2D CFT Partition Functions at Late Times

We consider the late time behavior of the analytically continued partition function $Z(\beta + it) Z(\beta - it)$ 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 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.