Shockwaves from the Operator Product Expansion

TL;DR

Shockwaves in AdS/CFT are shown to be dual to specific CFT states created by heavy local operators or by smeared, complexified light operators. The leading Regge OPE in the Regge limit is the bulk shockwave operator $\int du\, h_{uu}$, which either directly arises from a heavy insertion or emerges after smearing light operators, with double-trace contributions projected out. The authors connect this Regge data to boundary causality via the Engelhardt–Fischetti criterion and the chaos bound, reproducing known CFT constraints on three-point couplings and providing a unified framework for time delays, ANEC-like statements, and higher-spin consistency. The framework is extended to spinning operators, confirming that smeared OPEs purge double-trace contributions and yield gravity-consistent constraints on higher-derivative couplings, thereby strengthening the holographic dictionary for shockwaves and their CFT avatars.

Abstract

We clarify and further explore the CFT dual of shockwave geometries in Anti-de Sitter. The shockwave is dual to a CFT state produced by a heavy local operator inserted at a complex point. It can also be created by light operators, smeared over complex positions. We describe the dictionary in both cases, and compare to various calculations, old and new. In CFT, we analyze the operator product expansion in the Regge limit, and find that the leading contribution is exactly the shockwave operator, $\int du h_{uu}$, localized on a bulk geodesic. For heavy sources this is a simple consequence of conformal invariance, but for light operators it involves a smearing procedure that projects out certain double-trace contributions to the OPE. We revisit causality constraints in large-$N$ CFT from this perspective, and show that the chaos bound in CFT coincides with a bulk condition proposed by Engelhardt and Fischetti. In particular states, this reproduces known constraints on CFT 3-point couplings, and confirms some assumptions about double-trace operators made in previous work.

Figures (7)

• Figure 1: Planar shockwave \ref{['swmetric']} in Poincare coordinates. The source travels on a null geodesic parallel to the boundary, at fixed radial coordinate $z = z_0$. The shockwave is on the null plane $u=0$.
• Figure 2: Spherical shockwave \ref{['sw_HI']} in Poincare coordinates. The source travels on a radial null geodesic that hits the boundary at the origin, $u' = v'= 0$. The resulting shockwave is a null cone.
• Figure 3: Shockwave in global AdS, where the boundary is the Lorentzian cylinder. The dashed line is the source, which hits the boundary at the red dots. This source creates a shockwave on the shaded null surface. In the shaded-yellow Poincare patch, the solution is the spherical shockwave. In the shifted Poincare patch, shown as a thick blue outline, the solution is the planar shockwave. The yellow and blue patches in this figure correspond to the same color-coding as the other figures.
• Figure 4: Boundary version of figure \ref{['fig:cylinder']}, showing the action of the null inversion \ref{['fourshift']} on Minkowski patches. The left and right sides of the diagram are identified to make the Lorentzian cylinder. The $(u',v')$ coordinates cover the Minkowski spacetime shown as a yellow diamond. The coordinates $(u,v)$ cover the shifted patch, shown as a blue diamond. These overlap in the region $v'>0$, $v < 0$. The shockwave hits the boundary at the red dots; it is created at the origin of the yellow diamond, which is on $I^-$ of the blue diamond.
• Figure 5: The Regge limit: the operator product $\psi(u,v)\psi(-u,-v)$ can be replaced by a shockwave propagating along $v=0$ when $\Delta_\psi \gg 1$ .