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We study families of one-dimensional CFTs relevant for describing gapped QFTsin AdS$_2$. Using the Polyakov bootstrap as our main tool, we explain howS-matrices emerge from the flat space limit of CFT correlators. In this limitwe prove that the CFT OPE density matches that of a generalized free field, andthat this implies unitarity of the S-matrix. We establish a CFT dispersionformula for the S-matrix, proving its analyticity except for singularities onthe real axis which we characterize in terms of the CFT data. In particularpositivity of the OPE establishes that any such S-matrix must satisfy extendedunitarity conditions. We also carefully prove that for physical kinematics theS-matrix may be more directly described by a phase shift formula. Our resultscrucially depend on the assumption of a certain gap in the spectrum ofoperators. We bootstrap perturbative AdS bubble, triangle and box diagrams andfind that the presence of anomalous thresholds in S-matrices are preciselysignaled by an unbounded OPE arising from violating this assumption. Finally weclarify the relation between unitarity saturating S-matrices and extremal CFTs,establish a mapping between the dual S-matrix and CFT bootstraps, and discusshow our results help understand UV completeness or lack thereof for specificS-matrices.

From conformal correlators to analytic S-matrices: CFT$_1$/QFT$_2$