AdS3/RMT2 duality

A bstract We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS 3 quantum gravity. We present a 2d CFT trace formula, precisely analogous to the Gutzwiller trace formula for chaotic quantum systems, which originates from the SL(2 , ℤ ) spectral decomposition of the Virasoro primary density of states. An analogy to Berry’s diagonal approximation allows us to extract spectral statistics of individual 2d CFTs by coarse-graining, and to identify signatures of chaos and random matrix universality. This leads to a necessary and sufficient condition for a 2d CFT to display a linear ramp in its coarse-grained spectral form factor. Turning to gravity, AdS 3 torus wormholes are cleanly interpreted as diagonal projections of squared partition functions of microscopic 2d CFTs. The projection makes use of Hecke operators. The Cotler-Jensen wormhole of AdS 3 pure gravity is shown to be extremal among wormhole amplitudes: it is the minimal completion of the random matrix theory correlator compatible with Virasoro symmetry and SL(2 , ℤ )-invariance. We call this MaxRMT: the maximal realization of random matrix universality consistent with the necessary symmetries. Completeness of the SL(2 , ℤ ) spectral decomposition as a trace formula allows us to factorize the Cotler-Jensen wormhole, extracting the microscopic object Z RMT ( τ ) from the coarse-grained product. This captures details of the spectrum of BTZ black hole microstates. Z RMT ( τ ) may be interpreted as an AdS 3 half-wormhole. We discuss its implications for the dual CFT and modular bootstrap at large central charge.