  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.

High Energy Physics - Theory