Analytic Functional Bootstrap for CFTs in $d>1$

TL;DR

The paper develops analytic functionals that act on the conformal crossing equation in dimensions greater than one, unifying ideas from 1d functional bootstrap and Polyakov-style approaches. It constructs two main classes: HPPS functionals, meromorphic kernels with finite support on generalized free field data suitable for bootstrapping GFFs and certain AdS contact interactions, and d=2 product functionals built from tensoring 1d functionals to address 2d crossing, including Ising-model-related correlators. These functionals yield both analytic and numeric bounds on OPE data, such as optimal bounds on OPE density in 2d and dimension bounds saturated by known 2d CFTs, and provide a framework for exploring completeness and reconstruction of crossing from functional equations. The results offer a pathway to a higher-dimensional Polyakov bootstrap and suggest a rich structure in the space of functionals, with potential applications to numerical bootstrap and analytic bootstrap. The work also highlights key open questions about basis completeness, the role of double lightcone behavior, and how to extend product-functionality to higher dimensions beyond d=2.

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.

Figures (4)

• Figure 1: Contours of integration defined throughout this work. The vertical sections run along the line $\frac{1}{2}+i \mathbb R$. The horizontal sections wrap around the real line segments $(0,1/2)$ and $(1/2,1)$
• Figure 2: Schematic plots of the functional actions $\alpha_0^+(h)$ and $\beta_0^-(h)$ for ${\Delta_\phi}=1$. Since we only care about their positivity properties, we have rescaled them by non-negative functions of $h$ for clarity. For ${\Delta_\phi}<1$ the curves look similar replacing $4,8,\ldots$ by $2+2{\Delta_\phi}, 4+2{\Delta_\phi},\ldots$.
• Figure 3: Functional actions $\beta\alpha_{00}(\Delta,\ell)$ for various spins and ${\Delta_\phi}=1$. For $\ell=0$ the functional becomes positive when $\Delta>4$. Similarly there is a first order zero at $\Delta=\ell=2$. These imply that for ${\Delta_\phi}=1$ there must be at least one scalar operator with dimension smaller or equal to four, and higher than $\sim 1.215$.
• Figure 4: Scaling dimension bound. The thick red curve is the best bound with a basis of functionals $\alpha\alpha_{00}$, $\alpha\beta_{00}$, $\beta\alpha_{00}$ and $\beta\beta_{00}$. The blue dashed curves represent bounds obtained with the ordinary derivative basis, with 4 and 21 components. The crosses at (1/8,1) and (1,4) indicate the 2d Ising $\langle \sigma\sigma\sigma\sigma\rangle$ and $\langle \varepsilon\varepsilon\varepsilon\varepsilon\rangle$ correlators and the one at (1/2,2) the $c\to 1$ limit of minimal model's $\phi_{1,2}$ correlator. These are all points which we expect will saturate an optimal dimension bound.