BTZ dynamics and chaos

TL;DR

This work develops a boundary-diffeomorphism description of chaos in AdS$_3$/CFT$_2$ by deriving the effective action for Brown–Henneaux modes, revealing a left–right decoupled Schwarzian structure at leading order in $G_N$.The authors compute boundary 4pt functions to linear order in $G_N$, showing that AdS$_3$ Schwarzschild reproduces the standard Lyapunov exponent $λ_L=2π/β$, while rotating BTZ yields $λ_L=2π/β_+$ with $β_+=β(1-μ_L)$, indicating two distinct chaotic scales governed by angular momentum.The results imply a modified chaos bound in the presence of a chemical potential for angular momentum and motivate extensions to higher dimensions and near-extremal Kerr–AdS geometries, where similar Schwarzian-like soft modes may control chaos.Overall, the paper provides a boundary-action framework for maximal chaos in AdS$_3$ and highlights how rotation and chemical potentials reshape chaotic growth in holographic CFTs.

Abstract

We find an effective action for gravitational interactions with scalars in $AdS_3$ to first order in $G_N$ at the conformal boundary. This action can be understood as an action for the Brown-Henneaux modes and is given by the square-root product of right and left moving Schwarzian derivatives for conformal transformations of the boundary. We thus reproduce the result $λ_L=\frac{2π}β$ for OTOC computed first in arXiv:1412.6087 for a Schwarzchild black hole in $AdS_3$. Applying the same procedure to rotating BTZ we find the Lyapunov index to be $λ_L=\frac{2π}{β_+}>\frac{2π}β$ where $β_+=β(1-μ_L)$, with $μ_L=\frac{r_-}{r_+}$ being the chemical potential for angular momentum. We thus comment on a possible modification to a part of the proof given in arXiv:1503.01409 to accommodate this result.