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We study 3d quantum gravity with two asymptotically anti-de Sitter regions,in particular, using its relation with coupled Alekseev-Shatashvili theoriesand Liouville theory. Expressions for the Hartle-Hawking state, thermal$2n$-point functions, torus wormhole correlators and Wheeler-DeWittwavefunctions in different bases are obtained using the ZZ boundary states inLiouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity areuplifted to 3d gravity, with two copies of Liouville theory in 3d gravityplaying a similar role as Schwarzian theory in JT gravity. The connectionbetween 3d gravity and the Liouville ZZ boundary states are manifested byviewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormholeboundaries glued along the ZZ boundaries. In this work, we also study thefactorization problem of the Hartle-Hawking state in 3d gravity. With therelevant defect operator that imposes the necessary topological constraint forcontractibility, the trace formula in gravity is modified in computing theentanglement entropy. This trace matches with the one from von Neumann algebraconsiderations, further reproducing the Bekenstein-Hawking area formula fromentanglement entropy. Lastly, we propose a calculation for off-shellgeometrical quantities that are responsible for the ramp behavior in the latetime two-point functions, which follows from the understanding of the LiouvilleFZZT boundary states in the context of 3d gravity, and the identificationbetween Verlinde loop operators in Liouville theory and "baby universe"operators in 3d gravity.

Hartle-Hawking state and its factorization in 3d gravity