A note on the bulk interpretation of the Quantum Extremal Surface formula

TL;DR

This work provides a bulk canonical interpretation of the quantum extremal surface formula in AdS3 gravity by showing that the leading area term counts gravitational edge modes transforming under the quantum group SL_q^+(2,R) × SL_q^+(2,R). By combining bulk modular invariance, shrinkable boundary conditions, and an extended TQFT framework, the authors derive the QES entanglement entropy as S_gen = −Tr ρ_V log ρ_V, with the area contribution given by log dim_q p^* and the bulk entanglement accounting for S_bulk. They develop a bulk factorization map via the SL_q^+(2,R) co-product, connect boundary and bulk pictures through AdS/BCFT-like constructions, and compare to the c = 1 boson case to highlight edge-mode dominance for large central charge. The results support viewing AdS3 gravity as a topological phase with gravitational anyons and suggest a consistent, nonperturbative bulk path integral framework for cutting and gluing across co-dimension-2 surfaces. The extended TQFT perspective organizes edge data and fusion rules, offering a path toward a sharper canonical definition of bulk quantum information in gravity and guiding future generalizations to more complex states and higher dimensions.

Abstract

Defining quantum information quantities directly in bulk quantum gravity is a difficult problem due to the fluctuations of spacetime. Some progress was made recently in \cite{Mertens:2022ujr}, which provided a bulk interpretation of the Bekenstein Hawking formula for two sided BTZ black holes in terms of the entanglement entropy of gravitational edge modes. We generalize those results to give a bulk entanglement entropy interpretation of the quantum extremal surface formula in AdS3 gravity, as applied to a single interval in the boundary theory. Our computation further supports the proposal that AdS3 gravity can be viewed as a topological phase in which the bulk gravity edge modes are anyons transforming under the quantum group $\SL^{+}_{q}(2,\mathbb{R})$. These edge modes appear when we cut open the Euclidean path integral along bulk co-dimension 2 slices, and satisfies a shrinkable boundary condition which ensures that the Gibbons-Hawking calculation gives the correct state counting.

Figures (16)

• Figure 1: A path integral on geometries with the topology of a cigar can be given a trace interpretation by replacing the tip with a shrinkable boundary condition e. This effectively replaces the cigar with an annulus, which defines a trace in the "open string " channel.
• Figure 2: The lower half of the annulus with a shrinkable boundary condition $e$ inserted can be viewed as the path integral preparation of a factorized state in the bulk extended Hilbert space $\mathcal{H}_{V}\otimes \mathcal{H}_{\bar{V}}$. The subsequent Lorentzian evolution describes an entangled sum of one sided geometries
• Figure 3: On the left, we have the solid torus partition function of the bulk TQFT, with no Wilson lines inserted. This can be given a perturbative interpretation as the quantization of fluctuations around a BTZ saddle obtained by filling the Euclidean time cycle. On the right, we interpret $Z(\tau,\bar{\tau})$ in terms of the solid torus in the dual channel, where we fill in the spatial cycle with a disk punctured by a superposition of Wilson lines. The punctured disk is the spatial slice of a black hole geometry describing individual microstates of the thermal ensemble \ref{['solidtorus']}
• Figure 4: The shrinkable boundary condition
• Figure 5: On the left we show the Einsten Rosen bridge as the time refection symmetric slice of the Euclidean BTZ geometry. On the right, we show the two sided Lorentzian geometry of obtained by evolving the Hartle Hawking state produced by the path integral on the left