Hartle-Hawking state and its factorization in 3d gravity

TL;DR

This work develops a nonperturbative framework for 3d gravity with two asymptotic AdS boundaries by mapping it to two copies of Liouville theory via coupled Alekseev-Shatashvili theories. It identifies the Hartle-Hawking state with Liouville ZZ boundary states and uses wormhole slicing to connect gravity observables to Liouville overlaps and Virasoro blocks, reproducing BTZ partition functions, correlation functions, and Wheeler-DeWitt wavefunctions in various bases. A central result is the factorization of the two-sided Hartle-Hawking state into single-sided Hilbert spaces through a defect operator that enforces bulk contractibility, yielding a FLM-like entanglement entropy with an area term matching the Bekenstein-Hawking entropy. The paper also discusses off-shell quantities and late-time ramp behavior via baby universes and Verlinde loop operators, highlighting ensemble interpretations and universal structures in chaotic 2d CFTs, and outlines promising directions toward rotating BTZ, universal algebras, and topological recursion in 3d gravity.

Abstract

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.

Figures (33)

• Figure 1: On the annulus, the gravity Hilbert space is equivalent to a superposition of coupled chiral(anti-chiral) and anti-chiral(chiral) Alekseev-Shatashvili theories that lives on the inner and outer circle respectively.
• Figure 2: The spatial geometry that corresponds to fixed holonomies $K=\gamma L_0, \tilde{K}=\tilde{\gamma} \widetilde{L_0}$. The length of the waist is $\pi(\gamma+\bar{\gamma})$.
• Figure 3: (Left): On the Poincare disk, the ZZ boundary condition (indicated by orange line) is imposed at $|z| \rightarrow 1$. (Right): Alternatively, we can conformally map the disk to the upper half plane and impose the ZZ boundary condition on the real line $w=\bar{w}$.
• Figure 4: From the open-closed duality, we interpret the boundary condition in the open channel as a boundary state in the closed channelCardy:2004hm.
• Figure 5: The transition amplitude of the ZZ boundary states, as shown in \ref{['eq:Z_BTZ_ampltiude']}, has the topology of a cylinder and is equal to the identity character defined on the torus via the doubling trick.