The boundary correlation functions for a quantum field theory (QFT) in a fixed anti–de Sitter (AdS) background should reduce to S-matrix elements in the flat-space limit. We consider this procedure in detail for four-point functions. With minimal assumptions we rigorously show that the resulting S-matrix element obeys a dispersion relation, the nonlinear unitarity conditions, and the Froissart-Martin bound. QFT in AdS thus provides an alternative route to fundamental QFT results that normally rely on the LSZ axioms.

From a modern viewpoint the "S-matrix bootstrap" is the idea that general consistency conditions can be used to obtain quantitative constraints on scattering amplitudes. I will discuss the assumptions behind this approach, open questions about the structure of amplitudes, and discuss some fundamental results from the sixties and seventies. In the second part of the talk I will treat two modern approaches which were inspired by recent results on the conformal bootstrap, and show how they can be used to constrain scattering amplitudes in non-trivial ways.

In recent years we have witnessed a revival of the conformal bootstrap approach to CFTs. I will discuss the application of these ideas to six-dimensional conformal field theories with (2,0) supersymmetry, focusing on the universal four-point function of stress tensor multiplets. For these theories the program splits into an analytic and a numerical component. The analytic component yields exact results but in a protected subsector. The numerical component can be used to derive bounds on OPE coefficients and scaling dimensions from the constraints of crossing symmetry and unitarity. The principal numerical result is strong evidence that the A1 theory realizes the minimal allowed central charge (c=25) for any interacting (2,0) theory. This implies that the full stress tensor four-point function of the A1 theory is the unique unitary solution to the crossing symmetry equation at c=25. For this theory, we can estimate the scaling dimensions of the lightest unprotected operators appearing in the stress tensor operator product expansion. We also find rigorous upper bounds for dimensions and OPE coefficients for a general interacting (2,0) theory of central charge c. For large c, our bounds appear to be saturated by the holographic predictions obtained from eleven-dimensional supergravity.

In the past few years we have seen that the bootstrap approach to higher-dimensional conformal field theories (CFTs) can be surprisingly powerful. In particular, we are finally able to put the crossing symmetry equations to good use and extract nontrivial information about the spectrum and three-point functions in a generic CFT. In this talk I will discuss the application of these ideas to superconformal field theories, focussing on N=2 and N=4 theories in four dimensions. In those theories there exists a subsector where the crossing symmetry equations can be solved analytically. Together with the numerical analysis of the remaining constraints we can learn a great deal about the nonperturbative structure of these superconformal field theories.

We describe a new correspondence between four-dimensional conformal field theories with extended supersymmetry and two-dimensional chiral algebras. The meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four-dimensional theory. Infinite chiral symmetry has far-reaching consequences for the spectral data, correlation functions, and central charges of any four-dimensional theory with N=2 superconformal symmetry.