Dispersion relations and exact bounds on CFT correlators

TL;DR

This paper introduces new dispersion relations and sum rules for conformal field theory (CFT) correlators on the line, providing exact bounds and universal constraints that enhance understanding of crossing symmetry and the Regge limit.

Contribution

It develops a novel dispersion formula framework based on master functionals, offering a new formulation of crossing symmetry constraints and deriving exact bounds on CFT correlators.

Findings

Derived new crossing-symmetric dispersion relations for CFT correlators.

Established exact bounds on correlator values on the Euclidean section.

Provided universal constraints on the Regge limit and accurate representations of the 3d Ising correlator.

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.