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.

In dieser Dissertationsschrift werden Amplituden in QFTs und CFTs studiert. Zunächst wird mittels der »double copy«-Methode gezeigt, inwiefern Integranden von Gravitationsamplituden aus Integranden von Eichtheorien gewonnen werden können. Um diese Methode anzuwenden, bedarf es einer konkreten Darstellung der Eichtheorieintegranden, sodass die kinematischen Faktoren des Integranden die gleichen algebraischen Relationen erfüllen wie die Strukturkonstanten der Eichtheorie. Mithilfe dieser Methode werden Vierpunktsamplituden in N = 0 Supergravitation gekoppelt mit Yang-Mills in erster Ordnung der Störungsreihe berechnet, welche als asymptotische Zustände Gravitonen oder Gluonen positiver Helizität enthalten. Das Analogon der Amplituden in konform invarianten Theorien, genannt Mellinamplituden, wird anschließend im zweiten Teil diskutiert. Nicht nur durch ihre Beschreibung als Funktionen »lorentzinvarianter« Variablen, welche durch eine Art »LSZ-Reduktion« gewonnen werden, weisen Mellinamplituden eine formal heuristische Ähnlichkeit zu Amplituden auf, sondern auch können alle physikalischen Größen einer CFT aus ihnen berechnet werden. D.h., ebenso wie Amplituden einen Streuprozess vollständig charakterisieren, ist eine CFT eindeutig über ihre Mellinamplituden festgelegt. Fermionische Mellinamplituden wurden zum ersten Mal, in der Veröffentlichung worauf diese Dissertationsschrift basiert, studiert. Jede Komponente der fermionischen Mellinamplituden ist einer bestimmen Tensorstruktur zugeordnet, deren Polstruktur im einzelnen diskutiert wird. Es werden die analytische Eigenschaften der fermionischen Mellinamplituden der gemischten Vierpunktskorrelationsfunktion von zwei Fermionen und Skalaren, sowie von vier Fermionen studiert und darauffolgend werden diese Resultate durch störungstheoretische Rechnungen bei schwacher und starker Kopplung bestätigt. / In this thesis, amplitudes in QFT and CFT are studied. In the first chapter a modern technique to obtain integrands for gravity theories from gauge theory integrands is discussed. This formalism is called the doubly copy method and it can be applied if the gauge theory integrand is given in a specific representation where the kinematic numerator factors obey the same algebraic relations as the colour factors, e.g. the Jacobi identity. This method is applied to obtain the positive helicity sector of amplitudes in N = 0 supergravity coupled to Yang-Mills with external gravitons and gluons at one loop. Only the special case of four external particles is studied. Partial results are also obtained for pure Einstein-Yang-Mills amplitudes, where the axion and dilaton as virtual particles have been removed. In the second chapter, the natural analogue of amplitudes in CFTs is studied. These mathematical objects are called Mellin amplitudes. Mellin amplitudes can be understood as the CFT analogue of QFT amplitudes, because they are functions of “Lorentz invariant” quantities of their “momenta”. In addition all the CFT data is encoded in the Mellin amplitudes as all the data of a scattering process is included in usual amplitudes. The study of fermionic Mellin amplitudes has been carried out for the first time in the associated publication. These Mellin amplitudes have several components each associated to a certain tensor structure. The analytic properties of fermionic Mellin amplitudes corresponding to mixed four fermion-scalar conformal correlators and four fermion conformal correlators are deduced and finally these general results are confirmed by explicit perturbative calculations at weak and strong coupling.

The master´s thesis describes the basic information on the reporting of water losses. Means and methods to reduce water losses. Devices used to monitoring water supply, or detect malfunctioning of water supply line. Describes and evaluates the functionality of UFR on the laboratory model.