We study a number of

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2$, can we extract a boundary or defect entropy from EE? If so, can we prove whether it is monotonic along RG flows triggered by defect/boundary-localized operators? Can we prove whether it is monotonic along RG flows triggered by deformations of the ambient CFT? In short, does the $g$-theorem generalize to higher dimensions? These questions are difficult to answer, partly because EE is difficult to compute even in free theories. ]]>

2$. For CFTs dual to SUGRA, the prescription to compute EE in a time-independent state, conjectured in refs.~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef} and proven in ref.~\cite{Lewkowycz-ml-2013nqa}, is the following. On a fixed time slice in the bulk, we determine the codimension-one surface of minimal (Einstein-frame) area $\mathcal{A}_{\rm min}$ that approaches $\M$ at the $AdS_{d+1}$ boundary. The EE is then, with $G_N$ the bulk Newton's constant, \beq \label{rt} S = \frac{\mathcal{A}_{\rm min}}{4 G_N}. \eeq \looseness=-1 In principle, we would like to study holographic duals of RG flows between BCFTs and DCFTs. Many gravity solutions exist that describe RG flows between DCFTs, usually involving ``probe'' defects, meaning the defect's contributions to observables (including EE) are suppressed by factors of $N$ relative to the ambient CFT~\cite{Yamaguchi-ml-2002pa}. Few solutions exist describing conformal defects outside of the probe limit~\cite{Gutperle-ml-2012hy,Dias-ml-2013bwa,Korovin-ml-2013gha}. Some \emph{ad hoc} solutions for the holographic duals of BCFTs, and RG flows between BCFTs, appear in refs.~\cite{Takayanagi-ml-2011zk,Fujita-ml-2011fp,Nozaki-ml-2012qd,Gutperle-ml-2012hy}.\footnote{Despite the title of ref.~\cite{Gutperle-ml-2012hy}, the solutions there actually describe fixed points, not RG flows.} In some cases these are genuine solutions of SUGRA theories~\cite{Fujita-ml-2011fp}, and hence we have good reason to believe a pathology-free dual BCFT actually exists. In general, however, that is not guaranteed. Moreover, without a specific dual field theory, a comparison between results calculated on the two sides of the correspondence, gravity and field theory, is impossible.\footnote{\looseness=-1 The bottom-up models for BCFTs of refs.~\cite{Takayanagi-ml-2011zk,Fujita-ml-2011fp,Nozaki-ml-2012qd} also deviate in an essential way from almost all holographic BCFTs that arise in string theory: they are locally AdS. More precisely, in the bottom-up models of refs.~\cite{Takayanagi-ml-2011zk,Fujita-ml-2011fp,Nozaki-ml-2012qd}, the dual spacetime ends on a codimension-one brane which may support some localized matter content. The geometry is locally AdS everywhere away from the ``brane'' , and the shape of the brane is determined by the Israel junction condition involving the brane stress-energy tensor. Currently, the one and only example of such a holographic BCFT in string theory appears in ref.~\cite{Fujita-ml-2011fp}, where the dual spacetime ends on two separated O8$^{-}$ planes, together with two stacks of D8-branes. We do not expect such features to be generic in string theory. In particular, in all other known examples of holographic BCFTs in string theory, the dual spacetime caps off smoothly, rather than ending on a ``brane,'' and the metric only asymptotically approaches AdS. These examples include the $d=4$ BCFT arising from D3-branes ending on D5-branes~\cite{Aharony-ml-2011yc}, the $d=3$ BCFT arising from M2-branes ending on M5-branes~\cite{Bachas-ml-2013vza}, and the $d=2$ BCFTs of refs.~\cite{Chiodaroli-ml-2011fn,Chiodaroli-ml-2012vc}.} Our goal is a more general analysis, relying as little as possible on special limits such as the probe limit, and using genuine solutions of SUGRA, so that we have good reason to believe dual BCFTs and DCFTs exist. To our knowledge, no SUGRA solutions exist describing RG flows between BCFTs or DCFTs outside of the probe limit. We thus turn to known SUGRA solutions that describe fixed points rather than RG flows. We will be able to extract boundary and defect entropies from our holographic results for EE, but our arguments for their behavior along RG flows will be indirect. Such is the price we pay for working outside the probe limit and demanding that dual field theories exist. We focus exclusively on two CFTs that we will deform to obtain DCFTs and BCFTs. The first CFT is $(3+1)$-dimensional $\N=4$ supersymmetric (SUSY) $\SU(N)$ Yang-Mills (SYM) theory. The second CFT is the $(2+1)$-dimensional $\N=6$ SUSY $\U(N)_k\times \U(N)_{-k}$ Chern-Simons-matter theory of Aharony, Bergman, Jafferis, and Maldacena (ABJM)~\cite{Aharony-ml-2008ug}. In each theory we work in the 't Hooft large-$N$ limit, with large 't Hooft coupling, in which case the holographic dual is SUGRA on a background with an AdS factor. \looseness=-1 We choose these two CFTs for two reasons. First, in the dual SUGRA theories, many solutions are known that describe conformal boundaries and codimension-one defects~\cite{Bak-ml-2003jk,Clark-ml-2005te,D-ml-Hoker-ml-2006uu,D-ml-Hoker-ml-2007xy,D-ml-Hoker-ml-2007xz,D-ml-Hoker-ml-2009gg,Aharony-ml-2011yc,Suh-ml-2011xc,Berdichevsky-ml-2013ija,Bobev-ml-2013yra}. All of these solutions describe a boundary or defect that is planar. Second, not only are we confident that the dual DCFTs and BCFTs actually exist, in contrast to many bottom-up models, but also in many cases explicit Lagrangians are known for the dual DCFTs and BCFTs~\cite{DeWolfe-ml-2001pq,Erdmenger-ml-2002ex,Clark-ml-2004sb,D-ml-Hoker-ml-2006uv,Kim-ml-2008dj,Gaiotto-ml-2008sa,Honma-ml-2008un,Gaiotto-ml-2008ak,D-ml-Hoker-ml-2009gg}. We will perform a general calculation, applicable to essentially all of the solutions of refs.~\cite{Bak-ml-2003jk,Clark-ml-2005te,D-ml-Hoker-ml-2006uu,D-ml-Hoker-ml-2007xy,D-ml-Hoker-ml-2007xz,D-ml-Hoker-ml-2009gg,Aharony-ml-2011yc,Suh-ml-2011xc,Berdichevsky-ml-2013ija,Bobev-ml-2013yra}, however, we will present explicit results only for a representative sample of the SUGRA solutions in refs.~\cite{Bak-ml-2003jk,D-ml-Hoker-ml-2006uu,D-ml-Hoker-ml-2007xy,D-ml-Hoker-ml-2007xz,D-ml-Hoker-ml-2009gg,Aharony-ml-2011yc}, as we discuss below. \looseness=-1 For the entangling surface $\mathcal{M}$, for DCFTs we choose a sphere centered on the defect, as depicted in figure~\ref{fig:setup}. We do so for two reasons. First, for a special class of DCFTs we know the spherical EE provides a defect entropy monotonic under a defect RG flow, namely DCFTs in which the defect is a CFT in its own right, completely decoupled from the ambient CFT. In these cases, the spherical EE decomposes into a sum of two spherical EE's, one for the ambient CFT, $S_{\textrm{CFT}}$, and one for the defect CFT, $S_{\textrm{defect}}$, that is, $S=S_{\textrm{CFT}}+S_{\textrm{defect}}$. For a defect of spacetime dimension $2$, $3$, or $4$, and for RG flows triggered by defect-localized operators built out of defect fields, the $c$-, $F$-, and $a$-theorems, combined with eq.~\eqref{E:sphericalEE}, tell us that a certain term in the EE will change monotonically. For instance, if the defect has spacetime dimension $2$, then the defect entropy $S-S_{\textrm{CFT}}$ will include a logarithmic term whose coefficient always decreases under defect RG flows. Analogous statements apply for BCFTs, where we choose $\mathcal{M}$ to be a hemi-sphere centered on the boundary. \begin{figure}[t] \centering \includegraphics[width=.8\textwidth]{Surface.pdf} ]]>

2$, can we obtain $g$-functions, using EE or otherwise? If we wish to address these questions using holography, then we necessarily need gravity solutions describing RG flows between DCFTs or BCFTs, rather than just fixed points. Generically, holographic $c$-theorems invoke the null energy condition in the bulk~\cite{Freedman-ml-1999gp} to guarantee monotonicity of certain terms in the EE~\cite{Myers-ml-2010xs,Myers-ml-2010tj}: at fixed points these terms coincide with either an $a$-type central charge (for even $d$) or $(-1)^{(d-1)/2}$ times the free energy of the Euclidean theory on $\mathbb{S}^d$ (for odd $d$). Holographic $g$-functions have been proposed which invoke a null energy condition for the stress-energy tensor of a brane on the gravity side, either a probe brane dual to a defect~\cite{Yamaguchi-ml-2002pa} or the ``brane'' on which spacetime ends in the bottom-up holographic models of BCFTs of refs.~\cite{Takayanagi-ml-2011zk,Fujita-ml-2011fp,Nozaki-ml-2012qd}. What physical observables these $g$-functions are dual to in the field theory is not always clear. A natural question is whether they are dual to some contribution to an EE. Probe branes may provide the simplest way to address this question, since several techniques exist to calculate a probe brane's contribution to EE~\cite{Chang-ml-2013mca,Jensen-ml-2013lxa,Karch-ml-2014ufa}. \looseness=-1 In a $d=4$ CFT the coefficient of the $\ln\left(2R/\varepsilon\right)$ term in the EE is determined completely by $\mathcal{M}$ and the central charges $a$ and $c$. For spherical $\mathcal{M}$, the coefficient is $\propto a$, as in eq.~\eqref{E:sphericalEE}, while for cylindrical $\mathcal{M}$ it is $\propto c$~\cite{Casini-ml-2011kv}. The central charge $c$ obeys no monotonicity theorem: explicit examples show that $c$ can either increase or decrease under RG flows (see for example refs.~\cite{Anselmi-ml-1997am,Anselmi-ml-1997ys}). In this paper we focus on (hemi-)spherical $\mathcal{M}$, but what about other $\mathcal{M}$? Can we characterize the $\ln\left(2R/\varepsilon\right)$ terms in defect/boundary entropy by $\mathcal{M}$ and a finite set of ``central charges''? The results of ref.~\cite{Fursaev-ml-2013mxa} for $d=4$ BCFTs, for $\mathcal{M}$ that intersects the boundary, suggest that this may be the case. What about $\mathcal{M}$ that do not intersect the defect/boundary? Studying different $\mathcal{M}$ could be useful for identifying and studying candidates for defect/boundary entropies, for example by eliminating some candidates (like $c$ in $d=4$). \looseness=-1 There are proposals to use EE to detect topological order in $d=3$~\cite{Kitaev-ml-2005dm,Levin-ml-2006zz} and $d\geq 4$~\cite{Grover-ml-2011fa}. Holography can describe many topologically non-trivial phases, and so can help to test these proposals. For example, two kinds of holographic models exist for time-reversal invariant fractional topological insulators in $d=4$. The first kind uses probe branes~\cite{Maciejko-ml-2010tx,HoyosBadajoz-ml-2010ac}, for which EE could be computed using the methods of refs.~\cite{Chang-ml-2013mca,Jensen-ml-2013lxa,Karch-ml-2014ufa}. The second kind uses Janus solutions of SUGRA~\cite{Estes-ml-2012nx}, including some of the examples we study in sections~\ref{ssec:NSJanus} and~\ref{ssec:Janus}. (The two kinds of models may be closely related~\cite{Estes-ml-2012nx}.) A natural questions is: to what extent do our results in sections~\ref{ssec:NSJanus} and~\ref{ssec:Janus} characterize the topological order of these states? Lastly, SUSY localization has been used to compute a SUSY version of R\'enyi entropy for certain Chern-Simons-matter theories in $d=3$~\cite{Nishioka-ml-2013haa}. The EE may be extracted from this SUSY R\'enyi entropy~\cite{Nishioka-ml-2013haa}. Moreover, SUSY localization has also been used to compute the partition functions of SUSY theories on manifolds with boundaries~\cite{Sugishita-ml-2013jca,Honda-ml-2013uca,Hori-ml-2013ika}. Presumably these two things can be combined: for SUSY theories on manifolds with boundaries, SUSY localization could be used to compute EE. Such calculations could provide exact results for boundary entropies, which could be very useful for testing higher-dimensional $g$-theorems. \looseness=-1 This paper is organized as follows. In section~\ref{S:holo}, we discuss the calculation of spherical EE for general holographic DCFTs and BCFTs. We pay special attention to the regularization of the EE, so that we can perform the background subtractions in our definitions of $S_{\textrm{defect}}$ and $S_{\partial}$ in eq.~\eqref{E:defDefBdyS}. Section~\ref{S:examples} is a case-by-case study of spherical EE in our various examples of DCFTs and BCFTs. The appendix contains the technical details of our general analysis of spherical EE in $d=3$,$4$, including in particular the derivation of eqs.~\eqref{E:Sdefect} and~\eqref{E:Sbdy}. ]]>

0$ and $x<0$. In particular, from eq.~\eqref{AdSFG} we see that the $AdS_{d+1}$ boundary $z \to 0$ splits into two pieces at $x = \pm \infty$. These two pieces are glued together at the boundary of the $AdS_{d}$ slice, $u \to 0$, or equivalently at $x^{d-1} = 0$. In the dual CFT, we can think of $x^{d-1}=0$ as the location of a fictitious codimension-one planar defect. We now consider a spherical $\M$ of radius $R$ centered on the fictitious defect, or more precisely centered at the origin $\vec{x}=\vec{0}$. Following Ryu and Takayanagi~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef}, to compute this EE holographically we must compute the area of the minimal surface which lives on a fixed time-slice and approaches $\M$ as $z \to 0$. That minimal area surface wraps the $\mathbb{S}^{d-3}$ and so is described by a hypersurface in the $(x,u,r)$-space. If we describe that surface as $r(x,u)$, then the area functional becomes \beq \label{eq:AdSslicingarea} \mathcal{A} = \vol(\mathbb{S}^{d-3})L^{d-1} \int du \, dx \, r^{d-3} \, \frac{\cosh^{d-2}(x)}{u^{d-2}} \sqrt{1+ (\p_u r)^2 + \frac{\cosh^2(x)}{u^2} (\p_x r)^2}\,. \eeq We will discuss the endpoints of the $u$ and $x$ integrations in eq.~\eqref{eq:AdSslicingarea} momentarily. The Euler-Lagrange equation arising from eq.~\eqref{eq:AdSslicingarea} is a complicated partial differential equation for $r(x,u)$. However, the minimal area surface that we want has a simple description in Poincar\'e slicing~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef}: $z^2 + (x^{d-1})^2 + r^2 = R^2$, which at the $AdS_{d+1}$ boundary $z \to 0$ clearly describes a sphere of radius $R$ centered at the origin. Switching to $AdS_d$ slicing via eq.~\eqref{AdSFG}, the solution for the minimal area surface becomes \beq \label{eq:minsurface} u^2 + r^2 = R^2. \eeq A straightforward exercise shows that the solution for $r(x,u)$ given by eq.~\eqref{eq:minsurface} indeed solves the Euler-Lagrange equation arising from eq.~\eqref{eq:AdSslicingarea}. Notice that the solution for $r(x,u)$ given by eq.~\eqref{eq:minsurface} depends on $u$ but not on $x$. Let us now compute the value of the minimal area. To do so, we insert the solution in eq.~\eqref{eq:minsurface} into the area functional eq.~\eqref{eq:AdSslicingarea} and then integrate in $x \in (-\infty,\infty)$ and $u \in [0,R]$. The integrand in eq.~\eqref{eq:AdSslicingarea} diverges exponentially in the asymptotically $AdS_{d+1}$ regions at large $|x|$, and hence $\mathcal{A}$ is divergent. From the CFT perspective, these are the expected short-distance divergences from highly-entangled modes near $\mathcal{M}$. Again following Ryu and Takayanagi~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef}, we regulate the divergence by introducing a Fefferman-Graham (FG) cutoff: in the Poincar\'e-sliced coordinates we introduce a cutoff surface $z = \varepsilon$. In the $AdS_d$ slicing, the FG cutoff becomes a surface in the $(x,u)$-space. Via eq.~\eqref{AdSFG}, that surface is described as the union of two surfaces $\chi_{\pm}(\frac{\varepsilon}{u})$ given by \beq \chi_{\pm}\left(\frac{\varepsilon}{u}\right) \equiv \pm \text{arccosh}\left(\frac{u}{\varepsilon}\right)= \pm \ln \left( \frac{2u}{\varepsilon} \right) \pm \ln \left[ \frac{1}{2} + \frac{1}{2} \sqrt{1- \frac{\varepsilon^2}{u^2}} \right]\,, \eeq where $u\in [\varepsilon,R]$. Note that the cutoff surface is real and continuous for this choice of lower bound on $u$. Using these cutoffs and the solution for $r(x,u)$ in eq.~\eqref{eq:minsurface}, the integral for the minimal area becomes \beq \label{eq:sphereee1} \mathcal{A}_{\rm min} = \vol(\mathbb{S}^{d-3}) \, L^{d-1} \, R \int^R_{\varepsilon} du \, \frac{(R^2 - u^2)^\frac{d-4}{2}}{u^{d-2}} \int_{\chi_-(\frac{\varepsilon}{u})}^{\chi_+(\frac{\varepsilon}{u})} dx \, \cosh^{d-2}(x)\,. \eeq We are interested in the cases $d=3,4$, for which \beq \label{eq:minarearesult} \mathcal{A}_{\rm min} = \left\{ \begin{array}{ll} 2 \pi L^2 \left( \frac{R}{\varepsilon} - 1 \right)\,, & \quad d =3\,, \\ & \\ 2\pi \, L^{3} \left( \frac{R^2}{\varepsilon^2} - \ln\left( \frac{2R}{\varepsilon} \right) - \frac{1}{2} \right)\,, & \quad d=4\,. \end{array}\right. \eeq Following eq.~\eqref{rt}, we multiply eq.~\eqref{eq:minarearesult} by $1/(4G_N)$ to obtain the EE, which reproduces the results of refs.~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef}, as advertised. \looseness=-1 In this work, we study DCFTs and BCFTs where the ambient CFT is either the ABJM theory or $\mathcal{N}=4$ SYM. The holographic dual of $\U(N)_k\times \U(N)_{-k}$ ABJM theory is eleven-dimensional M-theory on $AdS_4 \times \mathbb{S}^7/\mathbb{Z}_k$, where the $AdS_4$ has radius $L$ and the $\mathbb{S}^7/\mathbb{Z}_k$ has radius $2L$. In the $N \gg k$ and $N\gg k^5$ limits, the M-theory is well-approximated by eleven-dimensional SUGRA. The minimal-area surface wraps the internal space $\mathbb{S}^7/\mathbb{Z}_k$, so the result for $\mathcal{A}_{\rm min}$ is the $d=3$ result in eq.~\eqref{eq:minarearesult} times the volume of $\mathbb{S}^7/\mathbb{Z}_k$, $\frac{\pi^4}{3k}(2L)^7$. In eleven dimensions the gravitational constant is given by $4 G_N = 2^6 \pi^7 l_p^9$ and the AdS radius is related to field theory quantities as $L^6 = \frac{\pi^2}{2} N k \, l_p^6$, where $l_p$ is the Planck length~\cite{Aharony-ml-2008ug}. The spherical EE then follows from eq.~\eqref{rt}, \beq \label{E:EEforABJM} S = \frac{\pi \sqrt{2}}{3} \, k^\frac{1}{2} \, N^\frac{3}{2} \left[ \frac{R}{\varepsilon} - 1 \right]\,. \qquad \textrm{(ABJM theory)} \eeq \looseness=-1 The holographic dual of $\N=4$ SYM theory is type IIB string theory on $AdS_5 \times \mathbb{S}^5$, where both the $AdS_5$ and $\mathbb{S}^5$ have radius $L$. In the $N,\lambda \gg 1 $ limits (with $\lambda\equiv g_{YM}^2N \ll N$), the string theory is well-approximated by type IIB SUGRA. The minimal-area surface wraps the internal space $\mathbb{S}^5$, so the result for $\mathcal{A}_{\rm min}$ is the $d=4$ result in eq.~\eqref{eq:minarearesult} times the volume of the $\mathbb{S}^5$, $\pi^3 L^5$. In ten dimensions and in Einstein frame, the gravitational constant is given by $4 G_N = 2^5 \pi^6 (\alpha')^4$ and the AdS radius is given in terms of SYM quantities as $L^4 = 4 \pi N (\alpha')^2$, where $\alpha'$ is the string length squared. The spherical EE then follows from eq.~\eqref{rt}, \beq \label{E:EEforSYM} S = N^2 \left[ \frac{R^2}{\varepsilon^2} - \ln\left( \frac{2R}{\varepsilon} \right) - \frac{1}{2} \right]\,. \qquad \textrm{($\N=4$ SYM theory)} \eeq ]]>

0$) and gauge group $SU\left(N_3^-\right)$ on the other side ($x^3<0$), with $N_3^+ \neq N_3^-$. Detailed discussions of these vacua appear in refs.~\cite{Gaiotto-ml-2008sa,Gaiotto-ml-2008ak}. At large $N_3^{\pm}$ and large coupling, the holographic duals (discussed below) indicate that this subset of Higgs vacua preserve defect conformal symmetry, which is perhaps counter-intuitive, since normally a scalar expectation value breaks scale invariance. As explained in ref.~\cite{McAvity-ml-1995zd}, however, defect conformal symmetry allows a primary scalar operator of dimension $\Delta$ to have a non-zero one-point function $\propto (x^3)^{-\Delta}$. To our knowledge, whether this subset of Higgs vacua preserves defect conformal symmetry for all values of $N_3^{\pm}$ and 't Hooft coupling is an open question. \looseness=-1 These DCFTs appear in string theory as the low-energy field theory living at the $(2+1)$-dimensional intersection of $N_3^{\pm}$ D3-branes and $N_5$ D5-branes, with $\Delta N_3\equiv N_3^+-N_3^-$ D3-branes ending on the D5-branes. When $N_3^{\pm}$ and $N_5$ are small, so that the D3- and D5-branes are probes in $(9+1)$-dimensional Minkowski space, the intersection is that of table~\ref{T:D3D5}, with the D5-branes at $x^3=0$ and with $N_3^+$ or $N_3^-$ D3-branes located in the half-spaces $x^3>0$ or $x^3<0$, respectively. \begin{table}[t] \centering \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline & $x^0$ & $x^1$ & $x^2$ & $x^3$ & $x^4$ & $x^5$ & $x^6$ & $x^7$ & $x^8$ & $x^9$ \\ \hline $N_3^{\pm}$ D3 & X & X & X & X & & & & & &\\ \hline $N_5$ D5 & X & X & X & & X & X & X & & &\\ \hline \end{tabular} ]]>

0$ region and $N_3^-$ D3-branes in the $x^3<0$ region, where $\Delta N_3\equiv N_3^+-N_3^-$ D3-branes end on the D5-branes.]]>

0$, and so that $e^{\delta}\geq 1$ for $\Delta N_3\geq 0$ while $e^{\delta}\in (0,1)$ for $\Delta N_3 < 0$, as dictated by eq.~\eqref{E:D3D5deltaMatch}. The bulk parameters $\{\alpha,\hat{\alpha},\delta\}$ are thus uniquely determined by the field theory parameters: given $\{g_{YM}^2,N_3^{\pm},N_5\}$, eq.~\eqref{eq:deltadef} gives us $\delta$, which we then insert into eq.~\eqref{E:alphasol} to determine $\alpha$, and from that $\hat{\alpha} = g_{YM}^2 \alpha/(4\pi)$. The explicit expressions for $\{\alpha,\hat{\alpha},\delta\}$ in terms of $\{g_{YM}^2,N_3^{\pm},N_5\}$ are cumbersome and unilluminating, so we will omit writing them in full generality. We will only present their explicit forms at leading order in the $\Delta N_3 \ll 1$ or equivalently $\delta \ll 1$ limit, \begin{align} \begin{split} \label{E:D3D5simpleBulk} \delta &= \frac{4\pi^2 \Delta N_3}{N_5}\frac{1}{ \xi-g_{YM}^2N_5 } + \mathcal{O}\left(\frac{\Delta N^3}{N_5^3\xi^3}\right)\,, \\ \alpha &=\frac{\xi-g_{YM}^2N_5}{4g_{YM}^2} - \frac{2\pi^3 \Delta N_3^2}{N_5}\frac{1}{\xi(\xi-g_{YM}^2N_5)} + \mathcal{O}\left( \frac{\Delta N_3^4}{N_5^3\xi^4}\right)\,, \\ \hat{\alpha} &= \frac{g_{YM}^2}{4\pi} \alpha\,, \end{split} \end{align} where for notational convenience we have defined \beq \label{E:xiDef} \xi^2 \equiv 16\pi^2 g_{YM}^2 N_3 + (g_{YM}^2N_5)^2\,. \eeq Our one and only example of a BCFT is the D3/D5 BCFT, obtained in string theory as the low-energy theory on $N_3$ coincident D3-branes that end on $N_5$ D5-branes. This BCFT is $\N=4$ SYM with gauge group $\SU(N_3)$ on a half space $x^3 \geq 0$ coupled to $N_5$ hypermultiplets localized at the boundary $x^3=0$. The D3/D5 BCFT preserves eight real supercharges and $\SO(3,2) \times \SO(3) \times \SO(3)$ bosonic symmetry, and at large $N_3$ and large 't Hooft coupling is dual to type IIB SUGRA in a background of the form in eqs.~\eqref{susymet},~\eqref{susydef}, and~\eqref{E:dilaton}, with harmonic functions~\cite{Aharony-ml-2011yc} \begin{align} \begin{split} \label{E:D3D5boundaryHarmonic} h_1(v,\bar{v}) & = \alpha'\left[- \frac{i\underline{\alpha}}{2}e^v - \frac{N_5}{4}\ln\left( \tanh\left( \frac{i\pi}{4}-\frac{v}{2}\right)\right)\right] + \text{c.c.}\,, \\ h_2(v,\bar{v}) & = \alpha' \frac{\hat{\underline{\alpha}}}{2}e^v + \text{c.c.}\,, \end{split} \end{align} with real parameters $\{\underline{\alpha},\hat{\underline{\alpha}}\}$. In the bottom-up holographic models of BCFTs of refs.~\cite{Takayanagi-ml-2011zk,Fujita-ml-2011fp,Nozaki-ml-2012qd}, the field theory's spatial boundary gives rise in the holographic dual to a ``brane'' on which the bulk spacetime ends. In contrast, in the dual of the D3/D5 BCFT the bulk spacetime does not end on a ``brane,'' but caps off smoothly~\cite{Assel-ml-2011xz}: if in eq.~\eqref{E:D3D5boundaryHarmonic} we change coordinates as \beq r^2 = \frac{2 N_5 e^{2(x-\delta)}}{N_5 + e^{\delta}\underline{\alpha}}, \eeq then as $x \to -\infty$ or equivalently $r \to 0$, the metric approaches \beq ds^2 = L_+^2 \left [ g_{AdS_4} \, + \, dr^2 \, + \, r^2 \left( dy^2 + \sin^2(y) \, g_{\mathbb{S}^2} + \cos^2(y) \, g_{\mathbb{S}^2}\right) \right], \eeq with $L_+^4 = 8 \hat{\underline{\alpha}} e^{\delta} N_5 (\alpha')^2$. Clearly the spacetime caps off smoothly as $r \to 0$. We can obtain the D3/D5 BCFT from the D3/D5 DCFT by sending the number of D3-branes on one side of the D5-branes to zero. To be concrete, we will take $N_3^-\to0$ while keeping $N_3^+$ fixed. In that limit the harmonic functions corresponding to the D3/D5 DCFT, eq.~\eqref{E:D3D5h}, reduce to those of the D3/D5 BCFT, eq.~\eqref{E:D3D5boundaryHarmonic}, as we will now show. The radius $L_-$ of the asymptotically $AdS_5 \times \mathbb{S}^5$ region at $x \to -\infty$ is related to the number of D3-branes there as $L_-^4 = 4\pi N_3^-(\alpha')^2$. The $N_3^- \to 0$ limit thus implies $L_-\to 0$, which by eqs.~\eqref{E:D3D5Match1} and~\eqref{E:alpha} means we must take $\alpha \to 0$ and $\hat{\alpha}\to 0$ while keeping fixed \begin{equation*} \alpha e^{\delta}=\frac{2\pi^2 N_3^+}{g_{YM}^2 N_5}\,, \qquad \textrm{and} \qquad \frac{\hat{\alpha}}{\alpha}=\frac{g_{YM}^2}{4\pi}\,. \end{equation*} In this limit, $\delta \to \infty$, and upon defining $\tilde{v}\equiv v - \delta = (x - \delta) + i y$, the harmonic functions corresponding to the D3/D5 DCFT, eq.~\eqref{E:D3D5h}, become \begin{align} \begin{split} \label{E:harmoniclimit} h_1(\tilde{v},\bar{\tilde{v}}) & = \alpha'\left[- \frac{i(\alpha e^{\delta})}{2}e^{\tilde{v}} -\frac{N_5}{4}\ln\left( \tanh\left( \frac{i\pi}{4}-\frac{\tilde{v}}{2}\right)\right) \right]+ \frac{i (\alpha e^{\delta})\alpha'}{2}e^{-\tilde{v}-2\delta}+\text{c.c.}\,, \\ h_2(\tilde{v},\bar{\tilde{v}}) & = \alpha'\frac{g_{YM}^2(\alpha e^{\delta})}{8\pi}e^{\tilde{v}} + \alpha' \frac{g_{YM}^2 (\alpha e^{\delta})}{8\pi}e^{-\tilde{v}-2\delta} + \text{c.c.}\,. \end{split} \end{align} Dropping the $e^{-\tilde{v}-2\delta}$ terms, which are exponentially suppressed as $\delta \to \infty$, and identifying \beq \alpha e^{\delta} = \underline{\alpha}, \qquad \frac{g_{YM}^2 \alpha e^{\delta} }{4\pi}= \hat{\underline{\alpha}}, \qquad \tilde{v} = v\,, \eeq we see that the harmonic functions in eq.~\eqref{E:harmoniclimit} are precisely those corresponding to the D3/D5 BCFT, eq.~\eqref{E:D3D5boundaryHarmonic}, as advertised. In what follows we will thus obtain results for the D3/D5 BCFT by working with the D3/D5 DCFT and then taking the limit above. The SUGRA duals of the D3/D5 DCFT and BCFT exhibit characteristic D5-brane singularities: both $\exp(2\phi)$ and the Einstein-frame metric go to zero at the D5-branes. As a result, near the D5-branes stringy corrections remain small but curvature corrections must become important. Currently the form of these curvature corrections is unknown, so for now we will simply work within the SUGRA approximation. Because the Einstein-frame metric vanishes at the D5-branes, the area density of the minimal surface (i.e.\ the integrand in eq.~\eqref{E:defectSEE}) is integrable at the D5-branes, so in practice the curvature singularity presents no obstruction to our holographic calculation of the EE. We hasten to emphasize, however, that we do not understand what role the curvature singularity plays, if any, when accounting for higher-derivative corrections in the holographic calculation of the EE. ]]>

\delta$, and in $\exp(x - \delta)$ for $x<\delta$. We next exchange the sum of the expansion with the $y$ integral, and then integrate in $y$ term-by-term. Finally, we re-sum the expansion, obtaining, for $x > \delta$, \begin{align} \label{eq:mathcals0int} \int_0^{\frac{\pi}{2}}\!\!dy \frac{(f_4f_1f_2\rho)^2}{2\pi \hat{\alpha}^2(\alpha')^4} \!&=\! 8 \alpha^2 \cosh^2(x) \!+\! N_5 \alpha\!\left[\! e^{2x+\delta}\!+\!4e^{\delta}\!+\!\left(\! 3\!+\!\frac{e^{2\delta}}{3}\!\right)\!e^{-2x+\delta}\!+\!e^{-6x+3\delta}\right] \\ \nonumber &\quad + 4N_5^2 \cosh (x)\cosh(2x\!-\!\delta)\left[ e^{-x+\delta}-2\text{arctanh}\left( e^{-2(x-\delta)}\right)\sinh (x\!-\!\delta)\right], \end{align} where we included the factor $1/(2\pi \hat{\alpha}^2(\alpha')^4)$ on the left-hand-side for convenience. The integration over $x$ is then straightforward.\footnote{In practice, for the integration over $x$ we found the choice $x_{\pm}^c \to \pm \infty$ the most convenient. We hasten to repeat, however, that the result for $S$ is independent of the choice of $x_{\pm}^c$, as mentioned below eq.~\eqref{eq:split2}.} For $x<\delta$, we find the same result as eq.~\eqref{eq:mathcals0int}, but with $\{x,\delta\}\to\{-x,-\delta\}$. Upon summing our results for $S_{\pm}$ and $S_0$, we find (ignoring terms that vanish as $\varepsilon \to 0$) \begin{align} \nonumber S &= \frac{\text{vol}(\mathbb{S}^1)\text{vol}(\mathbb{S}^2)^2}{4G_N} \int_0^{\frac{\pi}{2}} dy \left [ \left( A_{+}^{(-2)}(y)+A_-^{(-2)}(y)\right)\frac{2R^2}{\varepsilon^2} - \left( A_{+}^{(0)}(y)+A_-^{(0)}(y)\right) \ln\left( \frac{2R}{\varepsilon}\right) \right. \\ \label{E:SEEforD3D5} &\qquad\qquad \left. - \left(A_{+}^{(0)}(y)+A_-^{(0)}(y) + 4 A_{+}^{(-2)}(y)\mathcal{C}_{+}^{(2)}(y)+4A_-^{(-2)}(y)\mathcal{C}_-^{(2)}(y)\right)\right ] \\ \nonumber & \qquad\qquad \qquad \qquad + D_1 \frac{R}{\varepsilon} + \mathcal{S}_-+\mathcal{S}_0+\mathcal{S}_+\,, \end{align} where the term $D_1 \frac{R}{\varepsilon}$ is the sum of the ${\mathcal{O}}\left(\frac{R}{\varepsilon}\right)$ terms in eqs.~\eqref{eq:spm} and~\eqref{E:S0def}. We did not bother to compute $D_1$, which is non-universal. Upon performing the integration over $y$ in the first and second lines of eq.~\eqref{E:SEEforD3D5}, we find \beq \label{E:SEED3D5part1} S = \frac{\left(N_3^+\right)^2+\left(N_3^-\right)^2}{2}\left[\frac{R^2}{\varepsilon^2} -\ln\left( \frac{2R}{\varepsilon}\right) -\frac{1}{2}\right] + D_1 \frac{R}{\varepsilon} + \mathcal{S}_-+\mathcal{S}_0+\mathcal{S}_+\,. \eeq \looseness=-1 The term in brackets in eq.~\eqref{E:SEED3D5part1} is precisely half the spherical EE for $\mathcal{N}=4$ SYM theory with gauge group $\SU(N_3^+)$ plus half of the spherical EE for $\mathcal{N}=4$ SYM theory with gauge group $\SU(N_3^-)$. Following eqs.~\eqref{E:defDefBdyS} and~\eqref{E:Sdefect}, we thus identify the universal contribution to the defect entropy, \begin{align} \nonumber D_ 0 = & \, \mathcal{S}_-+\mathcal{S}_0+\mathcal{S}_+ \\ = & \frac{\pi^4}{4G_N}\left\{ \left(L_{+}^8c_++L_{-}^8c_-\right) + \frac{128 }{3} N_5\alpha\hat{\alpha}^2(\alpha')^4 \left[ \cosh(3\delta)-6\cosh(\delta)+12\delta\sinh(\delta)\right]\right. \nonumber \\ \label{E:SEED3D5part2} &\left. \phantom{\frac{1}{G}}+32N_5^2 \hat{\alpha}^2(\alpha')^4\left[(4\delta \sinh(2\delta) -3\cosh(2\delta)+8\ln 2 \sinh^2(\delta))\right]\right\}. \end{align} Using eq.~\eqref{E:D3D5Match1}, $L_{\pm}^4 = 4\pi N_3^{\pm}(\alpha')^2$, and $\hat{\alpha} = g_{YM}^2 \alpha/(4\pi)$, we can write our result for $D_0$ in terms of $g_{YM}^2$, $N_3^{\pm}$, $N_5$, $\alpha$, and $\delta$, \begin{align} \begin{split} \label{E:D3D5fullS} D_0 = & \frac{1}{4}\left[ \left(N_3^+\right)^2 \ln \left(\frac{g_{YM}^2\alpha^2}{\pi^2 N_3^+}\right) + \left(N_3^-\right)^2 \ln\left( \frac{g_{YM}^2\alpha^2}{\pi^2 N_3^-}\right)\right] \\ & \qquad + \frac{1}{12\pi^4}g_{YM}^4N_5\alpha^3 \left[ \cosh(3\delta) - 6 \cosh(\delta) + 12 \delta \sinh(\delta) \right] \\ & \qquad \qquad+ \frac{1}{16\pi^4}g_{YM}^4N_5^2 \alpha^2 \left[ 4\delta \sinh(2\delta) - 3\cosh(2\delta) + 8\ln2 \sinh^2(\delta)\right]\,, \end{split} \end{align} which is the main result of this subsection. In eq.~\eqref{E:D3D5fullS} we can translate $\alpha$ and $\delta$ to field theory quantities easily, using eqs.~\eqref{E:alphasol} and~\eqref{eq:deltadef}, but the result is cumbersome and unilluminating, so we will not present it in full generality. Instead, we will present the result in a few simplifying limits. When $N_5=0$, which via eq.~\eqref{E:D3D5deltaMatch} implies $\Delta N_3=0$, we find $D_0=0$, as expected. When $N_5 \neq 0$ and $\Delta N_3=0$, using eq.~\eqref{E:D3D5simpleBulk} we find \beq \label{E:D3D5simpleS} D_0 = \frac{N_3^2}{2}\ln\left( \frac{(\xi-g_{YM}^2N_5)^2}{16\pi^2 g_{YM}^2N_3} \right) - \frac{N_5(\xi-g_{YM}^2N_5)^2(5\xi+4g_{YM}^2N_5)}{768\pi^4g_{YM}^2}\,, \eeq where we recall $\xi^2 \equiv 16\pi^2 g_{YM}^2 N_3 + (g_{YM}^2N_5)^2$ from eq.~\eqref{E:xiDef}. If we additionally take the probe limit $N_5 \ll N_3$, then we find \beq \label{eq:D3D5EEprobe} D_0 = - \frac{2}{3\pi} \, \sqrt{\lambda} \, N_5 N_3 + \mathcal{O}\left( \lambda N_5^2\right)\,, \eeq where $\lambda\equiv g_{YM}^2N_3$ is the 't Hooft coupling. The order-$\sqrt{\lambda}$ term in eq.~\eqref{eq:D3D5EEprobe} agrees perfectly with that computed in refs.~\cite{Jensen-ml-2013lxa,Chang-ml-2013mca} using probe D5-branes in $AdS_5 \times \mathbb{S}^5$. As explained above, if we take $N_3^- \to 0$ with $N_3^+$ fixed, then the D3/D5 DCFT becomes the D3/D5 BCFT. In that limit the universal part of the defect entropy, $D_0$ in eq.~\eqref{E:D3D5fullS}, becomes the universal part of the boundary entropy, $B_0$, \beq \label{E:D3D5defectToBdy} \lim_{N_3^-\to 0} D_0 = B_0 = \frac{N_3^2}{8}\left( 2\ln\left(\frac{16\pi^2 N_3}{g_{YM}^2N_5^2}\right) -3\right) + \frac{\pi^2 N_3^3}{3g_{YM}^2N_5^2}\,. \eeq We also obtained the $B_0$ in eq.~\eqref{E:D3D5defectToBdy} directly, by plugging the harmonic functions corresponding to the D3/D5 BCFT, eq.~\eqref{E:D3D5boundaryHarmonic}, into eq.~\eqref{E:susyEE} and performing the integrations. ]]>

\! \delta, \\ \alpha \sin(y) e^{-x} \!+\! (\alpha \!+\! N e^{-\delta}) \sin(y) e^{+x} \!-\! \frac{N}{3} \sin(3y) e^{-3(\delta-x)} \!+\! {\cal{O}}\left(e^{-5(\delta -x)}\right) &x \!<\! \delta, \\ \end{cases} \cr \frac{h_2(v,\bar{v})}{\alpha'} \!&=\! \begin{cases} \hat{\alpha} \cos(y) e^{+x} \!+\! (\hat{\alpha} \!+\! N e^{\hat{\delta}}) \cos(y) e^{-x} \!+\! \frac{N}{3} \cos(3y) e^{3(\hat{\delta}-x)} \!+\! {\cal{O}}\left(e^{5(\hat{\delta}-x)}\right) & x \!>\! \hat{\delta}, \\ \hat{\alpha} \cos(y) e^{-x} \!+\! (\hat{\alpha} \!+\! N e^{-\hat{\delta}}) \cos(y) e^{+x} \!+\! \frac{N}{3} \cos(3y) e^{3(x-\hat{\delta})} \!+\! {\cal{O}}\left(e^{-5(x-\hat{\delta})}\right) & x \!<\! \hat{\delta}.\\ \end{cases} \end{align} We can argue that the terms of ${\cal{O}}(e^{\pm5(x-\delta)})$ and ${\cal{O}}(e^{\pm5(x-\hat{\delta})})$ and higher (henceforth the ``neglected terms'') do not contribute to the divergent or constant terms in the spherical EE, as follows. In the appendix we show explicitly that the $R^2/\varepsilon^2$ and $\ln(2R/\varepsilon)$ terms in the spherical EE receive contributions only from terms in eq.~\eqref{TSUNintg} that are non-vanishing in the $|x| \to \infty$ limit. The neglected terms vanish in that limit and hence do not contribute to the $R^2/\varepsilon^2$ and $\ln(2R/\varepsilon)$ terms in the spherical EE. The constant term in the spherical EE receives contributions of order $N^2$ and $N_3^2$ from the neglected terms, however these are not the leading contributions to the constant term: the biggest contribution comes from a term proportional to $N^2 \ln N$ or $N_3^2 \ln \left(N^2/N_3\right)$, as we will see below. These logarithmic contributions come from terms in eq.~\eqref{TSUNintg} that are independent of $x$. The neglected terms cannot contribute to a term independent of $x$, simply because eq.~\eqref{TSUNintg} involves a product of four harmonic functions, and so a term of order ${\cal{O}}(e^{\pm5(x-\delta)})$ or ${\cal{O}}(e^{\pm5(x-\hat{\delta})})$ would multiply a term of at most ${\cal{O}}(e^{\pm3x})$, coming from a product of the ${\cal{O}}(e^{\pm x})$ terms of three harmonic functions. In short, to obtain the leading divergent and constant contributions to the spherical EE, we only need the leading terms shown explicitly in eq.~\eqref{e:tsunh1h2}. \looseness=-1 Using eq.~\eqref{e:tsunh1h2} in eq.~\eqref{TSUNintg} and then performing the integrations in eq.~\eqref{E:susyEE}, we find that the universal part of the defect entropy, $D_0$, in the $N \gg 1$ limit depends on how we scale $N_3$ as we take $N \gg 1$: \begin{subequations} \label{eq:tsunresult} \begin{align} S &= N_3^2 \left[ \frac{R^2}{\varepsilon^2}-\ln\left( \frac{2R}{\varepsilon}\right)-\frac{1}{2}\right] + D_1 \frac{R}{\varepsilon} + D_0, \\ D_0 &= \begin{cases} - \frac{1}{2} N^2 \ln N + {\cal O}(N^2) & N \gg N_3 \gg 1, \\ - \frac{1}{2} N^2 \left(1 + 2 \frac{N_3}{N} + 2 \frac{N_2^3}{N^2} \right) \ln N + {\cal O}(N^2) & N \propto N_3 \gg 1\,, \\ - N_3^2 \ln \left(\frac{N^2}{N_3} \right) + {\cal O}(N_3^2) & N^2 \gg N_3 \gg N \gg 1\,, \end{cases} \end{align} \end{subequations} where once again we did not bother to compute the non-universal constant $D_1$. \looseness=-1 Our result for $D_0$ in eq.~\eqref{eq:tsunresult} offers a big hint for a higher-dimensional $g$-theorem: in the limit $N \gg N_3 \gg 1$ the leading contribution to $D_0$ is clearly minus the leading large-$N$ contribution to the free energy of the $T[\SU(N)]$ CFT on $\mathbb{S}^3$, $-F_{\mathbb{S}^3} = -\frac{1}{2} N^2 \ln N + {\cal{O}}(N^2)$~\cite{Nishioka-ml-2011dq}, precisely the quantity that obeys the F-theorem. Can the proof of the F-theorem in ref.~\cite{Casini-ml-2012ei}, based primarily on the strong sub-additivity of EE, be adapted to prove a higher-dimensional $g$-theorem? We will leave this important question for future research. ]]>

0$, we use a recursion relation for $\int d\theta \sin^n(\theta)$. We thus find, in the small-$\varepsilon$ limit, \begin{align*} {\cal I}_{-1} &= \frac{\pi R}{\varepsilon} - 2 u^c(y^a) +\mathcal{O}(\varepsilon), & {\cal I}_{0} &= - \ln \left(\frac{\varepsilon u^c(y^a)}{2 R} \right) +\mathcal{O}(\varepsilon), \nonumber \\ {\cal I}_{l} &= \frac{1}{l (2 u^c(y^a))^l} +\mathcal{O}(\varepsilon) & (l &> 0). \nonumber \end{align*} Using these integrals in eq.~\eqref{E:EEexpand3}, we find \beq \label{E:EEd3app} S \!=\! \frac{1}{2G_N} \!\!\int\!\! dy^a \!\left[\!\pi \coe_{-1} \frac{R}{\varepsilon} \!+\! \coe_0 \ln \lp \frac{2R}{\varepsilon} \rp \!-\! \coe_0 \ln \lp \frac{u^c(y^a)}{2}\rp \!+\! \sum_{\substack{l\neq0 \\ l=-1}}^\infty \frac{\coe_l}{l(2 u^c(y^a))^l}\!\right] \!+\! {\mathcal{O}}\left(\varepsilon\right). \eeq \looseness=-1 Here again we will not perform the integration over the $y^a$, since that will not change the $\varepsilon$-dependence of the terms. We thus find the expected leading divergent term, $\propto \coe_{-1} \frac{R}{\varepsilon}$. A straightforward exercise using eq.~\eqref{eq:Cns} shows that because the metric asymptotically approaches $AdS_4$, the value of $Y_{-1}$ is exactly the same as in $AdS_4$. The first sub-leading term, $\propto \coe_0 \ln \lp \frac{2R}{\varepsilon} \rp$, depends on $c_{\pm}$ , $A^{(1)}_{\pm}$, and $X_{\pm}^{(1)}$ via eq.~\eqref{eq:Cd3}, which ultimately depend only on the expansion coefficients in eq.~\eqref{eq:expansion}, and on $\mathcal{C}$, whose value depends on the entire geometry, not just the asymptotic regions or the part near the defect. We have thus shown that the coefficient $C^{(d=3)}_{log}$ in eq.~\eqref{app:sphericalEE} is independent of the choice of $u^c(y^a)$, and depends on the entire geometry, but depends on the $x$-cutoffs $\chi_{\pm}\left(\frac{\varepsilon}{u},y^a\right)$ only up to order $\left(\frac{\varepsilon}{u}\right)$, as advertised. On the other hand, the constant term in eq.~\eqref{E:EEd3app} depends on the $Y_l$ for all $l$, and explicitly depends on the choice of $u^c(y^a)$. In other words, $C^{(d=3)}_0$ in eq.~\eqref{app:sphericalEE} depends on $u^c(y^a)$, as advertised. ]]>