Abstract
We give a unified treatment of dispersive sum rules for four-point correlators in conformal field theory. We call a sum rule “dispersive” if it has double zeros at all double-twist operators above a fixed twist gap. Dispersive sum rules have their conceptual origin in Lorentzian kinematics and absorptive physics (the notion of double discontinuity). They have been discussed using three seemingly different methods: analytic functionals dual to double-twist operators, dispersion relations in position space, and dispersion relations in Mellin space. We show that these three approaches can be mapped into one another and lead to completely equivalent sum rules. A central idea of our discussion is a fully nonperturbative expansion of the correlator as a sum over Polyakov-Regge blocks. Unlike the usual OPE sum, the Polyakov-Regge expansion utilizes the data of two separate channels, while having (term by term) good Regge behavior in the third channel. We construct sum rules which are non-negative above the double-twist gap; they have the physical interpretation of a subtracted version of “superconvergence” sum rules. We expect dispersive sum rules to be a very useful tool to study expansions around mean-field theory, and to constrain the low-energy description of holographic CFTs with a large gap. We give examples of the first kind of applications, notably we exhibit a candidate extremal functional for the spin-two gap problem.
Highlights
The modern conformal bootstrap program was kindled by the observation [1] that the constraints of unitarity and crossing, even when applied to a small number of correlators, are surprisingly effective in carving out the space of consistent CFTs
The Mellin space sum rules of [49] can all be understood in the conventional language of analytic functionals, and follow from nothing more than the usual requirements of unitarity and crossing — there is no need to make the additional spectral assumptions encoded in the non-perturbative Polyakov conditions
What is its physical interpretation? We show that this is nothing but the fixed-u dispersion relation for the meromorphic function M (s, t)!
Summary
The modern conformal bootstrap program was kindled by the observation [1] that the constraints of unitarity and crossing, even when applied to a small number of correlators, are surprisingly effective in carving out the space of consistent CFTs. The authors of [49] have studied the validity of the Mellin representation for general CFTs and derived non-perturbative dispersion relations in Mellin space, analogous to the familiar dispersion relations obeyed by the S-matrix They have obtained interesting sum rules for CFT data by insisting (as befits a generic interacting theory) that the spectrum does not contain operators with exact double-twist quantum numbers. The Mellin space sum rules of [49] can all be understood in the conventional language of analytic functionals, and follow from nothing more than the usual requirements of unitarity and crossing — there is no need to make the additional spectral assumptions encoded in the non-perturbative Polyakov conditions. Appendix F includes the details of our numerical implementation of the twist gap maximization problem
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