\section{Introduction}
\label{sec:1}

    The AdS/CFT correspondence \cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj} maybe a useful tool to investigate the properties of quantum gravity. It is important to get a complete understanding of the AdS/CFT correspondence and to construct the systematical way of translation an arbitrary quantity of CFT to the dual one of the gravity side, vise versa. The most interesting issue among them maybe how to construct the gravity dual of the (reduced) density matrix of a CFT \cite{Czech:2012bh} because the (reduced) density matrix has all information of the (reduced) system. The Ryu-Takayanagi(RT) conjecture \cite{Ryu:2006bv,Ryu:2006ef} has played a fundamental role to work on this issue and shed light on the relation of the AdS/CFT correspondence and the information theory. Besides this issue, there are many information-theoretical analyses \cite{Swingle:2009bg,Pastawski:2015qua,Lashkari:2013koa,Dong:2016eik} of the AdS/CFT correspondence based on the RT conjecture. Thus, the RT conjecture is perhaps a most fundamental concept to grasp the AdS/CFT correspondence.

    The RT conjecture is believed to give a gravity dual of the entanglement entropy(EE) in a static system. If you take care of time-dependent systems, see \cite{Hubeny:2007xt}. In this paper, we deal with only static systems. First, we define EE. Let $\rho_A$ be the reduced density matrix of a spacelike region $A$ in CFT, then EE $S_A$ of a region $A$ is defined as
    \begin{align}
        S_A = - \mathrm{Tr}_A \left(\rho_A \log \rho_A \right).
     \label{eq:def of EE}
    \end{align}
    Then, the RT conjecture proposes that the homologous global minimal surface $\gamma_A$ in the dual bulk spacetime represents the EE of a region $A$ for the large $c$ limit static CFT as
    \begin{align}
        S_A = \frac{\mathrm{Area\ of\ }\gamma_A}{4G},
    \label{eq:RT conjecture}
    \end{align}
    where $G$ is the gravitational constant. The homologous condition means that there exists the region $\mathcal{R}_A$ satisfying $\partial \mathcal{R}_A = A \cup \gamma_A$ \cite{Headrick:2007km}. Surprisingly, in some systems, the RT conjecture eq.\eqref{eq:RT conjecture} is completely consistent with eq.\eqref{eq:def of EE}.

    There, however, exist counterexamples of the RT conjecture in the sense that the EE derived from it does not match that of CFT, as we will see later. Therefore, we have to find a refined gravity dual of the EE by improving the RT conjecture because the RT conjecture may yield to incorrect predictions not only as a tool of calculation for the EE of the dual CFT but also as a basis to unravel the mechanism of the AdS/CFT correspondence.

    To construct a better gravity dual of the EE, we will review how the RT conjecture has derived. Historically, the RT conjecture is proposed as an analog to the Bekenstein-Hawking entropy formula. Later, the homologous condition is imposed by hand to prevent the RT conjecture from contradiction to the strong subadditivity inequality \cite{Headrick:2007km}. Mathematically, the RT conjecture is derived by directly building a gravity dual of the EE eq.\eqref{eq:def of EE} due to the AdS/CFT correspondence \cite{Lewkowycz:2013nqa,Haehl:2014zoa}. These derivations can explain why the area of the \emph{extremal} surfaces is related to EE of dual CFT. In particular, Haehl et al. \cite{Haehl:2014zoa} derived the restriction of the configuration of the extremal surfaces as the intersection number of them, which is a general and rigorous condition of the homologous condition. However, no one has not justified the validity of the global minimality condition.

    In this paper, we will give a gravity dual of the EE by taking into account the multiple extremal surfaces satisfying some assumptions. We now consider that not only a single minimal or dominant surface but rather a bunch of extremal surfaces represent EE. We will justify this idea by generalizing the gravity action including a single cosmic brane \cite{Dong:2016fnf} to that including countless cosmic branes with positive and negative brane tension.

    The construction of this paper is as follows. In section~\ref{sec:2}, we will review the replica trick, which is a technique of calculation for EE and investigate the geometry of the replica manifold, and see counterexamples of the RT conjecture. In section~\ref{sec:3}, we will construct a gravity dual of the EE, modifying the existing derivations for the RT conjecture. In section~\ref{sec:4}, we will confirm that the gravity dual is consistent with the EEs mentioned as the counterexamples in section~\ref{sec:2} and calculate the mutual information. Finally, section~\ref{sec:5} is the conclusion.

\section{Comparison of RT conjecture and CFT result}
\label{sec:2}

    In this section, before we see counterexamples of RT conjecture, we give a quick review for the replica trick \cite{Calabrese:2004eu, Calabrese:2009qy} and investigate the geometry of the replica manifold to construct a gravity dual of it in the next section.

    \subsection{Replica trick and geometry of replica manifold}

    First, we will rewrite eq.\eqref{eq:def of EE} by the replica trick to clarify the geometry of the replica manifold. Applying the replica trick to eq.\eqref{eq:def of EE}, EE of $A$ is expressed as
    \begin{align}
        S_A = -\lim_{n \to 1} \frac{\partial}{\partial n} \log \mathrm{Tr}_A \left(\rho_A^{\ n} \right).
        \label{eq:replica trick}
    \end{align}
    For convenience, we also consider the Renyi entropy $S^{(n)}_A$ defined as
    \begin{align}
        S^{(n)}_A = - \frac{1}{n-1} \log \mathrm{Tr}_A \left(\rho_A^{\ n} \right).
        \label{eq:Renyi entropy}
    \end{align}
    Since $\mathrm{Tr}_A \left(\rho_A^{\ n=1} \right) =1$, the Renyi entropy $S^{(n)}_A$ goes to $S_A$ as taking $n \to 1$. Let $Z$ be a partition function on a manifold $\mathcal{B}$. We define the partition function $Z_A^{(n)}$ of the $n$ replicated theory on the replica manifold $\mathcal{B}_A^{(n)}$ as corresponding to a set of $\mathcal{B}$, $A$ and $Z$ in order that $\mathrm{Tr}_A \left(\rho_A^{\ n} \right) =  Z_A^{(n)}/Z^n$. Then, eq.\eqref{eq:Renyi entropy} is written by the partition functions as
    \begin{align}
        S^{(n)}_A = -\frac{1}{n-1} \left(  \log Z_A^{(n)} - n \log Z \right).
        \label{eq:replica trick2}
    \end{align}
    Note that in this paper, $Z^n$ means $Z$ to the $n$-th powered and the quantity with the superscript $(n)$ like $Z_A^{(n)}$ represents the $n$ replicated quantity of the original one.

    For example, consider a partition function $Z$ of a massless free scalar filed on the $1+1$ dimensional Minkowski spacetime $\mathcal{B}$. Let $A$ be an interval $[u,v]$ on a time slice, where $v-u = \ell \geq \epsilon$ and $\epsilon$ is the UV-cutoff. Note that it is irrational to consider the EE of a region of which length is smaller than $\epsilon$. The replica manifold $\mathcal{B}_A^{(n)}$ is the $n$ branched covering of $\mathcal{B}$ of which a branch cut is from $u$ to $v$. In this case, we can write down the $n$ replicated partition function $Z_A^{(n)}$ as
    \begin{align}
        Z_A^{(n)} = \langle \mathcal{T}^{(n)}(u)\mathcal{\tilde{T}}^{(n)}(v) \rangle_{\mathcal{B}_A^{(n)}},
        \label{eq:n replicated partition function}
    \end{align}
    where $\mathcal{T}^{(n)}(u)$ and $\mathcal{\tilde{T}}^{(n)}(v)$ are the twist operators which connect the fields on branch cut from $u$ to $v$ in an appropriate manner. As the scaling dimensions of the twist operators are $c\,(n^2-1)/12n$ where $c$ is the central charge, we can evaluate eq.\eqref{eq:n replicated partition function} analytically. Then eq.\eqref{eq:replica trick} is
    \begin{align}
        S_{A}
        = \lim_{n \to 1} \frac{c(n+1)}{6n} \log \frac{\ell}{\epsilon}
        = \frac{c}{3} \log \frac{\ell}{\epsilon}.
        \label{eq:EE of a interval}
    \end{align}
    In the large $c$ limit, this is a well known universal term of EE and the Renyi entropy of $1+1$ dimensional CFT on an infinitely long line at zero temperature.

    Next, we will study the geometry of $\mathcal{B}_A^{(n)}$. Consider the scale transformation of the Renyi entropy for the above example. Let $g_{\mu \nu},\, g^{(n)}_{A\,\mu \nu}$ be the metric tensor of the manifold $\mathcal{B},\, \mathcal{B}_A^{(n)}$ and $T,\,T^{(n)}_{A}$ be the trace of the energy-momentum tensors from $Z,\,Z_A^{(n)}$, respectively. Applying the Ward-Takahashi identity for the scale transformation to eq.\eqref{eq:replica trick2}, we get
    \begin{align}
        \ell \frac{\partial}{\partial \ell} S_{A}^{(n)}
        &= - \frac{2}{n-1} \int d^2x \left( g^{(n)}_{A\,\mu \nu} \frac{\partial}{\partial g^{(n)}_{A\, \mu \nu}}  \log Z_A^{(n)} - n g_{\mu \nu}\frac{\partial}{\partial g_{\mu \nu}}\log Z \right) \nonumber \\
        &= - \frac{1}{n-1} \left( \int d^2x \sqrt{\left| g^{(n)}_A \right|} \langle T^{(n)}\rangle_{\mathcal{B}_A^{(n)}} -n \int d^2x \sqrt{\left|g\right|} \langle T \rangle_{\mathbb{R}^2}\right) \nonumber \\
        &= - \frac{1}{n-1} \int d^2x \frac{c}{24\pi}\sqrt{\left|g^{(n)}_A \right|}R_A^{(n)}.
        \label{eq:scale trans for EE}
    \end{align}
    where $g^{(n)}_A$ is the determinant of $g^{(n)}_{A\, \mu \nu}$ and $R_A^{(n)}$ is the Ricci scalar from the metric $g^{(n)}_{A\, \mu \nu}$. Recall that the replica manifold $\mathcal{B}_A^{(n)}$ is a $n$ branched covering of $\mathcal{B}$. There are the conical singularities at $\partial A := \{ (0,u) \cup (0,v) \}$. From comparison for eq.\eqref{eq:EE of a interval} and eq.\eqref{eq:scale trans for EE}, the conical defect angle is $\pi(1-n^2)/n$. Then, $R_A^{(n)}$ is almost flat but singular at $\partial A$ as
    \begin{align}
        R^{(n)}_A = 2\pi \frac{1-n^2}{n} \delta(t)\left[ \delta(x-u) + \delta(x-v) \right].
        \label{eq:conical deficit}
    \end{align}
    Actually, substituting eq.\eqref{eq:conical deficit} into eq.\eqref{eq:scale trans for EE}, we get

    \begin{align}
        \ell \frac{\partial}{\partial \ell} S^{(n)}_A = \frac{c(n+1)}{6n}.
        \label{eq:2-point}
    \end{align}
    Integrating this, we can restore eq.\eqref{eq:EE of a interval}. Remind that the replica manifold $\mathcal{B}_A^{(n)}$ has singularities of which defect angle is $\pi(1-n^2)/n$ made by the twist operators at $\partial A$.

    \subsection{Counterexamples of RT conjecture}

    First, consider the EE of two disjoint space-like interval $A$ in $1+1$ dimensional CFT at zero temperature. Without loss of generality thanks to conformal symmetry, it is enough to consider the intervals as $A = [0, \ell] \cup [1, \infty],\, \ell \in (\epsilon,1-\epsilon)$. The trace of $\rho_A^{\ n}$ is described by $4$-point function of the twist operators as follows.
    \begin{align}
        \mathrm{Tr}_A \left(\rho_A^{\ n} \right) &= \langle \mathcal{T}^{(n)} (0)\tilde{\mathcal{T}}^{(n)}(\ell)\mathcal{T}^{(n)}(1)\tilde{\mathcal{T}}^{(n)}(\infty) \rangle
    \label{eq:4-point}
    \end{align}
    In the large $c$ limit, according to the paper \cite{Hartman:2013mia} with the crossing symmetry we can evaluate eq.\eqref{eq:4-point} as
    \begin{align}
        S_A
        = \lim_{n \to 1} \frac{c(n+1)}{6n}\left[  \log \frac{\ell}{\epsilon} + \log \left( \frac{1-\ell}{\epsilon} \right) \right] + c_1
        = \frac{c}{3}\left[  \log \frac{\ell}{\epsilon} + \log \left( \frac{1-\ell}{\epsilon} \right) \right] + c_1.
    \label{eq:EE of 2 disjoint interval}
    \end{align}
    where $\epsilon$ is the UV cut off, $c_1$ is determined by the IR cut off.

    Second, consider the EE of a space-like interval $A$ in $1+1$-dimensional CFT on the circle of which circumference $1$ at the inverse temperature $\beta$ \cite{Azeyanagi:2007bj,Datta:2013hba}. EE of an interval $A = [-\ell/2, \ell/2],\, \ell \in (\epsilon,1-\epsilon)] $ in the large $c$ limit is
    \begin{align}
        S_A
        &= \lim_{n \to 1} \frac{c(n+1)}{6n} \left\{ \log\left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi\ell}{\beta}\right)\right]
        + \sum_{m=1}^{\infty}\log \frac{(1-e^{2\pi \ell/\beta}e^{-2\pi m/\beta})(1-e^{-2\pi \ell/\beta}e^{-2\pi m/\beta})}{(1-e^{-2\pi m/\beta})^2} \right\} \nonumber\\
        &=\frac{c}{3} \log\left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi\ell}{\beta}\right)\right]
        + \frac{c}{3} \sum_{m=1}^{\infty}\log \frac{\sinh \left[\frac{\pi}{\beta}(m-\ell)\right]\sinh \left[\frac{\pi}{\beta}(m+\ell)\right]}{\sinh^2 \left(\frac{\pi m}{\beta}\right)} .
    \label{eq:EE of finite circle}
    \end{align}
    Note that the above calculations are complete within CFT and irrelevant from the AdS/CFT correspondence.

    On the other hand, we will see EEs derived from the RT conjecture applying to the corresponding spacetimes as the AdS spacetime and BTZ spacetime \cite{Banados:1992wn}, respectively. The EE by the RT conjecture corresponding to eq.\eqref{eq:EE of 2 disjoint interval} is
    \begin{align}
        S_A = \min \left\{  \frac{c}{3}\log \frac{\ell}{\epsilon},\ \frac{c}{3}\log \left( \frac{1-\ell}{\epsilon} \right) \right\}
    \label{eq:EE from RT of 2 disjoint interval}
    \end{align}
    and to eq.\eqref{eq:EE of finite circle} is
    \begin{align}
        S_A
        =\min \left\{ \frac{c}{3} \log \left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi\ell}{\beta}\right)  \right],\ \frac{c \pi}{3\beta} + \frac{c}{3} \log \left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi}{\beta}(1-\ell)\right)  \right]\right\}
    \label{eq:EE from RT of finite circle}
    \end{align}
    If and only if $\ell \sim \epsilon$ or $1-\ell \sim \epsilon$, they are consistent with eq.\eqref{eq:EE of 2 disjoint interval} and eq.\eqref{eq:EE of finite circle} respectively. In this limit, one extremal surface contributes dominantly to EE. However, no longer they have the consistency with each other if we consider EE of a more general interval.

    How do we improve the RT conjecture? Looking at eq.\eqref{eq:EE of 2 disjoint interval} and eq.\eqref{eq:EE of finite circle}, we think intuitively that they are described by multiple extremal surfaces in the corresponding bulks. In section~\ref{sec:3}, we will use this idea as a clue to improve the RT conjecture. Then, in section~\ref{sec:4}, we will confirm that our proposal is consistent with eq.\eqref{eq:EE of 2 disjoint interval} and eq.\eqref{eq:EE of finite circle}.

\section{Refinement of RT conjecture}
\label{sec:3}

    In this section, based on the derivation \cite{Lewkowycz:2013nqa,Dong:2016fnf}, we will formulate a gravity dual of EE eq.\eqref{eq:replica trick} and the Renyi entropy eq.\eqref{eq:Renyi entropy} by modifying three assumptions about the geometry of the bulk replica manifold, the $n$-replicated gravity action, and the homologous condition.

    \subsection{Gravity dual of replica manifold}

    First, we consider the geometry of the gravity dual manifold of the replica boundary manifold. Consider the $d+1$-dimensional replica bulk spacetime $\mathcal{M}_A^{(n)}$ of which boundary is the $d$ dimensional replica manifold $\mathcal{B}^{(n)}_A$ in the sense of $\partial \mathcal{M}_A^{(n)} = \mathcal{B}_A^{(n)}$. Since $\mathcal{B}_A^{(n)}$ has the conical singularities as mentioned in the previous section, we assume that the replica bulk spacetime $\mathcal{M}_A^{(n)}$ is a singular manifold. This is the first modification. We restrict the configuration of the singularities $\mathcal{C}_A^{(n)}$ in $\mathcal{M}_A^{(n)}$ so that $\mathcal{M}^{(n)}_A$ has the same topology as $\mathcal{B}_A^{(n)}$ has. We call this condition as the topological consistency condition; TCC according to \cite{Haehl:2014zoa}. In the next subsection, we will see that the TCC restricts the configuration of the extremal surfaces that can contribute to the EE. Notice that the first modification does not matter to the net result of this discussion, but this assumption makes the argument clear and leads to the natural bulk dual of the twist operators as we will see later.

    \begin{figure}[h]
    \vspace{10pt}
    \centering
    \includegraphics[width=0.6\linewidth]{CosmicBranes.pdf}
    \caption{The red and blue dotted lines represent the cosmic branes $\mathcal{C}_{A,\,\pm}^{(n),\, i}$ on a time-slice with the defect angle $\pm \pi(1-n^2)/n$, respectively. The total defect angle is $\pi(1-n^2)/n$ at both $\partial A$.}
    \label{fig:Cosmic branes}
    \end{figure}

    Second, we will build the gravity action which realizes the $n$ replicated bulk manifold $\mathcal{B}^{(n)}_A$ as a solution. By the AdS/CFT correspondence, the partition function $Z$ of a CFT can be described by the path integral as $Z = \int \mathcal{D}\phi \exp (-I[\phi])$, where $I[\phi]$ is the gravity action. Then, what is a gravity action $I_A^{(n)}[\phi]$ corresponding to $Z_A^{(n)}$? Since the partition function $Z_A^{(n)}$ is the vacuum expectation value of the twist operators, the corresponding gravity action $I_A^{(n)}[\phi]$  should have the somehow sources which make the conical singularities of which defect angle is $\pi(1-n^2)/n$ at $\partial A$. The following cosmic brane $\mathcal{C}_A^{(n)}$ anchored on $\partial A$ makes the singularities with defect angle $\pi(1-n^2)/n$.
    \begin{align}
        I_{\mathrm{brane}}^{(n)} = -\frac{1-n^2}{8n G} \int_{\mathcal{C}^{(n)}_A} d^{d-1} y \sqrt{h},
    \label{eq:Action for cosmic brane}
    \end{align}
    where $y$ is the $d-1$ dimensional coordinate on $\mathcal{C}^{(n)}_A$ and $h$ is the determinant of the induced metric on $\mathcal{C}^{(n)}_A$. However, this is not a unique action satisfying the boundary condition on $\partial A$. As shown in figure~\ref{fig:Cosmic branes}, if we also introduce cosmic branes with the brane tension $-(1-n^2)/8nG$, we can generalize eq.\eqref{eq:Action for cosmic brane} involving infinitely many cosmic branes as follows.
    \begin{align}
        I_{\mathrm{branes}}^{(n)} = \sum_i &-\frac{1}{16 \pi G} \int_{\mathcal{M}^{(n)}_A} d^{d+1} x \sqrt{g}\, \frac{2\pi(1-n^2)}{n} \delta(t)\delta(\xi-\xi^{(n),\,i}_{A,\,+}) \nonumber \\
        &+ \sum_j \frac{1}{16 \pi G} \int_{\mathcal{M}^{(n)}_A} d^{d+1} x \sqrt{g}\, \frac{2\pi(1-n^2)}{n} \delta(t)\delta(\xi-\xi^{(n),\,j}_{A,\,-}),
    \label{eq:Action for cosmic brane2}
    \end{align}
    where $\xi^{(n),\,i}_{A,\,\pm}$ represents the location of a $d-1$ dimensional singularities anchored on $\partial A$ with defect angle $\pm \pi(1-n^2)/n$, respectively.

    As the second modification, we use eq.\eqref{eq:Action for cosmic brane2} instead of eq.\eqref{eq:Action for cosmic brane}. That is, we regard the $n$-replicated bulk gravity action as $I_A^{(n)}[\phi] = n I[\phi] + I_{\mathrm{branes}}^{(n)}$. Since eq.\eqref{eq:Action for cosmic brane2} is generalization of eq.\eqref{eq:conical deficit}, we can regard the cosmic branes as a gravity dual of the twist operators. In the large $N$ limit, we can use the saddle point approximation to evaluate the path integral as $Z^{(n)}_A = \int \mathcal{D}\phi \exp (-I^{(n)}_A[\phi]) \sim \exp (-\tilde{I}^{(n)}_{A}[\phi])$ where $\tilde{I}^{(n)}_{A}[\phi]$ is the solutions for the Eular-Lagrange equation. If $n \sim 1$, the difference of the Ricci scalar of $\tilde{I}^{(n)}_{A}[\phi]$ and $n \tilde{I}[\phi]$ is only the cosmic branes action. Then, the Renyi entropy \eqref{eq:replica trick2} in $N \to \infty, n \to 1$ is
    \begin{align}
        S_A^{(n)} =  -\frac{1}{n-1}\,\tilde{I}_{\mathrm{branes}}^{(n)}.
        \label{eq:Area of cosmic brane?}
    \end{align}
    Since $\tilde{I}_{\mathrm{branes}}^{(n)}$ is the saddle point, it is the sum of signed areas of extremal surfaces. In the next subsection, we will study what a combination of extremal surfaces can satisfy the TCC.

    \subsection{Intersection number of cosmic branes}

    We discuss the third modification that is the refinement of the homologous condition. Recall that the homologous condition is required for consistency with the strong subadditivity of the EE. However, the homologous condition that there exist the region $\mathcal{R}_A$ satisfying $\partial \mathcal{R}_A = A \cup \gamma_A$ makes sense only when we deal with the minimal surface. Thus, we need to refine the homologous condition to treat a self intersecting cosmic brane and multiple cosmic branes.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.78\linewidth]{Intersection.pdf}
    \caption{An example of 2-dimensional surface $\mathcal{D}$. Thanks to the condition $\partial \mathcal{D} \subset \mathcal{B}- \partial A$, the intersection number is $0$ for an arbitrarily $\mathcal{D}$ if we assign the orientation of an interval $A$ and a cosmic brane $\mathcal{C}_{A}^{(n)}$ like this figure.}
    \label{fig:Intersection}
    \end{figure}

    Fortunately, there is such a refinement of the homologous condition thanks to \cite{Haehl:2014zoa}. Let $\mathcal{D}$ be a 2-dimensional surface satisfying $\partial \mathcal{D} \subset \mathcal{B}-\partial A$ like figure~\ref{fig:Intersection}. The definition of the intersection number is the number of times that the oriented cosmic branes $\mathcal{C}_{A,\,\pm}^{(n),\, i}$ in $\mathcal{M}_A^{(n)}$ and the oriented region $A$ on $\mathcal{B}$ penetrate $\mathcal{D}$. Notice that $\mathcal{M}_A^{(n)}$ satisfies TCC if and only if the intersection number is 0. Let $E_A$ be the set of all extremal surfaces of which intersection number is 0 against a region $A$ and an arbitrary $\mathcal{D}$ and which satisfy the boundary condition at $\partial A$.

    In $1+1$ dimensional CFT system, we impose one more assumption. Consider a cosmic string connects two twist operators of the same type. For example, a string connects $\mathcal{T}^{(n)}(u_1)$ with $\mathcal{T}^{(n)}(u_2)$ or $\tilde{\mathcal{T}}^{(n)}(v_1)$ with $\tilde{\mathcal{T}}^{(n)}(v_2)$. We assume that such a cosmic string has a negative defect angle as $-\pi (1-n^2)/n$. If not, eq.\eqref{eq:Area of cosmic brane?} would reproduce incorrect EE and contradict the SSA inequality as we will see. Then, in this paper, we say that the extremal surfaces satisfy the refined homologous condition(RHC) if the extremal surfaces are elements of $E_A$ and cosmic branes $\mathcal{C}_{A,\,+}^{(n),\, i}$ with a positive defect angle do not connect the same type of twist operators.

    As the conclusion of the discussion of this whole section, the gravity dual of the Renyi entropy of the large $N$ limit static CFT is as follows.
    \begin{align}
        S_A^{(n)} = \frac{n+1}{8n G} \sum_{ \mathcal{C}_{A,\,\pm}^{(n),\, i} \in  \tilde{E}_A} \left(  \mathrm{Area\ of\ }\mathcal{C}_{A,\,+}^{(n),\, i} -  \mathrm{Area\ of\ } \mathcal{C}_{A,\,-}^{(n),\, i} \right)
        \label{eq:the gravity dual}
    \end{align}
    where $n \sim 1$ and $\tilde{E}_A$ is the set of all cosmic branes $\mathcal{C}_{A,\,\pm}^{(n),\, i}$ satisfying the refined homologous condition. Notice that we have to introduce the UV cutoff for CFT to prevent the IR divergence of the gravity partition function. Then, the areas of the cosmic brane take finite value.

\section{Holographic EE from extremal surfaces}
\label{sec:4}

    We will confirm for our gravity dual of the Renyi entropy eq.\eqref{eq:the gravity dual} to be consistent with eq.\eqref{eq:EE of 2 disjoint interval} and eq.\eqref{eq:EE of finite circle}.

    \subsection{EE and mutual information of two intervals}

    We will apply eq.\eqref{eq:the gravity dual} to the calculation for EE of the two intervals as $A = [u_1, v_1] \cup [u_2, v_2],\, u_1<v_1<u_2<v_2$. Since the dual bulk spacetime is planar AdS spacetime, there only exists the cosmic string connecting the two points at the AdS boundary. Therefore, the cosmic strings in $\tilde{E}_A$ is depicted in figure~\ref{fig:2interval} and the Renyi entropy is
    \begin{align}
        S_A^{(n)} = \frac{c(n+1)}{6n} \log \frac{(v_1-u_1)(v_2-u_2)(u_2-v_1)(v_2-u_1)}{\epsilon\, (u_2-u_1)(v_2-v_1)}
    \label{eq:4-point2}
    \end{align}
    If $(u_1,v_1,u_2,v_2)=(0,\ell,1,\infty)$, $\epsilon \leq \ell \leq 1-\epsilon$, eq.\eqref{eq:4-point2} goes to eq.\eqref{eq:EE of 2 disjoint interval}. Notice that general behavior of $4$-point function of the twist operators in $1+1$ dimensional CFT is \cite{Calabrese:2009qy}
    \begin{align}
        \mathrm{Tr}_A \left(\rho_A^{\ n} \right)
        = \left[\frac{(v_1-u_1)(v_2-u_2)(u_2-v_1)(v_2-u_1)}{(u_2-u_1)(v_2-v_1)}\right]^{-\frac{c}{6}\left(n-\frac{1}{n}\right)}\mathcal{F}_n(x)
    \label{eq:4-point3}
    \end{align}
    where $x = (v_1-u_1)(v_2-u_2)/(u_2-u_1)(v_2-v_1)$ is the cross ratio and the function $\mathcal{F}_n$ depends on the detail of CFT. In the large $c$ limit, since the spectrum involved in a theory become universal, eq.\eqref{eq:4-point2} is consistent with eq.\eqref{eq:4-point3}.
    \begin{figure}[h]
    \centering
    \includegraphics[width=0.7\linewidth]{2interval.pdf}
    \caption{The extremal surfaces satisfying RHC for $2$ intervals case. }
    \label{fig:2interval}
    \end{figure}

    Next, consider the mutual information of two intervals $A = [u_1, v_1]$ and $B = [u_2, v_2]$, $u_1<v_1<u_2<v_2 $. The mutual information $I_{AB}$ is
    \begin{align}
        I_{AB}
        = S_A + S_B -S_{A+B}
        = \frac{c}{3} \log \frac{(u_2-v_1)(v_2-u_1)}{ (u_2-u_1)(v_2-v_1)} \geq 0.
    \label{eq:mutual information}
    \end{align}
    As explained in \cite{Headrick:2007km}, this is non negative thanks to the relation between the size of the extremal surface area, and we can regard it as a result of the strong sub additivity inequality $S_{A+B+C} + S_{B} \geq S_{A+B} + S_{B+C}$. If $A = [u_1, v_1], B = [v_1, u_2]$ and $C = [u_2, v_2]$, then eq.\eqref{eq:mutual information} is non negative.

    Revisit the condition that a cosmic string with a positive defect angle doesn't connect the same type of twist operators. Suppose that the positive extremal surfaces connect ($u_1$,$v_1$), ($u_2$,$v_2$), ($u_1$,$u_2$) and ($v_1$,$v_2$), and the negative ones connect ($u_1$,$v_2$) and ($v_1$,$u_2$). Although there is no problem with the boundary condition and the intersection number of them, eq.\eqref{eq:mutual information} becomes negative. On the other hand, notice that we permit for cosmic strings with a negative defect angle connecting $\mathcal{T}$ and $\tilde{\mathcal{T}}$. Consider that such surfaces are added in figure~\ref{fig:2interval}. Then, we have to add again those with a positive defect angle as many as with a negative defect angle to satisfy RHC. These contributions are canceled each other and the only contribution of original surfaces is left. In particular, eq.\eqref{eq:the gravity dual} is equivalent to the RT conjecture if there exists the only extremal surface corresponding a region $A$ in the spacetime.

    \subsection{BTZ black hole}

    The BTZ black hole spacetime eq.\eqref{eq:BTZmetric} is the dual gravity spacetime of $1+1$ dimensional CFT on the circle at the finite temperature. If the inverse temperature of CFT be $\beta$, the black hole mass satisfies $M =1/\beta^2$. Let a subregion $A$ be a single interval, then there are infinitely many extremal surfaces anchored on the $\partial A$ in the BTZ black hole spacetime, as shown in figure~\ref{fig:ExtremalsInBTZ}. See also appendix~\ref{Appendix}.
    \begin{figure}[h]
    \vspace{4pt}
    \centering
    \includegraphics[width=0.8\linewidth]{Extremalsurfaces.pdf}
    \vspace{-4pt}
    \caption{The extremal surfaces in the BTZ black hole spacetime. These red and blue extremal surfaces are anchored on $\theta = \pm \pi/3$ and $\theta= - \pi/3$, respectively. The black hole mass $M=1$. Each disk represents a time slice of the spacetime with compactification for the radial direction and the outer circles represent the AdS boundary. Note that the black hole is not written out.}
    \label{fig:ExtremalsInBTZ}
\end{figure}

    First, we confirm how RHC works in this system. Since the time coordinate is periodic in a finite temperature system, there can exist a special $2$-surface $\mathcal{D}$. For example, the surface $\mathcal{D}$ has the boundary $\partial \mathcal{D} = \{(t,\theta)| 0\leq t \leq \beta,\,\theta=\mathrm{const.} \}$ shown in figure~\ref{fig:TCC}. Furthermore, some surface can connect between $\partial A$ avoiding such a surface $\mathcal{D}$ as the topology of the boundary manifold $\mathcal{B}$ is the torus.

    \begin{figure}[h]
    \centering
    \includegraphics[width=0.35\linewidth]{TCC.pdf}
    \caption{An example of the extremal surfaces satisfying RHC in the BTZ black hole spacetime. The condition for the intersection number forbid to replace a surface to one that goes around the black hole a lot.}
    \label{fig:TCC}
    \end{figure}

    Noting that, we construct the combination of the extremal surfaces satisfying RHC as follows. See figure~\ref{fig:TCC} and \ref{fig:ThesurfacesInBTZ} and specify the signed surfaces by the function $\pm s(m \pm \ell)$. First, the surface $s(\ell)$ satisfies RHC. Next, from the boundary condition consider the surfaces $+s(m_1-\ell)$, $+s(m_2+\ell)$, $-s(m_3)$ and $-s(m_4)$. The condition for the intersection number imposes $m_1 = m_2 = m_3 =m_4$. To confirm this, note that we can transform the pair of the surfaces $+s(m-\ell)$ and $-s(m)$ to $A$ homotopically, and do so the pair of the surfaces $s(m+\ell)$ and $s(m)$ too. Then, the surfaces $+s(\ell)$, $+s(m-\ell)$, $+s(m+\ell)$, $-s(m)$ and $-s(m)$ satisfies RHC. If we replace a surface out of them to another surface, it always changes the intersection number. Therefore, all the combination of the extremal surfaces satisfying RHC is $+s(\ell)$ and all the pair of $4$ surfaces $+s(m-\ell)$, $+s(m+\ell)$, $-s(m)$ and $-s(m)$ for $m \geq 1$.

    \begin{figure}[h]
    \centering
    \includegraphics[width=0.74\linewidth]{TheExtremals.pdf}
    \caption{The diagrams of the extremal surfaces which contribute to the EE of a single interval in the BTZ black hole spacetime. The function $s$ defined as eq.\eqref{eq:DefofAreaFunction} denotes the area$/4G$ of each surface.}
    \label{fig:ThesurfacesInBTZ}
\end{figure}

    Then, EE is the following sum of signed area of all surfaces.
    \begin{align}
        S_A
        &= s(\ell) + \sum_{m=1}^{\infty} \left[ s(m - \ell) + s(m + \ell) - 2 s(m) \right] \nonumber \\
        &= \frac{c}{3} \log\left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi\ell}{\beta}\right)\right]
        + \frac{c}{3} \sum_{m=1}^{\infty}\log \frac{\sinh \left[\frac{\pi}{\beta}(m-\ell)\right]\sinh \left[\frac{\pi}{\beta}(m+\ell)\right]}{\sinh^2 \left(\frac{\pi m}{\beta}\right)}.
        \label{eq:BTZ}
    \end{align}
    This is consistent with eq.\eqref{eq:EE of finite circle}.

    We discuss a few geometrical aspects of the configuration of the extremal surfaces corresponding to eq.\eqref{eq:BTZ}. First, the most striking difference in the geometry between the RT conjecture's eq.\eqref{eq:EE from RT of finite circle} and ours eq.\eqref{eq:BTZ} is the region where the surfaces can sweep. If the homologous minimal surface can describe EE completely, we face to the plateaux problem \cite{Freivogel:2014lja}. That is, EE is independent of the property of the bulk region slightly outside the black hole horizon because the minimal surface cannot reach such a region even considering any boundary region. Therefore, if RT conjecture is correct in this spacetime, it may be impossible to reconstruct the bulk geometry from the boundary information. On the other hand, since the surfaces which represent eq.\eqref{eq:BTZ} can sweep all bulk region outside the black hole horizon, the plateaux problem is solved in the BTZ spacetime.  In the same way, since in the higher dimensional Schwarzschild-AdS black hole there are surfaces that can approach the black hole horizon infinitesimally, the plateaux problem may not happen.

    Second, the configuration of the surfaces of eq.\eqref{eq:BTZ} is consistent with the restriction of causality \cite{Headrick:2014cta}. That is, all the surface of eq.\eqref{eq:BTZ} exist on the entanglement shadow region. In this sense, RHC is more strong condition than the causality constraint since it determines not only the position but also the combination of the extremal surfaces, although the causality cannot restrict the combination of them.

    Finally, we point out the relation between surfaces of eq.\eqref{eq:BTZ} and the black hole horizon. Although eq.\eqref{eq:BTZ} seems not to include the black hole horizon explicitly, if $A = 1-\delta,\, \epsilon < \delta \ll 1$, eq.\eqref{eq:BTZ} goes
    \begin{align}
        S(1-\delta) &= \frac{c}{3} \log\left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi \delta}{\beta}\right)\right]
        + \frac{c}{3} \lim_{n \to \infty} \log \frac{\sinh \left(\frac{\pi}{\beta}(m+1)\right)}{\sinh \left(\frac{\pi m}{\beta}\right)} + O(\delta) \nonumber \\
        &= \frac{c}{3} \log\left[ \frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi \delta}{\beta}\right)\right] + \frac{c \pi}{3 \beta}.
        \label{eq:BTZ large interval}
    \end{align}
    Note that taking $\delta, \epsilon \to 0$, the Araki-Lieb is saturated and the difference of the EEs corresponds to $c$ times the black hole entropy $S_{BH}$:
    \begin{align}
        \lim_{\delta,\epsilon \to 0} S(1-\delta)- S(\delta) = \frac{c \pi}{3 \beta} = c S_{BH}.
        \label{eq:Araki-Lieb}
    \end{align}
    See eq.\eqref{eq:BTZ large interval}, since the black hole horizon emerges as a result of the difference of the surface wrapped $m+1$ times around the black hole and one wrapped $m$ times taking $m$ to the infinity, both sides of eq.\eqref{eq:Araki-Lieb} are equivalent as the surfaces not only values.

\section{Conclusion}
\label{sec:5}

    We provided a gravity dual of the Renyi entropy eq.\eqref{eq:the gravity dual} as the sum of the signed areas of cosmic branes satisfying the refined homologous condition. We saw that the RT conjecture has contradictions for the cases that EE of two intervals on an infinitely long line at zero temperature and that of an interval on a circle at finite temperature. Our holographic EE eq.\eqref{eq:the gravity dual} is consistent with them.

    In the course of the derivation of eq.\eqref{eq:the gravity dual}, we first investigated the geometry of the replica manifold on a CFT side. On the corresponding gravity side, we gave the gravity action with infinitely many cosmic branes that is consistent with the geometry of the boundary replica manifold. After that, we impose the refined homologous condition on the configuration of the cosmic branes. In $1+1$-dimensional system, we imposed a non-trivial assumption as for the connection of twist operators by the cosmic strings. Then, this assumption may have to be revisited and justified on a somehow natural assumption.

    As future works, we have to verify our conjecture eq.\eqref{eq:the gravity dual} in various systems by confirming the consistency with CFT calculation, for example, in higher-dimensional black holes, the AdS soliton spacetime, the D$p$-brane spacetime, and things like that.

    We should revisit existing arguments relying on the homologous minimality of the RT conjecture. In a general system, since a region on the AdS boundary has many corresponding extremal surfaces in the bulk, the gravity dual of EE should contain the contribution of multiple extremal surfaces. For example, regarding the HRT conjecture\cite{Hubeny:2007xt} and the quantum correction of the RT conjecture\cite{Faulkner:2013ana}, we have to consider the dynamics and the quantum correction of multiple cosmic branes.

\acknowledgments{
I would like to thank Yasusada Nambu and Chulmoon Yoo for useful discussions and comments.
}