Analytic Functional Bootstrap for CFTs in $d>1$

TL;DR

This paper develops analytic functionals to analyze crossing symmetry in conformal field theories across dimensions, providing new tools for bootstrap bounds and insights into the structure of CFTs.

Contribution

It introduces a new class of analytic functionals acting on crossing equations in arbitrary dimensions, enhancing the bootstrap approach and connecting to known solutions in 2D.

Findings

Functional kernels constrain crossing symmetry

New functionals bootstrap AdS contact interactions

Optimal bounds on OPE density in 2D CFTs

Abstract

We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE, (double) lightcone and Regge limits. Compatibility with the crossing equation imposes constraints on the functional kernels which we study in detail. We then introduce two simple classes of functionals. The first class has a simple action on generalized free fields and their deformations and can be used to bootstrap AdS contact interactions in general dimension. The second class is obtained by tensoring holomorphic and antiholomorphic copies of $d=1$ functionals which have been considered recently. They are dual to simple solutions to crossing in $d=2$ which include the energy correlator of the Ising model. We show how these functionals lead to optimal bounds on the OPE density of $d=2$ CFTs and argue that they provide an equivalent rewriting of the $d=2$ crossing equation which is better suited for numeric computations than current approaches.