2D CFT partition functions at late times

Abstract

We consider the late time behavior of the analytically continued partition function Z(β + it)Z(β − it) in holographic 2d CFTs. This is a probe of information loss in such theories and in their holographic duals. We show that each Virasoro character decays in time, and so information is not restored at the level of individual characters. We identify a universal decaying contribution at late times, and conjecture that it describes the behavior of generic chaotic 2d CFTs out to times that are exponentially large in the central charge. It was recently suggested that at sufficiently late times one expects a crossover to random matrix behavior. We estimate an upper bound on the crossover time, which suggests that the decay is followed by a parametrically long period of late time growth. Finally, we discuss gravitationally-motivated integrable theories and show how information is restored at late times by a series of characters. This hints at a possible bulk mechanism, where information is restored by an infinite sum over non-perturbative saddles.

Supersymmetric SYK model and random matrix theory

“…Some other related issues and similar models are discussed in . In the recent discussions, people have discovered that the SYK hamiltonian has a clear correspondence with the categories of the three fold standard Dyson ensembles, unitary, orthogonal and sympletic ensembles, in the random matrix theory [54][55][56][57]. In the recent work, [56,57], it is understood that the time-dependent quantum dynamics of the temperature-dependent spectral form factor, namely, the combinations of partition functions with a special analytic continuation in SYK model, is computable in the late time by form factors in the random matrix theory with the same analytic continuation, as a probe of the discrete nature of the energy spectrum in a quantum black hole, and also a solid confirmation on the three-fold classification [57].…”

“…It is established that SYK model is classified by random matrix theory in that the random interacting SYK hamiltonian fall into one of the three standard Dyson ensembles in the eight-fold way [54][55][56][57]. It is natural to believe that the supersymmetric extension can also be described by random matrix theory.…”

“…Now we apply the Altland-Zirnbauer classification theory (see appendix A for some necessary knowledges) to the original SYK model [54][55][56][57]. This is accomplished by finding the symmetry of the theory (and has been already discussed in other works, see [54,57]).…”

A numerical approach to Virasoro blocks and the information paradox

“…Roughly speaking, the decay should stop when the correlator reaches ∼ e −S and begins to oscillate chaotically. At a more detailed level, the time dependence can change qualitatively on timescales of order S and e S as different features of λ 2 (E) come into play [41][42][43][44].…”

“…which is related to the first channel by the bootstrap equation (or by modular invariance in the case of the partition function [43]). In this channel we are sensitive to the exchange of states between the heavy and light operators.…”