In dieser Dissertation stellen wir moderne analytische Methoden zur Untersuchung von konformen Feldtheorien (CFTs) in mehr als zwei Dimensionen vor. Mit Hilfe dieser Methoden können das Spektrum der Theorie und die Operatorprodukt-Koeffizienten (OPE-Koeffizienten) ermittelt werden. Zunächst untersuchen wir das Spektrum lokaler Operatoren in CFTs auf einem Defekt mit Kodimension größer eins. Wir zeigen, dass für großen transversalen Spin s das Spektrum jeder Theorie abzählbar unendlich viele Häufungspunkte aufweist. Der Spin s ist die Quantenzahl, die zu der Untergruppe der Lorentzgruppe gehört, welche den Defekt invariant lässt. Des Weiteren finden wir die OPE-Koeffizienten und die anomalen Dimensionen der zu den Häufungspunkten gehörenden Operatoren in einer Entwicklung in 1/s mit Hilfe von Lichtkegel-Bootstrap-Techniken. Außerdem leiten wir aus der Diskontinuität der kausalen Zweipunktfunktion die Operatordimensionen und OPE-Koeffizienten als analytische Funktionen von s her. Im zweiten Teil dieser Arbeit führen wir die Mellindarstellung von konformen Korrelationsfunktionen ein. In dieser Darstellung sind das Spektrum und die OPE-Koeffizienten manifest enthalten. Wir legen den Fokus auf die Beschreibung von Vierpunktfunktionen in drei Dimensionen von entweder ausschließlich Spin 1/2 Operatoren oder einer Mischung aus Spin 1/2 und skalaren Operatoren. Nachdem wir für diese Vierpunktfunktionen die Mellinamplituden definieren, untersuchen wir die Polstruktur dieser genauer. Im Anschluss illustrieren wir die Analyse an konkreten Mellinamplituden von fermionischen Wittendiagrammen und konformen fermionischen Feynmandiagrammen. Im letzten Teil untersuchen wir die OPE im Kontext der Holographie. Hierbei leiten wir theorieunabhängige Beziehungen zwischen den OPE-Koeffizienten der Weltflächen-CFT einer Stringtheorie in Anti-de-Sitter-Raumzeit und der dualen CFT her. / In this thesis, we discuss some modern analytical approaches to studying conformal field theories (CFTs) in dimensions greater than two. The results thus derived pertain to the dynamical data that define a generic CFT, namely the spectrum of operators and the coefficients in the operator product expansion (OPE). We begin with an investigation of the spectrum of local operators supported on conformal defects of codimension greater than one and establish the existence therein of a countably infinite number of universal accumulation points at large transverse spin s. Here, s is a quantum number associated with the symmetry under the Lorentz transformations that preserve the defect. Using lightcone bootstrap techniques, we calculate the anomalous dimensions and OPE coefficients of the operators that populate these accumulation points in a large s expansion. Furthermore, we derive an integral formula to obtain the CFT data associated with the defect theory from the discontinuity in the causal two-point function of scalar operators in the ambient theory, thereby inverting the expansion of this correlator in the defect channel. This formula extracts the operator dimensions and OPE coefficients in an analytic function in s and also enables us to resum the large s expansion obtained using lightcone bootstrap. Thereafter we move on to a discussion of the Mellin representation of fermionic conformal correlators. The dynamical data in CFTs is manifest in the analytic properties of Mellin amplitudes. We define, concretely for three spacetime dimensions, the Mellin amplitudes associated with the four-point function of spin-half operators and the mixed four-point function of spin-half and scalar operators. We analyze the pole structure of these Mellin amplitudes and illustrate the general features thus unraveled with some explicit computations of Mellin amplitudes associated with Witten diagrams and conformal Feynman integrals with fermionic legs. Finally we look at the OPE in the context of holography and derive a set of theory independent relations between OPE coefficients in the worldsheet CFT of a string theory in anti-de Sitter spacetime and those in the dual CFT.

Conformal Field Theories (CFT) are Quantum Field Theories characterized by enhanced (conformal) symmetries. They are interesting to Theoretical Physicists because they occur at critical points in phase transitions of various systems and also in the world sheet formulation of String Theory. CFTs allow Operator Product Expansion (OPE) in their correlators. The idea of Conformal Bootstrap is to solely use the conformal symmetries and crossing symmetry in the OPE to solve a conformal led theory and not explicitly use a lagrangian. Solving a CFT is equivalent to obtaining the anomalous dimensions and OPE coe client’s of the operators. The work presented in this thesis shows how ideas of bootstrap can be used to get analytic results for dimensions and OPE coe client’s of various operators in CFTs. In the conventional bootstrap program, the OPE in the direct (s-) channel is compared with the OPE of a crossed (t-) channel. This requirement of crossing symmetry is called the bootstrap equation. The flow of logic is somewhat reversed in the \new" idea that is formulated in this thesis. The trick is to expand a CFT correlator in terms of Witten diagrams, in all channels. This is a manifestly crossing symmetric description, and is in contrast to the usual expansion in terms of conformal blocks, which is in only one channel. For convenience we work with the Mellin transforms of Witten diagrams. For consistency of the Witten diagrams expansion with the conformal block expansion in a certain channel, we require the satisfaction of some equations, which we call the bootstrap equations in Mellin space. This scheme was rest chalked out by Polyakov in 1973, where he proposed the use of \unitary amplitudes" to expand a correlator. The unitary amplitudes had similar symmetry and analytic properties as the Witten diagrams. Even though he did not take his idea forward, replacing unitary amplitudes with Witten diagrams seems to work very well for obtaining analytic results. The working of bootstrap equations in Mellin space is demonstrated for the 4 Wilson-Fisher fixed point in d = 4 , O(N) theory at Wilson-Fisher point (in d = 4 ), as well as with large N (in general d), and large spin operators in strongly coupled and weakly coupled theories. For the case of global symmetry we have also analysed the somewhat unexplored case of cubic anisotropy. The results are obtained as perturbative series in , 1=N, or 1=` as applicable, and they are consistent with known results in literature. We also obtain various new results, for instance the OPE coe client’s of general higher spin operators. These results are otherwise very di cult to end from Feynman diagrams, but in this approach they come out very simply, essentially by solving some algebraic equations. We also show the use of the conventional bootstrap strategy, for analytically obtaining anomalous dimensions of large spin operators having higher twists, in a O(N) theory, by working in the light cone limit. One can question the validity of the proposal of using Witten diagrams to expand a correlator. One such issue is convergence of the sum over Witten diagrams. Convergence can be shown to hold for the operator spectrum we have worked with. Also there are operators that might upset convergence under some conditions. Resolutions of such cases, and ways to improve convergence have also been discussed. The conventional bootstrap method has been very successful in giving numerical results in nonpertur-bative CFTs, like the 3 dimensional Ising model. Numerical analysis can also be made possible with the new bootstrap in Mellin space approach. Having a convergent basis of expansion improves the prospect of numeric. The goal is to formulate a bootstrap scheme that, under a single framework, can make most of all the CFT properties. It should be systematic, so that one can obtain anomalous dimensions and OPE coe client’s of all operators up to any desired order, and works for all strongly/weakly coupled and perturbative/nonpertur-bative CFTS, both analytically and numerically. Finally, the use of Witten diagrams also indicates the possibility of Ising CFT or weakly coupled CFTs having connections with AdS/CFT, and hence String Theory. It does seem we have a right direction towards achieving our goal.

In dieser Arbeit betrachten wir zwei Ansätze zur Untersuchung von Korrelationsfunktionen in eindimensionalen konformen Feldtheorien mit Defekten (dCFT1), insbesondere solche, die durch 1/2-BPS-Wilson-Linien-Defekte in den drei- und vierdimensionalen superkonformen Theorien definiert sind, die für die AdS/CFT-Korrespondenz relevant sind. Zunächst verwenden wir den analytischen konformen Bootstrap, um zwei Beispiele von Defektkorrelatoren auszuwerten. Der Vier-Punkt-Korrelator des Verschiebungs-Supermultipletts, das auf der 1/2-BPS-Wilson-Linie in der ABJM-Theorie eingefügt ist, wird bis zur dritten Ordnung in einer starken Kopplungsexpansion berechnet und reproduziert die expliziten Witten-Diagramm-Berechnungen erster Ordnung. Anschließend wird der Fünf-Punkt-Korrelator von 1/2-BPS-Operatoren, die auf der 1/2-BPS-Wilson-Linie in N=4 Super-Yang-Mills eingefügt sind, untersucht und in einer starken Kopplungsexpansion bis zur ersten Ordnung gebootstrapped. Anschließend werden die CFT1-Daten extrahiert, die bestätigen, dass das Mischen von Operatoren die anomale Dimension erster Ordnung nicht beeinflusst. Der zweite Ansatz betrachtet die allgemeine Struktur von Korrelatoren in effektiven Theorien in AdS2. Es werden alle skalaren n-Punkt-Kontakt-Witten-Diagramme für externe Operatoren mit ganzzahligem konformem Gewicht berechnet. Effektive Theorien in AdS2, die durch eine Wechselwirkungslagrange mit einer beliebigen Anzahl von Ableitungen definiert sind, werden dann betrachtet und mit Hilfe eines neuen Formalismus der Mellin-Amplituden für 1d-CFTs bis zur ersten Ordnung gelöst. Schließlich wird die diskretisierte Wirkung der Cusped-Wilson-Linie als alternative Möglichkeit zur Gewinnung nicht-perturbativer Daten vorgestellt: durch die Gitterfeldtheorie. / In this thesis, we consider two approaches to the study of correlation functions in one-dimensional defect Conformal Field Theories (dCFT1), in particular those defined by 1/2-BPS Wilson line defects in the three- and four-dimensional superconformal theories relevant in the AdS/CFT correspondence. In the first approach, we use the analytic conformal bootstrap to evaluate two examples of defect correlators. The four-point correlator of the displacement supermultiplet inserted on the 1/2-BPS Wilson line in ABJM theory is computed to the third order in a strong-coupling expansion and reproduces the explicit first-order Witten diagram calculations. The CFT1 data are then extracted from this correlator, and the operator mixing is solved at first order. Consequently, all-order results are derived for the part of the correlator with the highest logarithm power, uniquely determining the double-scaling limit. Then, the five-point correlator of 1/2-BPS operators inserted on the 1/2-BPS Wilson line in =4 super Yang-Mills are studied. The superblocks are derived for all channels of the OPE, and the five-point correlator is bootstrapped to first order in a strong coupling expansion. The CFT1 data are then extracted, confirming that operator mixing does not affect the first-order anomalous dimension. The second approach considers the general structure of correlators in effective theories in AdS2. All scalar n-point contact Witten diagrams for external operators of integer conformal weight are computed. Effective theories in AdS2 defined by an interaction Lagrangian with an arbitrary number of derivatives are then considered and solved to first order using a new formalism of Mellin amplitudes for 1d CFTs. Finally, the cusped Wilson line discretised action is presented as an alternative way to obtain non-perturbative data: through Lattice Field Theory.