Bootstrapping superconformal field theories in four dimensions

The conformal bootstrap is a non-perturbative technique designed to study conformal field theories using only first principles, such as unitarity, crossing symmetry and the existence of an Operator Product Expansion. In this thesis we discuss an application of the bootstrap method in four dimensional conformal field theories. We also consider in detail the special case where the theory is supersymmetric. In particular we focus on the case study of four abelian currents. The non-supersymmetric setup applies to all conformal field theories with a global abelian symmetry group. When we include the assumption of supersymmetry, the current is taken to be the generator of the R-symmetry, which is tied to the stress tensor due to the superconformal algebra. The supersymmetric setup therefore applies to all local superconformal field theories. We start by introducing all the necessary ingredients. In particular, we discuss the formalism of the embedding space and of the conformal frame to study conformal kinematics. We also give a supersymmetric generalization of the conformal frame formula to count three-point tensor structures. Then we address the important problem of expanding superspace correlators in their components. To this aim we introduce a set of differential operators that act in superspace. Using this formalism we are able to compute the linear relations among the operators in the same superconformal multiplet. This is a necessary step in the computation of superconformal blocks, but it will also be useful for other purposes that we discuss before passing to the bootstrap analysis. First we use it to impose the averaged null energy condition on arbitrary superconformal field theories. This will lead to interesting consequences on their local operator spectrum. Next we focus on the case of local superconformal field theories with eight supercharges and we prove that a certain class of operators termed ``exotic primaries'' cannot exist. Finally, after a pedagogical introduction to the notion of the conformal bootstrap, we carry out a detailed study of the correlator of four conserved currents. In particular, we compute the conformal and superconformal blocks and the crossing equations. We conclude by proposing several numerical studies and strategies and by showing some preliminary results for the non-supersymmetric case.