We compute the holographic entanglement entropy for the anomaly polynomial

Article funded by SCOAP3

1$, $Z_n$ and $Z_1$ are the partition functions of the CFTs on $M_n$ and $M_1$, respectively. We are interested in the von Neumann entropy of the CFT on $M_1$. Formally, this is computed by the analytical continuation of~(\ref{eq:Renyi}) to $n\rightarrow 1$. It is worth noting, however, that $M_n$ doesn't have a geometric interpretation for non-integer values of $n$ and so it is not clear what the above analytic continuation means geometrically for the boundary manifold. It was pointed out in~\cite{Lewkowycz} that for theories that have holographic duals, the bulk geometry $B_n$ (which is the bulk geometry associated to the replica manifold $M_n$) does indeed have a geometric meaning even for non-integer values of $n$. $B_n$ is completely regular but the action of $\mathbb{Z}_n$ on it has a fixed point set $C_n$ which is a codimension-2 surface. The orbifold bulk geometry $\hat{B}_n = B_n/\mathbb{Z}_n$, whose boundary is our original boundary manifold $M_1$, thus has a singular surface $C_n$ with a conical deficit of $\epsilon = 1 - \nicefrac{1}{n}$. One can regularize this cone by introducing a smoothing parameter $a$, and the metric of the manifold near this surface is given by~\cite{Loganayagam} \begin{align} ds^2 &= e^{2 A} \left[dz d\bar z +e^{2 A} T (\bar z dz-z d\bar z)^2\right] +(g_{ij} + 2 K_{aij} x^a+Q_{abij} x^a x^b ) dy^i dy^j \nonumber \\ &\quad + 2 i e^{2 A} (U_i +V_{ai} x^a ) (\bar z dz -z d\bar z)dy^i+\dots \, , \label{eq:cone} \end{align} where $x^a= \{z, \bar z\}$ are the complexified coordinates transverse to the codimension-2 surface and $y^i, \mbox{ with }i=1,2,\ldots d$, are the coordinates along this surface. The functions, $T,g_{ij}$, $K_{aij}$, $Q_{abij}$, $U_i$, $V_{ai}$ all depend on $y^i$. $A= -\frac{\epsilon}{2} \ \text{log} (|z|^2+a^2)$ is the regularization function that smooths out the squashed cone. The regularization parameter $a$ keeps track of the contribution from the singular limit of the cone (when $a\rightarrow 0$), which is subtracted before taking the $a\rightarrow 0$ limit. The holographic formula (in the large $N$ limit) for the entanglement entropy then is given by \begin{align} S_{\mathrm{EE}}=\lim_{n\rightarrow 1} \frac{n}{n-1}\left(S\left[\hat{B}_n\right] - S\left[\hat{B}_1\right]\right)=\left.\partial_n S\left[\hat{B}_n\right]\right|_{n=1} \, , \end{align} where $S[\hat{B}_n]$ and $S[\hat{B}_1]$ are the classical bulk action evaluated on the orbifolds $\hat{B}_n$ and $\hat{B}_1$, respectively. As noted above, since the fixed point set $C_n$ is singular on the orbifolds, we smooth out the geometry near the tip of the cone by excising a small region and replacing the tip of the cone by a smoothed-out tip. Calling this new smoothed-out region near the tip as the `inside' region, it can then be shown that~\cite{Dong} \begin{align} S_{\mathrm{EE}} = -\left. \partial_\epsilon S\left[\hat{B}_n\right]_{\mathrm{inside}}\right|_{\epsilon=0}. \end{align} Applying this to theories whose bulk gravitational action contains only the Einstein-Hilbert term (in addition to the usual cosmological constant) yields the usual Wald term in the expression for the entanglement entropy. It was shown by Dong~\cite{Dong} that for theories with higher derivative coordinate-invariant terms one gets a correction to the Wald term. The total entanglement entropy then is given by~(\ref{eq:Dong-covariant}) when expressed in a coordinate-invariant way. For explicit computations, however, it is convenient to express~(\ref{eq:Dong-covariant}) in the coordinate system implicit in~(\ref{eq:cone}). One then gets the following expression for the holographic entanglement entropy \begin{align} S_{\mathrm{EE}}=2\pi \int_{\Sigma} d^dy \sqrt{g}\left\{\frac{\partial L}{\partial R_{z\bar{z}z\bar{z}}}+\sum_\alpha\left(\frac{\partial^2 L}{\partial R_{zizj} \partial R_{\bar{z}k\bar{z}l}} \right)\frac{8K_{zij} K_{\bar{z}kl}}{q_\alpha+1}\right\} \label{eq:Dong}\, , \end{align} where the integral is taken over a codimension-2 surface $\Sigma$ that is homologous to the entangling surface in the dual CFT on the boundary. The extra terms derived by Dong arise from would-be logarithmic divergences that come from the squashed cone method. Consequently, these are naturally interpreted as anomaly terms and the coefficients $q_\alpha$ can be thought of as `anomaly coefficients.' In our case, however, the anomaly coefficient is trivial. We refer the reader to~\cite{Dong} for a more detailed discussion on this issue. A new issue that arises with the addition of the new terms is how to determine the entangling surface in the bulk. The rigorous way of deriving the entangling surface is to solve the equations of motion. But this could be too difficult in practice and so Dong~\cite{Dong} conjectures that the appropriate surface is the one that extremizes~(\ref{eq:Dong}). Since in our case, the anomaly polynomial does not add any new term to the equations of motion, the bulk entangling surface will be same as the Ryu-Takayanagi surface. In order to understand how Dong's prescription gives the expression for holographic entanglement entropy, let us denote the two relevant part of the Lagrangian~(\ref{eq:action1}) by \be L_1 = - \frac{1}{16\pi G} R \, , \ L_2 = \frac{\kappa}{64\pi G} *RR\, . \ee According to~\cite{Dong}, the contribution that $L_1$ makes to the entanglement entropy is given by \be \lim_{\epsilon\rightarrow 0} \frac{1}{4G} \int d^4x \sqrt{G}\left(\frac{\delta^2(x^1,x^2)}{(\rho^2 + a^2)^\epsilon} - \frac{\epsilon \log(\rho^2+a^2)}{(\rho^2 +a^2)^\epsilon} \delta^2(x^1,x^2)\right) \, , \ee where $\rho$ is the polar coordinate defined by $\rho=|z|$. In the $\epsilon \rightarrow 0 $ limit the second term drops out and it is easy to see that the first term is nothing but the Ryu-Takayanagi formula \be \left. S^{(1)}_{\mathrm{EE}}\right|_{L_1}= \frac{\mathcal{A}}{4G}\, , \ee where $\mathcal{A}= \int_\Sigma d^2y \sqrt{g}.$ Since the first term $L_1$ is first order in curvature it doesn't make any contribution to the entanglement entropy coming from the second term in~(\ref{eq:Dong}). $L_2$, on the other hand, contributes to both. Its contribution to the Wald term is computed to be \begin{align} \left. S^{(1)}_{\mathrm{EE}}\right|_{L_2}= - \frac{\kappa}{4G}\int d^2y \sqrt{g}\left(\frac{\partial U_j}{\partial y_i}-\frac{\partial U_i}{\partial y_j}+2 g^{kl} K_{z\; jk} K_{\bar{z}\;il} \right) \epsilon^{ij}. \end{align} The contribution from the $L_2$ term to the second term in~(\ref{eq:Dong}) can be shown to be \be \left. S^{(2)}_{\mathrm{EE}}\right|_{L_2}= -\frac{\kappa}{2G}\int d^2y\sqrt{g} K_{z\; ij} K_{\bar{z}\; kl} \epsilon^{jl} g^{ki}, \ee which cancels out the second term in $L_2$'s contribution to $S^{(1)}_{\mathrm{EE}}$. Thus the final result is (continuing back to Lorentzian signature) \be S_{\mathrm{EE}} = \frac{1}{4G_N}\int d^2 y \sqrt{g} - \frac{\kappa}{4G}\int d^2 y \sqrt{g}\left[\partial_i U_j - \partial_j U_i\right] \epsilon^{ij}. \label{eq:ee-coordinates} \ee This expression is computed in the particular coordinate system given above. This can be expressed in the following coordinate-invariant way \be S_{\mathrm{EE}} = \frac{1}{4G} \int_\Sigma d^2y \sqrt{g} + \frac{\kappa}{8G}\int_\Sigma F_{ij} dy^i \wedge dy^j \label{eq:entropyformula} \, , \ee where $F_{ij} = \partial_i A_j - \partial_j A_i$ is the curvature of the normal bundle of $\Sigma$. The Abelian connection $A_i$ of the normal bundle is given by \be A_i = - \frac{1}{2} \epsilon_a{}^b\, \Gamma^a_{i b}. \label{eq:gauge} \ee The early Latin indices denote the normal direction to the surface $\Sigma$ and in four spacetime dimensions they take on two values. The quantity $\Gamma^a_{ib}$ is given by \begin{align} \Gamma^a_{ib} = \left(\partial_i n^\mu_b+ \hat{\Gamma}^\mu_{\sigma \nu} e^\sigma_i n^\nu_b\right) n^a_\mu \, , \end{align} where $\hat{\Gamma}^\mu_{\sigma\nu}$ are the Christoffel symbols of the bulk spacetime and $e^\sigma_i =\frac{\partial x^\sigma}{\partial y^i}$ is the pull-back map. $n^a_\mu$ for $a=0, 1$ are the unit normal vectors. In our convention $n^0$ is time-like, while $n^1$ is space-like. Happily, this computation reproduces the results given in~\cite{Loganayagam}. ]]>

|b|$. We work with metric expressed in~(\ref{eq:Kerr}). Then the normalized vectors which lie normal to the entangling circle are given by \begin{align} n^0_\mu &= (n^0_t, n^0_r, n^0_\theta) = \left(1/\sqrt{1 - b^2/r^2}, 0, 0\right)\\ n^1_\mu &= (n^1_t, n^1_r, n^1_\theta) = \left(0, 1, 0\right)\, . \end{align} Note that our time-like normal vector becomes imaginary at $r=|b|$. It is then straightforward to compute the boundary gauge field $\hat{a}_\mu$, \begin{align} \hat{a}_\mu = \left( 0, 0, -\frac{b}{r\sqrt{1 - b^2/r^2}}\right)\, . \end{align} Thus, we get the following contribution to the entanglement entropy \begin{align} \Delta S_{\mathrm{EE}}= -\frac{\pi\kappa}{4 G}\frac{b}{\sqrt{R^2 - b^2}}\, . \label{eq:kerrentropy} \end{align} For comparison with our field theory computation in in the next section we note the small angular momentum or large $R$ limit: \begin{align} \Delta S_{\mathrm{EE}}\approx \frac{\pi\kappa b}{4 GR}\, , \label{eq:leadingorder} \end{align} where we have dropped the minus sign by replacing $b$ by $-b$. As expected the contribution vanishes in the non-rotating limit. ]]>