We use entanglement entropy to define a central charge associated to
a two-dimensional defect or boundary in a conformal field theory (CFT).
We present holographic calculations of this central charge for several
maximally supersymmetric CFTs dual to eleven-dimensional supergravity in
Anti-de Sitter space, namely the M5-brane theory with a Wilson surface
defect and three-dimensional CFTs related to the M2-brane theory with a
boundary. Our results for the central charge depend on a partition of

Article funded by SCOAP3

4$ is surprising, since power counting rules out any local Lagrangian. Two additional challenges arise in writing a local Lagrangian for the M5-brane theory. First, imposing self-duality at the level of the Lagrangian in any dimension is difficult. Second, generalizing $\mathfrak{u}(M)$ gauge transformations to a higher-form gauge field, such as the two-form $A_2$, is challenging. Remarkably, for the Abelian case, $M=1$, a classical action for the M5-brane worldvolume fields that has $\mathfrak{u}(1)$ gauge invariance, $\N=(2,0)$ superconformal symmetry, and locality, as well as 6d self-duality of $F_3$ enforced by auxiliary scalar field (acting as a Lagrange multiplier), is known~\cite{Pasti-ml-1997gx,Bandos-ml-1997ui}. However, whether a classical action can be written for $M>1$ remains unknown.
Despite the challenges, the M5-brane theory is extremely important to study, not only because M5-branes are key ingredients of M-theory, but also because the M5-brane worldvolume theory holds a uniquely privileged place among QFTs. In 6d, $\N=(2,0)$ is the maximal amount of SUSY possible, and 6d is the maximal dimension in which superconformal symmetry is possible~\cite{Nahm-ml-1977tg}. A 6d $\N=(2,0)$ SUSY CFT cannot be reached as the infra-red (IR) fixed point of a local renormalization group (RG) flow from a free UV fixed point. The M5-brane theory has no known dimensionless parameters besides $M$ that could be tuned to allow a perturbative expansion. In short, the dimensionality, symmetries, and $M$ determine the M5-brane theory completely. The M5-brane theory is thus an isolated, intrinsically strongly-interacting fixed point. Via compactification the M5-brane theory can describe many lower-dimensional SUSY QFTs and dualities among them, and could potentially be a ``master theory'' containing information about all lower-dimensional QFTs.
In this paper, we will use holography to study the M5-brane theory with a 2d conformal defect as well as 3d CFTs with boundaries (BCFTs) related to the ABJM BCFT, using the following 1/4-BPS intersection of M-branes:
\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{ |c | c || c | c | c | c | c | c | c | c | c | c | c|}
\hline
& & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_8$ & $x_9$ & $x_{10}$ \\
\hline
$N$ & M2 & X & X & X & & & & & & & & \\
$M$ & M5 & X & X & & X & X & X & X & & & & \\ \hline
$M'$ & M5$'$ & X & X & & & & & & X & X & X & X \\ \hline
\end{tabular}
\end{center}
Schematically, this table represents a stack of $N$ coincident M2-branes, a stack of $M$ coincident M5-branes, and another stack of $M'$ coincident M5-branes that we label M5$'$ to distinguish them from the first stack. Most importantly, the M2-branes end on the M5- and M5$'$-branes at $x_2=0$.
Consider first setting $M' =0$, that is, the intersection of $N$ M2-branes ending on $M$ M5-branes, with no M5$'$-branes. Recall that the endpoint of a semi-infinite string ending on a D-brane gives rise to an infinitely massive (s)quark in the D-brane's worldvolume. Integrating out the heavy (s)quark yields a Wilson line, i.e.\ the holonomy of the D-brane worldvolume gauge field along the (s)quark worldline~\cite{Maldacena-ml-1998im,Rey-ml-1998ik}. Similarly, the end of a semi-infinite M2-brane ending on an M5-brane produces a self-dual string with infinite tension, which can be integrated out to yield a ``Wilson surface'' operator~\cite{Ganor-ml-1996nf}. In the Abelian case, $M=1$, a precise form is known for the 1/2-BPS Wilson surface operator, $\exp\left( i\int A_2 + \ldots \right)$, with the integral over the surface spanned by the self-dual string and the ellipsis denoting terms required by SUSY, involving the M5-branes' worldvolume scalars, see for example ref.~\cite{Bullimore-ml-2014upa}. In the non-Abelian case, $M>1$, a precise form is unknown but is believed to be schematically $\tr_{\cal{R}} \exp \left( i\int A_2 + \ldots\right)$, with the trace in representation $\cal{R}$ of $\mathfrak{su}(M)$. The representation $\cal{R}$ describes how the $N$ M2-branes are partitioned among the $M$ M5-branes, as we discuss in detail below.
\looseness=-1 We will consider only flat Wilson surfaces, i.e.\ Wilson surfaces extended along the $\mathbb{R}^{1,1}$ spanned by $x_0$ and $x_1$. From the M5-brane theory's perspective the Wilson surface is a 1/2-BPS 2d superconformal soliton. The M5-brane theory's bosonic symmetry is $\SO(6,2) \times \SO(5)_R$, where $\SO(6,2)$ is the 6d conformal group. The table above shows that the Wilson surface preserves an $\SO(2,2) \times \SO(4)_R \times \SO(4)_R$ subgroup, where the global 2d conformal group $\SO(2,2) \subset \SO(6,2)$ leaves invariant the Wilson surface, and the first $\SO(4)_R$ rotates $(x_3,x_4,x_5,x_6)$ while the second rotates $(x_7,x_8,x_9,x_{10})$. The R subscripts indicate that these act as R-symmetries on the supercharges preserved by the Wilson surface, forming a 2d ``large'' $\N=(4,4)$ SUSY (``small'' $\N=(4,4)$ has a single $\SO(4)_R$).
We will compute a central charge associated with the Wilson surface using holography. The holographic description's geometry is an ${\rm AdS}_3$ and two $S^3$'s fibered over a Riemann surface~\cite{D-ml-Hoker-ml-2008wc,DHoker-ml-2008rje,Estes-ml-2012vm,Bachas-ml-2013vza} and hence has the expected isometry $\SO(2,2)\times \SO(4)_R \times \SO(4)_R$. The holographic description represents a large $M$ limit with arbitrary $N$. The geometry is asymptotically locally ${\rm AdS}_7 \times S^4$ for all $N$, and becomes precisely ${\rm AdS}_7 \times S^4$ when $N=0$.
As is well-known, 2d large $\N=(4,4)$ super-groups actually come in a one-parameter family called $D(2,1;\gamma)\times D(2,1;\gamma)$, where $\gamma$ is the free parameter~\cite{Sevrin-ml-1988ew,Frappat-ml-1996pb}. The most general solutions of 11d SUGRA that have super-isometry $D(2,1;\gamma)\times D(2,1;\gamma)$ and locally asymptote to ${\rm AdS}_7 \times S^4$ are known~\cite{Bachas-ml-2013vza}. The bosonic subgroup of $D(2,1;\gamma)\times D(2,1;\gamma)$ is $\SO(2,2)\times \SO(4) \times \SO(4)$, and hence these solutions all involve an ${\rm AdS}_3$ and two $S^3$'s fibered over a Riemann surface~\cite{DHoker-ml-2008wvd}. The solutions describing Wilson surfaces in the M5-brane theory at large $M$ and arbitrary $N$ have $\gamma=-1/2$.
\sloppy
Additionally, the most general solutions of 11d SUGRA with super-isometry $D(2,1;\gamma)\times D(2,1;\gamma)$ are known that locally asymptote to ``half'' of ${\rm AdS}_4 \times S^7$, in a sense we explain below~\cite{Bachas-ml-2013vza}. These solutions are holographically dual to 3d maximally SUSY BCFTs. The exact BCFTs are not yet known, though some properties are clear from the 11d SUGRA solutions. In particular, in these solutions generically both $M$ and $M'$ are non-zero. The solutions thus describe M2-branes ending on M5- and M5$'$-branes, and hence the BCFTs must be cousins of the maximally SUSY ABJM BCFT.\footnote{Solutions of 11d SUGRA that are candidates for the holographic dual of the maximally SUSY ABJM BCFT appear in ref.~\cite{Bachas-ml-2013vza}, but have potentially dangerous singularities.} Presumably these BCFTs are obtained from the ABJM BCFT by couplings to 2d SUSY multiplets at the boundary, and/or by sources or expectation values of scalar operators away from the boundary, similar to the superconformal interfaces between ABJM theories in refs.~\cite{DHoker-ml-2009lky,Bobev-ml-2013yra}. We will henceforth refer to these theories as ``cousins of the ABJM BCFT.''
\fussy
For $k=1$ or $2$, ABJM's bosonic symmetry is enhanced to $\SO(3,2)\times \SO(8)_R$, where $\SO(3,2)$ is the 3d conformal group, and in the intersection above the $\SO(8)_R$ acts on $(x_3,\ldots,x_{10})$. Maximally superconformal boundary conditions at $x_2=0$ preserve the $\SO(2,2) \subset \SO(3,2)$ that leaves the boundary invariant, and $\SO(4)_R \times \SO(4)_R \subset \SO(8)_R$ R-symmetry. The maximally SUSY ABJM BCFT's bosonic symmetry is thus $\SO(2,2) \times \SO(4)_R \times \SO(4)_R$, and the super-group of the theory is $D(2,1;\gamma)\times D(2,1;\gamma)$ with $\gamma=1$. The cousins of the maximally SUSY ABJM BCFT that we will study have arbitrary $\gamma<0$, and their holographic duals again involve an ${\rm AdS}_3$ and two $S^3$'s fibered over a Riemann surface~\cite{Bachas-ml-2013vza}. These solutions correspond to a limit with large $N$ and a large number of M5- and M5$'$-branes. However the values of $M$ and $M'$ are in fact undetermined, intuitively because the M2-branes do not ``know'' how many M5- and M5$'$-branes have zero M2-branes ending on them, and so cannot know the total numbers of M5- and M5$'$-branes. Additionally, sending both $M$ and $M'$ to zero produces a singular solution, presumably because removing a BCFT's boundary is a singular operation.
Using these 11d SUGRA solutions, we will compute a central charge associated with the Wilson surface or 3d BCFT boundary. Crucially, the Wilson surface or 2d boundary is not a 2d CFT, but a 2d defect in, or boundary of, an ambient CFT that has $d>2$, which implies in general that the full Virasoro symmetry is not present. To be more precise, the Wilson surface or boundary breaks the ambient $\SO(6,2)$ or $\SO(3,2)$ conformal symmetry down to $\SO(2,2)$, i.e.\ the global part of the Virasoro symmetry, in which the usual central charge does not appear. We must therefore define a central charge some other way.
We will use entanglement entropy (EE)\@. Following refs.~\cite{Jensen-ml-2013lxa,Estes-ml-2014hka,Gentle-ml-2015jma}, we will compute holographically the EE of a spherical region centered on the 2d defect or of a semi-circle centered on the 2d boundary. This EE has UV divergences, as expected, so we introduce a UV cutoff and subtract the EE of the ambient CFT\@. What remains is a term logarithmic in the cutoff, a constant term, and terms that vanish as the cutoff is removed. In analogy with the EE of a single interval in a 2d CFT~\cite{Holzhey-ml-1994we,Calabrese-ml-2004eu}, we identify the coefficient of that logarithmic term as $1/3$ times the central charge. In short, we compute the \emph{change} in the coefficient of the EE's logarithmic term due to the 2d defect or boundary, and use the result to define a central charge. We will denote the resulting Wilson surface or 2d boundary central charge as $b_{6d}$ or $b_{3d}$, respectively. Again in analogy with 2d CFT, we will interpret these as counting massless degrees of freedom supported on the Wilson surface or 2d boundary.\footnote{In a 3d BCFT the boundary central charge defined from EE is proportional to a central charge that appears in the trace anomaly~\cite{Fursaev-ml-2013mxa,Fursaev-ml-2016inw} and obeys a c-theorem, strictly decreasing in a boundary RG flow~\cite{Jensen-ml-2015swa}. Our $b_{3d}$ thus counts massless degrees of freedom at the 2d boundary. However, a similar interpretation of $b_{6d}$ may not always be justified. In particular, examples of defects in higher-d CFTs are known in which central charges defined from EE do \emph{not} decrease along defect RG flows~\cite{Kumar-ml-2016jxy,Kumar-ml-2017vjv,Kobayashi-ml-2018lil}. These examples include certain RG flows on a Wilson surface in a totally symmetric representation ${\cal R}$~\cite{Rodgers-ml-2018mvq}.} Whether and how these central charges defined from EE are related to other potential definitions, for example via the thermodynamic entropy, stress tensor correlators, and so on, we leave as an important open question.
Our main result, for $b_{6d}$, takes a remarkably simple form,
\begin{equation}
\label{eq:b6dcompact}
b_{6d} = \frac{3}{5} \left [ 16 \left(\lambda,\varrho\right)-\left(\lambda,\lambda\right)\right],
\end{equation}
where $\lambda$ is the highest weight vector of the Wilson surface's representation $\cal{R}$, $\varrho$ is the $\mathfrak{su}(M)$ Weyl vector, and $(\cdot\,,\,\cdot)$ is defined with the Killing form on the weight space. The inner product $(\cdot,\,\cdot)$ is invariant under the action of the Weyl group, hence so is $b_{6d}$. In particular, $b_{6d}$ is invariant under complex conjugation of a representation, $\cal{R} \to \overline{\cal{R}}$, which acts as a Weyl reflection. Eq.~\eqref{eq:b6dcompact} is also reminiscent of the results for the M5-brane theory's own central charges, $a$ and $c$, which can similarly be written in terms of purely group theoretic data~\cite{Beem-ml-2014kka,Cordova-ml-2015vwa}. However, eq.~\eqref{eq:b6dcompact} obscures $b_{6d}$'s dependence on $N$ and $M$. In section~\ref{sec:cc} we present $b_{6d}$ for some specific $\cal{R}$, such as the rank $N$ totally symmetric and anti-symmetric representations, to explore the dependence on $N$ and $M$. Our results for $b_{3d}$ cannot be written so neatly as eq.~\eqref{eq:b6dcompact}, largely because we do not know the total number of M5- and M5$'$-branes and thus do not know $\mathfrak{su}(M)$ or $\mathfrak{su}(M')$.
However, a key observation about our results for both $b_{6d}$ and $b_{3d}$ is that neither naturally scales as $N^{3/2}$, characteristic of M2-branes at large $N$~\cite{Klebanov-ml-1996un,Drukker-ml-2010nc}, or as $M^3$, characteristic of M5-branes at large $M$ ($a \propto c \propto M^3$ at large $M$)~\cite{Freed-ml-1998tg,Harvey-ml-1998bx,Henningson-ml-1998gx}. In other words, in general for the M-brane intersections we consider, the number of massless degrees of freedom on the Wilson surface or 2d boundary does not scale, in any obvious way, with the total number of degrees of freedom of the M2- or M5-brane theory.
This paper is organized as follows. In section~\ref{sec:sugra} we review the 11d SUGRA solutions describing M5-branes with Wilson surfaces or cousins of the ABJM BCFT with $\gamma<0$. In section~\ref{sec:hc} we derive an integral for the holographic EE, and then evaluate the integral to extract $b_{6d}$ and $b_{3d}$. In section~\ref{sec:cc} we summarize our results, including for specific $\cal{R}$, and compare to existing results. (Readers interested only in our results can skip directly to section~\ref{sec:cc}.) We conclude in sec~\ref{sec:discussion} with discussion and suggestions for future research.
\sloppy
The companion paper ref.~\cite{Rodgers-ml-2018mvq} reproduces our results for $b_{6d}$ for the fundamental, totally anti-symmetric, and totally symmetric representations, using probe branes in \mbox{${\rm AdS}_7 \times S^4$}, namely a probe M2-brane when $N\ll M$ or M5-brane when $N \ll M^2$. Ref.~\cite{Rodgers-ml-2018mvq} also uses the probe M5-branes to explore RG flows on Wilson surfaces.
\fussy
\paragraph{Note added.} After this paper appeared on the arxiv, ref.~\cite{Jensen-ml-2018rxu} clarified how our central charge $b$ obtained from the EE of a (hemi-)sphere is related to central charges in the defect's contribution to the Weyl anomaly. This clarifies the relationship between unitarity and positivity of $b$ discussed in section~\ref{sec:wilson-surface}, among other things.
For a CFT in $\mathbb{R}^{1,d-1}$ with $d\geq 3$ and a 2d conformal defect, the trace of the stress tensor splits into two terms, $\left< T_{~\mu}^{\mu} \right> = \left< T_{~\mu}^{\mu}\right>_{\textrm{bulk}} + \delta^{d-2}(x) \left

0$ and $|G|<1$ everywhere on the interior of the upper half $w$ plane (all $\textrm{Im}\left(w\right)>0$) and that $h=0$ and $G = \pm i$ on the boundary of the upper half $w$ plane ($\textrm{Im}\left(w\right)=0$). All the solutions that we consider below will obey these conditions. The four-form $F_4 = dC_3$ of these solutions appears in ref.~\cite{Bachas-ml-2013vza}. We will not present the solution for $F_4$ explicitly, but in section~\ref{sec:fluxes} we will discuss the M2- and M5-brane charges determined by the solution for $F_4$. The invariance of $D(2,1;\gamma)\times D(2,1;\gamma)$ under $\gamma \to 1/\gamma$ and swapping of $\SO(3)$ sub-groups appears in these 11d SUGRA solutions as invariance under $\gamma \to 1/\gamma$ and exchange of $W_+ \leftrightarrow W_-$. As clear from eq.~\eqref{eq:metfuncs}, that leaves $f_1$ and $\Omega$ invariant but trades $f_2 \leftrightarrow f_3$, thus effectively interchanging the geometry's two $S^3$ factors. \boldmath ]]>

0$ then describe Wilson surfaces in the M5-brane theory. \boldmath ]]>

N_2 > N_3 \ldots$. In other words, the $N_a$ are $n$ \emph{distinct non-zero} integers, each with degeneracy $M_a$. The final entries in $\rho$ are $N_{n+1}=0$, with degeneracy $M_{n+1}$, representing the number of M5-branes with no M2-branes ending on them. We thus specify $\rho$ by specifying the set of integers $\{N_a\}$ and the set of their degeneracies $\{M_a\}$. The partition $\rho$ has a total of $2n+1$ parameters, the $n$ distinct integers in $\{N_a\}$ plus the $n+1$ distinct integers in $\{M_a\}$. In this parametrization the total numbers $N$ of M2-branes and $M$ of M5-branes are \begin{equation} \label{eq:totnm} N = \sum_{a=1}^{n+1} M_a N_a, \qquad M = \sum_{a=1}^{n+1} M_a. \end{equation} As in the first parametrization, when writing $\rho$ we will omit all zero entries, that is, we omit the entries $N_{n+1}$. As examples, figure~\ref{fig:part2} shows the sets $\{N_a\}$ and $\{M_a\}$ for $\rho=\{2,1\}$ with $M=4$ (figure~\ref{fig:part2} left) and $\rho=\{3,3\}$ with $M=3$ (figure~\ref{fig:part2} right). \begin{figure} \centering \includegraphics[width=.8\textwidth]{figs/fig2} ]]>

0$. In ${\rm AdS}_3$ eq.~\eqref{eq:rt} reproduces known results for 2d CFTs~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef,Nishioka-ml-2009un,Holzhey-ml-1994we,Calabrese-ml-2004eu}. For example, when the entangling surface consists of two points a distance $\ell$ apart, the minimal surface in ${\rm AdS}_3$ is a semi-circle at fixed $t$ with diameter $\ell$ centered on the ${\rm AdS}_3$ boundary. In that case eq.~\eqref{eq:rt} gives \begin{equation} \label{eq:ee2d} \see^{2d} = \frac{c}{3} \ln \left(\frac{\ell}{\varepsilon}\right) + \mathcal{O}\left(\varepsilon^0\right), \end{equation} with CFT central charge $c$. Henceforth, we will use a superscript to distinguish $\see$ in different dimensions, such as the superscript $2d$ on $\see^{2d}$. Crucially, re-scaling the cutoff $\varepsilon$ changes the $\mathcal{O}\left(\varepsilon^0\right)$ terms, while the coefficient of $\ln \left(\frac{\ell}{\varepsilon}\right)$, namely $c/3$, is cutoff-independent and hence physical. In ${\rm AdS}_4$ eq.~\eqref{eq:rt} produces the form expected for a 3d CFT~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef,Nishioka-ml-2009un}. For example when the entangling surface is a circle of radius $\ell$, the minimal surface in ${\rm AdS}_4$ is a hemisphere at fixed $t$ with radius $\ell$ centered on the ${\rm AdS}_4$ boundary. In that case eq.~\eqref{eq:rt} gives \begin{equation} \label{eq:ee3d} \see^{3d} = c_1 \, \frac{\ell}{\varepsilon} + c_0 + \mathcal{O}\left(\varepsilon\right), \end{equation} where $c_1$ and $c_0$ are constants. Re-scalings of the cutoff change $c_1$ but not $c_0$, so only the latter is physical. Indeed $c_0$ is proportional to minus the logarithm of the Euclidean CFT partition function on $S^3$~\cite{Casini-ml-2011kv}. The ${\rm AdS}_4 \times S^7$ solution of 11d SUGRA gives $c_0 \propto - N^{3/2}$~\cite{Klebanov-ml-1996un,Drukker-ml-2010nc}. In ${\rm AdS}_7$ eq.~\eqref{eq:rt} produces the form expected for a 6d CFT~\cite{Ryu-ml-2006bv,Ryu-ml-2006ef,Nishioka-ml-2009un}. For example, when the entangling surface is an $S^4$ of radius $\ell$, the minimal surface in ${\rm AdS}_7$ is a five-dimensional hemisphere at fixed $t$ with radius $\ell$ centered on the ${\rm AdS}_7$ boundary. In that case eq.~\eqref{eq:rt} gives \begin{equation} \label{eq:ee6d} \see^{6d} = c_4 \, \frac{\ell^4}{\varepsilon^4} + c_2 \, \frac{\ell^2}{\varepsilon^2} + c_L \ln \left(\frac{\ell}{\varepsilon}\right) + \mathcal{O}\left(\varepsilon^0\right), \end{equation} where $c_4$, $c_2$ and $c_L$ are constants. Only $c_L$ is invariant under re-scalings of the cutoff, hence only $c_L$ is physical. Indeed, $c_L\propto - a$, where $a$ is a central charge of the 6d CFT~\cite{Casini-ml-2011kv}. The ${\rm AdS}_7 \times S^4$ solution of 11d SUGRA gives $c_L \propto - a \propto -M^3$~~\cite{Freed-ml-1998tg,Harvey-ml-1998bx,Henningson-ml-1998gx}. Following refs.~\cite{Jensen-ml-2013lxa,Estes-ml-2014hka,Gentle-ml-2015jma}, in our (B)CFTs we choose (hemi-)spherical entangling surfaces centered on the 2d defect or boundary, as follows. For the 11d SUGRA solutions reviewed in section~\ref{subsec:ads7metric}, dual to the M5-brane theory with a Wilson surface, our entangling surface will be an $S^4$ of radius $\ell$ centered on the Wilson surface, as shown in figure~\ref{fig:eesurf}(a). For the 11d SUGRA solutions reviewed in section~\ref{subsec:ads4metric}, dual to cousins of the ABJM BCFT, our entangling surface will be a semi-circle centered on the CFT's boundary, as shown in figure~\ref{fig:eesurf}(b). \begin{figure} \centering \includegraphics[width=.4\textwidth]{figs/eesurf} ]]>

0$ to $u=\ell$, \begin{equation} \label{eq:uint} \int_{\varepsilon_u}^{\ell} du \frac{\ell}{u \sqrt{\ell^2 - u^2}} = \ln\left(\frac{2\ell}{\varepsilon_u}\right)+{\cal{O}}\left(\varepsilon_u^2\right), \end{equation} so that the integral for $\see$ in eq.~\eqref{eq:integral} becomes \begin{equation} \label{eq:integral2} \see = \frac{2 \, \textrm{Vol}\left(S^3\right)^2}{4 \gn} \int dw \, d\overline{w} \left( \frac{\Omega^2}{f_1^2} \, f_1^3 f_2^3 f_3^3 \right) \left[\ln\left(\frac{2\ell}{\varepsilon_u}\right)+{\cal{O}}\left(\varepsilon_u^2\right)\right]. \end{equation} We define ${\cal{I}}$ as the remaining integral, \begin{equation}\label{eq:Iastintegral} {\cal{I}}\equiv \frac{2 \, \textrm{Vol}\left(S^3\right)^2}{4 \gn} \int dw \, d\overline{w} \left( \frac{\Omega^2}{f_1^2} \, f_1^3 f_2^3 f_3^3 \right). \end{equation} To write ${\cal I}$ explicitly we plug in $\textrm{Vol}\left(S^3\right)=2\pi^2$, \begin{align} \label{eq:metproducts} &\frac{\Omega^2}{f_1^2} = \frac{1}{2} \frac{c_1^2}{c_2 c_3} \frac{|\partial_w h|^2}{h^2} (G \overline{G} - 1) \,, & &f_1 f_2 f_3 = \pm \frac{h}{c_1 c_2 c_3}\,, \end{align} and in the second equation of eq.~\eqref{eq:metproducts} we choose the sign to guarantee a positive integrand, given that $|G|<1$ as mentioned below eq.~\eqref{eq:wpm}. Eq.~\eqref{eq:Iastintegral} then becomes \begin{equation} \label{} {\cal I} = \frac{2 (2 \pi^2)^2}{4 \gn} \frac{1}{c_1 c_2^4 c_3^4}\frac{1}{2}\int dw \, d\overline{w} \, |\partial_w h|^2 \, h \left(1-G \overline{G}\right). \end{equation} Using $h = - i \left(w-\overline{w}\right)$ from eqs.~\eqref{eq:ads7metric} and~\eqref{eq:ads4metric}, we have $|\partial_w h|^2 = 1$. Introducing polar coordinates $w \equiv r e^{i \theta}$, so that $dw\,d\overline{w} = 2\,d\theta \,dr \, r$ and $h = -i (w - \overline{w})=2r \sin \theta$, we find \begin{align} \label{eq:integral3} {\cal I} &= \frac{2 (2 \pi^2)^2}{4 \gn} \frac{2}{c_1 c_2^4 c_3^4} \int_0^{\pi} d \theta \sin\theta \int_0^{r_c} dr \, r^2 (1 - G \overline{G}), \end{align} where we have made the endpoints of integration explicit, including a large-$r$ cutoff, $r_c$. The prescription of refs.~\cite{Estes-ml-2014hka,Gentle-ml-2015jma} is to choose $r_c$ in a way that preserves the subgroup of the Poincar\'e group that leaves the 2d defect or boundary invariant. Crucially, a constant $r_c$ does not preserve those symmetries, rather $r_c$ must be a more complicated function whose form depends on the details of the 11d SUGRA solution. In the next two subsections we will compute $r_c$ and then extract ${\cal I}$ in eq.~\eqref{eq:integral3}. As mentioned above, in principle we would like to extract $b_{6d}$ or $b_{3d}$ from a term in $\see$ that is $\propto \ln(\ell/\varepsilon)$, with FG cutoff $\varepsilon$. However, how do we do so using the cutoffs $\varepsilon_u$ and $r_c$? The result for the integral ${\cal I}$ will be a sum of terms, including terms with positive powers of $r_c$, a term independent of $r_c$, and terms with negative powers of $r_c$. In eq.~\eqref{eq:integral2} these all multiply $\ln(\ell/\varepsilon_u)$. The terms with positive powers of $r_c$, which are clearly cutoff-dependent and hence unphysical, will turn out to be identical to those of the undeformed ${\rm AdS}_7 \times S^4$ or (half of) ${\rm AdS}_4 \times S^7$ solutions, and so will cancel in the background subtraction $\see - \see^{6d}$ or $\see - \frac{1}{2} \see^{3d}$. The terms with negative powers of $r_c$ clearly vanish as $r_c \to \infty$ and so can be safely ignored. We will thus be left with the term independent of $r_c$, or more precisely what remains of that term after the background subtraction, which still multiplies $\ln(\ell/\varepsilon_u)$. Applying $3 \ell \frac{d}{d\ell}$, as in eqs.~\eqref{eq:b6ddef} and~\eqref{eq:b3ddef}, then extracts this coefficient of $\ln(\ell/\varepsilon_u)$, which is thus our $b_{6d}$ or $b_{3d}$. In short, we will apply eqs.~\eqref{eq:b6ddef} and~\eqref{eq:b3ddef} as advertised, though the form of divergences will look very different in terms of $\varepsilon_u$ and $r_c$ as compared to the usual FG cutoff $\varepsilon$. For a more detailed comparison of these cutoffs, see ref.~\cite{Gentle-ml-2015jma}. \boldmath ]]>

0$, which translates to a cutoff $r_c$ that depends on $\varepsilon_v$ and $\theta$. Explicitly, in eq.~\eqref{eq:AdS7FGtrans} we set $v = \varepsilon_v$ and then invert to find $r_c(\varepsilon_v,\theta)$ in a small $\varepsilon_v$ expansion,
{\rdmathspace
\begin{align}
\label{eq:AdS7cutoff}
r_c(\varepsilon_v,\theta) &= \frac{2 (1+ \gamma)^2 m_1}{-\gamma} \frac{1}{\varepsilon_v^2} + \frac{(1+ 2 \gamma)m_1}{-8 \gamma} + \frac{m_2}{2m_1} \cos\theta - \frac{(1+2 \gamma) m_1}{-24 \gamma} \cos(2\theta) \nonumber\\
& + \left\{ \frac{89 m_1^4+212 \gamma(1 + \gamma)m_1^4 + 729 \gamma^2 m_2^2 - 576 \gamma^2 m_1 m_3}{9216 \gamma(1+\gamma)^2 m_1^3}
+ \frac{(1+2\gamma)m_2}{24 (1 + \gamma^2)m_1}\cos\theta \sin^2\theta \right . \nonumber\\&
+ \left . \frac{3m_1^4 + 8 \gamma m_1^4 + 8 \gamma^2 m_1^4 - 36 \gamma^2 m_2^2 + 80 \gamma^2 m_1 m_3}{768 \gamma(1+\gamma)^2 m_1^3} \cos(2\theta) + \frac{(1+2\gamma)^2 m_1}{3072 \gamma(1+\gamma)^2} \cos(4\theta) \right\} \varepsilon_v^2 \nonumber \\ & + {\cal O}(\varepsilon_v^4),
\end{align}}\relax
where we have kept an additional order as compared to eq.~\eqref{eq:AdS7FGtrans}, which will be necessary to extract $b_{6d}$ from $\see$. When $\gamma = - 1/2$, eq.~\eqref{eq:AdS7cutoff} simplifies considerably,
{\rdmathspace
\begin{align}
\label{eq:rcads7m12}
r_c(\varepsilon_v,\theta) &= \frac{m_1}{\varepsilon_v^2} + \frac{m_2}{2m_1} \cos(2\theta) +\left( \frac{4 m_1 m_3 - m_1^4 - 5 m_2^2}{32 m_1^3} + \frac{20 m_1 m_3 - 9 m_2^2 + m_1^4}{96 m_1^3} \cos(2\theta) \right) \varepsilon_v^2 \nonumber \\ & \quad + {\cal O}(\varepsilon_v^4).
\end{align}}\relax
Plugging the expression for $G$ in eq.~\eqref{eq:ads7metric} into eq.~\eqref{eq:integral3} gives
{\rdmathspace
\begin{align}
\label{eq:ads7int}
{\cal I} &= \frac{2 (2 \pi^2)^2}{4 \gn} \frac{2}{c_1 c_2^4 c_3^4} \sum_{j=1}^{2n+2} (-1)^j \int_0^{\pi} d \theta \sin\theta
\int_0^{r_c(\varepsilon_v,\theta)} dr \, r^2 \frac{2(r \cos(\theta) - \xi_j)}{\sqrt{r^2 + \xi_j^2 - 2 r \xi_j \cos(\theta)}} \nonumber\\
&\quad + \frac{2 (2 \pi^2)^2}{4 \gn} \frac{2}{c_1 c_2^4 c_3^4} \sum_{j,k=1}^{2n+2} (-1)^{j+k} \int_0^{\pi} d \theta \sin\theta
\int_0^{r_c(\varepsilon_v,\theta)} dr \, r^2 \frac{r e^{i \theta} - \xi_j}{|r e^{i \theta} - \xi_j|}\frac{r e^{-i \theta} - \xi_k}{|r e^{i \theta} - \xi_k|},
\end{align}}\relax
where $r_c(\varepsilon_v,\theta)$ is the cutoff in eq.~\eqref{eq:AdS7cutoff}. The integrals in eq.~\eqref{eq:ads7int} are performed in ref.~\cite{Gentle-ml-2015jma},\footnote{In ref.~\cite{Gentle-ml-2015jma} the first and second lines of eq.~\eqref{eq:ads7int} are denoted $J_1$ and $J_2$, respectively.} and are very similar to those in the asymptotically locally ${\rm AdS}_4 \times S^7$ case performed in the appendix, so here we only quote the result:
\begin{align}
\label{eq:ads7inttemp}
{\cal I} &= -\frac{8(2 \pi^2)^2 L_{S^4}^9}{4 \gn \gamma (1 + \gamma)}
\Bigg(\frac{8 (1+\gamma)^4}{3\gamma^2}\frac{1}{\varepsilon_v^4}+\frac{2(1+\gamma)^2(3+16\gamma)}{15\gamma^2}\frac{1}{\varepsilon_v^2} -\frac{85-8\gamma(52+115\gamma)}{5040\gamma^2}\nonumber\\
&\quad +\frac{m_2^2}{10m_1^4}-\frac{2m_3}{15m_1^3} -\frac{1}{3m_1^3}\sum_{\underset{j

8M$, eq.~\eqref{eq:b6dsymm} clearly shows that $b_{6d}<0$. In a standard 2d CFT, unitarity and normalizability of the ground state require the central charge to be non-negative. However, whether unitarity imposes a lower bound on $b_{6d}$ is currently unknown. In similar cases, such as 3d BCFTs, unitarity allows negative values of the boundary central charge. For example, in the free massless scalar 3d BCFT with a Dirichlet boundary condition --- a perfectly unitary theory --- the boundary central charge is negative~\cite{Nozaki-ml-2012qd,Jensen-ml-2015swa,Fursaev-ml-2016inw}. More generally, for unitary 3d BCFTs ref.~\cite{Herzog-ml-2017kkj} conjectured a lower bound on the boundary central charge that was negative. The fact that Wilson surfaces in the M5-brane theory and their holographic duals have no other known violations of unitarity leads us to suspect strongly that $b_{6d}<0$ does not necessarily signal unitarity violation. ]]>

1$, BH move onto the Coulomb branch, separating some M5-branes out of the stack of $M$ coincident M5-branes. In that case, the M5-branes' effective action includes the Ganor-Intriligator-Motl term~\cite{Ganor-ml-1998ve,Intriligator-ml-2000eq}, $\alpha \int dA_2 \wedge \phi(A_T)$, where $\alpha$ is known. Starting from a stack of M5-branes with ADE Lie algebra $\mathfrak{g}$ and breaking to a subgroup with ADE Lie algebra $\mathfrak{h}$ times $\mathfrak{u}(1)$, $\alpha = \frac{1}{4}(\textrm{dim}\,\mathfrak{g} - \textrm{dim}\,\mathfrak{h}-1)$. The pullback of the Ganor-Intriligator-Motl term to the 2d self-dual string worldsheet produces a term with an $\SO(4)_{R_N}$ anomaly that must cancel the 2d contribution, which allows BH to identify \begin{equation} \bbh = \frac{1}{2} N \alpha. \end{equation} \looseness=-1 If we separate a single M5-brane from the stack then $\mathfrak{g} =\mathfrak{su}(M)$ and $\mathfrak{h}=\mathfrak{su}(M-1)$ (as mentioned in section~\ref{sec:intro}, we ignore the overall center-of-mass $\mathfrak{u}(1)$), so $\alpha = \frac{1}{2} \left(M-1\right)$ and hence \begin{equation} \label{eq:bbhone} \bbh = \frac{1}{4} N \left(M-1\right) \qquad \textrm{for} \quad \mathfrak{su}(M) \to \mathfrak{su}(M-1) \times \mathfrak{u}(1). \end{equation} If we separate all $M$ M5-branes from each other such that $\mathfrak{g} =\mathfrak{su}(M)$ and $\mathfrak{h}=\mathfrak{u}(1)^{M-1}$, then $\alpha = \frac{1}{4} \left(M^2 - M - 1\right)$ and hence \begin{equation} \label{eq:bbhmax} \bbh = \frac{1}{8} N \left(M^2-M-1\right) \qquad \textrm{for} \quad \mathfrak{su}(M) \to \mathfrak{u}(1)^{M-1}. \end{equation} If we compare $\bbh$ in eqs.~\eqref{eq:bbhone} and~\eqref{eq:bbhmax} to $b_{6d}$ of the totally symmetric or anti-symmetric representations in eq.~\eqref{eq:b6dsymm} or~\eqref{eq:b6dantisymm}, respectively, then the only obvious similarities are terms scaling as $MN$ and $N$, with different numerical coefficients. However, we can identify at least four reasons why $b_{6d}$ and $\bbh$ need not agree. First, whether $b_{6d}$ and $\bbh$ are the same quantity is unclear. In a 2d CFT the chiral R-symmetry anomaly coefficient is proportional to the central charge $c$~\cite{Benini-ml-2012cz}, and BH assume the same remains true for a 2d defect in a higher-d CFT\@. However, that has not been demonstrated, and moreover, whether and how either quantity is related to the defect's EE is unclear. Second, we calculated $b_{6d}$ for any representation ${\cal R}$. However, $\bbh$ seems to involve no data about a representation. Which representation(s) are appropriate in comparing $b_{6d}$ and $\bbh$ (if any) is unclear. Third, our $b_{6d}$ was computed at the conformal point where all M5-branes are coincident, whereas $\bbh$ was computed on the Coulomb branch. Fourth, we computed $b_{6d}$ in the SUGRA limit of large $M$, whereas the calculation of $\bbh$ is in principle valid for any $M$. At best we might expect agreement between $b_{6d}$ and $\bbh$'s large-$M$ limit, and indeed, both our results for $b_{6d}$ in eqs.~\eqref{eq:b6dsymm} and~\eqref{eq:b6dantisymm} and the results for $\bbh$ in eqs.~\eqref{eq:bbhone} and~\eqref{eq:bbhmax} scale as $MN$ at large $M$ but with different numerical coefficients. \sloppy If we compare $b_{3d}$ of the totally ``symmetric'' or ``anti-symmetric'' partitions in eqs.~\eqref{eq:b3d-cousins-symm} or~\eqref{eq:b3d-cousins-asymm}, then the only obvious similarities are terms scaling as $N$, with different numerical coefficients. However, as before, BH compute a different quantity, which appears to involve no information about a partition. We can also identify two further reasons why $b_{3d}$ and $\bbh$ need not agree. First, as discussed in sections~\ref{sec:intro} and~\ref{sec:ads4part}, $b_{3d}$ arises from a case with both M5- and M5$'$-branes, while $\bbh$ arises from a case with no M5$'$-branes. Second, as discussed in section~\ref{sec:intro} and~\ref{subsec:ads4metric}, $b_{3d}$ arises in a case with super-group $D(2,1;\gamma)\times D(2,1;\gamma)$ with $\gamma\in(-1,0)$, while BH presumably have $\gamma = 1$. \fussy ]]>