Holographic Reconstruction of AdS Exchanges from Crossing Symmetry

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 $\tau$ 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 $\gamma_{n,\ell}$ 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 $\gamma_{n,\ell}$, relevant for the Regge and bulk-point regimes.