Islands and Page curves in 4d from Type IIB

TL;DR

This work provides a UV-complete string-theory realization of entanglement islands and Page curves for black holes in four-dimensional gravity by constructing Type IIB AdS4×S^2×S^2×Σ solutions sourced by D3/D5/NS5 branes. It analyzes 8d minimal (Ryu-Takayanagi) surfaces that wrap internal spheres and Σ, distinguishing Hartman-Maldacena surfaces from island surfaces for both non-gravitating and gravitating baths, and demonstrates Page curves and phase structures controlled by brane-setup parameters such as N5/K and δ. The results connect to Karch/Randall-type physics, identify 10d analogs of left/right entanglement entropy, and reveal how critical brane parameters govern the existence and dominance of islands, with finite temperature regulating divergences. The findings offer a concrete bridge between island physics and dual BCFT/3d SCFT descriptions, enabling future QFT-based investigations of information paradox resolutions in higher-dimensional, UV-complete frameworks.

Abstract

Variants of the black hole information paradox are studied in Type IIB string theory setups that realize four-dimensional gravity coupled to a bath. The setups are string theory versions of doubly-holographic Karch/Randall brane worlds, with black holes coupled to non-gravitating and gravitating baths. The 10d versions are based on fully backreacted solutions for configurations of D3, D5 and NS5 branes, and admit dual descriptions as $\mathcal N=4$ SYM on a half space and 3d $T_ρ^σ[SU(N)]$ SCFTs. Island contributions to the entanglement entropy of black hole radiation systems are identified through Ryu/Takayanagi surfaces and lead to Page curves. Analogs of the critical angles found in the Karch/Randall models are identified in 10d, as critical parameters in the brane configurations and dual field theories.

Figures (12)

• Figure 1: Left: Karch/Randall model for non-gravitating bath. The figure shows part of $AdS_5$ with the ETW brane cutting off the shaded region. The dashed curve is the black hole horizon and $R$ is the radiation region (blue). The green curve ending on the horizon represents the HM surface; the green curve extending from the boundary of $R$ to the ETW brane is the island surface. $I$ is the island (red). Right: For a gravitating bath a second ETW brane is introduced, leaving only a 3-dimensional part of the conformal boundary.
• Figure 2: Left: Geometry of $AdS_4\times S^2\times S^2\times\Sigma$ solutions with $\Sigma=\lbrace x+iy\in\mathds{C}\vert \,0\leq y\leq \frac{\pi}{2}\rbrace$ for non-gravitating baths. On each boundary component an $S^2$ collapses, so the 10d geometry is closed. D5/NS5 brane sources are located on the $y=0$/$y=\frac{\pi}{2}$ boundaries. The limit $x\rightarrow -\infty$ is a regular point of the internal space. For $x\rightarrow\infty$ the solutions approach locally $AdS_5\times S^5$; this region corresponds to the conformal boundary in fig. \ref{['fig:KR-nongrav']}. The ETW brane in fig. \ref{['fig:KR-nongrav']} can be seen as effective description for the remaining 10d geometry. Right: Associated configuration of D5, NS5 and D3 branes, with D3-branes suspended between 5-branes and semi-infinite D3-branes emerging in one direction. The distribution of 5-brane sources in the supergravity solution encodes how many D5/NS5 branes there are and how the D3-branes end on them.
• Figure 3: Left: $AdS_4\,{\times}\, S^2\,{\times}\, S^2\,{\times}\,\Sigma$ solutions for gravitating baths. The $AdS_5\times S^5$ region is closed off; the limits $x\rightarrow \pm\infty$ both lead to regular points in the internal space. This leaves the 3d conformal boundary of $AdS_4$, corresponding to the remaining point of the conformal boundary in fig. \ref{['fig:KR-grav']}. Right: The associated brane configurations have no semi-infinite D3-branes, only D3-branes suspended between 5-branes.
• Figure 4: Brane configurations for representative non-gravitating bath solutions (left) and gravitating bath solutions (right). Hanany-Witten transitions can be used to make the 3d quiver gauge theories more apparent, as in figs. \ref{['fig:AdS4-sol']}, \ref{['fig:AdS4-sol-grav']}. The numbers of D3-branes on the right are controlled by $\delta$ through $\Delta=\frac{1}{2}+\frac{2}{\pi}\arctan e^{2\delta}$.
• Figure 5: Island surfaces from top left to bottom right anchored at $r_R\in\lbrace 5,3,2.1,1\rbrace$. The horizon is at $r_h=0$ and $N_5/K=2$. The $AdS_5\times S^5$ region emerges at $\tanh x=1$, the 5-brane sources are at $\tanh x=0$ and $\tanh x=-1$ is a regular point in the internal space. For smaller $r_R$ (smaller radiation region) the surfaces stay closer to the horizon. Near the 5-brane sources the surfaces reach to the horizon for all $r_R$.