Abstract

We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fields. Besides bounds on correlators, we apply this technology to determine new universal constraints on the Regge limit of arbitrary CFTs and obtain very simple and accurate representations of the 3d Ising spin correlator.

Highlights

  • Are CFT correlators free to take values as they please? Alas, the present work says no: even they are confined and allowed to wander only within a limited range, see figure 1

  • The action of a given master functional on the crossing equation leads to a one-parameter family of sum rules which is equivalent to a dispersion relation for a CFT correlator

  • Using the dispersion relations and the positivity properties of the master functionals we prove bounds on CFT correlators restricted to the line z = z

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Summary

Introduction

Are CFT correlators free to take values as they please? Alas, the present work says no: even they are confined and allowed to wander only within a limited range, see figure 1. The action of a given master functional on the crossing equation leads to a one-parameter family of sum rules which is equivalent to a dispersion relation for a CFT correlator. The dispersion relations express the behaviour of a CFT correlator on the complexified line z = zin terms of its double discontinuity, up to a finite set of low energy data. This allows us to take a plunge deep into the complex plane to explore conformal Regge kinematics [19], tethered to the relative safety of the Euclidean line. We turn to the outline of this work and some of the main results

Outline and summary of results
Kinematics
Functional bases
Master functionals
Motivation: correlator bounds
Equations and boundary conditions
General solution
Subtleties in the master functional definitions
The Polyakov bootstrap and completeness
From functionals to dispersion relations
From dispersion relations to functionals
Application: correlator bounds
Definitions and properties
General bounds
Dispersion relation analysis
Application
Ising correlator and interacting Polyakov blocks
Comments on higher dimensions
Summary and outlook
A Computing functional kernels
Computing from basis expansion
B Asymptotics of Polyakov blocks

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