Dispersion relations and exact bounds on CFT correlators

TL;DR

This work develops dispersion relations for CFT correlators restricted to the line z=bar z by introducing master functionals that generate complete, crossing symmetric sum rules. These functionals reveal a dispersion form equivalent to the Polyakov bootstrap, with built in positivity yielding exact bounds on Euclidean correlators that are saturated by generalized free fields. The framework also yields universal Regge constraints and provides an efficient 1d representation of the 3d Ising spin correlator using a small number of Polyakov blocks. The results offer a robust, positivity driven approach to bounding and reconstructing CFT data, with potential extensions to higher dimensions and new numerical bootstrap strategies.

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.

Figures (4)

• Figure 1: Upper and lower bounds on CFT correlators $\mathcal{G}(w):=\mathcal{G}(w,w)$ of a field of dimension ${\Delta_\phi}$, shown for the case ${\Delta_\phi}=\Delta_\sigma^{\hbox{\tiny Ising}}\sim 0.518$. Here $\mathcal{G}^{B,F}$ stand for the Bosonic/Fermionic generalized free field correlators, $\mathcal{G}^{B,F}(w)=\pm 1+w^{-2{\Delta_\phi}}+(1-w)^{-2{\Delta_\phi}}$. Any unitary CFT correlator must live inside the shaded region, with the caveat that the upper bound is only valid for correlators where the first operator has scaling dimension $\Delta_g\geq 2{\Delta_\phi}$. In red the 3d Ising spin field correlator computed from the CFT data provided in Simmons-Duffin:2016wlq.
• Figure 2: Functional kernels for values of ${\Delta_\phi}$ ranging from ${\Delta_\phi}\sim 0.25$ to $\infty$, evaluated for $w=1/3$. For clarity we plot the kernels in terms of $\tilde{g}^{B,F}_w(z)\equiv (1-z)^{-2{\Delta_\phi}}\hat{g}^{B,F}_w(z)$. For special values of ${\Delta_\phi}$ the numerical curves above match with analytic expresions. The kernels are positive for all $z\in(0,1)$. The rough shape of the curves does not change with $w$
• Figure 3: Comparison between the 3d Ising spin field correlator and its approximation as a sum of two Polyakov blocks. The results are normalized relative to the generalized free fermion correlator $\mathcal{G}^F(w)$. The line is the correlator as computed from the conformal block expansion up to dimension 10, plotted in the interval $w\in(0,1/2)$ which is then mirrored to get a crossing symmetric shape. This is not necessary for the Polyakov block approximation which is automatically crossing symmetric and shown as the red dots.
• Figure 4: Comparison between the 3d Ising spin field correlator and its approximations. The results are normalized relative to the generalized free fermion correlator $\mathcal{G}^F(w)$. The blue line (the lowest) is the correlator as computed from the conformal block expansion up to dimension 10. The red dots represent the two Polyakov block approximation of section \ref{['sec:approxIsing']}. The green line is the identity interacting Polyakov block described in the text, computed from its conformal block expansion.