Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space

TL;DR

The paper analyzes nearly AdS2 gravity through Jackiw–Teitelboim dilaton theory, showing that a spontaneously broken reparametrization symmetry (broken to SL(2)) is explicitly broken by the Schwarzian action. This Schwarzian dynamics governs the boundary modes (boundary gravitons), fixes the near-extremal entropy behavior (linear in temperature), and determines gravitational corrections to correlation functions, including the out-of-time-ordered correlators that diagnose chaos. By coupling to matter and performing perturbative and nonperturbative analyses, the authors derive explicit four-point corrections, loop effects, and a full resummation in the chaos regime, while clarifying the Lorentzian SL(2) structure and the role of charges as horizon data and edge modes. The framework provides a compact, hydrodynamics-like effective action for IR dynamics in NAdS2 and offers a controlled setting to study backreaction and scrambling in two-dimensional gravity.

Abstract

We study a two dimensional dilaton gravity system, recently examined by Almheiri and Polchinski, which describes near extremal black holes, or more generally, nearly $AdS_2$ spacetimes. The asymptotic symmetries of $AdS_2$ are all the time reparametrizations of the boundary. These symmetries are spontaneously broken by the $AdS_2$ geometry and they are explicitly broken by the small deformation away from $AdS_2$. This pattern of spontaneous plus explicit symmetry breaking governs the gravitational backreaction of the system. It determines several gravitational properties such as the linear in temperature dependence of the near extremal entropy as well as the gravitational corrections to correlation functions. These corrections include the ones determining the growth of out of time order correlators that is indicative of chaos. These gravitational aspects can be described in terms of a Schwarzian derivative effective action for a reparametrization.

Figures (5)

• Figure 1: (a) Hyperbolic space or Euclidean $AdS_2$. The orbits of $\tau$ translations look like circles. Orbits of $t$ are curves that touch the boundary at $t=\pm\infty$. (b) Lorentzian $AdS_2$. The $\nu, \sigma$ coordinates cover the whole strip. The ${\hat{t}}, z$ coordinates describe the Poincare patch denoted here in yellow. The red region is covered by the ${\hat{\tau} } , \rho$ coordinates. There are different choices for how to place the ${\hat{\tau} } , \rho$ region that are generated by SL(2) isometries. In (b) and (c) we show two choices and give the relation between the Poincare time ${\hat{t}}$ and the ${\hat{\tau} }$ at the boundary of the space. In (d) we show a generic pair of Rindler wedges.
• Figure 2: In (a) we see the full $AdS_2$ space. In (b) we cut it off at the location of a boundary curve. In (c) we choose a more general boundary curve. The full geometry of the cutout space does depend on the choice of the boundary curve. On the other hand, the geometry of this cutout region remains the same if we displace it or rotate it by an SL(2) transformation of the original $AdS_2$ space.
• Figure 3: The black contour at left is the minimal time contour needed to compute $\langle V_1W_3V_2W_4\rangle$. Horizontal is real time, vertical is imaginary time, the ends are identified. It is convenient to extend the folds to infinity, as shown at right. The blue and red indicate which sheets have the $X^-$ and $X^+$ perturbations turned on.
• Figure 4: The correlation function (\ref{['resummed']}) in the configuration (\ref{['equalspacing']}), with $8C = 10^6$. Scaling up $C$ simply translates all of the curves to the right without changing their shape. The initial descent from the plateau is characterized by the $e^{\frac{2\pi}{\beta}\ul}$ behavior. The final approach to zero is determined by quasinormal decay.
• Figure 5: In (a) we show the trajectories of the $V,W$ quanta without backreaction. In (b) we show the backreaction of the $V$ particle. This is still a piece of $AdS_2$, but it is a smaller piece. In (c) we add back $W$ in the arrangement appropriate for the operator ordering $V(-\ul)W(0)$. The trajectories are (almost) the same as (a) relative to the fixed $AdS_2$ coordinates of the diagram, but they change relative to the physical $\ul$ coordinate. In (d) we show the other ordering $W(0)V(-\ul)$. Now the red line touches the boundary at time $\ul = 0$. Although it is difficult to see in this frame, the $V$ line has moved down slightly, so that it no longer reaches the boundary.