Symmetries and spectral statistics in chaotic conformal field theories

We discuss spectral correlations in coarse-grained chaotic two-dimensional CFTs with large central charge. We study a partition function describing the dense part of the spectrum of primary states in a way that disentangles the chaotic properties of the spectrum from those which are a consequence of Virasoro symmetry and modular invariance. We argue that random matrix universality in the near-extremal limit is an independent feature of each spin sector separately; this is a non-trivial statement because the exact spectrum is fully determined by only the spectrum of spin zero primaries and those of a single non-zero spin (spectral determinacy). We then describe an argument analogous to the one leading to Cardy's formula for the averaged density of states, but in our case applying it to spectral correlations: assuming statistical universalities in the near-extremal spectrum in all spin sectors, we find similar random matrix universality in a large spin regime far from extremality.

Automated summary

We discuss the spectral correlations in two-dimensional chaotic CFTs with large central charge. By studying the partition function, we separate the chaotic properties of the spectrum from those governed by Virasoro symmetry and modular invariance. We find that random matrix universality is independently observed in each spin sector, even though the exact spectrum is determined by only the spectrum of spin zero primaries and a single non-zero spin. We also show the existence of random matrix universality in a large spin regime far from extremality.