Cut-touching linear functionals in the conformal bootstrap

TL;DR

This work addresses the challenge of justifying swapping between an infinite conformal-block sum and the action of cut-touching linear functionals in the 1d conformal bootstrap. It develops general finiteness and swapping criteria grounded in analytic continuation in the cut plane, via the ρ-coordinate, and applies them to functionals of Mazác's type to confirm swapping holds at least for low $${\Delta_\phi}$$. The authors provide a concrete blueprint for rigorous use of cut-touching functionals, with explicit checks for Mazác-style functionals up to ${\Delta_\phi}\le 9/2$, and discuss extensions to continuous spectra and higher dimensions. The results underpin the mathematical solidity of analytic bootstrap bounds derived with cut-touching functionals and widen the toolkit for deriving optimal CFT bounds.

Abstract

The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial "swapping" property, allowing to swap infinite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving finite sums of derivatives. However, it is far from obvious for "cut-touching" functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazac in his work on analytic derivation of optimal bootstrap bounds. We derive general swapping criteria for the cut-touching functionals, and check in a few explicit examples that Mazac's functionals pass our criteria.

Figures (7)

• Figure 1: A region where the series (\ref{['CBexp']}) converges uniformly (the image of the disk $|\rho|\leqslant 1-\epsilon$ in the $z$ plane).
• Figure 2: A region of uniform convergence of the series in the crossing relation (\ref{['cross']}).
• Figure 3: Support of integration in the functional of Example 1.
• Figure 4: Contour $\Gamma_z$ . Also shown are the two cuts of the cut plane.
• Figure 5: Contour $\Gamma_x$ . Also shown are the images of the two cuts of the cut plane under the transformation from $z$ to $x$. The function $h(x)$ and the functions $f(z(x))$ on which the functional is evaluated will be analytic in the upper half-plane.