La correspondance AdS/CFT est la première réalisation précise de la dualité jauge/gravité. Jusqu’à maintenant la correspondance AdS/CFT reste une conjecture. La dualité de N = 4 SYM et la théorie des cordes est un exemple le plus notable de correspondance AdS/CFT. Un des obstacles principaux à l’explorer est le fait que le régime de couplage faible pour la théorie de jauge est le régime de couplage fort pour la théorie des cordes et vice versa. Par conséquent, aussi longtemps que les méthodes perturbatives sont appliquées, on ne peut pas comparer les observables de deux cotés de la correspondance directement en dehors de quelques cas particuliers. A ce stade, l’énorme symétrie de N = 4 SYM joue un rôle important en permettant le calcul exact des observables de la théorie au moins dans la limite planaire. Cette thèse est consacrée au calcul des fonctions à trois, l’un des principaux observables de N = 4 SYM, et est composée de deux parties. Dans la première partie nous considérons l’approche générale pour le calcul des fonctions à trois points sur la base de soi-disant vertex de spin, qui est inspiré de la théorie de champs des cordes. Dans la deuxième partie, nous considérons un type spécifique de fonctions à trois points appelés lourd-lourd-léger, qui sont caractérisés par la propriété que la longueur de l’un des opérateurs est beaucoup plus petite des longueurs de deux autres. Il s’avère que ces fonctions de corrélations peuvent être identifiées à des facteurs de forme diagonaux et ainsi on peut appliquer les résultats concernant les facteurs de forme. / N=4 SYM theory has been drawing the attention of a lot of physicists during two last decades mainly due to the two aspects: AdS/CFT correspondence and integrability. AdS/CFT correspondence is the first precise realization of the gauge/string duality whose history starts in the 60's, when a string theory was considered as a candidate for describing the strong interactions. In 1997 Maldacena made a proposal about the duality between certain conformal field theories (CFT) and string theories defined on the product of AdS space and some compact manifold, which implies a one to one map between the observables of the gauge and string counterparts. Up to now AdS/CFT correspondence still remains a conjecture. The duality of N=4 SYM and the appropriate string counterpart is the most notable example of the AdS/CFT correspondence. One of the main obstructions to exploring it is the fact that weak coupling regime for the gauge theory is the strong coupling regime for the string theory and vice versa. Therefore as long as perturbative methods are applied, one can not compare the observables of dual counterparts directly apart from some specific cases. At this point the huge symmetry of N=4 SYM plays an important role allowing exact computation of the theory observables at least in the planar limit. This property of the theory is called integrability. The observables of the N=4 SYM are Wilson loops and correlation functions built out of gauge invariant operators. The space-time dependence of the two- and three-point correlators is fixed by the conformal symmetry up to some parameters: dimensions of the operators in the case of two-point functions and dimensions of the operators and structure constants in the case of three-point functions. It's commonly accepted to refer to the problem of finding the dimensions of the operators as the spectral problem. On the classical level the operator dimension is equal to the sum of the dimensions of the fundamental fields out of which the operator is composed. When the interaction is turned on, the conformal dimension gets quantum correction. In order to compute three-point functions, apart from the conformal dimensions of corresponding operators one needs to compute the structure constants. In CFT computation of the higher-point correlators eventually can be reduced to computation of two- and three-point functions by means of the operator product expansion. Therefore two- and three-point functions appear to be building blocks of any correlator of the theory. This thesis is devoted to computation of three-point functions and consists of two parts. In the first part we consider the general approach for computing three-point functions based on the so-called spin vertex, which is inspired from the string field theory. In the second part we consider a specific kind of three-point functions called heavy-heavy-light, which are characterized by the property that the length of one of the operators is much smaller the lengthes of other two. It happens that this kind of correlators can be considered as diagonal form factors which supposes that in this case one can apply the results obtained in the form factor theory.
| Identifer | oai:union.ndltd.org:theses.fr/2016SACLS233 |
| Date | 14 September 2016 |
| Creators | Petrovskii, Andrei |
| Contributors | Université Paris-Saclay (ComUE), Kostov, Ivan, Serban, Didina |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text, Image, StillImage |
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