressed using the parametrisation in terms of the gauge group.Comment: 17 pages. Latex, 1 figure. v2: minor changes, published versio

Rôle(s) du champ de fond antisymétrique en théorie des cordes.

The guideline of this thesis has been to understand the role of the antisymmetric background B-field in string theory. As a natural companion of the curvature g, it is indeed one of the most promising leads in many of recent developments, to which I hope this work will honourably contribute. The first chapter is dedicated to the study of the Seiberg-Witten map, by which ordinary branes in a non trivial B-field are linked to non commutative branes. Searching an explicit expression of it in the gauge sector leads to a better understanding of its meaning. Beyond describing the couplings of D-branes to non trivial background fields, this would allow a new insight on their non abelian dynamics, through Morita equivalence. This non abelian dynamics is also the aim of chapter 2, this time in the context of M-theory. There I try to describe the field content of non-abelian M5 branes, by different methods including supersymmetry and non-abelian 2-forms. Some geometrical aspects may also help to reproduce the expected N^3 anomaly. Finally SU(3)-manifolds are relevant for compactification when a non trivial B-field is turned on. Mirror symmetry, defined on less general Calabi-Yau manifolds, can then be extended as T-duality on T^3-fibered SU(3)-manifolds, in the spirit of the Strominger-Yau-Zaslow conjecture. Its geometrical description involve mixings of the components of the intrinsic torsion with those of the curvature H=dB, as detailed in chapter 3.Cette thèse s'attache à comprendre le rôle du champ de fond antisymétrique B en théorie des cordes. Nouveauté essentielle et prometteuse par rapport à la théorie des champs, puisqu'il accompagne naturellement la courbure de l'espace-temps g, son importance a été soulignée ces dernières années dans différents domaines, auxquels j'ai tenté de contribuer. Le premier chapitre étudie la transformation de Seiberg-Witten, qui relie des branes ordinaires plongées dans un champ B à des branes non-commutatives. A la recherche d'une expression explicite sur le secteur de jauge, il tente d'en éclaircir la signification. Le chapitre 2 s'attaque à la dynamique non abélienne des branes M5 en M-théorie. Par différentes approches, supersymétrique ou plus géométrique, je tente d'y proposer un contenu en champs pour un paquet de N M5-branes, expliquant leur anomalie en N^3. Ces champs formeraient alors une version non-abélienne des théories de jauge à connexion tensorielle. Enfin, la présence d'un champ B autorise des variétés de compactification plus générales que les espaces de Calabi-Yau, dites variétés à structure SU(3). La symétrie-miroir peut être étendue dans ce cadre, en la décrivant comme une T-dualité le long d'une fibration toroïdale. Sa description géométrique met alors en jeu les composantes de la torsion intrinsèque, qui sont mélangées à celles de la courbure H=dB, ainsi que je le détaille dans le chapitre 3