A CFT perspective on holographic correlators

Abstract: Conformal symmetry is ubiquitous in physics. It emerges in a variety of situations from critical phenomena, through condensed matter systems and jets at the LHC to string theory. Conformal field theories (CFTs) describe the universal behavior underlying these very different setups. An efficient and powerful way to study CFTs is represented by the conformal bootstrap. This non-perturbative technique allows constraining a CFT just relying on its own symmetries and a set of consistency conditions such as unitarity, crossing symmetry and the operator product expansion.In this thesis, we discuss applications of analytic bootstrap techniques to holographic superconformal field theories, which possess an additional space-time symmetry, known as supersymmetry, besides the conformal one. Through the AdS/CFT correspondence, these theories are dual to quantum gravity in AdS such that the analysis of CFT correlators gives access to gravitational amplitudes in curved space-time. We will see how the existence of a protected sector in such theories greatly simplifies the problem and allows us to bootstrap these observables.In this work, we devote our attention to the study of four-point functions in N=4 super Yang Mills (SYM) and in N=2 theories in four dimensions. In the first part of the thesis, we review the basics of superconformal algebra and superspace and then we introduce the main analytic bootstrap tool, the Lorentzian inversion formula. In the second part, after a brief description of the spectrum of N=4 SYM and its holographic realization in AdS, we focus on the correlator of four gravitons. We thoroughly analyze this four-point function in the supergravity approximation and as an expansion at large central charge. We conclude with a discussion of less supersymmetric correlators involving so-called quarter-BPS operators. In the last part instead, we change the setting and we study correlators of spinning operators belonging to the flavor current multiplet in N=2 superconformal theories.  By using analytic superspace techniques, we build the four-point function of gluon superfields and, from that, we extract correlators of all component fields, including the four-current one. In the end, we comment on the existence of an AdS double copy connecting these gluon amplitudes with their gravitational counterpart in N=4 SYM.