Entanglement entropy in three dimensional gravity

TL;DR

This work develops an algebraic framework to compute holographic entanglement entropy in three-dimensional gravity by expressing spacetimes as quotients of AdS$_3$ and relating regulated geodesic lengths to traces of group elements. The method handles both Lorentzian and Euclidean descriptions, enabling precise analyses of BTZ, RP$^2$ geon, and multi-boundary wormholes, including behind-horizon effects and time-dependent entanglement. It also clarifies how covariant HRT entropy may emerge from Euclidean replica calculations and highlights the role of complexified geodesics in certain phases. Collectively, the results provide a fast, coordinate-free toolkit for exploring entanglement in highly nontrivial topologies and bolster understanding of RT/HRT in lower-dimensional gravity with potential implications for higher-dimensional holography and CFT interpretations.

Abstract

The Ryu-Takayanagi and covariant Hubeny-Rangamani-Takayanagi proposals relate entanglement entropy in CFTs with holographic duals to the areas of minimal or extremal surfaces in the bulk geometry. We show how, in three dimensional pure gravity, the relevant regulated geodesic lengths can be obtained by writing a spacetime as a quotient of AdS3, with the problem reduced to a simple purely algebraic calculation. We explain how this works in both Lorentzian and Euclidean formalisms, before illustrating its use to obtain novel results in a number of examples, including rotating BTZ, the RP2 geon, and several wormhole geometries. This includes spatial and temporal dependence of single-interval entanglement entropy, despite these symmetries being broken only behind an event horizon. We also discuss considerations allowing HRT to be derived from analytic continuation of Euclidean computations in certain contexts, and a related class of complexified extremal surfaces.

Figures (8)

• Figure 1: The choices of geodesics to compute $S(\mathcal{A}\cup\mathcal{B})$ in the $\mathbb{R}\mathbb{P}^2$ geon. Options (a) and (b) would be admissible in BTZ, and (c) would additionally require inclusion of the event horizon. The geodesics in (d) pass through the crosscap, the arrows indicating that the antipodal points are identified.
• Figure 2: Phases of mutual information of two intervals in the $\mathbb{R}\mathbb{P}^2$ geon, of length $l$, plotted vertically, with centres separated by $\Delta$, plotted horizontally, for various values of $r_+$. The blue, yellow, green and red regions indicate where phases 1,2,3 and 4 respectively dominate, as labelled in subfigure (c).
• Figure 3: The $t=0$ slice of $AdS_3$, showing a fundamental region and event horizons for the three boundary wormhole. The blue curves, identified by $g_1$, and the orange, identified by $g_2$, mark the edge of the fundamental domain. The dashed, dotted, and dot-dashed lines mark the event horizons of the three exterior BTZ regions.
• Figure 4: The geodesics giving the four possible phases of entanglement entropy of a single interval, in green, along with the event horizons added to satisfy the homology constraint, marked by dashed lines.
• Figure 5: Phase diagram for single-interval entanglement as a function of the moduli of the spacetime, in the case when two of the horizon lengths are equal. Below and to the right of the solid line, one of phases 3 or 4 dominates in some region of moduli space, and $S(\mathcal{A})$ has nontrivial dependence on space and time. This is always below the dashed line $\ell_1=\ell_2+\ell_3$, where there is a phase transition associated to the closed geodesics.