Kinematics of 4D Conformal Field Theories

In this thesis we develop a framework for performing bootstrap computations in 4-dimensional conformal field theories. We use the conformal symmetry to construct generic 2-, 3- and 4-point functions and in turn generic bootstrap equations. An emphasis is made on the unification of all the obtained theoretical results and on their implementation into a Mathematica package “CFTs4D” for an easy and convenient use. The two main conceptual problems one faces are the construction of generic n-point tensor structures and the construction of generic conformal blocks. We address the first problem using 2 alternative methods: the covariant (embedding space) formalism and the non-covariant (conformal frame) formalism. Both have their advantages and disadvantages. We establish a precise connection between them which allows their interchangeable use depending on the situation. We address the second problem by reducing generic conformal blocks to an (infinite) set of seed conformal blocks. This is done using the so called spinning differential operators. We first construct explicitly a suitable (finite) set of such operators. We then introduce a new formalism which provides an (infinite) set of conformally covariant differential operators. The spinning operators are obtained as their invariant products. This heavily enlarges the original list of spinning differential operators. Finally we compute the seed conformal blocks in two different ways: by directly solving the Casimir equation and by using the shadow formalism augmented with group-theoretic properties of our new covariant differential operators.