Relaxation in Conformal Field Theory, Hawking-Page Transition, and Quasinormal/Normal Modes

TL;DR

The paper investigates how 1+1D conformal field theories relax to thermal equilibrium at finite temperature and finite volume. Using linear response, torus correlators, and AdS3/CFT2 duality, it connects the decay of perturbations to bulk quasinormal modes in the BTZ black hole and the oscillatory normal modes of thermal AdS, with a Hawking-Page transition at large k demarcating these regimes. In the strong coupling limit, the transition reflects a switch from exponential decay to oscillations as the dominant bulk geometry changes, while finite-k corrections are expected to restore Poincaré recurrences and spectrum discreteness. The work highlights a deep link between bulk gravitational dynamics and boundary CFT relaxation, and outlines future directions for nonperturbative finite-k analyses and extensions to higher dimensions.

Abstract

We study the process of relaxation back to thermal equilibrium in $(1+1)$-dimensional conformal field theory at finite temperature. When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time. On the other hand, when the inverse temperature is large, the relaxation is oscillatory with characteristic period set by the size of the system. We then analyse the intermediate regime in two specific models, namely free fermions, and a strongly coupled large $\tt k$ conformal field theory which is dual to string theory on $(2+1)$-dimensional anti-de Sitter spacetime. In the latter case, there is a sharp transition between the two regimes in the ${\tt k}=\infty$ limit, which is a manifestation of the gravitational Hawking-Page phase transition. In particular, we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behaviour of the conformal field theory.