Dispersive CFT sum rules

A bstract We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. call rule “dispersive” if it has double zeros at all double-twist operators above fixed twist gap. Dispersive have their conceptual origin Lorentzian kinematics and absorptive physics (the notion discontinuity). They been discussed using three seemingly different methods: analytic functionals dual to operators, dispersion relations position space, Mellin space. show that these approaches can be mapped into one another lead completely equivalent rules. central idea our discussion is fully nonperturbative expansion the correlator as over Polyakov-Regge blocks . Unlike usual OPE sum, utilizes data two separate channels, while having (term by term) good Regge behavior third channel. construct which are non-negative gap; they physical interpretation subtracted version “superconvergence” expect very useful tool study expansions around mean-field theory, constrain low-energy description holographic CFTs with large examples first kind applications, notably we exhibit candidate extremal functional spin-two gap problem.