CFTs and the Bootstrap

This thesis deals with investigations in the field of higher dimensional CFTs. The first part is focused on the technology neccessary for the calculation of general conformal blocks in 4D CFTs. These special functions are neccessary for general boostrap analysis in 4D CFTs. We show how to reduce the calculation of arbitrary conformal blocks to the calculation of a minimal set of "seed" conformal blocks through the use of differential operators. We explicitly write the set of operators necessary and show a general basis for the case of external traceless symmetric operators. We then compute in closed analytical form this set of seeds. We write in a compact form the set of quadratic Casimir equations and proceed to solve them in closed form with the use of an educated Ansatz. Various details on the form of the ansatz are deduced with the use of the so called shadow formalism. The second part of this thesis deals with numerical investigations of the bootstrap equation for external scalar operators. We compute bounds on the OPE coefficients in 4D CFTs for theories with and without global symmetries, and write the bootstrap equations for theories with SO(N ) × SO(M ) and SU (N ) × SO(M ) symmetries. The last part of the thesis presents the Multipoint bootstrap, a conformal-bootstrap method advocated in ref. [25]. In contrast to the most used method based on derivatives evaluated at the symmetric point z = z = 1/2, ̄ we can consistently “integrate out" higher-dimensional operators and get a reduced, simpler, and faster to solve, set of bootstrap equations. We test this “effective" bootstrap by studying the 3D Ising and O(n) vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the O(n) vector models, with n = 2, 3, 4, which have not yet been computed using bootstrap techniques.