Relaxation in conformal field theory, Hawking-Page transition, and quasinormal or normal modes

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 the characteristic period set by the size of the system. We then analyze the intermediate regime in two specific models: namely, free fermions, and a strongly coupled large 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 k = infinity 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 behavior of the conformal field theory.