Microcanonical states in higher dimensions, and WdW wavefunction & observables in 3d gravity.

In this talk, I will first describe on obtaining a microcanonical thermofield double state at fixed energy and angular momentum from the gravitational path integral. The corresponding boundary value problem and gravitational action are analyzed. The overlap of this state with the canonical thermofield double state, which is interpreted as the Hartle-Hawking wavefunction of an eternal black hole in a mini-superspace approximation, is calculated semiclassically. The relevant saddlepoint is a higher-dimensional, rotating generalization of the wedge geometry that has been studied in two-dimensional JT gravity. For the remaining part of this talk, I will talk about obtaining the off-shell Hartle-Hawking wavefunction in 3d gravity from overlaps with Liouville ZZ boundary states, and matching the large c limit with the gravitational on-shell action. Similar to JT gravity, there are two different bases to the Hartle-Hawking wavefunction in 3d gravity in the mini-superspace approximation. Finally, I will talk about obtaining thermal 2n-correlation functions from insertion of Liouville primary operators between Hartle-Hawking states. The large c limit reproduces semiclassical BTZ coupled to point particles. The correlation functions in terms of Liouville overlaps also match with the 2d CFT ensemble-averaged result from previous work. This talk is based on 2309.05041 and 2309.05126.