Chaos and the reparametrization mode on the AdS$_2$ string

TL;DR

This paper analyzes holographic correlators in AdS$_2$ string theory, revealing chaos saturation, an exact double-scaling limit solution, and clarifying the role of boundary reparametrizations, with applications to defect CFTs in ${ m f N}=4$ SYM.

Contribution

It introduces a conformal gauge approach to AdS$_2$ strings, deriving the reparametrization mode's non-local action and connecting it to chaos and boundary correlators, differing from JT gravity.

Findings

OTOCs exhibit Lyapunov growth saturating chaos bound.

Exact double-scaling limit correlator matches JT gravity results.

Reparametrization mode governed by a non-local action distinct from Schwarzian.

Abstract

We study the holographic correlators corresponding to scattering of fluctuations of an open string worldsheet with AdS$_2$ geometry. In the out-of-time-order configuration, the correlators display a Lyapunov growth that saturates the chaos bound. We show that in a double-scaling limit interpolating between the Lyapunov regime and the late time exponential decay, the out-of-time-order correlator (OTOC) can be obtained exactly, and it has the same functional form found in the analogous calculation in JT gravity. The result can be understood as coming from high energy scattering near the horizon of a AdS$_2$ black hole, and is essentially controlled by the flat space worldsheet S-matrix. While previous works on the AdS$_2$ string employed mainly a static gauge approach, here we focus on conformal gauge and clarify the role of boundary reparametrizations in the calculation of the correlators. We find that the reparametrization mode is governed by a non-local action which is distinct from the Schwarzian action arising in JT gravity, and in particular leads to $SL(2,\mathbb{R})$ invariant boundary correlators. The OTOC in the double-scaling limit, however, has the same functional form as that obtained from the Schwarzian, and it can be computed using the reparametrization action and resumming a subset of diagrams that are expected to dominate in the limit. One application of our results is to the defect CFT defined by the half-BPS Wilson loop in ${\cal N}=4$ SYM. In this context, we show that the exact result for the OTOC in the double-scaling limit is in precise agreement with a recent analytic bootstrap prediction to three-loop order at strong coupling.