The Ads/CFT correspondence of technology.
Systèmes intégrables non commutatifs et la correspondance Ads/CFT en cosmologie.
Résumé
My argument unfolds along two main lines of work. Both arguments are treated a relationship between string theory and phenomenological aspects / cosmological. On the one hand, the noncommutative geometry (NC) is a natural consequence of the presence of branes and fluxes in string theory. The non-commutativity distorts some fundamental properties ordinary theories describing such electro-weak interactions and strong or statistical models. It is in this sense that the geometry represents a CN application to the phenomenology of strings. On the other hand, the branes are the key ingredient in models of brane worlds. The model Randall-Sundrum (RS) in particular, offers new perspectives both from the point of view of cosmology, creating scenarios of cosmological evolution unconventional from the point of view of holography. The first part of the thesis is dedicated to the NC geometry and, in particular, theories NC integrable fields. The main purpose of the work was to study the consequences of non commutativit'e compared al'integrabilite. More precisely, we wanted to verify or refute in a theorem that links the NC, in two dimensions, the integrability is factoring the matrix S. With integrability we speak of the existence of an infinite number of local currents preserved, associated the symmetries of the theory of champs.Le starting point has been to ensure the presence of such currents, using the formalism of the double complex. This method allows to obtain the equations of motion as conditions of int 'egrabilite a linear system of differential equations. has from solutions of linear system even after the current canned. In Weyl operator formalism, the procedure is immediately generalized to the NC geometry. An algebra of functions (Weyl operators) defined on a space is associated with a CN CN algebra or number of functions is executed by an NC Moyal product: the product. By introducing the product in the linear system and deducting the equations of motion CN, we obtain the generalization NC Bicomplex. It was the first model infers in generalizing the double complex of modeledesine Gordon-(SG) geometry NC. We have deduced (in collaboration with Grisaru, Penati, Tamassia) action corresponding to equations of motion previously established by Grisaru and Penati. The calculation of scattering amplitudes and production has determined the characteristics of the S matrix of the model. Acausal behavior have been identified for diffusion process. In addition, the processes of production have a nonzero amplitude: where the non-validity of the theorem of integrability vs.. factorization for this version of the NC model of SG. Other properties have been put in evidence, as the relation with string theory and bosonization. The second model was proposed SG CN in collaboration with Lechtenfeld, Penati, Popov, Tamassia. The equations of motion have been derived from dimensional reduction of the sigma model in 2 +1 dimensional NC, which in turn is the reduction of the theory of self-dual Yang-Mills in 2 +2 dimensions CN (N = describing superstrings 2avecchamps B). The action was calculated that the amplitudes of the same. Production processes possessing zero amplitudes and diffusion ones do not dependent parameter of NC, and lead to causal behavior. The second model seems to obey SG CN has the equivalent between integrability and factorization matrix S. The reduction of string theory retains its validity even at the level of action contrary to the previous model. The second part of my thesis deals with models of brane universes, or more precisely of SAR models. The model proposed by Randall and Sundrumse located in a bulk 5-dimensional, characterized per a symmetry of Z2 orbifold wrt the position of the 4-dimensional brane. Thanks to the warp factor that multiplies the subspace 4-dimensional parallel to the brane, we obtain the location of the modes of the graviton. Therefore, the Newtonian gravitational potential is effective to energies are smaller than Planck mass. By introducing a further term in the bulk material and considering the exchange of energy between brane and bulk, a variety of new cosmologies in drift. In the first part of my work on RS we have proposed a similar model is in a 7-dimensional bulk. The 6-dimensional brane - compactification with two dimensions - is placed at the orbifold fixed point Z2. To study the cosmological evolution when we are in relationship with the observations, we have introduced the exchange of energy between brane and bulk. The possible scenarios are many and depend on the explicit form of the parameter of exchange of energy. Among other things, the fixed points possess a positive acceleration, which may well represent the recent acceleration of the universe. There are also stable for a wide range of parameters values. Finally we can draw scenarios that start from an initial accelerated, passing a succession era of deceleration, to end on a stable fixed point of inflation. The models of universe brane has the RS possess a dual via holographic AdS / CFT. The AdS / CFT establishes a theory of supergravity (or plusg 'en' eral, strings) in a background field of anti de Sitter (AdS) in d + 1 dimensions is dual to a theory of field line (CFT) in d dimensions. Taking into account the differences presented in the two descriptions, this correspondence has been made more precise in the formulation of the holographic renormalization. If space in AdS is regularized by an IR Cutoff, the corresponding CFT RESULT regularized by an ultraviolet cutoff and coupled to the gravity d-dimensional. In analogy al'analyse effected in five dimensions by Kiritsis, we constructed the dual theory to cosmological model SAR in seven dimensions. To capture the dynamics dictated by the exchange of energy between brane and bulk, the six-dimensional holographic theory was generalized to the interacting case (between material and CFT) and non-compliant. The result is the relationship between the mass parameters belonging to two descriptions and the exchange of energy between, on one side, and the interaction parameter, on the other. In addition, the parameter of failure to comply is associated with self-interaction parameter of bulk in the description of seven-dimensional supergravity. The research therefore includes results which may be implemented in the phenomenology and cosmology of the string. On the one hand we investigate the influence of bound al'integrabilite noncommutative model of SG. On the other hand, the cosmological consequences of the location of the RS model in seven dimensions have been studied and the AdS / CFT has been applied to derive information on the theory dual, coupled to gravity.
Ma these se deroule suivant deux principales lignes de recherche. Les deux arguments traites constituent une relation entre la theorie des cordes et les aspects phenomenologiques/cosmologiques. D'une part, la geometrie noncommutative (NC) est une consequence naturelle de la presence de branes et flux dans la theorie des cordes. La non commutativite deforme certaines proprietes fondamentales des theories ordinaires decrivant par exemple les interactions electro–faibles et fortes ou les modeles statistiques. C'est dans ce sens que la geometrie NC represente une application a la phenomenologie des cordes. D'autre part, les branes sont l'ingredient clé des modeles d'univers branaires. Le modele de Randall–Sundrum (RS) en particulier offre de nouvelles perspectives tant du point de vue de la cosmologie, ouvrant des scenarios d'evolution cosmologique non conventionnelle, que du point de vue de l'holographie. La premiere partie de la these est dediee a la geometrie NC et, en particulier, aux theories de champs NC integrables. Le but principal du travail a ete d'etudier les consequences de la non commutativit´e par rapport a l'integrabilite. Plus precisement, on a voulu verifier ou refuter dans un contexte NC le theoreme qui lie, en deux dimensions, l'integrabilite a la factorisation de la matrice S. Avec integrabilite on parle de l'existence d'un nombre infini de courants locaux conserves, associes aux symetries de la theorie de champs.Le point de depart a donc ete de garantir la presence de tels courants, au moyen du formalisme du bicomplexe. Cette methode permet d'obtenir les equations du mouvement en tant que conditions d'int´ egrabilite d'un systeme d'equations differentielles lineaires. a partir des solutions du meme systeme lineaire suivent les courants conserves. En exploitant le formalisme de Weyl, la procedure est immediatement generalisable a la geometrie NC. Une algebre de fonctions (operateurs de Weyl) definie sur un espace NC est associee a une algebre NC de fonctions ou la multiplication est executee au moyen d'un produit NC de Moyal: le produit. En introduisant le produit au niveau du systeme lineaire et en en deduisant les equations du mouvement NC, on obtient la generalisation NC du bicomplexe. On a infere le premier modele en generalisant le bicomplexe du modeledesine–Gordon(SG)a la geometrie NC. Nous avons deduit (en collaboration avec Grisaru, Penati, Tamassia) l'action correspondante aux equations du mouvement precedemment etablies par Grisaru et Penati. Le calcul des amplitudes de diffusion et production a determine les caracteristiques de la matrice S du modele. Des comportements acausaux ont ete releves pour les processus de diffusion. En outre, les processus de production possedent une amplitudes non nulle: d'ou la non validite du theoreme d'integrabilite vs. factorisation pour cette version NC du modele de SG. D'autres proprietes ont ete mises en evidence, comme la relation avec la theorie des cordes et la bosonisation. Le deuxieme modele de SG NC a ete propose en collaboration avec Lechtenfeld, Penati, Popov, Tamassia. Les equations du mouvement ont ete tirees de la reduction dimensionnelle du modele sigma NC en 2+1 dimensions, qui a son tour est la reduction de la theorie de self–dual Yang–Mills NC en 2+2 dimensions (decrivant les supercordes N =2avecchamps B). L'action a ete calculee de meme que les amplitudes. Les processus de production possedant des amplitudes nulles et ceux de diffusion ne dependant pas du parametre de NC, entraınent ainsi un comportement causal. Le deuxieme modele de SG NC semble donc obeir a l'equivalence entre integrabilite et factorisation de la matrice S. La reduction de la theorie des cordes garde sa validite meme au niveau de l'action contrairement au modele precedent. La deuxieme partie de ma these traite des modeles d'univers branaires, ou plus precisement des modeles de RS. Le modele propose par Randall et Sundrumse situe dans un bulk 5–dimensionnel, caracterise per une symetrie d'orbifold Z2 par rapport `a la position de la brane 4–dimensionnelle. Grace au facteur de warp qui multiplie le sous–espace 4–dimensionnel parallele a la brane, on obtient la localisation des modes du graviton. Par consequent, le potentiel gravitationnel efficace est newtonien aux energies inferieures a la masse de Planck. En introduisant en outre un terme de matiere dans le bulk et en considerant l'echange d'energie entre brane et bulk, une variete de nouvelles cosmologies en derive. Dans la premiere partie de mon travail sur RS nous avons propose un modele analogue situe dans un bulk 7–dimensionnel. La brane 6–dimensionnelle — ayant compactifie deux dimensions — est placee au point fixe de l'orbifold Z2. Afin d'etudier l'evolution cosmologique en nous mettant en relation avec les observations, nous avons introduit l'echange d'energie entre brane et bulk. Les scenarios possibles sont nombreux et dependent de la forme explicite du parametre d'echange d'energie. Entre autres, les points fixes possedent une acceleration positive, pouvant ainsi representer la recente acceleration de l'univers. Il sont egalement stables pour un large ensemble des valeurs des parametres. Finalement, on peut tracer des scenarios qui partent d'une phase initiale acceleree, en passant successivement a une ere de deceleration, pour terminer sur un point fixe stable d'inflation. Les modeles d'univers branaires a la RS possedent un dual holographique via AdS/CFT. La correspondance AdS/CFT etablit qu'une theorie de supergravite(ou,plusg´ en´ eralement, de cordes) dans un champ de fond d'anti de Sitter (AdS) en d + 1 dimensions est duale a une theorie de champs conforme (CFT) en d dimensions. Tenant compte des divergences presentes dans les deux descriptions, cette correspondance a ete rendue plus precise par la formulation de la renormalisation holographique. Si l'espace de AdS est regularise au moyen d'un cutoff infrarouge, la correspondante CFT resulte regularisee par un cutoff ultraviolet et couplee a la gravite d–dimensionnelle. En analogie a l'analyse effectuee en cinq dimensions par Kiritsis, nous avons construit la theorie duale au modele cosmologique de RS en sept dimensions. Pour capturer les dynamiques dictees par l'echange d'energie entre brane et bulk, la theorie holographique en six dimensions a ete generalisee au cas interagissant (entre matiere et CFT) et non conforme. Le resultat sont les relations entre les parametres de masse appartenant aux deux descriptions et entre l'echange d'energie, d'un cote, et le parametre d'interaction, de l'autre. De plus, le parametre de rupture conforme est associe au parametre d'auto–interaction du bulk dans la description de supergravite 7–dimensionnelle. Le travail de recherche inclut donc des resultats pouvant trouver leur application dans la phenomenologie et cosmologie des cordes. D'une part on a enqueter sur l'influence de la noncommutativite liee a l'integrabilite du modele de SG. D'autre part, les consequences cosmologiques de l'emplacement du modele de RS en sept dimensions ont ete etudiees et la correspondance AdS/CFT a ete appliquee afin d'en tirer des informations sur la theorie duale, couplee a la gravite.
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