Complex Entangling Surfaces for AdS and Lifshitz Black Holes?

TL;DR

This work investigates whether complex codimension-2 extremal surfaces in analytically continued AdS and Lifshitz black holes can participate in holographic entanglement entropy as proposed by RT/HRT. It develops a contour-based method to locate complex surfaces, revealing multiple secondary-sheets saddles with real parts of the renormalized area that can match or undercut real extremal-surface areas in certain spacetimes (notably Schwarzschild-AdS and Lifshitz). A straw-man prescription using Re$A_{ ext{ren}}$ is explored, with BTZ serving as a baseline and higher-dimensional AdS/Lifshitz cases showing richer complex-geometry structures. The findings suggest complex surfaces could plausibly govern CFT entropies in some regimes, but also highlight substantial conceptual and technical challenges in selecting physical saddles and ensuring consistency with replica and causality constraints. Further work is needed to connect these complex saddles to concrete CFT entropies and to establish robust selection criteria across spacetimes.

Abstract

We discuss the possible relevance of complex codimension-two extremal surfaces to the the Ryu-Takayanagi holographic entanglement proposal and its covariant Hubeny-Rangamani-Takayanagi (HRT) generalization. Such surfaces live in a complexified bulk spacetime defined by analytic continuation. We identify surfaces of this type for BTZ, Schwarzschild-AdS, and Schwarzschild-Lifshitz planar black holes. Since the dual CFT interpretation for the imaginary part of their areas is unclear, we focus on a straw man proposal relating CFT entropy to the real part of the area alone. For Schwarzschild-AdS and Schwarzschild-Lifshitz, we identify families where the real part of the area agrees with qualitative physical expectations for the appropriate CFT entropy and, in addition, where it is smaller than the area of corresponding real extremal surfaces. It is thus plausible that the CFT entropy is controlled by these complex extremal surfaces.

Figures (11)

• Figure 1: A conformal diagram of our spacetimes. The asymptotic regions are located in the left and right regions. The imaginary part of the time coordinate $t$ is constant in each wedge, and $t$ has period $t \sim t + i\beta$. We consider extremal surfaces anchored at the points indicated on each boundary.
• Figure 2: The branching structure of the integrands of \ref{['eq:deltat']} and \ref{['eq:areacontour']} in the complex $R$-plane, and sample contours of integration $\gamma$. The number of branch points depends on the precise form of $V_\mathrm{eff}$; here we draw four, as for geodesics in $d = 3$ AdS-Schwarzschild. The branch points correspond to zeros of $V_\mathrm{eff}$ and often an additional branch point at $R=0$. We introduce branch cuts in order to draw figures; the solid and dashed portions of $\gamma$ indicate segments that run on different sheets of the associated Riemann surface. For convenience we choose the branch cuts to run radially inward, connecting all other branch points $R_{\mathrm{branch}}$ directly to $R=0$. We adopt this convention even when $R=0$ is not a branch point -- in effect momentarily introducing an artificial branch point whose effects must disappear from the final results. Figure (a) shows the generic (complex $E$) case in which all the branch points lie at complex $R$. Figure (b) shows the special case in which $E$ is real, in which case at least one of the branch points lies on the positive $R$-axis. The extremal surface corresponding to the indicated contour $\gamma$ is then equivalent to a real extremal surface which may be described as having a turning point at the encircled branch point. The integrand for $\Delta t$ may also have poles at other values of $R$, but these are not shown.
• Figure 3: Sample integration contours $\gamma_1', \gamma_2'$ for \ref{['eq:deltat']} and \ref{['eq:areacontour']} that define secondary Riemann sheets of $\Delta t(E)$. Both contours are obtained from $\gamma$ in figure \ref{['fig:integrationcontour']} by exchanging the branch points in quadrants $1$ and $3$. For $\gamma_1'$ the originally-encircled branch point passes below the other during the exchange, while for $\gamma_2'$ it passes above. At each step, the contour must be deformed to keep it smooth on the associated Riemann surface; it must avoid both branch points and poles, though for simplicity we show only the former.
• Figure 4: A sample choice of branch cut structure used to define a single sheet of $\Delta t$ and $A_\mathrm{ren}$ in the complex $E$-plane; the particular structure shown here is that of e.g. geodesics in Reissner-Nordström AdS$_5$ or codimension-2 extremal surfaces in Schwarzschild-AdS$_7$. The branch points shown here correspond to the critical energies $E_c$ at which the contour of integration $\gamma$ for \ref{['eq:deltat']} and \ref{['eq:areacontour']} becomes pinched between two roots of $V_\mathrm{eff}$ that coincide, and are therefore energies at which $|\Delta t|$ and $|A|$ diverge.
• Figure 5: The structure of the $t_I = \beta/2$ contours for geodesics in Schwarzschild-AdS$_{d+1}$; arrows denote the direction of increasing $t_b$. From left to right, the figures show $d = 3$, $d = 4$, and $d \geq 5$. Note that there is always a contour along the real $E$-axis, which for $d \geq 5$ is disconnected from the two complex ones. The complex contours spiral into the branch points.