Analytical Methods for Conformal Field Theory

Abstract

In this dissertation, we present modern analytical methods for studying conformal field theories (CFTs) in more than two dimensions. Using these methods, the spectrum of theory and the operator product coefficients (OPE coefficients) can be determined. First, we examine the spectrum of local operators in CFTs on a defect with a codimension greater than one. We show that for large transverse spin s, the spectrum of each theory has countably infinitely many accumulation points. The spin s is the quantum number belonging to the subgroup of the Lorentz group, which leaves the defect invariant. Furthermore, we find the OPE coefficients and the anomalous dimensions of the clustering operators in a 1 / s evolution using light cone bootstrap techniques. In addition, we derive the operator dimensions and OPE coefficients as analytic functions of s from the discontinuity of the causal two-point function. In the second part of this work, we introduce the Mellind representation of conformal correlation functions. In this representation, the spectrum and the OPE coefficients are manifest. We focus on the description of four-point functions in three dimensions of either exclusively spin 1/2 operators or a mixture of spin 1/2 and scalar operators. After defining the mellin amplitudes for these four-point functions, we examine the pol- We then illustrate the analysis of concrete mellin amplitudes of fermionic Wittendiagrams and conformal fermionic Feynman diagrams. In the last part, we examine the OPE in the context of holography. Here we derive theory-independent relations between the OPE coefficients of the world surface CFT of a string theory in anti-de-sitter space-time and the dual CFT. In this thesis, we discuss some of the more advanced analytical approaches to studying conformal field theories (CFTs) in terms of more than two. The CFT, which is the spectrum of operators and the coefficients in the operator product expansion (OPE). Therein about a countable 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 scale expansion. Furthermore, the additive theory of the discontinuity in the causal two-point function of scalar operators in the ambient theory, inverts 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 thus enables us to resume the large expansion obtained using lightcone bootstrap. Thereafter we move to a discussion of the Mellin representation of fermionic conformal correlators. The dynamic 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. Feynman integrals with fermionic legs. Mellin amplitudes and illustrate the general features. Finally we look at the OPE in the context of holography and derive a set of theory. OPE coefficients in the worldsheet CFT of a string theory in anti-time Sitter spacetime and those in the dual CFT. Feynman integrals with fermionic legs. Mellin amplitudes and illustrate the general features. Finally we look at the OPE in the context of holography and derive a set of theory. OPE coefficients in the worldsheet CFT of a string theory in anti-time Sitter spacetime and those in the dual CFT. Feynman integrals with fermionic legs. Mellin amplitudes and illustrate the general features. Finally we look at the OPE in the context of holography and derive a set of theory. OPE coefficients in the worldsheet CFT of a string theory in anti-time Sitter spacetime and those in the dual CFT.