Quantum Extremal Islands Made Easy, PartIII: Complexity on the Brane

TL;DR

The work investigates holographic complexity in the island paradigm within a doubly holographic setup, deriving leading island contributions to the subregion CV via a Fefferman-Graham expansion near the brane. It proposes a generalized CV functional on the brane, consisting of a generalized volume $W_{ m gen}$ and an extrinsic-curvature term $W_K$, to encode higher-curvature corrections; the coefficients are fixed by matching near-brane expansions and consistency checks. The authors test the proposal in Gauss-Bonnet and $f(\mathcal{R})$ bulk gravity, showing that the island complexity on the brane derived from the induced higher-curvature actions agrees with the bulk results, thereby linking brane-induced and bulk holographic complexities. The findings illuminate how geometric and quantum contributions partition in island setups and suggest avenues to extend CV to broader higher-curvature theories, with implications for the interplay between complexity, gravity, and quantum information in holography.

Abstract

We examine holographic complexity in the doubly holographic model introduced in [arXiv:2006.04851][arXiv:2010.00018] to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the induced higher-curvature gravity action on the brane. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(R) gravity in the bulk.

Figures (5)

• Figure 1: The choice of RT surfaces for the boundary subregion $\mathbf{R}=\mathbf{R}_{\textrm{\tiny L}}\cup\mathbf{R}_{\textrm{\tiny R}}$ on a constant time slice in the presence of the brane (coloured green), showing the island and no-island phases in the right and left panels, respectively. The complexity ${\cal C}^{\rm sub}_{\textrm{\tiny V}}(\mathbf{R})$ in eqs. \ref{['eq:CVsubregion']} and \ref{['eq:CVsubregion2']} is determined by the extremal surface $\mathcal{B}=\mathcal{B}_{L}\cup \mathcal{B}_R$. In the island phase, the intersection of this surface with the brane defines the 'island' $\widetilde{\mathcal{B}}=\mathcal{B}\,\cap\,$brane.
• Figure 2: The holographic setup with islands in AdS$_{d+1}$. The two AdS$_{d+1}$ geometries are cut off at $\theta=\theta_\textrm{\tiny B}$ (or $z=z_{\textrm{\tiny B}}$) and glued together with the brane at the junction between the two. The island region emerges on the brane when the RT surfaces $\Sigma_{\mathbf{R}}$ of the boundary subregion $\mathbf{R}=\mathbf{R}_{\textrm{\tiny L}} \cup \mathbf{R}_{\textrm{\tiny R}}$ cross the brane. The maximal volume bulk slice $\mathcal{B} = \mathcal{B}_\textrm{\tiny L} \cup \mathcal{B}_\textrm{\tiny R}$ crosses the brane, and the intersection of these two surfaces determines the island $\tilde{\cal B} = {\cal B} \cap {\rm brane} = {\cal B}_\textrm{\tiny L} \cap {\cal B}_\textrm{\tiny R}$.
• Figure 3: The full asymptotically AdS$_{d+1}$ geometry from the right side of the construction in figure \ref{['twoAdS']}. The time slice $\mathbf{S}$ is introduced in the left panel and detailed in the right panel. We explicitly show various metrics for the different regions.
• Figure 4: Different hypersurfaces in the doubly holographic system and their corresponding extrinsic curvatures.
• Figure 5: Different boundary subregions, $\mathbf{R}=\mathbf{R}_{\textrm{\tiny L}} \cup \mathbf{R}_{\textrm{\tiny R}}$ and $\mathbf{R}'=\mathbf{R}'_{\textrm{\tiny L}} \cup \mathbf{R}'_{\textrm{\tiny R}}$ with the same boundaries, i.e.,$\partial\mathbf{R} =\partial\mathbf{R}'$. The entanglement entropy and the RT surface remains the same for both subregions. However, the extremal surfaces $\mathcal{B}$ and $\mathcal{B}'$ (denoted by the orange regions) are different and hence they produce different islands $\widetilde{\mathcal{B}}$ and $\widetilde{\mathcal{B}}'$ on the brane (represented by the blue slice). The QES on the brane is unchanged and hence $\partial\widetilde{\mathcal{B}}=\partial\widetilde{\mathcal{B}}' =\sigma_{\mathbf{R}}$. The red shaded regions on the asymptotic boundary represent the causal domain of $\mathbf{R}$ ($\widetilde{\mathcal{B}}$). The subregion $\mathbf{R}'$ may be any spacelike surface in this causal domain. Similarly, $\widetilde{\mathcal{B}}'$ will always lie within the causal domain of the brane (denoted by the pink region).