We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 1+1 dimensions due to crossing symmetry and unitarity. In this way we establish rigorous bounds on the cubic couplings of a given theory with a fixed mass spectrum. In special cases we identify interesting integrable theories saturating these bounds. Our analytic bounds match precisely with numerical bounds obtained by considering massive QFT in an AdS box and studying the boundary correlators using the technology of the conformal bootstrap. The methods can be generalized to QFT in higher dimensions and I will comment on work underway in 3+1 dimensions. Talk based on 1607.06109 and 1607.06110 with Miguel Paulos, Joao Penedones, Balt van Rees and Pedro Vieira.
