Maximal Chaos from Strings, Branes and Schwarzian Action

Abstract

In this article, we explicitly demonstrate that, for a sufficiently generic class of examples, an open string embedded in an AdS-background yields an effective Schwarzian action and, in the semi-classical description, the fluctuation modes of the open string couple to this Schwarzian sector. This leads to a maximal chaos, observed in the open string degrees of freedom, irrespective of the gravitational background. This corresponds to the dynamics of quark-like degrees of freedom in a strongly coupled large N_c gauge theory. We also demonstrate a maximal chaos, resulting from an inherent D-brane horizon, by computing the four-point out-of-time-ordered correlator of spin-one operators. We also offer some observations and comments regarding a class of effective theories that is described by a generic functional of a Schwarzian derivative.

Figures (6)

• Figure 1: The maximally extended geometry for an eternal black hole in AdS. This represents a pure state in which the complete dual CFT is a tensor product of the two CFTs, defined on the left and the right boundaries.
• Figure 2: A pictorial representation of gluing two arbitrary states along a Schwinger-Keldysh contour, after the insertion of an operator at real time $t$. The genus $3$ surface in the picture is supposed to represent an arbitrary initial state. The choice of genus $=3$ is simply representative. This picture is inspired from the discussions and figures in SonnerLecture.
• Figure 3: A demonstration of the Schwinger-Keldysh contour on which the four-point OTOC is calculated. The solid dots denote an insertion of the corresponding operator.
• Figure 4: A cartoon demonstrating how Diff$_2$ can be obtained from a Diff$_3$. The left dashed curve represents a fluctuating worldsheet, embedded in an AdS$_3$. If the fluctuation belongs to the soft sector, then a Diff$_2$ can be used to map the fluctuating worldsheet back to the classical profile.
• Figure 5: A Penrose diagram representation of the $2-2$ elastic scattering which is used to calculate the phase shift. This is identical to the discussion in Shenker:2014cwa. The Kruskal extension was obtained in Banerjee:2016qeu.