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.
