Non-Local Effects of Multi-Trace Deformations in the AdS/CFT Correspondence

TL;DR

The paper addresses how multi-trace deformations in the AdS/CFT correspondence alter bulk locality and causality. It develops a framework for bulk fields on $AdS_{d+1}\times M$ and analyzes how double-trace and marginal deformations modify boundary conditions, affecting bulk-to-bulk propagators and commutators. The main finding is that while the undeformed theory maintains vanishing commutators for certain pairs connected only by spacelike bulk geodesics, the multi-trace deformations can render these commutators nonzero, revealing bulk non-locality consistent with causality. The work provides a concrete, calculable demonstration that multi-trace deformations create non-local bulk dynamics in a controlled setting, with implications for non-local string theories and the holographic understanding of locality.

Abstract

The AdS/CFT correspondence relates deformations of the CFT by "multi-trace operators" to "non-local string theories". The deformed theories seem to have non-local interactions in the compact directions of space-time; in the gravity approximation the deformed theories involve modified boundary conditions on the fields which are explicitly non-local in the compact directions. In this note we exhibit a particular non-local property of the resulting space-time theory. We show that in the usual backgrounds appearing in the AdS/CFT correspondence, the commutator of two bulk scalar fields at points with a large enough distance between them in the compact directions and a small enough time-like distance between them in AdS vanishes, but this is not always true in the deformed theories. We discuss how this is consistent with causality.

Figures (6)

• Figure 1: Diagrams of AdS space. The dependence on the $\Omega_{d-1}$ coordinates is suppressed. The regions (marked by G) in which points can be connected to the origin by a time-like geodesic are shown in (a). A time-like geodesic in AdS connecting the origin with a point which is also in the 'boundary affected' region (see below) is shown in (b).
• Figure 2: Diagrams of AdS space. The distinction between regions for which the commutator is potentially affected or unaffected by the boundary conditions is shown in (a). A causal curve in AdS connecting the origin with a point in the 'boundary affected' region is shown in (b). The proper time along this curve is larger than $\delta/\cos(\theta_0)$. By taking $\theta_0$ to $\pi/2$ it can be made arbitrarily long.
• Figure 3: The branch cuts of the function $\cosh^{-\Delta_1}(\tau'-\tau_E(\eta))\cosh^{-\Delta_2}(\tau')$ during the $\tau_E$ rotation. The dashed curve shows the trajectory followed by the point $\tau_E(\eta)$, and the branch cuts are drawn for a particular point on this trajectory.
• Figure 4: The $\tau'$ contour after $\tau_E$ rotation for $0<\tau<\pi$. The relevant branch cuts of the function $\cosh^{-\Delta_1}(\tau'+i\tau)\cosh^{-\Delta_2}(\tau')$ are shown on the figure. The dashed curves show the last part of the trajectory followed by the points $\tau_E(\eta)$ and $\tau_E(\eta)+i\pi/2$ during the rotation.
• Figure 5: The $\tau'$ contour after $\tau_E$ rotation for $\pi<\tau<2\pi$. The relevant branch cuts of the function $\cosh^{-\Delta_1}(\tau'+i\tau)\cosh^{-\Delta_2}(\tau')$ are shown on the figure. The dashed curve shows the last part of the trajectory followed by the point $\tau_E(\eta)+i\pi/2$ during the rotation.