Hartle-Hawking state and its factorization in 3d gravity

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal $2n$-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and "baby universe" operators in 3d gravity.