Flowery Horizons & Bulk Observers: $sl^{(q)}(2,\mathbb{R})$ Drive in $2d$ Holographic CFT

TL;DR

This paper investigates the geometric evolution of the event horizon in a 2D holographic CFT driven by an $sl^{(q)}(2, {R})$ Hamiltonian, revealing distinct behaviors and symmetry-breaking patterns related to the drive.

Contribution

It introduces a geometric classification of horizon evolution under $sl^{(q)}(2, {R})$ driving, including symmetry-breaking structures and connections to the CFT modular Hamiltonian.

Findings

Horizon evolution exhibits oscillatory, exponential, and power-law behaviors.

Explicit symmetry breaking manifests as a $U(1) o Z_q$ transition.

Fixed points of asymptotic Killing vectors relate to bulk Ryu-Takayanagi surfaces.

Abstract

We explore and analyze bulk geometric aspects corresponding to a driven two-dimensional holographic CFT, where the drive Hamiltonian is constructed from the $sl^{(q)}(2,\mathbb{R})$ generators. In particular, we demonstrate that starting with a thermal initial state, the evolution of the event horizon is characterized by distinct geometric transformations in the bulk which are associated to the conjugacy classes of the corresponding transformations on the CFT. Namely, the bulk evolution of the horizon is geometrically classified into an oscillatory (non-heating) behaviour, an exponentially growing (heating) behaviour and a power-law growth with an angular rotation (the phase boundary), all as a function of the stroboscopic time. We also show that the explicit symmetry breaking of the drive is manifest in a flowery structure of the event horizon that displays a $U(1) \to {\mathbb Z}_q$ symmetry breaking. In the $q\to \infty$ limit, the $U(1)$ symmetry is effectively restored. Furthermore, by analyzing the integral curves generated by the asymptotic Killing vectors, we also demonstrate how the fixed points of these curves approximate a bulk Ryu-Takayanagi surface corresponding to a modular Hamiltonian for a sub-region in the CFT. Since the CFT modular Hamiltonian has an infinitely many in-equivalent extensions in the bulk, the fixed points of the integral curves can also lie outside the entanglement wedge of the CFT sub-region.