Holographic Reconstruction of AdS Exchanges from Crossing Symmetry

TL;DR

The paper shows how crossing symmetry in 4d large-N CFTs uniquely fixes the AdS5 exchange amplitude corresponding to a single-trace operator. By leveraging twist conformal blocks and a Mellin-space framework, it separates the anomalous dimensions γ_{n,ℓ} into an asymptotic large-spin piece and a finite-spin piece, providing explicit results for twist-two exchanges (notably the stress tensor) and revealing a general deviation from the derivative relation in OPE coefficients. The authors verify that γ_{n,ℓ} is negative, monotone, and convex in ℓ for ℓ > s, tying these CFT data to bulk causality and gravitational locality, and they derive Regge and bulk-point limits that reproduce known AdS results and predict new corrections due to AdS curvature. The work enables a complete CFT-based reconstruction of tree-level AdS exchanges and lays the groundwork for future loop-level holographic computations and constraints on holographic CFTs via bulk causality and Reggeization.

Abstract

Motivated by AdS/CFT, we address the following outstanding question in large $N$ conformal field theory: given the appearance of a single-trace operator in the ${\cal O}\times{\cal O}$ OPE of a scalar primary ${\cal O}$, what is its total contribution to the vacuum four-point function $\langle {\cal O}{\cal O}{\cal O}{\cal O}\rangle$ as dictated by crossing symmetry? We solve this problem in 4d conformal field theories at leading order in $1/N$. Viewed holographically, this provides a field theory reconstruction of crossing-symmetric, four-point exchange amplitudes in AdS$_5$. Our solution takes the form of a resummation of the large spin solution to the crossing equations, supplemented by corrections at finite spin, required by crossing. The method can be applied to the exchange of operators of arbitrary twist $τ$ and spin $s$, although it vastly simplifies for even-integer twist, where we give explicit results. The output is the set of OPE data for the exchange of all double-trace operators $[{\cal O}{\cal O}]_{n,\ell}$. We find that the double-trace anomalous dimensions $γ_{n,\ell}$ are negative, monotonic and convex functions of $\ell$, for all $n$ and all $\ell>s$. This constitutes a holographic signature of bulk causality and classical dynamics of even-spin fields. We also find that the "derivative relation" between double-trace anomalous dimensions and OPE coefficients does not hold in general, and derive the explicit form of the deviation in several cases. Finally, we study large $n$ limits of $γ_{n,\ell}$, relevant for the Regge and bulk-point regimes.

Figures (5)

• Figure 1: The form of the CFT conformal block decomposition of the complete crossing-symmetric AdS amplitude due to $\varphi_{\tau,s}$ exchange. ${\cal O}$ is the boundary operator on the external legs, and ${\cal O}_{\tau,s}$ is the operator dual to $\varphi_{\tau,s}$. In this paper, we solve for the right-hand side of this equation using CFT crossing symmetry at large $N$.
• Figure 2: The schematic form of $\gamma_{n,\ell}$ in a large $N$ CFT with a weakly coupled, local gravity dual, valid for all $n$. The contribution to $\gamma_{n,\ell}$ from individual single-trace operators ${\cal O}_{\tau,s}$ is a negative, monotonic, convex function of $\ell$ for $\ell>s$. In holographic CFTs with $\Delta_{\rm gap}\rightarrow\infty$, there is a finite number of such contributions, all with spin $s\leq 2$, thus yielding the above behavior. For $\ell=0,2$, various behaviors are possible, due to non-analytic contributions.
• Figure 3: The asymptotic contribution due to massless scalar exchange, $\gamma_{n,\ell}^{\rm as}|_{(4,0)}$, at various values of $n$ and $\Delta$. In both plots, $\Delta=4+2m$ for $0\leq m \leq 10$; the red line is $\Delta=4$, with increasing $\Delta$ as we move downwards through the rainbow. The left plot is at $n=1$. The right plot is at integer $100\leq n \leq 104$, where each thick line is comprised of five individual lines. In all cases, the result is negative, monotonic and convex.
• Figure 4: The same setup as in Figure \ref{['f1']}, but now for $\tau=4, s=2$ exchange, $\gamma_{n,\ell}^{\rm as}|_{(4,2)}$. In both cases, the result is negative and convex as a function of $\ell$. Notice that in this case, $\gamma_{n,\ell}|_{(4,2)}$ increases as a function of $\Delta$ in the right plot.
• Figure 5: The complete contributions to $\ell=0$ anomalous dimensions due to stress tensor exchange, $\gamma_{n,0}|_{T}$, for $1\leq n \leq 10$, plotted as a function of $\Delta\geq 1$. The red line is $n=1$, with increasing $n$ as we move through the rainbow. All results are negative. In the right plot, we show only $1\leq n \leq 5$ to make clear that these lines sit below the $x$-axis.