11institutetext: STAG Research Centre, Physics and Astronomy, University of
Southampton,
Highfield, Southampton SO17 1BJ, United Kingdom22institutetext: Department of
Chemistry and Physics, SUNY Old Westbury, Old Westbury, NY, United
States33institutetext: INFN, sezione di Milano-Bicocca, Piazza della Scienza
3, I-20126 Milano, Italy

# From Large to Small $\mathcal{N}=(4,4)$ Superconformal Surface Defects in
Holographic 6d SCFTs

Pietro Capuozzo 2 , John Estes 3 , Brandon Robinson 1 , and Benjamin
Suzzoni<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two-dimensional (2d) $\mathcal{N}=(4,4)$ Lie superalgebras can be either
“small” or “large”, meaning their R-symmetry is either $\mathfrak{so}(4)$ or
$\mathfrak{so}(4)\oplus\mathfrak{so}(4)$, respectively. Both cases admit a
superconformal extension and fit into the one-parameter family
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$,
with parameter $\gamma\in(-\infty,\infty)$. The large algebra corresponds to
generic values of $\gamma$, while the small case corresponds to a degeneration
limit with $\gamma\to-\infty$. In 11d supergravity, we study known solutions
with superisometry algebra
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$
that are asymptotically locally AdS${}_{7}\times\mathds{S}^{4}$. These
solutions are holographically dual to the 6d maximally superconformal field
theory with 2d superconformal defects invariant under
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$.
We show that a limit of these solutions, in which $\gamma\to-\infty$,
reproduces another known class of solutions, holographically dual to small
$\mathcal{N}=(4,4)$ superconformal defects. We then use this limit to generate
new small $\mathcal{N}=(4,4)$ solutions with finite Ricci scalar, in contrast
to the known small $\mathcal{N}=(4,4)$ solutions. We then use holography to
compute the entanglement entropy of a spherical region centered on these small
$\mathcal{N}=(4,4)$ defects, which provides a linear combination of defect
Weyl anomaly coefficients that characterizes the number of defect-localized
degrees of freedom. We also comment on the generalization of our results to
include $\mathcal{N}=(0,4)$ surface defects through orbifolding.

##  1 Introduction

Six dimensional (6d) superconformal field theories (SCFTs) hold a special
place among quantum field theories (QFTs). Owing to the classification
discovered in the seminal work by Nahm Nahm:1977tg , superconformal symmetry
is only possible in six and fewer spacetime dimensions. Moreover,
$\mathcal{N}=(2,0)$ is the maximal amount of supersymmetry (SUSY) that a 6d
theory can have. Combining this amount of SUSY with conformal symmetry
constrains a 6d $\mathcal{N}=(2,0)$ SCFT to such a degree that the only
additional information that is necessary to completely determine the theory is
the choice of a gauge algebra.

The study of the 6d $\mathcal{N}=(2,0)$ theory is thus of fundamental
importance in QFT, for many reasons. For example, the 6d $\mathcal{N}=(2,0)$
SCFT determines the physics of many other QFTs in 6d, via SUSY-breaking
deformations Antoniadis:1998ki ; Antoniadis:1998ep ; Cordova:2016xhm  such as
orbifolds Blum:1997fw ; Blum:1997mm ; Brunner:1997gf ; Intriligator:1997dh .
By suitable (partial) topological twisting, the 6d theory compactified on,
e.g., a Riemann surface Gaiotto:2009we  or a 3-manifold Dimofte:2011ju  can
also determine the physics of infinite families of QFTs in $d<6$.

The 6d $\mathcal{N}=(2,0)$ SCFT is also of fundamental importance in quantum
gravity. Currently, the leading candidate for an ultra-violet (UV)-complete
theory of quantum gravity is M-theory. M-theory’s fundamental objects are
M2-branes Duff:1990xz  and M5-branes Gueven:1992hh , and the low-energy
worldvolume theory on $M$ coincident M5-branes is the 6d $\mathcal{N}=(2,0)$
SCFT with gauge algebra $A_{M-1}$ Strominger:1995ac . Understanding the 6d
$\mathcal{N}=(2,0)$ SCFT is thus essential to understanding M-theory in
general. In particular, via the Anti-de Sitter/CFT (AdS/CFT) correspondence,
the 6d $\mathcal{N}=(2,0)$ SCFT can provide a fully non-perturbative
definition of M-theory on an asymptotically 7d AdS spacetime, AdS7, times a
four-sphere, $\mathds{S}^{4}$ Witten:1996hc ; Maldacena:1997re ; Witten:1998wy
.

Strongly interacting SCFTs constructed in string- and M-theory, including the
non-Abelian 6d $\mathcal{N}=(2,0)$ SCFT, are prohibitively difficult to study,
for many reasons, of which we will mention only three. First, the
$\mathcal{N}=(1,0)$ and $\mathcal{N}=(2,0)$ SUSY multiplets include a chiral
two-form gauge field, and writing a local, gauge-invariant Lagrangian for a
non-Abelian higher-form gauge field remains a major open problem. These SCFTs
thus have no known Lagrangian descriptions. Second, in the space of
renormalization group (RG) flows, these SCFTs are isolated fixed points, and
in particular they cannot be reached as infra-red (IR) fixed points of RG
flows from free ultra-violet (UV) fixed points. Third, these SCFTs are
intrinsically strongly interacting. For example, the 6d $\mathcal{N}=(2,0)$
SCFT has no dimensionless parameter besides $M$ that can be tuned to allow a
perturbative expansion.

As a result, practically all of our direct knowledge111Indirect methods such
as dimensional reduction to 5d $\mathcal{N}\leq 2$ SUSY QFTs have also been
used to great effect to study these theories, e.g. by using the resulting
lower dimensional Lagrangian description together with supersymmetric
localization techniques Bullimore:2014upa . of interacting 6d SCFTs comes from
non-perturbative methods, such as the superconformal bootstrap Beem:2015aoa ,
F-theory Heckman:2015bfa , and especially AdS/CFT Apruzzi:2013yva , where
holographic computations of quantities like Weyl anomalies Henningson:1998gx
and entanglement entropy (EE) Hung:2011xb  are used to great effect to
characterize 6d SCFTs at large $M$.

An aspect of 6d SCFTs, and generally of QFTs in three and higher dimensions,
that requires particularly careful treatment to characterize is the spectrum
of 2d, string-like or surface, defects. In the co-dimension one case, 2d
defects in 3d QFTs arise as boundaries or interfaces, and so are more easily
studied and, thus, more familiar than their higher co-dimension realizations.
Despite being somewhat more exotic in standard treatments of QFTs, 2d defects
of co-dimension two and greater show up in a number of settings222This is by
no means an exhaustive list of the work done on 2d defects. For a recent
review of boundaries and defects in QFTs and further references on the topic,
see Andrei:2018die .: from free field theories Soderberg:2017oaa ;
Lauria:2020emq ; Giombi:2021uae ; Bianchi:2021snj  to strongly interacting and
non-Lagrangian 4d QFTs, e.g. Gukov:2006jk ; Gukov:2008sn ; Alday:2009fs , to
being fundamental objects in 6d SCFTs Ganor:1996nf  and in the study of EE and
Renyi entropies Hung:2014npa ; Lewkowycz:2014jia . As such, the last few
decades have seen tremendous advancements in characterizing Billo:2016cpy  and
constraining Jensen:2015swa ; Kobayashi:2018lil  the properties of 2d defects,
and it is vitally important in the study of QFTs, generally, to continue this
effort by finding novel constructions of surface defects and examining their
unique physics.

Of interest to us in the current work are the holographic descriptions,
afforded by AdS/CFT, of 6d SCFTs and the defects that they support. In
particular, we will primarily focus our attention on solutions to 11d
supergravity (SUGRA) that are asymptotically locally
AdS${}_{7}\times\mathds{S}^{4}$ and are contained in a one-parameter family of
solutions with superisometry given by the exceptional Lie superalgebra
$\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$ Bachas:2013vza .
Crucially, an asymptotically AdS${}_{7}\times\mathds{S}^{4}$ solution is
possible only at certain values of $\gamma$.

The most well studied case is $\gamma=-1/2$, which holographically describes
1/2-BPS Wilson surface operators in the 6d $A_{M-1}$ $\mathcal{N}=(2,0)$ SCFT
that preserve a large $\mathcal{N}=(4,4)$ 2d SUSY. Indeed, these BPS Wilson
surface operators have a long history in M-theory descriptions of 6d SCFTs
Strominger:1995ac ; Ganor:1996nf ; Howe:1997ue ; Maldacena:1998im ;
Berenstein:1998ij , and there has been a recent resurgence of interest in
these defects where holographic Gentle:2015jma ; Estes:2018tnu ;
Jensen:2018rxu  and field theoretic Chalabi:2020iie ; Drukker:2020atp ;
Drukker:2020dcz  computations have characterized these defect CFTs through
their EE and Weyl anomalies.

Recently, new solutions in 11d SUGRA have been constructed that are proposed
to be holographically dual to 2d BPS surface defects in 6d $\mathcal{N}=(1,0)$
SCFTs preserving “small” $\mathcal{N}=(4,4)$ or $\mathcal{N}=(0,4)$ 2d SUSY
Faedo:2020nol . In the sections below, we will clearly demonstrate that these
new solutions also fit into the one-parameter family of solutions in
Bachas:2013vza  in the limit where $\gamma\to-\infty$. It will turn out that
the solutions in Faedo:2020nol  are, in fact, a particular case within a
broader class of solutions that can be realized in the $\gamma\to-\infty$
limit, which we will characterize by computing their contributions to the EE
of a spherical region.

A crucial point that we emphasize in our construction of the
$\gamma\to-\infty$ solutions is that the superisometry algebra of the near-
horizon limit of a stack of M5-branes, $\mathfrak{osp}(8^{*}|4)$, does not
contain
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$
as a sub-superalgebra frappat1996dictionary ; D_Hoker_2008 ; D_Hoker_2008b .
In the supergravity solution, this is manifested by the appearance of
additional terms in the four-form flux as compared to that due to pure
M5-branes. Nevertheless, the geometry is still locally asymptotically
AdS${}_{7}\times\mathds{S}^{4}$, which is in line with the fact that the
bosonic subalgebra of
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$
is a subalgebra of $\mathfrak{osp}(8^{*}|4)$.

One upshot of our analysis that follows from the identification of the global
symmetries in the $\gamma\to-\infty$ limit is that it allows for a suitable
regulation scheme, which we will employ when we compute the defect sphere EE.
Lacking a generalized program of holographic renormalization for SUGRA
solutions dual to defects on the field theory side, we will use a background
subtraction scheme in order to remove the ambient degrees of freedom and
isolate contributions to physical quantities coming from the defect. The key
step in this background subtraction scheme is the correct identification of
the vacuum solution, and as will be made clear, the ambient theory with a
“trivial defect” is a deformed 6d $\mathcal{N}=(2,0)$ SCFT preserving the
bosonic superconformal subalgebra
${\mathfrak{so}}(2,2)\oplus{\mathfrak{so}}(4)\oplus{\mathfrak{so}}(5)\subset{\mathfrak{osp}}(8*|4)$.
Since the solutions in Faedo:2020nol  belong to the class of solutions that we
study below, in finding a vacuum solution that manifests the corresponding
isometries and carefully carrying out background subtraction, we will also
resolve a puzzle in Faedo:2020nol , where physical quantities like the “defect
central charge” were divergent.

Ultimately, we will see that that universal part of the defect sphere EE,
$S_{\rm EE}^{\tiny\rm(univ)}$, i.e. the coefficient of its log-divergent part,
is determined in terms of the highest weight vector, $\varpi$, of an $A_{M-1}$
irreducible representation determined by a Young diagram that encodes the
partition of M5-branes that specifies the defect. Explicitly, we will show
that

$\displaystyle S_{\rm EE}^{\tiny\rm(univ)}=-\frac{(\varpi,\varpi)}{5}~{},$ (1)

where $(\cdot,\cdot)$ is the scalar product on the weight space. This result
is similar to the Wilson surface sphere EE Estes:2018tnu  in that both are
expressible in terms of scalar quantities derived from representation data but
differ in that eq. (1) is negative definite333Unlike in an ordinary 2d CFT,
eq. (1) being strictly negative does not necessarily signal that the theory
may be non-unitary. Indeed, the 2d defect sphere EE is expressible as a signed
linear combination of defect Weyl anomaly coefficients Jensen:2018rxu , which
is not bounded from below. and is completely determined by the highest weight
vector.

In section 2, we begin by reviewing the 11d supergravity solutions dual to 2d
small $\mathcal{N}=(4,4)$ defects in 6d $\mathcal{N}=(1,0)$ SCFTs found in
Faedo:2020nol . In section 3 we briefly review the
$\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$-invariant solutions
to 11d supergravity found in Bachas:2013vza , and we show that by orbifolding
the solutions in the $\gamma\to-\infty$ limit, we can recover the solutions in
Faedo:2020nol . We then use the $\gamma\to-\infty$ limit to construct new 2d
small $\mathcal{N}=(4,4)$ defects with finite Ricci scalar in section 4.
Further in section 4, we demonstrate that the naïve
AdS${}_{7}\times\mathds{S}^{4}$ vacuum is inappropriate to use in a background
subtraction scheme for regulating holographic computations in the
$\gamma\to-\infty$ limit, and we identify the correct background to use in
this scheme. In section 5, we utilize the new $\gamma\to-\infty$ supergravity
solutions and correct regulating scheme in a computation of the contribution
of a flat 2d small $\mathcal{N}=(4,4)$ superconformal defect to the EE of a
spherical region in a 6d SCFT. We then summarize our findings and discuss
remaining issues and open questions surrounding these new defect solutions in
section 6.

In addition, there are two appendices that detail technical aspects of the
computations in the main text. First, in appendix A, we analyze the asymptotic
expansion of the supergravity data that specify the new solutions in the
$\gamma\to-\infty$ limit and construct the map to Fefferman-Graham (FG) gauge.
Lastly, in appendix B, we carefully treat the integral in the area functional
of the Ryu-Takayanagi (RT) surface in the computation of the holographic EE of
the defect in the dual field theory.

##  2 Review: small ${\mathcal{N}=(4,4)}$ surface defects

| $\mathbb{R}^{1,1}$ | $r$ | $\theta^{1}$ | $\theta^{2}$ | $\chi$ | $z$ | $\rho$ | $\varphi^{1}$ | $\varphi^{2}$ | $\phi$
---|---|---|---|---|---|---|---|---|---|---
KK′ |  | – | –
---|---
– | – | – | – | – | $\cdot$ | $\cdot$ | $\cdot$ | ISO
M5′ |  | – | –
---|---
– | – | – | – | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
M2 |  | – | –
---|---
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | – | $\sim$ | $\sim$ | $\sim$ | $\sim$
M5 |  | – | –
---|---
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\sim$ | – | – | – | –
KK |  | – | –
---|---
$\cdot$ | $\cdot$ | $\cdot$ | ISO | – | – | – | – | –
Table 1: The 1/8-BPS brane setup of Faedo:2020nol , with M2-M5 defect branes
intersecting orthogonal M5′-branes, and with both stacks of 5-branes probing
A-type singularities. In our conventions, –, $\cdot$ and $\sim$ denote
directions along which a brane is extended, localized, and smeared,
respectively, while ISO(metric) denotes the compact direction of the KK-
monopoles.

We begin with a brief summary of the results of Faedo:2020nol . The particular
11d supergravity metric constructed therein is the uplift of a 7d charged
AdS${}_{3}\times\mathds{S}^{3}$ domain wall initially found in
Dibitetto:2017tve , and is given by

$\displaystyle\text{d}s^{2}=4kQ_{\text{M5}}H_{\text{M5}^{\prime}}^{-1/3}\left(\text{d}s^{2}_{\text{AdS}_{3}}+\text{d}s^{2}_{\mathds{S}^{3}/\mathbb{Z}_{k}}\right)+H_{\text{M5}^{\prime}}^{2/3}\left(\text{d}z^{2}+\text{d}\rho^{2}+\rho^{2}\text{d}s^{2}_{\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}}}\right),$
(2)

for some parameter $Q_{\text{M5}}$ and a function $H_{\text{M5}^{\prime}}$
defined over a 4d space parametrized by the coordinates
$\\{z,\rho,\varphi^{1},\varphi^{2}\\}$. The solution above captures the near-
horizon geometry of the brane intersection depicted in table 1. Namely, a
“bound state” (in the sense discussed in footnote 4) of M2- and M5-branes,
with charges $Q_{\text{M2}}$ and $Q_{\text{M5}}$, intersects an orthogonal
stack of M5′-branes, thus forming a 1/4-BPS brane setup. In 2d notation, this
corresponds to $\mathcal{N}=(4,4)$ supersymmetry, with the large R-symmetry
realized geometrically as the isometry of the two 3-spheres $\mathds{S}^{3}$
and $\tilde{\mathds{S}}^{3}$ with coordinates
$\\{\chi,\theta^{1},\theta^{2}\\}$ and $\\{\phi,\varphi^{1},\varphi^{2}\\}$,
respectively. In addition, the two stacks of 5-branes can be made to probe ALE
singularities by introducing two Kaluza-Klein (KK) monopoles, with charges $k$
and $k^{\prime}$ and Taub-NUT directions $\partial_{\chi}$ and
$\partial_{\phi}$, respectively. The inclusion of any one of the two KK
monopoles results in a further breaking of the preserved supersymmetries and a
degeneration to an 1/8-BPS setup, which in 2d language corresponds to (large)
$\mathcal{N}=(0,4)$ supersymmetry. The presence of the second KK monopole does
not incur a further loss of supersymmetry, so the final brane configuration is
always at least a $1/8$-BPS supergravity solution.

Furthermore, the defect M2- and M5-branes are fully localized within a 2d
submanifold of the worldvolume of the M5′-branes. This is to be expected from
the holographic realization of a surface defect in a 6d SCFT; this
interpretation was first attached to the 7d domain wall in Dibitetto:2017klx
and to the full 11d SUGRA background in Faedo:2020nol . On the other hand, the
defect branes are smeared in the directions transverse to the worldvolume of
the M5′-branes444It is this property – that the M2- and M5-branes do not share
transverse directions with the M5′-branes, other than those along which the
former are smeared – which we are implicitly referring to when we describe the
M2- and M5-branes as forming a “bound state”. This aligns with the terminology
used in Faedo:2020nol , and should not to be confused with the dyonic
supermembrane Izquierdo:1995ms , which is a rather different multimembrane
solution of 11d supergravity. Indeed, in the setup described by table 1, there
is no M2-brane charge dissolved within the M5-brane worldvolume, nor do the
M2-branes polarize into a fuzzy 3-sphere via the Myers effect., so that their
charge is localized within $\mathds{S}^{3}/\mathbb{Z}_{k}$, but not
$\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}}$. Therefore, while the metric
in eq. (2) manifests the isometry groups of both (orbifolded) 3-spheres, the
R-symmetry is partially broken, giving rise to small $\mathcal{N}=(0,4)$
supersymmetry. For $k^{\prime}=1$, the solution above fits into the
classification of $\mathcal{N}=(0,4)$
AdS${}_{3}\times\mathds{S}^{3}/\mathbb{Z}_{k}\times\operatorname{CY}_{2}$
backgrounds foliated over an interval performed in Lozano:2020bxo , where
$\operatorname{CY}_{2}=\mathbb{R}^{4}$ contains the (round)
$\tilde{\mathds{S}}^{3}$. In particular, the solution above corresponds to
taking the M5′-branes to be completely localized in their transverse space.

On shell, the defect brane charges are constrained to be equal,
$Q_{\text{M2}}=Q_{\text{M5}}$, while the function $H_{\text{M5}^{\prime}}$
satisfies Faedo:2020nol

$\displaystyle\nabla^{2}_{\mathbb{R}^{3}_{\hat{\rho}}}H_{\text{M5}^{\prime}}(z,\hat{\rho})+\frac{k^{\prime}\partial_{z}^{2}H_{\text{M5}^{\prime}}(z,\hat{\rho})}{\hat{\rho}}=0,$
(3)

where we rescaled $\hat{\rho}=\rho^{2}/(4k^{\prime})$, and denoted by
$\mathbb{R}^{3}_{\hat{\rho}}$ the three-dimensional subspace which is
transverse to the M5-branes, parallel to the M5′-branes, and along which the
M2-branes are smeared. Following the parametrization adopted in table 1,
$\mathbb{R}^{3}_{\hat{\rho}}$ is then the space spanned by
$\\{\partial_{\hat{\rho}},\partial_{\varphi^{1}},\partial_{\varphi^{2}}\\}$,
with $\hat{\rho}$ being the radial coordinate and
$\\{\varphi^{1},\varphi^{2}\\}$ parametrizing a 2-sphere.

In terms of the brane setup described above, then, the spacetime in eq. (2) is
reached by approaching the brane intersection locus from within the
worldvolume of the M5′-branes in a radial fashion, i.e. by taking $r\to 0$. In
this limit, the $ISO(1,1)$ isometry group gets promoted to $SO(2,2)$, and the
M5′-brane worldvolume becomes AdS${}_{3}\times\mathds{S}^{3}/\mathbb{Z}_{k}$.

A particular solution to eq. (3) is, for any $\alpha\in\mathbb{R}$,

$H_{\text{M5}^{\prime}}(z,\rho)=\frac{4\sqrt{2}}{g^{3}}\frac{1}{P_{+}P_{-}}\frac{\sqrt{P_{+}^{2}+P_{-}^{2}-4\alpha^{2}+2P_{+}P_{-}}}{P_{+}^{2}+P_{-}^{2}+2P_{+}P_{-}},$
(4)

where $P_{\pm}=\sqrt{z^{2}+(\rho\pm\alpha)^{2}}$. By redefining

$\displaystyle\rho$ $\displaystyle=\alpha\frac{\cos\xi}{\sqrt{1-\mu^{5}}}$
(5a) $\displaystyle z$
$\displaystyle=\alpha\mu^{\frac{5}{2}}\frac{\sin\xi}{\sqrt{1-\mu^{5}}}$ (5b)

and setting $\alpha=(2^{7/4}g^{3/2}kQ_{\text{M5}})^{-1}$, the particular
solution in eq. (4) can be matched to the one found in eq. (2.17) of
Faedo:2020nol , which we now reproduce for clarity555To see this, it is
convenient to first rationalize the product $P_{+}P_{-}$ as
$P_{+}P_{-}=\alpha^{2}\big{(}1+\mu^{5}-(1-\mu^{5})\cos(2\xi)\big{)}/2(1-\mu^{5})$.:

$H_{\text{M5}^{\prime}}(\mu,\xi)=2^{27/4}(\sqrt{g}kQ_{\text{M5}})^{3}\frac{\mu^{5/2}(1-\mu^{5})^{3/2}}{\mu^{5}\cos^{2}\xi+\sin^{2}\xi}~{}.$
(6)

Furthermore, in Faedo:2020nol  it was argued that, as $\mu\to 1$ (which
corresponds to a non-linear limit in the original coordinates $\hat{\rho}$ and
$z$), the near-horizon geometry in eq. (2) locally recovers the
AdS${}_{7}/\mathbb{Z}_{k}\times\mathds{S}^{4}/\mathbb{Z}_{k^{\prime}}$ vacuum
of M-theory. This was shown by realizing the 11d line element as the uplift of
the domain wall solution to $\mathcal{N}=1$, $d=7$ supergravity (whose gauge
coupling constant $g$ appears in eq. (6) above) found in Dibitetto:2017tve ,
which interpolates between AdS7 (as $\mu\to 1$) and an infrared singularity
(at $\mu=0$).

While the singular nature of the solution is not immediately obvious, it can
be made manifest by studying the Ricci scalar, $\mathcal{R}$, in the $z\to 0$
limit. For metrics generally of the form of eq. (2), and following
D_Hoker_2008b , we can write the expression for the Ricci scalar for an
arbitrary harmonic function $H_{\text{M5}^{\prime}}$ as

$\displaystyle\mathcal{R}$ $\displaystyle=H_{\rm
M5^{\prime}}^{-2/3}\left[\frac{1}{6}\frac{(\partial_{z}H_{\rm
M5^{\prime}})^{2}+(\partial_{\rho}H_{\rm M5^{\prime}})^{2}}{H_{\rm
M5^{\prime}}^{2}}-\frac{2}{3}\frac{\partial_{z}^{2}H_{\rm
M5^{\prime}}+\partial_{\rho}^{2}H_{\rm M5^{\prime}}}{H_{\rm
M5^{\prime}}}-2\frac{\partial_{\rho}H_{\rm M5^{\prime}}}{\rho H_{\rm
M5^{\prime}}}\right].$ (7)

The function $H_{\rm M5^{\prime}}$ has a branch point located at $z=0$ and
$\rho=\alpha$ and correspondingly admits two different expansions as
$z\rightarrow 0$, depending on whether $\rho$ is larger or smaller than
$\alpha$. The choice of sign for the branch cut can be determined by the
requirement $P_{+}P_{-}\geq 0$. For $\rho>\alpha$, this leads to

$\displaystyle\mathcal{R}=\frac{g^{2}(2\alpha^{2}-3\rho^{2})^{2}}{12\rho^{2/3}(\rho^{2}-\alpha^{2})^{5/3}}+\mathcal{O}(z^{2})$
(8)

which has a pole as $\rho\rightarrow\alpha$. This corresponds to setting
$\xi=0$, with the pole appearing as $\mu\rightarrow 0$. For $\rho<\alpha$, we
find

$\displaystyle\mathcal{R}=\frac{g^{2}\alpha^{2/3}(\alpha^{2}-\rho^{2})}{12z^{8/3}}+\mathcal{O}\bigg{(}\frac{1}{z^{2/3}}\bigg{)}$
(9)

which corresponds to taking $\mu\rightarrow 0$ with $\xi\neq 0$. Thus for all
values of $\xi$, we find that the Ricci scalar diverges as $\mu\rightarrow 0$.

##  3 From large to small $\mathcal{N}=(4,4)$ surface defects

To begin, we will briefly review the classification due to D_Hoker_2008b ;
Bachas:2013vza  of the $d=11$ supergravity solutions with superisometry given
by two copies of the exceptional Lie superalgebra
${\mathfrak{d}}(2,1;\gamma)$. This classification represents the foundation
for the new defect solutions we present below.

In particular, we will be interested in the real form
$\mathfrak{d}(2,1;\gamma;0)$ which arises as a real subsuperalgebra of the
fixed points of an involutive semimorphism of $\mathfrak{d}(2,1;\gamma)$
frappat1996dictionary . Recall that the 9-dimensional maximal bosonic
subalgebra of the real form ${\mathfrak{d}}(2,1;\gamma;0)$ is
${\mathfrak{so}}(2,1)\oplus{\mathfrak{so}}(3)\oplus{\mathfrak{so}}(3)$.
Labelling each factor in this subalgebra by an index $a\in\\{1,2,3\\}$, the
generators $T^{(a)}$ satisfy

$\displaystyle[T^{(a)}_{i},\,T^{(b)}_{j}]=i\delta^{ab}\varepsilon_{ijk}\eta^{kl}_{a}T^{(a)}_{l}\quad\text{for
}i,j\in\\{1,2,3\\}~{},$ (10)

where $\varepsilon_{ijk}$ and $\eta^{kl}_{a}$ are the totally anti-symmetric
tensor (with $\varepsilon_{123}=1$) and the canonical metric induced by the
Killing form, respectively. In addition to the bosonic sector,
${\mathfrak{d}}(2,1;\gamma)$ contains an 8-dimensional fermionic generator
with components $F_{A_{1}A_{2}A_{3}}$, where the indices $A_{a}\in\\{\pm\\}$
transform in the spinorial representation $\boldsymbol{2}$ of the $a^{\rm th}$
factor in the even subalgebra. Furthermore, the components
$F_{A_{1}A_{2}A_{3}}$ obey the following anti-commutation relation

$\displaystyle\\{F_{A_{1}A_{2}A_{3}},F_{B_{1}B_{2}B_{3}}\\}=\ $
$\displaystyle\beta_{1}C_{A_{2}B_{2}}C_{A_{3}B_{3}}(C\sigma^{i})_{A_{1}B_{1}}T_{i}^{(1)}$
(11)
$\displaystyle+\beta_{2}C_{A_{1}B_{1}}C_{A_{3}B_{3}}(C\sigma^{i})_{A_{2}B_{2}}T_{i}^{(2)}$
$\displaystyle+\beta_{3}C_{A_{1}B_{1}}C_{A_{2}B_{2}}(C\sigma^{i})_{A_{3}B_{3}}T_{i}^{(3)}~{},$

where $C\equiv i\sigma^{2}$ is the charge conjugation matrix,
$\\{\sigma^{i}\\}_{i=1}^{3}$ are the Pauli matrices, and
$\\{\beta_{i}\\}_{i=1}^{3}$ are real parameters satisfying
$\sum_{a=1}^{3}\beta_{a}=0$, which follows from the (generalized) Jacobi
identity. This last constraint, together with the possibility of absorbing any
rescaling $\\{\lambda\beta_{1},\lambda\beta_{2},\lambda\beta_{3}\\}$ for
$\lambda\in\mathbb{C}$ into a redefinition of the normalization of the
fermionic generator, implies that ${\mathfrak{d}}(2,1;\gamma)$ is entirely
specified by the choice of a ratio of any two of the three $\beta$ parameters;
here, we take $\gamma\equiv\beta_{2}/\beta_{3}$. Note that
${\mathfrak{d}}(2,1;\gamma)$ is the only (finite-dimensional) Lie superalgebra
admitting a continuous parametrization.

Amongst the possible values that $\gamma$ can take, there are clearly three
special values corresponding to the vanishing of any one of the $\beta$
parameters. Specifically, choosing $\beta_{1}=0$ fixes $\gamma=-1$. The more
interesting case, and the one relevant to our analysis, is $\beta_{3}=0$,
which corresponds to $\gamma\to\pm\infty$. The case $\beta_{2}=0$ corresponds
to $\gamma=0$ and is equivalent under a group involution as discussed at the
end of this section. In the limit $\beta_{3}=0$, the anticommutator in eq.
(11) degenerates to

$\\{F_{A_{1}A_{2}A_{3}},F_{B_{1}B_{2}B_{3}}\\}=\beta_{1}\left(C_{A_{2}B_{2}}C_{A_{3}B_{3}}(C\sigma^{i})_{A_{1}B_{1}}T_{i}^{(1)}-C_{A_{1}B_{1}}C_{A_{3}B_{3}}(C\sigma^{i})_{A_{2}B_{2}}T_{i}^{(2)}\right)~{}.$
(12)

Consequently, the
$\mathfrak{so}(4)_{R}\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ R-symmetry
of the large superalgebra is broken into the single $\mathfrak{so}(3)_{R}$
factor which constitutes the R-symmetry of a small superalgebra. Note that the
other $\mathfrak{so}(3)$ factor remains a bosonic symmetry of the supergravity
solution; however, it is now realized as a flavor symmetry, rather than as an
outer automorphism of the supersymmetry algebra.

In addition, there are isolated points of interest in the $\gamma$-parameter
space where the real form $\mathfrak{d}(2,1;\gamma;0)$ reduces to classical
Lie superalgebras:

$\displaystyle\mathfrak{d}(2,1;\gamma;0)$
$\displaystyle=\mathfrak{osp}(4^{*}|2)\quad$ $\displaystyle\text{for
}\gamma\in\\{-2,-1/2\\}$ (13a) $\displaystyle\mathfrak{d}(2,1;\gamma;0)$
$\displaystyle=\mathfrak{osp}(4|2;\mathbb{R})\quad$ $\displaystyle\text{for
}\gamma=1~{}.$ (13b)

In particular, $\gamma=-1/2$ is the only value (up to algebra involutility, as
we will discuss shortly) for which
$\mathfrak{d}(2,1;\gamma;0)\oplus\mathfrak{d}(2,1;\gamma;0)$ admits a
canonical inclusion into the superisometry algebra $\mathfrak{osp}(8^{*}|4)$
of AdS${}_{7}\times\mathds{S}^{4}$. This case was studied extensively in
Estes:2018tnu ; it is the holographic realization of Wilson surfaces, 1/2-BPS
codimension-4 superconformal solitons, within the 6d $\mathcal{N}=(2,0)$ SCFT.
In this case, the ambient 6d SCFT is undeformed, in the sense that the
supergroup of symmetries preserved by the defect is a subgroup of the 6d
superconformal symmetry group. This is a common feature throughout the study
of defects embedded in higher-dimensional theories.

For generic values of $\gamma$, including the limit $\gamma\to\pm\infty$, the
superisometry algebra
$\mathfrak{d}(2,1;\gamma;0)\oplus\mathfrak{d}(2,1;\gamma;0)$ is not a
subalgebra of the $\mathfrak{osp}(8^{*}|4)$ superconformal Lie superalgebra
Estes:2018tnu . As a result, we expect the 6d ambient theory to be deformed as
we tune $\gamma$ away from the special value $\gamma=-1/2$. In the
supergravity solutions we construct below, this deformation will appear at
leading order in the asymptotic expansion of the four-form flux. Additionally,
the warp factors of the symmetric spaces corresponding to $T^{(1)}$ and
$T^{(2)}$ have the correct normalization to create an AdS7 space only when
$|\beta_{1}|=|\beta_{2}|$, which can happen only when $\gamma=-1/2$ or
$\gamma\to\pm\infty$.

Finally, we note that the complex Lie superalgebra $\mathfrak{d}(2,1;\gamma)$
enjoys a triality symmetry generated by $\gamma\mapsto\gamma^{-1}$,
$\gamma\mapsto-(\gamma+1)$, and $\gamma\mapsto-\gamma/(\gamma+1)$, any two of
which are linearly independent. However, only the $\gamma\mapsto\gamma^{-1}$
involutility survives at the level of the real form
$\mathfrak{d}(2,1;\gamma;0)$, due to the distinguished nature of the $T^{(1)}$
generator. Therefore, our analysis below can be equivalently carried out in
the $\gamma\to\pm\infty$ limits, or even in the $\gamma\to 0$ limit (albeit
with a permutation of the two ${\mathfrak{so}}(3)\oplus{\mathfrak{so}}(3)$
factors contained in two copies of ${\mathfrak{d}}(2,1;\gamma;0)$)666In spite
of the involution relating the $\gamma\to-\infty$ and $\gamma\to 0$ limits,
these two descriptions cannot be smoothly deformed into one another by varying
$\gamma$. Indeed, the two regimes are separated by the decompactification
point $\gamma=-1$.. The physical interpretation of the choice of either limit
corresponds to fixing the orientation of the M5-branes that engineer the
ambient 6d SCFT.

###  3.1 Supergravity solutions for general $\gamma$

In this section, we will review the structure of supergravity solutions with
${\mathfrak{d}}(2,1;\gamma;0)\oplus{\mathfrak{d}}(2,1;\gamma;0)$ superisometry
algebra, for generic $\gamma$, as first described in D_Hoker_2008b ;
Estes:2012vm ; Bachas:2013vza . The odd subspace of this superalgebra is
16-dimensional for all $\gamma$, so that all supergravity backgrounds
discussed below are $1/2$-BPS. Furthermore, the maximal bosonic subalgebra of
${\mathfrak{d}}(2,1;\gamma;0)\oplus{\mathfrak{d}}(2,1;\gamma;0)$ is

$\mathfrak{so}(2,2)\oplus\mathfrak{so}(4)\oplus\mathfrak{so}(4)~{}.$ (14)

To realize this superisometry, the supergravity metric must be of the
form777For the previously mentioned special values $\gamma=-1$ and $\gamma=0$,
AdS3 Wigner-İnönü contracts to $\mathbb{R}^{2,1}$ and one of the 3-spheres
decompactifies into $\mathbb{R}^{3}$, respectively.

$(\text{AdS}_{3}\times\mathds{S}^{3}\times\tilde{\mathds{S}}^{3})\ltimes\Sigma_{2}~{},$
(15)

for a Riemann surface $\Sigma_{2}$.

The Killing spinor equations on such manifolds can be recast into conditions
on two auxiliary functions, $h$ and $G$, defined over the Riemann surface
$\Sigma_{2}$ D_Hoker_2008b ; Estes:2012vm ; Bachas:2013vza . Specifically, if
we employ local complex888The SUGRA orientation and Riemannian metric
automatically endow the Riemann surface $\Sigma_{2}$ with a complex structure.
coordinates $\\{w,\bar{w}\\}$ on $\Sigma_{2}$, so that the metric on the
Riemann surface is given by
$\text{d}s^{2}_{\Sigma_{2}}=\text{d}w\text{d}\bar{w}$, the function $h$ is
constrained to be $\mathbb{R}$-valued and harmonic

$\partial_{w}\partial_{\bar{w}}h=0~{},$ (16)

while $G$ is $\mathbb{C}$-valued and satisfies the conformally covariant
equation

$\partial_{w}G=\frac{1}{2}\left(G+\bar{G}\right)\partial_{w}\ln h~{}.$ (17)

Note that both constraints above hold for generic $\gamma$.

Regularity conditions on the auxiliary functions $h$ and $G$ have been
discussed in Bachas:2013vza . Here, we will consider Riemann surfaces with a
boundary, $\partial\Sigma_{2}\neq\emptyset$. In short, regularity of the
supergravity solution constrains $G=\pm i$ on $\partial\Sigma_{2}$, while the
(holomorphic part of the) function $h$ must either vanish or have a simple
pole at any point of $\partial\Sigma_{2}$.

The triple $(h,G,\gamma)$ uniquely specifies a bosonic background within the
space of 11d supergravity solutions with superisometry algebra
${\mathfrak{d}}(2,1;\gamma;0)\oplus{\mathfrak{d}}(2,1;\gamma;0)$. In
particular, the metric field in the supergravity solution can be written in
terms of this data as

$\text{d}s^{2}=f_{\text{AdS}_{3}}^{2}\text{d}s^{2}_{\text{AdS}_{3}}+f_{\mathds{S}^{3}}^{2}\text{d}s^{2}_{\mathds{S}^{3}}+f_{\tilde{\mathds{S}}^{3}}^{2}\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}+f_{\Sigma_{2}}^{2}\text{d}s^{2}_{\Sigma_{2}}~{},$
(18)

where the metric factors are functions over $\Sigma_{2}$ and can be expressed
in terms of the auxiliary functions

$\displaystyle W_{\pm}:=|G\pm i|^{2}+\gamma^{\pm 1}(G\bar{G}-1)$ (19)

as Bachas:2013vza

$\displaystyle f_{\text{AdS}_{3}}^{6}$
$\displaystyle=\frac{h^{2}W_{+}W_{-}}{\beta_{1}^{6}(G\bar{G}-1)^{2}}~{},$
(20a) $\displaystyle f_{{\mathds{S}}^{3}}^{6}$
$\displaystyle=\frac{h^{2}(G\bar{G}-1)W_{-}}{\beta_{2}^{3}\beta_{3}^{3}W_{+}^{2}}~{},$
(20b) $\displaystyle f_{\tilde{\mathds{S}}^{3}}^{6}$
$\displaystyle=\frac{h^{2}(G\bar{G}-1)W_{+}}{\beta_{2}^{3}\beta_{3}^{3}W_{-}^{2}}~{},$
(20c) $\displaystyle f_{\Sigma_{2}}^{6}$
$\displaystyle=\frac{|\partial_{w}h|^{6}}{\beta_{2}^{3}\beta_{3}^{3}h^{4}}(G\bar{G}-1)W_{+}W_{-}~{}.$
(20d)

The bosonic sector of the 11d supergravity solution is completed by a three-
form gauge potential $\mathcal{C}_{(3)}$, which can be written in terms of the
$(h,G,\gamma)$ data as

$\displaystyle\mathcal{C}_{(3)}=\sum_{i=1}^{3}b_{\mathcal{M}_{i}}\operatorname{vol}_{\mathcal{M}_{i}}$
(21)

where $\operatorname{vol}_{\mathcal{M}_{i}}$ denotes the volume form on the
manifold
$\mathcal{M}_{i}=\\{\text{AdS}_{3},\mathds{S}^{3},\tilde{\mathds{S}}^{3}\\}_{i}$.
We also introduced the gauge potentials

$\displaystyle b_{\text{AdS}_{3}}$
$\displaystyle:=\frac{\tau_{1}}{\beta_{1}^{3}}\left[-\frac{h(G+\bar{G})}{1-G\bar{G}}+(2+\gamma+\gamma^{-1})\Phi-(\gamma-\gamma^{-1})\tilde{h}+b_{1}^{0}\right]~{},$
(22a) $\displaystyle b_{\mathds{S}^{3}}$
$\displaystyle:=\frac{\tau_{2}}{\beta_{2}^{3}}\left[-\frac{\gamma
h(G+\bar{G})}{W_{+}}+\gamma(\Phi-\tilde{h})+b_{2}^{0}\right]~{},$ (22b)
$\displaystyle b_{\tilde{\mathds{S}}^{3}}$
$\displaystyle:=\frac{\tau_{3}}{\beta_{3}^{3}}\left[\frac{h(G+\bar{G})}{\gamma
W_{-}}-\frac{\Phi+\tilde{h}}{\gamma}+b_{3}^{0}\right]~{},$ (22c)

where $\\{b^{0}_{i}\\}_{i=1}^{3}$ are integration constants, while the dual
harmonic function $\tilde{h}$ and the real auxiliary function $\Phi$ are
defined in terms of $h$ and $G$ via

$\displaystyle\partial_{w}\tilde{h}$ $\displaystyle=-i\partial_{w}h~{},$ (23a)
$\displaystyle\partial_{w}\Phi$ $\displaystyle=\bar{G}\partial_{w}h~{}.$ (23b)

Finally, in eq. (22) we made use of the signs $\tau_{i}=\pm 1$ for
$i\in\\{1,2,3\\}$, which are subject to the constraint

$\prod_{i=1}^{3}\beta_{i}f_{\mathcal{M}_{i}}+h\prod_{i=1}^{3}\tau_{i}=0~{}.$
(24)

As discussed above, regularity conditions force $h$ and $G$ to have prescribed
behavior on $\partial\Sigma_{2}$. The solutions that we are interested in
studying are asymptotically AdS${}_{7}\times\mathds{S}^{4}$, which are indeed
described by $G=\pm i$ and $h$ having a single simple pole, which corresponds
to having a single asymptotic AdS${}_{7}\times\mathds{S}^{4}$ region.
Generically at any point on $\partial\Sigma_{2}$, the regularity constraint on
$h$ implies that its Laurent expansion reduces to

$h=-ih_{0}w+\operatorname{c.c.}=\frac{2h_{0}\sin\vartheta}{\varrho}~{},$ (25)

for some real constant $h_{0}$. In the second equality, we have introduced new
convenient set of coordinates $\\{\varrho,\vartheta\\}$ on $\Sigma_{2}$
defined by $w=e^{i\vartheta}/\varrho$. Adopting these new polar coordinates
and using eq. (25) allows us to perturbatively solve eq. (17) in small
$\varrho$ to find

$G=-i+a_{1}\varrho e^{i\vartheta}\sin\vartheta+\mathcal{O}(\varrho^{2})$ (26)

for some constant $a_{1}$. If we then insert eq. (26) into eqs. (19) and (27),
we can perturbatively expand eq. (18) in small $\varrho$ to find

$\text{d}s^{2}=L^{2}\frac{\text{d}\varrho^{2}}{\varrho^{2}}-\frac{2\gamma
L^{2}}{a_{1}(1+\gamma)^{2}\varrho}\left(\text{d}s^{2}_{\text{AdS}_{3}}+\frac{(1+\gamma)^{2}}{\gamma^{2}}\text{d}s^{2}_{\mathds{S}^{3}}\right)+L^{2}\left(\text{d}\vartheta^{2}+\sin^{2}\vartheta\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}\right)+\dots$
(27)

with

$L^{6}=\frac{a_{1}^{2}h_{0}^{2}(1+\gamma)^{6}}{\beta_{1}^{6}\gamma^{2}}~{}.$
(28)

We can see the AdS${}_{7}\times\mathds{S}^{4}$ asymptotic geometry in eq. (27)
clearly for certain values of $\gamma$. The obvious case is if we choose
$\gamma=-1/2$, which was studied extensively in Bachas:2013vza ;
Gentle:2015jma ; Estes:2018tnu . The other possibility, namely the limits
$\gamma\to\pm\infty$, will be discussed in the next section.

For $\gamma<0$ and for the function $h$ given in eq. (25), the general
solution to eq. (17) for the function $G$ with an even number $2n+2$ of branch
points $\\{\xi_{i}\\}_{i=1}^{2n+2}\subset\partial\Sigma_{2}$ was found in
DHoker:2008rje  to be

$G=-i\left(1+\sum_{j=1}^{2n+2}(-1)^{j}\frac{w-\xi_{j}}{|w-\xi_{j}|}\right)~{},$
(29)

where $G$ flips sign $\pm i\to\mp i$ upon crossing each branch point
$\xi_{i}$.

###  3.2 The $\gamma\to-\infty$ limit

Of particular interest to us in the following sections are solutions that are
constructed by taking the scaling limit

$\displaystyle\gamma\to\pm\infty~{},$ $\displaystyle\gamma
a_{1}\to\text{constant}~{},$ $\displaystyle L\to\text{constant}~{}.$ (30)

From eq. (27) and the requirement $\gamma a_{1}\to\text{constant}$, we can see
that this scaling limit realizes an AdS${}_{7}\times\mathds{S}^{4}$ asymptotic
geometry. In this and the following subsections, we will consider the
$\gamma\to-\infty$ limit in greater detail and show that the solutions
engineered in this limit contain the class of solutions found in Faedo:2020nol
which we reviewed in section 2. Below, we will expand upon these solutions and
construct new surface defects with small $\mathcal{N}=(4,4)$ SUSY.

To begin, we introduce a complex function $F$ via

$G=-i\left(1+\gamma^{-1}F\right)~{},$ (31)

and we rescale $\beta_{1}=\hat{\beta}(-\gamma)^{1/3}$. This ensures that $L$
remains finite as required by the limiting procedure of eq. (30). In this
limit, the metric in eq. (27) becomes

$\displaystyle\text{d}s^{2}$
$\displaystyle=\left[\frac{4h^{2}}{\hat{\beta}^{6}(F+\bar{F})}\right]^{\frac{1}{3}}\left(\text{d}s^{2}_{\text{AdS}_{3}}+\text{d}s^{2}_{\mathds{S}^{3}}\right)+\left[\frac{h^{2}(F+\bar{F})^{2}}{16\hat{\beta}^{6}}\right]^{\frac{1}{3}}\left(\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}+\frac{4|\partial_{w}h|^{2}}{h^{2}}\text{d}w\text{d}\bar{w}\right)~{}.$
(32)

In addition, the gauge potentials are given by

$\displaystyle
b_{\text{AdS}_{3}}=b_{\mathds{S}^{3}}=-\frac{2\tilde{h}}{\hat{\beta}^{3}},\qquad
b_{\tilde{\mathds{S}}^{3}}=\frac{h}{2\hat{\beta}^{3}}\frac{-i(F-\bar{F})}{2}-\hat{\Phi}~{},$
(33)

where the function $\hat{\Phi}$ is defined via

$\displaystyle\partial_{w}\hat{\Phi}$
$\displaystyle=\bar{F}\frac{\partial_{w}\tilde{h}}{\hat{\beta}^{3}}~{}.$ (34)

Let us introduce a new set of coordinates $\\{z,\rho\\}$ on $\Sigma_{2}$
defined via $w=z+i\rho$, and choose $h_{0}$ in eq. (25) such that the harmonic
function $h=\hat{\beta}^{3}h_{1}\rho$ for some constant $h_{1}$. Let us also
parametrize the real and imaginary parts of the complex function $F$ such that

$F=\frac{2\rho^{2}}{h_{1}}H+iF_{I}~{},$ (35)

for real functions $H$ and $F_{I}$. Note that the latter does not enter the
metric field; indeed, we can rewrite the line element in eq. (32) entirely in
terms of $H$ as

$\text{d}s^{2}=h_{1}H^{-\frac{1}{3}}\left(\text{d}s^{2}_{\text{AdS}_{3}}+\text{d}s^{2}_{\mathds{S}^{3}}\right)+H^{\frac{2}{3}}\left(\text{d}z^{2}+\text{d}\rho^{2}+\rho^{2}\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}\right)~{}.$
(36)

The condition in eq. (17) implies that the complex function $F$ satisfies the
first order equation

$\partial_{w}F=\frac{1}{2}(F-\bar{F})\partial_{w}\ln h~{},$ (37)

which in turn can be recast into a second order equation for its real part,

$\partial_{\rho}^{2}H+\frac{3}{\rho}\partial_{\rho}H+\partial_{z}^{2}H=0~{}.$
(38)

The supergravity solution in the $\\{z,\rho\\}$ parametrization of
$\Sigma_{2}$ is completed by the following expressions for the gauge
potentials,

$\displaystyle b_{\text{AdS}_{3}}=b_{\mathds{S}^{3}}=2h_{1}z,\qquad
b_{\tilde{\mathds{S}}^{3}}=\frac{h_{1}\rho}{2}F_{I}+\hat{\Phi}~{}.$ (39)

In particular, a more explicit form can be given for the derivatives of
$b_{\tilde{\mathds{S}}^{3}}$ as

$\displaystyle\partial_{\rho}b_{\tilde{\mathds{S}}^{3}}=\rho^{3}\partial_{z}H,\qquad\partial_{z}b_{\tilde{\mathds{S}}^{3}}=-\rho^{3}\partial_{\rho}H~{}.$
(40)

These solutions have a scaling symmetry under the transformation

$\displaystyle z\rightarrow\lambda z,$
$\displaystyle\rho\rightarrow\lambda\rho,$ $\displaystyle
H\rightarrow\lambda^{-3}H,$ $\displaystyle h_{1}\rightarrow\lambda^{-1}h_{1},$
(41)

for which the metric and flux are invariant. This transformation could be used
to fix the value of $h_{1}$ so that the solution is uniquely given by choice
of function $H$.

###  3.3 Inclusion of KK-monopoles and recovering known solutions

At the level of the local geometry described by eq. (36), KK-monopoles can be
included in a straightforward fashion by replacing $\mathds{S}^{3}$ and
$\tilde{\mathds{S}}^{3}$ with the lens spaces $\mathds{S}^{3}/\mathbb{Z}_{k}$
and $\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}}$, respectively, where $k$
and $k^{\prime}$ are the orbifold charges. The lens spaces can be realized as
the total spaces of circle bundles over 2-spheres, where the orbifold acts on
the Hopf fiber; this amounts to the substitutions

$\displaystyle\text{d}s^{2}_{\mathds{S}^{3}}\to$ $\displaystyle\
\text{d}s^{2}_{\mathds{S}^{3}/\mathbb{Z}_{k}}\
=\frac{1}{4}\left[\left(\frac{\text{d}\chi}{k}+\omega\right)^{2}+\text{d}s^{2}_{\mathds{S}^{2}}\right]~{},$
(42a) $\displaystyle\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}\to$ $\displaystyle\
\text{d}s^{2}_{\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}}}=\frac{1}{4}\left[\left(\frac{\text{d}\phi}{k^{\prime}}+\eta\right)^{2}+\text{d}s^{2}_{\tilde{\mathds{S}}^{2}}\right]~{},$
(42b)

where $\text{d}\omega=\operatorname{vol}_{\mathds{S}^{2}}$ and
$\text{d}\eta=\operatorname{vol}_{\tilde{\mathds{S}}^{2}}$. The inclusion of
either KK-monopole incurs the loss of $1/2$ of the existing supersymmetry
generators, resulting in a small $\mathcal{N}=(0,4)$ supersymmetry algebra.
Indeed, dimensional reduction along the Taub-NUT direction produces a D6-brane
which breaks half of the supersymmetries of the massless type IIA AdS7 vacuum.
The second KK-monopole can be added without bringing about any further
breaking of the supersymmetries, and provides a second isometric direction
upon which the background can be dimensionally reduced to massless type IIA
supergravity. Therefore, the final solution is an $1/8$-BPS configuration. At
the level of the superconformal symmetry algebra, this corresponds to a
reduction from
$\mathfrak{d}\left(2,1;\gamma\right)\oplus\mathfrak{d}\left(2,1;\gamma\right)$
to $\mathfrak{d}\left(2,1;\gamma\right)$.

The inclusion of orbifolds allows us to match the $\gamma\to-\infty$ solutions
of section 3.2 to those found in Faedo:2020nol  and reviewed in section 2. In
particular, we see that the $\gamma\to-\infty$ metric in eq. (36), subject to
the inclusion of KK-monopoles as in eq. (42), matches the M2-M5-KK-M5′-KK′
near-horizon in eq. (2) under the identifications $h_{1}=4kQ_{\text{M5}}$ and
$H=H_{\text{M5}^{\prime}}$. In particular, the functions $H$ and
$H_{\text{M5}^{\prime}}$ solve the same equation, since eq. (38) maps to eq.
(3) under the rescaling $\hat{\rho}=\rho^{2}/4$. We have thus managed to fully
recover the solutions described in Faedo:2020nol  as a limiting case of those
in Bachas:2013vza .

##  4 New small $\mathcal{N}=(4,4)$ surface defects

In this section, we will construct a new explicit family of solutions
$H(z,\rho)$ to eq. (38). In particular, without loss of generality we will
specialize to the $\gamma\to-\infty$ limit. We recall that for $\gamma<0$, a
general solution with an even number of branch points
$\\{\xi_{i}\\}_{i=1}^{2n+2}\subset\partial\Sigma_{2}$ was given by eq. (29).
We rescale the singular points $\xi_{j}$ as

$\xi_{j}=\nu_{j}-\gamma^{-1}\hat{\xi}_{j}\in\partial\Sigma_{2}\quad\text{for}\quad
j\in\\{1,2,\dots,2n+2\\}~{},$ (43)

where the collapse points satisfy $\nu_{j}\leq\nu_{j+1}$ for all $j$, so as to
preserve the total order of the set $\\{\xi_{j}\\}$ following the
$\gamma\to-\infty$ limit. Furthermore, to ensure finiteness of the complex
function $F$ in this limit, we must also demand that all points $\nu_{j}$, for
$1\leq j\leq 2n+2$, correspond to the collapse of even-dimensional clusters
$\\{\xi_{k},\xi_{k+1},\ldots,\xi_{k+2m+1}\\}$ of singular points, where $k\leq
j\leq k+2m+1$. This can be implemented by identifying

$\nu_{k}\equiv\nu_{k+1}\equiv\ldots\equiv\nu_{k+2m+1}$ (44)

for each cluster. Finally, in order to preserve the ordering of the singular
points throughout the $\gamma\to-\infty$ limit, we must also arrange the
corresponding collapse parameters within each cluster so that
$\hat{\xi}_{k}<\hat{\xi}_{k+1}<\ldots<\hat{\xi}_{k+2m+1}$. Without loss of
generality, then, it suffices to consider a pairwise collapse of neighboring
singular points, which can be realized by identifying

$\nu_{2i-1}\equiv\nu_{2i}\quad\text{for}\quad i\in\\{1,2,\dots,n+1\\}~{}.$
(45)

However, we will later comment on certain phenomena which appear only when the
singular points collapse in clusters of 4 or more.

$\Sigma_{2}$$z$$\xi_{1}$$\nu_{1}$$\xi_{2}$$\xi_{3}$$\nu_{3}$$\xi_{4}$$\cdots$$\xi_{2n+1}$$\nu_{2n+1}$$\xi_{2n+2}$$\gamma\rightarrow-\infty$
Figure 1: A particular choice of collapse points
$\nu_{j}\in\partial\Sigma_{2}$ and the behavior of the singular points
$\xi_{i}\in\partial\Sigma_{2}$ as $\gamma\to-\infty$ under pairwise collapse.
The collapse points $\nu_{j}$ can be chosen arbitrarily, while maintaining
$\nu_{j}<\nu_{j+1}$, but for clarity in the figure have been depicted at the
midpoint in the region $[\xi_{j},\,\xi_{j+1}]$ to demonstrate pairwise
collapse.

An illustration of the pairwise collapse described by eq. (43) and eq. (45) is
shown in figure 1. Under this collapse dynamic, the complex function $F$
becomes, in the $\gamma\to-\infty$ limit,

$F(w,\bar{w})=\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}\frac{\bar{w}-w}{2(\bar{w}-\nu_{j})|w-\nu_{j}|}~{}.$
(46)

The corresponding function $H$ is given by

$H(z,\rho)=\frac{h_{1}}{2}\sum_{j=1}^{2n+2}\frac{(-1)^{j}\hat{\xi}_{j}}{(\rho^{2}+(z-\nu_{j})^{2})^{3/2}}~{}.$
(47)

For pure AdS${}_{7}\times\mathds{S}^{4}$, this procedure produces what we
refer to as the single-pole vacuum (1PV) solution with

$\displaystyle F_{\text{1PV}}(w,\bar{w})$
$\displaystyle=\hat{\xi}\frac{\bar{w}-w}{\bar{w}|w|}~{},$ (48a) $\displaystyle
H_{\text{1PV}}(z,\rho)$
$\displaystyle=\frac{h_{1}\hat{\xi}}{(z^{2}+\rho^{2})^{\frac{3}{2}}}~{},$
(48b)

which can be obtained by taking $n=0$, $\xi_{2}=-\xi_{1}=\xi$, and
$\nu_{1}=\nu_{2}=0$. Alternatively, this solution can also be obtained by
collapsing all singular points to the origin, i.e. $\nu_{j}=0$ for all $j$.
Note that eq. (38) is linear in $H$ and admits a translation symmetry under
shifts of $z$. Using these two properties, the general solutions given in eq.
(47) can be reconstructed from the 1PV solution by taking linear combinations
and making use of the translation symmetry.

One may worry that since the solution for the potential $H$ in eq. (47) is
generically singular at the points $(z,\rho)=(\nu_{i},0)$ for all
$i\in\\{1,2,\dots,2n+2\\}$, the resulting supergravity metric may have a
singularity or these points may simply correspond to horizons. In order to
investigate the regularity of the spacetime geometry, we compute the scalar
curvature at these points. Without loss of generality we can choose to
evaluate the Ricci scalar at the point $(z,\rho)=(\nu_{2j},0)$. Recall that
for metrics generally of the form in eq. (36), the scalar curvature is given
by eq. (7) with $H_{\rm M5^{\prime}}$ replaced by $H$. Note that since all
collapse points $\nu_{2k-1}$ with odd labels are identified with even-labelled
ones $\nu_{2k}$ via eq. (45), the point $(z,\rho)=(\nu_{2j},0)$ is indeed
generic within the set of collapsed points. Ultimately, we find

$\mathcal{R}\bigr{|}_{(z,\rho)=(\nu_{2j},0)}=\frac{3}{2L_{\mathds{S}^{4}}^{2}}\left(\frac{\hat{\xi}_{2j}-\hat{\xi}_{2j-1}}{\hat{m}_{1}}\right)^{-2/3}~{},$
(49)

which is indeed finite, where $L_{\mathbb{S}^{4}}$ and $\hat{m}_{1}$ are
strictly positive constants introduced below. This suggests that the geometry
is regular at these points.

We also note that the 1PV solution corresponds to a spacetime with a constant
scalar curvature given by

$\mathcal{R}=\frac{3}{2L_{\mathds{S}^{4}}^{2}}~{},$ (50)

with $L_{\mathds{S}^{4}}=(h_{1}\hat{\xi})^{1/3}$.

###  4.1 Asymptotic local behavior

We will now show explicitly that the solutions described by eq. (47) are
asymptotically locally AdS${}_{7}\times\mathds{S}^{4}$. As described in
greater detail in appendix A, the Riemann surface $\Sigma_{2}$ admits a
parametrization by Fefferman-Graham (FG) coordinates $\\{v,\phi\\}$ in terms
of which the line element takes the following asymptotic form for small $v$,

$\text{d}s^{2}=\frac{4L_{\mathds{S}^{4}}^{2}}{v^{2}}\left[\text{d}v^{2}+\alpha_{1}\left(\text{d}s^{2}_{\text{AdS}_{3}}+\text{d}s^{2}_{\mathds{S}^{3}}\right)\right]+L_{\mathds{S}^{4}}^{2}\left[\alpha_{3}\text{d}\phi^{2}+\alpha_{4}\sin^{2}\phi\
\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}\right],$ (51)

where the metric factors are of the form $\alpha_{i}=1+\mathcal{O}(v^{4})$ for
$i\in\\{1,3,4\\}$. They are given explicitly, together with the asymptotic
mapping to FG coordinates on $\Sigma_{2}$, in appendix A. The asymptotic
$\mathds{S}^{4}$ radius $L_{\mathds{S}^{4}}$ can be expressed in terms of the
moments

$\hat{m}_{k}:=\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}^{k}$ (52)

as

$L_{\mathds{S}^{4}}^{3}=\frac{h_{1}\hat{m}_{1}}{2}~{}.$ (53)

As claimed above, we may recognize within eq. (51), at leading order in $v$,
the large-$x$ limit of the line element of AdS7 (with radius
$2L_{\mathds{S}^{4}}$) written in AdS3 slicing,

$\text{d}s^{2}_{\text{AdS}_{7}}=4L^{2}_{\mathds{S}^{4}}\left[\text{d}x^{2}+\cosh^{2}x\
\text{d}s^{2}_{\text{AdS}_{3}}+\sinh^{2}x\
\text{d}s^{2}_{\mathds{S}^{3}}\right],$ (54)

where the coordinate normal to the AdS3 foliation is related to the FG
coordinate via $x=\log(2/v)$. The large-$x$ limit trivializes the relative
warping $\coth^{2}x$ between the AdS3 and the $\mathds{S}^{3}$ subspaces,
matching the behavior seen in eq. (51).

In the $\gamma\to-\infty$ limit, the auxiliary functions $\Phi$ and
$\tilde{h}$ admit the following FG expansions,

$\displaystyle\Phi=-\tilde{h}$
$\displaystyle=\frac{4\hat{m}_{1}\cos\phi}{v^{2}}+\frac{2\hat{n}_{1}}{\hat{m}_{1}}+\frac{\hat{m}_{1}\hat{n}_{2}-\hat{n}_{1}^{2}}{\hat{m}_{1}^{3}}\cos\phi~{}v^{2}+\mathcal{O}(v^{4})~{},$
(55)

where the moments $\hat{n}_{i}$ are defined in eq. (96). Using these
expansions, we find that the asymptotic geometry in eq. (51) is supported by a
four-form flux999In the following, we have made a choice on the signs
$\tau_{1,2,3}$ in line with the constraint $\prod_{i=1}^{3}\tau_{i}=1$ which
follows from evaluating eq. (24) on the $\gamma\to-\infty$ solutions.

$\displaystyle\frac{\mathcal{F}_{(4)}}{L_{\mathds{S}^{4}}^{3}}$
$\displaystyle=-\frac{16\cos\phi}{v^{3}}\text{d}v\wedge(\operatorname{vol}_{\mathds{S}^{3}}+\operatorname{vol}_{\text{AdS}_{3}})-\frac{8\sin\phi}{v^{2}}\text{d}\phi\wedge(\operatorname{vol}_{\mathds{S}^{3}}+\operatorname{vol}_{\text{AdS}_{3}})$
(56) $\displaystyle\quad+3\sin^{3}\phi\
\text{d}\phi\wedge\operatorname{vol}_{\tilde{\mathds{S}}^{3}}+\ 4\cos\phi\
\frac{\hat{m}_{1}\hat{n}_{2}-\hat{n}_{1}^{2}}{\hat{m}_{1}^{4}}v\
\text{d}v\wedge(\operatorname{vol}_{\mathds{S}^{3}}+\operatorname{vol}_{\text{AdS}_{3}})+\mathcal{O}(v^{2}).$

The first line in the equation above manifests the deformation of the 6d
ambient SCFT by the insertion of a source, as already hinted at by the
superalgebra structure discussed in section 3. This deformation takes the form
of an S-wave over the internal $\tilde{\mathds{S}}^{3}$. It can be compared to
the undeformed theory, which corresponds to $\gamma=-1/2$. In that case, the
four-form field strength at the conformal boundary $v=0$ is
$\mathcal{F}_{(4)}=3L^{3}_{\mathds{S}^{4}}\operatorname{vol}_{\mathds{S}^{4}}$,
where $\mathds{S}^{4}$ is the internal 4-sphere spanned by $\phi$ and
$\tilde{\mathds{S}}^{3}$. We can recognize this term as the first term in the
second line of eq. (56).

###  4.2 Single-pole vacuum

$\Sigma_{2}$$z$$\nu$$\xi_{1}$$\xi_{2}$$\mathfrak{C}_{a}$$\mathfrak{C}_{a}$$\Sigma_{2}$$\xi_{1}$$\nu$$\xi_{2}$
Figure 2: The 1PV solution described in section 4.2 is characterized by two
singular points $\xi_{\\{1,2\\}}$, which collapse to the same
$\nu\in\partial\Sigma_{2}$ in the $\gamma\to-\infty$ limit. As discussed in
section 4.3, the basis of non-contractible four-cycles for this solution is
one-dimensional. The profile of a representative cycle $\mathfrak{C}_{a}$
along $\Sigma_{2}$ is shown; note that the same 3-sphere collapses at both of
its endpoints on $\partial\Sigma_{2}$. On the right, the 1PV on the upper half
plane is mapped to a semi-infinite strip.

For later reference, we now identify a vacuum solution within the family of
supergravity backgrounds presented above. An appropriate choice of vacuum is
necessary for computing properties associated to a defect embedded in a
holographic CFT. Indeed, in order to isolate quantities which are intrinsic to
a defect, one must ensure that the contributions due to the ambient degrees of
freedom are taken into account. The gravitational analogue of this operation,
upon recasting a field theory computation into a bulk one, is vacuum
subtraction. In particular we must use a vacuum solution which is
characterized by the same bulk deformation as the general solutions discussed
above. As discussed above, pure AdS${}_{7}\times\mathds{S}^{4}$ with
$\gamma=-1/2$ does not satisfy this criteria, as the solutions we consider
here with $\gamma\rightarrow-\infty$ contain a bulk deformation as can be seen
in the asymptotic expression for the flux given in eq. (56).

We take the vacuum to be the 1PV we identified in eq. (48), which is the
solution corresponding to the $\gamma\rightarrow-\infty$ of pure
AdS${}_{7}\times\mathds{S}^{4}$, i.e. two singular points $\xi_{\\{1,2\\}}$
collapsing to a single point $\nu$, as shown in figure 2. More precisely, for
general $\gamma$, the 1PV metric is given in FG form by

$\displaystyle\allowdisplaybreaks\frac{\text{d}s^{2}_{\text{1PV}}(\gamma)}{L_{\mathds{S}^{4}}^{2}}=\frac{4}{v^{2}}$
$\displaystyle\left[\text{d}v^{2}+\left(1+\frac{2\gamma+3-(2\gamma+1)c_{2\phi}}{16(\gamma+1)^{2}}v^{2}\right)\text{d}s^{2}_{\text{AdS}_{3}}\right.$
(57) $\displaystyle\
+\left.\left(\frac{(\gamma+1)^{2}}{\gamma^{2}}+\frac{2\gamma-1-(2\gamma+1)c_{2\phi}}{16\gamma^{2}}v^{2}\right)\text{d}s^{2}_{\mathds{S}^{3}}+\mathcal{O}(v^{4})\right]$
$\displaystyle+$
$\displaystyle\left[\left(1+\frac{(2\gamma+1)(2c_{2\phi}+1)}{12(\gamma+1)^{2}}v^{2}\right)s^{2}_{\phi}\text{d}s_{\tilde{\mathds{S}}^{3}}^{2}+\left(1+\frac{(2\gamma+1)c^{2}_{\phi}}{4(\gamma+1)^{2}}v^{2}\right)\text{d}\phi^{2}+\mathcal{O}(v^{4})\right],$

where in order to keep the expression manageable, we have adopted the abusive
notation

$\displaystyle\cos x\equiv c_{x},\quad\text{and}\quad\sin x\equiv s_{x},$ (58)

which will be employed from this point forward. For $\gamma=-1/2$, the 1PV
recovers the AdS${}_{7}\times\mathds{S}^{4}$ vacuum in FG gauge. In the
$\gamma\to-\infty$ limit which is of relevance here, the line element of the
1PV becomes instead

$\text{d}s^{2}_{\text{1PV}}(\gamma\to-\infty)=\frac{4L_{\mathds{S}^{4}}}{v^{2}}\left[\text{d}v^{2}+\text{d}s^{2}_{\text{AdS}_{3}}+\text{d}s^{2}_{\mathds{S}^{3}}+\mathcal{O}(v^{4})\right]+L_{\mathds{S}^{4}}^{2}\left[s^{2}_{\phi}\text{d}s_{\tilde{\mathds{S}}^{3}}^{2}+\text{d}\phi^{2}+\mathcal{O}(v^{4})\right].$
(59)

Furthermore, as a quick sanity check, we can take the 1PV limit of eq. (49),
which recovers $\mathcal{R}|_{\rm 1PV}=3/(2L_{\mathds{S}^{4}}^{2})$ as
expected.

As remarked above, the 1PV contains no explicit defect data – as we will see
later, no Young Tableau can be associated to it. However, it does fully
capture the bulk S-wave deformation discussed previously. This follows from
the fact that all terms in eq. (56) which do not vanish at the conformal
boundary $v=0$ are independent of the moments $\\{\hat{m}_{i},\hat{n}_{j}\\}$.
Therefore, the 6d ambient theory dual to the 1PV is deformed by the same
sources as the theories dual to completely generic $\gamma\to-\infty$ bulk
solutions, and so, in this limit, there is no smooth deformation of the field
theory parameters that restores the ambient conformal symmetry in full. This
is to be contrasted with the global AdS${}_{7}\times\mathds{S}^{4}$ solution,
which enjoys the full $SO(6,2)$ conformal symmetry. Lastly, in taking the 1PV
limit, the solution exhibits a flavor symmetry enhancement $SO(4)\to SO(5)$.

In the construction of Faedo:2020nol  reviewed in section 2, the 1PV solution
can be obtained from eq. (3) by taking $\alpha\rightarrow 0$ and setting
$g^{3}=2\sqrt{2}/h_{1}\hat{\xi}$. Alternatively, it can also be obtained in
the limit $g\rightarrow 0$. To see this, first make a scale transformation
using the scaling symmetry given by eq. (41), take $g$ to scale with $\lambda$
as $g^{3}=2\sqrt{2}/h_{1}\hat{\xi}\lambda^{3}$, and then take
$\lambda\rightarrow\infty$.

###  4.3 Partition data

In this subsection, our aim is to identify within the $\gamma\to-\infty$
solutions of eq. (47) a basis of independent, non-contractible cycles threaded
by four-form flux. Integrating the flux along these cycles will allow us to
compute the integral M-brane charges that label a supergravity solution. In
turn, this characterization will enable us to recast the specification of a
supergravity solution in the form of a partition containing the representation
data associated to the defect string in the dual gauge theory, in analogy with
the Wilson surfaces of the $\gamma=-1/2$ solutions Estes:2018tnu . We begin by
searching for independent, non-contractible four-cycles through the 11d
geometry of the pairwise-collapsed $\gamma\to-\infty$ solutions. Later, we
will comment on how the cycles are modified if less generic collapse dynamics
are considered.

It is straightforward to see that, in the $\gamma\rightarrow-\infty$ limit,
the volume of $\mathds{S}^{3}$ vanishes on
$\partial\Sigma_{2}-\\{\nu_{j}\\}_{j=1}^{2n+2}$, i.e. all along the boundary
of the Riemann surface, except at the locations of the collapse points
$\nu_{j}$. In turn, any open curve on $\Sigma_{2}$ with end points on
$\partial\Sigma_{2}-\\{\nu_{j}\\}_{j=1}^{2n+2}$ will be a closed curved in the
full 11d geometry. Therefore, any curve encircling at least one of the
distinct collapse midpoints $\nu_{j}$ will not be contractible to a point.
This observation allows us to build a basis for non-contractible four-cycles
$\mathfrak{C}_{a}$, by taking the product of such curves on $\Sigma_{2}$ with
$\tilde{\mathds{S}}^{3}$, i.e.

$\mathfrak{C}_{a}\equiv\\{Re^{i\theta}-\nu_{2a-1}\,|\,0\leq\theta\leq\pi\\}\times\tilde{\mathds{S}}^{3}\,,$
(60)

for $1\leq a\leq n+1$. Note that any curve enveloping multiple $\nu_{a}$’s can
be decomposed into a linear combination of curves enclosing a single collapse
point; hence, to build an irreducible basis of cycles, we take the radius $R$
above such that $\mathfrak{C}_{a}$ encircles only a single collapse point
$\nu_{a}$.

The $\mathfrak{C}_{a}$ cycles alone do not exhaust the set of all non-
contractible four-cycles. Indeed, we can consider a four-cycle constructed
from a curve on $\Sigma_{2}$ connecting singular points $\nu_{a}$. For
irreducibility, we only consider curves connecting neighboring collapse
points. While the singular nature of their endpoints might make such curves
appear problematic at first glance, building a regular four-cycle from such a
curve is possible as the volume of $\mathds{S}^{3}$ vanishes at both
endpoints, while the volume of $\tilde{\mathds{S}}^{3}$ remains finite.
Indeed, the metric factor of $\mathds{S}^{3}$ contains $h_{1}H^{-1/3}$, while
the metric factor of $\tilde{\mathds{S}}^{3}$ has $\rho^{2}H^{2/3}$, and from
eq. (47), the function $H$ goes as $\rho^{-3}$ near any collapse point
$\nu_{a}$. Thus, the cycle

$\mathfrak{C}^{\prime}_{a}\equiv\left\\{\frac{1}{2}(\nu_{2a+1}-\nu_{2a-1})e^{i\theta}+\frac{1}{2}(\nu_{2a+1}+\nu_{2a-1})\,\Big{|}\,0\leq\theta\leq\pi\right\\}\times\mathds{S}^{3}~{},$
(61)

has the desired behavior of a non-contractible four-cycle. The distinction
between the $\mathfrak{C}_{a}$ and $\mathfrak{C}_{a}^{\prime}$ cycles is
illustrated in figure 3. Given that, by construction, they connect neighboring
collapse points, there are $n$ distinct such cycles
$\mathfrak{C}_{a}^{\prime}$. Combining them with the $n+1$ cycles
$\mathfrak{C}_{a}$, we can thus build a basis consisting of $2n+1$ non-
contractible four-cycles in the solutions defined by eq. (47).

$\Sigma_{2}$$z$$\rho$$\nu_{2a-3}$$\nu_{2a-1}$$\nu_{2a+1}$$\mathfrak{C}^{\prime}_{a}$$\mathfrak{C}_{a}$
Figure 3: The profile of the non-contractible four-cycles $\mathfrak{C}_{a}$
and $\mathfrak{C}^{\prime}_{a}$ in $\Sigma_{2}$. Every point on the red and
blue curves is a 3-sphere.

Having identified a basis for non-contractible four-cycles, we are now in a
position to derive the M-brane charges by integrating the four-form flux over
the $\mathfrak{C}_{a}$. To ease the computation, we abstractly write the flux
as Faedo:2020nol

$\begin{split}\mathcal{F}_{(4)}&=2h_{1}\operatorname{vol}_{\text{AdS}_{3}}\wedge\text{d}z+2h_{1}\operatorname{vol}_{\mathds{S}^{3}}\wedge\text{d}z\\\
&+\partial_{z}H\rho^{3}\text{d}\rho\wedge\operatorname{vol}_{\tilde{\mathds{S}}^{3}}-\partial_{\rho}H\rho^{3}\text{d}z\wedge\operatorname{vol}_{\tilde{\mathds{S}}^{3}}~{}.\end{split}$
(62)

The pull-back of the four-form field strength $\mathcal{F}_{(4)}$ onto any of
the $\mathfrak{C}_{a}$ cycles eliminates the first two terms in eq. (62).
Integrating along a given $\mathfrak{C}_{a}$ then yields

$\int_{\mathfrak{C}_{a}}P_{\mathfrak{C}_{a}}[\mathcal{F}_{(4)}]=2h_{1}\operatorname{Vol}({\tilde{\mathds{S}}^{3}})(\hat{\xi}_{2a}-\hat{\xi}_{2a-1})\,.$
(63)

The choice of orientation we made while defining the cycles $\mathfrak{C}_{a}$
ensures that the integral above is positive. This enables us to define the
following charge

$\displaystyle M_{a}$
$\displaystyle=\frac{1}{2(4\pi^{2}G_{N})^{1/3}}\int_{\mathfrak{C}_{a}}P_{\mathfrak{C}_{a}}[\mathcal{F}_{(4)}]\,,$
(64)

which can be interpreted as the number of M5-branes in the $a^{\rm th}$ stack
Bachas:2013vza , which obey $\sum_{a}M_{a}=M$ with $M$ being the total number
of M5-branes.

It is also possible to generalize the construction above. As we mentioned
previously, we can in fact build four-cycles surrounding more than one
collapse point $\nu_{a}$. The four-cycle defined by
$\mathfrak{C}_{bc}\equiv\sum_{a=b}^{c}\mathfrak{C}_{a}$, with $1\leq b\leq
c\leq n+1$, is also non-contractible by construction, and is characterized by
a charge

$\int_{\mathfrak{C}_{bc}}P_{\mathfrak{C}_{bc}}[\mathcal{F}_{(4)}]=2h_{1}\operatorname{Vol}({\tilde{\mathds{S}}^{3}})\sum_{a=b}^{c}(\hat{\xi}_{2a}-\hat{\xi}_{2a-1})\,,$
(65)

under the four-form field strength.

We can follow a similar analysis for the flux threading the other set of non-
contractible four-cycles, which we labelled $\mathfrak{C}^{\prime}_{a}$
earlier. In this case, only the second term in eq. (62) provides a non-
vanishing contribution after pulling back the four-form field strength
$\mathcal{F}_{(4)}$ to the cycle $\mathfrak{C}^{\prime}_{a}$. Integrating the
flux through $\mathfrak{C}^{\prime}_{a}$ gives

$\int_{\mathfrak{C}^{\prime}_{a}}P_{\mathfrak{C}^{\prime}_{a}}[\mathcal{F}_{(4)}]=2h_{1}\operatorname{Vol}({\mathds{S}^{3}})(\nu_{2a+1}-\nu_{2a-1})~{}.$
(66)

Similar to the integrated fluxes through the $\mathfrak{C}_{a}$ cycles, we
define the charge

$\displaystyle M^{\prime}_{a}$
$\displaystyle=\frac{1}{2(4\pi^{2}G_{N})^{1/3}}\int_{\mathfrak{C}^{\prime}_{a}}P_{\mathfrak{C}^{\prime}_{a}}[\mathcal{F}_{(4)}]\,,$
(67)

which is read as the number of M5′-branes in the $a^{\rm th}$ stack (see Table
1).

Again, we can easily generalize this analysis to four-cycles that connect non-
neighboring collapse points $\nu_{a}$. Denoting the sum of four-cycles as
$\mathfrak{C}^{\prime}_{bc}\equiv\sum_{a=b}^{c}\mathfrak{C}^{\prime}_{a}$,
where given the construction of the $\mathfrak{C}_{a}^{\prime}$ in eq. (66)
$1\leq b\leq c\leq n$, the integral of the flux through this cycle is simply

$\int_{\mathfrak{C}^{\prime}}P_{\mathfrak{C}^{\prime}}[\mathcal{F}_{(4)}]=2h_{1}\operatorname{Vol}({\mathds{S}^{3}})\sum_{a}(\nu_{2a+1}-\nu_{2a-1})\,.$
(68)

In addition, following Bachas:2013vza  we can deduce the number of M2-branes
ending on the $a^{\text{th}}$ stack of M5-branes, which we denote

$N_{a}=\sum_{b=a}^{n}M^{\prime}_{b}=\frac{h_{1}\operatorname{Vol}({\mathds{S}^{3}})}{(4\pi^{2}G_{N})^{1/3}}\sum_{b=a}^{n}(\nu_{2b+1}-\nu_{2b-1})\,.$
(69)

One may notice how the definition of the collapse points $\nu_{a}$ influences
the properties of $N_{a}$. Indeed, since $\\{\nu_{a}\\}$ is an ordered set, we
see that $N_{a}\geq N_{b}$ for $a\leq b$. In other words, the set
$\\{N_{a}\\}$ forms a partition of the total number of M2-branes
$N=\sum_{a}N_{a}$, and so it is possible to define a Young diagram to encode
the brane charges as illustrated in figure 4(a).

$N_{1}$$M_{1}$$M_{2}$$M_{3}$$M_{4}$$N_{4}$ (a)

$N_{1}$$M_{1}$$M_{2}$$M_{3}$$N_{3}$ (b)

Figure 4: (a) Young diagram corresponding to the partition specifying a
$\gamma\to-\infty$ solution with $5$ distinct $\nu_{a}$ constructed by
pairwise collapse of 10 different $\xi_{j}$. (b) Young diagram obtained by
“multiwise” collapse of 10 different $\xi_{j}$ to $4$ distinct $\nu_{a}$.
Another way to realize the construction of (b) starts from the partition in
(a) and collapses $\nu_{2}=\nu_{3}$.

We can now recast the various moments defined in eqs. 52 and 96 in terms of
this partition data. Since only $\hat{m}_{1}$ appears in the asymptotic
expansions in appendix A, and so also in the physical quantities computed
using those expansions, we will not treat the higher $\hat{m}_{j}$ moments and
simply write

$\hat{m}_{1}=\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}=\frac{(4\pi^{2}G_{N})^{1/3}}{h_{1}\operatorname{Vol}({\tilde{\mathds{S}}^{3}})}M\,,$
(70)

which gives back the usual relation between $M$ and the length scale on the
$\mathds{S}^{4}$,

$L^{3}_{\mathds{S}^{4}}=\frac{h_{1}\hat{m}_{1}}{2}=\frac{(G_{N})^{1/3}}{(2\pi)^{4/3}}M\,.$
(71)

The $\hat{n}_{j}$ moments are a bit trickier to re-express in terms of $M_{a}$
and $N_{a}$, but one can show that the following relation holds for any $j$

$\displaystyle\sum_{k=0}^{j}(-1)^{j+k}\frac{j!}{k!(j-k)!}\nu_{2n+1}^{k}~{}\hat{n}_{j-k}=\left(\frac{G_{N}^{1/3}}{h_{1}\pi(2\pi)^{1/3}}\right)^{j+1}~{}\sum_{a=1}^{n+1}M_{a}N_{a}^{j}~{},$
(72)

where it is understood that $\hat{n}_{0}=\hat{m}_{1}$. However, we will only
require relations up to $j=2$ moving forward. Using the expressions above, we
can write

$\displaystyle\hat{n}_{1}^{2}-\hat{m}_{1}\hat{n}_{2}=\frac{G_{N}^{4/3}}{h_{1}^{4}\pi^{4}(2\pi)^{4/3}}\left[\left(\sum_{a=1}^{n+1}M_{a}N_{a}\right)^{2}-M\sum_{a=1}^{n+1}M_{a}N_{a}^{2}\right]~{},$
(73)

which will be useful in the following section.

Before moving on, we recall from our previous discussion that it is also
possible to consider solutions where more than two branch points $\xi_{j}$
collapse to a single point $\nu_{a}$ and, of course, build non-contractible
four-cycles around or connecting them. Let us again index the $p$ distinct
loci of collapse as $\nu_{a}$, where now $1\leq a\leq p+1$ with $p\leq n$. The
limiting case $p=n$ recovers the pairwise collapse described above. It will be
useful to label $I_{a}$ and $K_{a}$ respectively as the smallest and largest
$j$-indices of the $\xi_{j}$ branch points which collapse to a given
$\nu_{a}$, i.e. $\nu_{a}:=\nu_{I_{a}}=\nu_{I_{a}+1}=\cdots=\nu_{K_{a}}$. The
ordering of the collapse points $\nu_{a}$ and of the branch points $\xi_{j}$
was discussed previously in section 4; in particular, we recall that the
parameters $\hat{\xi}_{j}$’s associated to the branch points collapsing to the
same $\nu_{a}$ are ordered amongst themselves. In the end, the analysis for
these “multiwise” collapse solutions is identical to the one presented above
for the pairwise collapse scenario, up to the replacements

$\displaystyle\hat{\xi}_{2a}-\hat{\xi}_{2a-1}\quad\longrightarrow\quad\sum_{j=I_{a}}^{K_{a}}(-1)^{j}\hat{\xi}_{j}~{}$
(74)

and $n\to p$ throughout the expressions above. The net effect is that the
partition data yields a differently shaped Young diagram, illustrated in
figure 4(b). In particular, since the multiwise collapse can greatly reduce
the number of singular points on the boundary of the Riemann surface, we see
that the Young tableaux specifying the $\gamma\to-\infty$ solutions have at
most $p$ rows. This is a striking difference compared to the Young tableau
construction of Estes:2018tnu  for the Wilson surface solutions with
$\gamma=-1/2$, whose associated Young tableaux always have $n$ rows. This
difference is maximal in the 1PV solution described in section 4.2, to which
one cannot attach any Young tableau interpretation at all. This is due to the
fact that the 1PV solution features a single collapse point $\nu$, so that the
construction of the ${\mathfrak{C}}_{a}^{\prime}$ cycle fails, thus preventing
the definition of the $M_{a}^{\prime}$ and $N_{a}$ charges.

##  5 Entanglement entropy of small $\mathcal{N}=(4,4)$ surface defects

The entanglement entropy $S_{\text{EE}}$ of a spatial subregion $\mathcal{B}$
within a QFT is defined as the von Neumann entropy of the reduced density
matrix obtained by tracing out the states in the complementary region
$\overline{\mathcal{B}}$ of the QFT. For CFTs with weakly coupled gravity
duals, the RT prescription Ryu:2006bv ; Ryu:2006ef ; Nishioka:2021uef
holographically recasts the computation of $S_{\text{EE}}$ into the following
Plateau problem in the asymptotically AdS bulk,

$S_{\text{EE}}=\min_{\zeta}\frac{\mathcal{A}[\zeta]}{4G_{\text{N}}}~{},$ (75)

where $\mathcal{A}[\zeta]$ is the area functional evaluated on a bulk
hypersurface $\zeta$ which is homologous to the chosen spatial subregion in
the dual CFT: $\zeta\cup\mathcal{B}=\partial b$ for some static bulk subregion
$b$. We will denote this extremal bulk hypersurface as $\zeta_{\text{RT}}$. In
the computations below, we choose the spatial subregion to be an Euclidean
5-ball, $\mathcal{B}=\mathbb{B}_{R}^{5}\hookrightarrow\mathbb{R}^{5}$, with
radius $R$. We take $\mathcal{B}$ to be centered on the spatial extent of the
surface defect, which has a Lorentzian worldvolume
$\Upsilon_{2}=\mathbb{R}^{1,1}$.

In an ordinary QFT, the presence of highly entangled UV degrees of freedom
induces short-distance divergences near the surface
$\partial\mathcal{B}=\mathds{S}^{4}$. Most of the divergences in the EE of a
general QFT are non-universal and shape-dependent. However, in even
dimensional theories, there are universal log-divergent contributions to the
EE, which are generically related at conformal fixed points to the Weyl
anomalies of the CFT.

This is true in the presence of a defect as well, but we see additional
divergences that arise from the defect degrees of freedom near
$\Upsilon_{2}\cap\partial\mathcal{B}$. In order to isolate these defect-
localized contributions, we will adopt a scheme where we subtract off the EE
of the undeformed, vacuum ambient CFT, $S_{\rm EE}[\emptyset]$, from the EE
computed in the presence of the defect $S_{\rm EE}[\Upsilon_{2}]$. We can then
extract the coefficient of the universal, log-divergent part of the defect
contribution to the sphere EE by

$\displaystyle S_{\rm EE}^{\rm(univ)}$
$\displaystyle=R\frac{\text{d}}{\text{d}R}\left(S_{\text{EE}}[\Upsilon_{2}]-S_{\text{EE}}[\emptyset]\right)\big{|}_{R\to
0}~{}.$ (76)

In Jensen:2018rxu , it was shown that contribution to the log-divergent part
of the EE of a spherical region coming from a flat, $2$d conformal defect
embedded in a $d$-dimensional flat-space ambient CFT takes the form of a
linear combination of defect localized Weyl anomalies. Explicitly, for a $6$d
ambient CFT

$\displaystyle S_{\rm
EE}^{\rm(univ)}=\frac{1}{3}\left(a_{\Upsilon}-\frac{3}{5}d_{2}\right)~{}.$
(77)

where $a_{\Upsilon}$ is the A-type defect Weyl anomaly coefficient–i.e.
appearing with the intrinsic Euler density–and $d_{2}$ is the B-type anomaly
that enters with the trace of the pullback of the ambient Weyl tensor.
Importantly, while Jensen:2018rxu  demonstrated that $d_{2}\geq 0$ based on
energy conditions, $a_{\Upsilon}$, though obeying a defect “c-theorem”, has no
positivity constraints. This means that as opposed to an ordinary, unitary
$2$d CFT where the universal part of the EE is is proportional to the central
charge Holzhey:1994we  and, hence, is non-negative, it is clear from eq. (77)
that $S_{\rm EE}^{\rm(univ)}$ is not similarly bounded nor is it RG monotonic.

In completing the holographic computation, we also need to contend with the
fact that the FG expansion is not globally defined as it typically breaks down
in a region near the AdS submanifold that is dual to the insertion of the
defect in the field theory. However, we have a full analysis of the asymptotic
expansions of the data specifying the $\gamma\to-\infty$ solutions in appendix
A, and we have the general prescription for the FG transformation suitable for
defect EE in Estes:2014hka . Together, we will be able to unambiguously
holographically compute $S_{\rm EE}^{\rm(univ)}$.

To begin the holographic computation of the defect EE, we choose the following
parametrization for the AdS3 subspace in eq. (51),

$\text{d}s^{2}_{\text{AdS}_{3}}=\frac{1}{u^{2}}\left(\text{d}u^{2}-\text{d}t^{2}+\text{d}x^{2}_{\parallel}\right).$
(78)

Furthermore, we take the RT hypersurface $\zeta$ to wrap both $\mathds{S}^{3}$
and $\tilde{\mathds{S}}^{3}$, and its profile in the remaining subspace to be
described by $x_{\parallel}(u,\rho,z)$. The area of $\zeta$ as measured
against the metric in eq. (51) is then

$A[\zeta]=\operatorname{Vol}(\mathds{S}^{3})\operatorname{Vol}(\tilde{\mathds{S}}^{3})\int\text{d}u\int_{\Sigma_{2}}\
\text{d}\rho\text{d}z\ \mathcal{L}~{},$ (79)

where the Lagrangian is

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{h_{1}^{2}\rho^{3}}{u^{2}}\left[h_{1}\left((\partial_{\rho}x_{\parallel})^{2}+(\partial_{z}x_{\parallel})^{2}\right)H(z,\rho)+u^{2}\left(1+(\partial_{u}x_{\parallel})^{2}\right)H(z,\rho)^{2}\right]^{1/2}.$
(80)

As shown in Jensen:2013lxa , the minimal area surface wraps the Riemann
surface $\Sigma_{2}$ too, so that
$\partial_{\rho}x_{\parallel}=\partial_{z}x_{\parallel}=0$. The Lagrangian is
thus minimized by

$x_{\parallel}^{2}+u^{2}=R^{2},$ (81)

for a constant $R$, so that the area of the extremal hypersurface
$\zeta_{\text{RT}}$ is

$A[\zeta_{\text{RT}}]=h_{1}^{3}\operatorname{Vol}(\mathds{S}^{3})\operatorname{Vol}(\tilde{\mathds{S}}^{3})\log\left(\frac{2R}{\epsilon_{u}}\right)\int_{\Sigma_{2}}\text{d}\rho\text{d}z\
\rho^{3}\sum_{j=1}^{2n+2}\frac{(-1)^{j}\hat{\xi}_{j}}{\left(\rho^{2}+(z-\nu_{j})^{2}\right)^{3/2}}+\mathcal{O}(\epsilon_{u}^{2}),$
(82)

where we introduced a small-$u$ cut-off $\epsilon_{u}>0$.

The evaluation of the integral above is performed in detail in appendix B. The
resulting holographic entanglement entropy is

$S_{\text{EE}}[\Upsilon_{2}]=\frac{\pi^{4}L_{\mathds{S}^{4}}^{9}}{G_{\text{N}}}\log\left(\frac{2R}{\epsilon_{u}}\right)\left[\frac{64}{3}\frac{1}{\epsilon_{v}^{4}}+\frac{16}{5}\frac{\hat{n}_{1}^{2}}{\hat{m}_{1}^{4}}-\frac{16}{5}\frac{\hat{n}_{2}}{\hat{m}_{1}^{3}}+\mathcal{O}(\epsilon_{v}^{2})\right]+\mathcal{O}(\epsilon_{u}^{2}),$
(83)

where $\epsilon_{v}>0$ is a small-$v$ cutoff in the FG parametrization.

Subtracting off the 1PV contribution to the entanglement entropy,
$S_{\text{EE}}^{\text{1PV}}$, precisely removes the $\epsilon_{v}^{-4}$
divergence from eq. (83), and leaves the $\mathcal{O}(\epsilon_{v}^{0})$ term
unchanged. To see this, we recall that the 1PV limit takes $\hat{n}_{k}\to 0$,
which in eq. (83) gives

$\displaystyle S_{\text{EE}}^{\rm
1PV}=\frac{\pi^{4}L_{\mathds{S}^{4}}^{9}}{G_{\text{N}}}\log\left(\frac{2R}{\epsilon_{u}}\right)\left[\frac{64}{3}\frac{1}{\epsilon_{v}^{4}}+\mathcal{O}(\epsilon_{v}^{2})\right]+\mathcal{O}(\epsilon_{u}^{2}).$
(84)

Therefore, we can at once compute the coefficient of the universal part of the
defect sphere EE using eq. (76) and plugging in eqs. (83) and (84) with
$S_{\rm EE}[\emptyset]=S_{\rm{EE}}^{\rm 1PV}$ to find

$\displaystyle S_{\rm EE}^{\rm(univ)}$
$\displaystyle=\frac{16\pi^{4}L_{\mathds{S}^{4}}^{9}}{5G_{\text{N}}}\frac{\hat{n}_{1}^{2}-\hat{m}_{1}\hat{n}_{2}}{\hat{m}_{1}^{4}}$
(85a)
$\displaystyle=\frac{1}{5M}\left[\left(\sum_{a=1}^{n+1}M_{a}N_{a}\right)^{2}-M\sum_{a=1}^{n+1}M_{a}N_{a}^{2}\right],$
(85b)

where we have mapped to field theory quantities using
$L_{\mathds{S}^{4}}^{3}=\frac{G_{N}^{1/3}}{(2\pi)^{4/3}}M$ and used the
definitions of moments $\hat{n}_{j}$, $\hat{m}_{j}$ in terms of the numbers of
branes in eqs. (70) and (73). In terms of the highest weight $\varpi$ of the
$A_{M-1}$ irreducible representation encoded in the Young diagrams that
specify the defect discussed in the previous section, we can re-express the
defect sphere EE as Estes:2018tnu

$\displaystyle S_{\rm EE}^{\rm(univ)}=-\frac{(\varpi,\varpi)}{5}~{},$ (86)

where $(\cdot,\cdot)$ is the scalar product on the weight space induced by the
Killing form.

Finally, note that we can trivially compute the same quantity in the
orbifolded theory as described in section 3.3 simply by rescaling

$\displaystyle S_{\rm
EE}^{\rm(univ)}\longrightarrow\frac{\operatorname{Vol}(\mathds{S}^{3}/\mathbb{Z}_{k})\operatorname{Vol}(\tilde{\mathds{S}}^{3}/\mathbb{Z}_{k^{\prime}})}{\operatorname{Vol}(\mathds{S}^{3})\operatorname{Vol}(\tilde{\mathds{S}}^{3})}S_{\rm
EE}^{\rm(univ)}=\frac{S_{\rm EE}^{\rm(univ)}}{kk^{\prime}}~{}.$ (87)

##  6 Summary and Outlook

In this work, we have constructed a novel class of solutions in 11d SUGRA that
are holographically dual to 2d superconformal defects preserving small
$\mathcal{N}=(4,4)$ and $\mathcal{N}=(0,4)$ SUSY in 6d SCFTs at large $M$.
These solutions fit into the one-parameter family organized in a general
classification scheme of 11d SUGRA solutions with superisometry
$\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$ Bachas:2013vza ;
specifically, they are obtained in the $\gamma\to-\infty$ limit. There are
several features of these new solutions that separate them from the more
familiar $\gamma=-1/2$ case that holographically corresponds to $1/2$-BPS
Wilson surface type defects in the 6d $\mathcal{N}=(2,0)$ $A_{M-1}$ SCFT.

Within the one-parameter family of solutions labelled by $\gamma$, the
$\gamma\to-\infty$ limit is slightly unusual from the superalgebra
perspective. Despite producing an AdS${}_{7}\times\mathds{S}^{4}$ asymptotic
geometry as shown in section 3, taking the $\gamma\to-\infty$ limit means that
$\mathfrak{d}(2,1;\gamma)\oplus\mathfrak{d}(2,1;\gamma)$ is not a subalgebra
of the $\mathfrak{osp}(8^{*}|4)$ superisometry of
AdS${}_{7}\times\mathds{S}^{4}$. On the field theory side of the holographic
duality, this means that the ambient theory into which the defects are
inserted is some deformation of the 6d $\mathcal{N}=(2,0)$ SCFT.

We have seen that choosing all of the singular loci in the internal space to
collapse to a single point – a configuration which we label 1PV – destroys the
data that specifies the defect, i.e. the Young diagram corresponding to the
arrangement of M5-branes. However, as is clear from the discussion in section
4.2, the vacuum that we arrive at has an isometry group of $SO(2,2)\times
SO(3)\times SO(5)$, as opposed to the vacuum solution at $\gamma=-1/2$, which
instead enjoys the full $SO(6,2)$. In the latter case, the trivial defect
corresponds to a Wilson surface transforming in the $\bf{1}$ of $A_{M-1}$, and
the conformal symmetry of the ambient 6d $\mathcal{N}=(2,0)$ theory is
restored. For $\gamma\to-\infty$, the trivial defect dual to the 1PV still
possesses what looks like the ‘defect’ conformal symmetry despite being the
vacuum solution, which renders giving a precise definition for and
interpretation of the defect CFT difficult. The 1PV does, however, enable us
to correctly employ a background subtraction scheme and arrive at a finite
result for $S_{\rm EE}^{\rm(univ)}$ and, we believe, resolves the puzzling
appearance of divergences in the “defect central charge” computed in
Faedo:2020nol .

On a more fundamental level, the small $\mathcal{N}=(4,4)$ defects at
$\gamma\to-\infty$ cannot be viewed as a smooth deformation of the Wilson
surface defects at $\gamma=-1/2$. Indeed, the $\gamma\to-\infty$ solutions
cannot even be smoothly deformed into the solution at $\gamma=0$, to which
they are related by the involution $\gamma\mapsto 1/\gamma$ with an exchange
of the $\mathfrak{so}(3)\oplus\mathfrak{so}(3)$ factors in
$\mathfrak{d}(2,1;\gamma;0)\oplus\mathfrak{d}(2,1;\gamma;0)$. The reason is
that there is a special point at $\gamma=-1$ where the real form
$\mathfrak{d}(2,1;\gamma;0)$ becomes $\mathfrak{osp}(4|2;\mathbb{R})$. At this
value of $\gamma$, $SO(2,2)$ Wigner-İnönü contracts to $ISO(1,2)$, and AdS3
becomes $\mathbb{R}^{2,1}$. Therefore, the $\gamma\to-\infty$ solutions are
isolated from the other class of asymptotically
AdS${}_{7}\times\mathds{S}^{4}$ geometries.

In light of the new small $\mathcal{N}=(4,4)$ solutions that we have
constructed and holographically studied, there are a number of open questions
that remain to be answered.

Firstly, as we discussed at the start of section 5 , the contribution from a
flat 2d conformal defect to the log-divergent, universal part of the EE of a
spherical region in a $d\geq 4$ ambient CFT is built from a linear combination
of two defect Weyl anomaly coefficients, $a_{\Upsilon}$ and $d_{2}$ that
characterize the defect theory. In order to disentangle these two fundamental
defect quantities, we would compute a second holographic quantity that
contains either $a_{\Upsilon}$ or $d_{2}$. For instance, $d_{2}$ controls the
normalization of the one-point function $\left<T_{\mu\nu}\right>$ of the
stress-energy tensor, which can be readily computed for most 10d or 11d
supergravity solutions by dimensional reduction on the internal space
deHaro:2000vlm ; Skenderis:2006uy . Therefore, it is natural to try to compute
$d_{2}$ and $S_{\rm EE}^{\rm(univ)}$ in order to isolate the independent
defect Weyl anomalies101010In fact, $a_{\Upsilon}$ and $d_{2}$ are the only
independent defect Weyl anomaly coefficients for superconformal defects
preserving at least 2d $\mathcal{N}=(0,2)$ supersymmetry. This was shown for
co-dimension four defects in 6d SCFTs in Drukker:2020atp  and co-dimension two
defects in 4d SCFT in Bianchi:2019sxz . . This was successfully done for the
Wilson surfaces at $\gamma=-1/2$ in Estes:2018tnu  and for codimension-2
defects in Capuozzo:2023fll . However, the $\gamma\to-\infty$ solutions are
more subtle, and a naïve application of dimensional reduction techniques would
be inappropriate. Namely, in reducing the 11d solutions to 7d, the presence of
the non-trivial four-form flux modifies the gravitational equations of motion
at the conformal boundary of AdS7, which violates the assumptions in
deHaro:2000vlm . Therefore, computing $d_{2}$ holographically from
$\left<T_{\mu\nu}\right>$ requires a generalization to account for flux
contributions, which is the subject of ongoing work.

Furthermore, it is natural to look for physical observables which can be
reliably computed on the field theory side and employed to test the
holographic predictions made above. For 2d BPS conformal defects in 6d
$A_{M-1}$ and $D_{M}$ $\mathcal{N}=(2,0)$ SCFTs at large $M$, recent
developments in analytic bootstrap methods have enabled the computation of
correlations functions in the presence of 2d defects that are controlled by
anomalies Drukker:2020atp ; Meneghelli:2022gps . Further, despite the lack of
a Lagrangian description and of supersymmetric localization methods for 6d
SCFTs at large $M$, chiral algebra methods have also been shown to give exact
results for defect correlators Meneghelli:2022gps  and the defect SUSY Casimir
energy Bullimore:2014upa ; Chalabi:2020iie . Currently, only Wilson surface
type defects, i.e. the holographically dual theories to the $\gamma=-1/2$
solutions, have been studied using these field theory techniques. It is
reasonable to wonder whether any of these methods are applicable to the types
of defects in the deformed 6d theory that we have constructed in this work.

The biggest hurdle to clear in trying to generalize bootstrap or chiral
algebra methods for use in the dual to the 1PV of the $\gamma\to-\infty$
solutions is clarifying the precise role of the deformation parameter
$\gamma$. As we have explained in the $\gamma\to-\infty$ limit, the ambient 6d
theory has reduced global and conformal symmetries, and there is no smooth
path in field theory space as $\gamma$ is varied through $\gamma=-1$ to get
from the 6d $A_{M-1}$ $\mathcal{N}=(2,0)$ theory to the dual of the 1PV. It is
unclear at the moment precisely what symmetry breaking operators are sourced
on the field theory side in the $\gamma\to-\infty$ limit.

## Acknowledgments

The authors would like to thank Yolanda Lozano and Nicolò Petri for useful
discussions during the completion of this work and for their comments on a
draft of this manuscript. The authors would also like to thank Andy O’Bannon
for his contributions during the early phase of this research. The work of PC
is supported by a Mayflower studentship from the University of Southampton.
The work of BR is supported by the INFN. The work of BS is supported in part
by the STFC consolidated grant ST/T000775/1. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under Award Number DE-SC0024557.

Disclaimer: “This report was prepared as an account of work sponsored by an
agency of the United States Government. Neither the United States Government
nor any agency thereof, nor any of their employees, makes any warranty,
express or implied, or assumes any legal liability or responsibility for the
accuracy, completeness, or usefulness of any information, apparatus, product,
or process disclosed, or represents that its use would not infringe privately
owned rights. Reference herein to any specific commercial product, process, or
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necessarily constitute or imply its endorsement, recommendation, or favoring
by the United States Government or any agency thereof. The views and opinions
of authors expressed herein do not necessarily state or reflect those of the
United States Government or any agency thereof.”

##  Appendix A Fefferman-Graham parametrization

In this Appendix, we will be concerned with finding a reparametrization
$\\{w,\bar{w}\\}\to\\{v,\phi\\}$ of the Riemann surface $\Sigma_{2}$ under
which the 11d SUGRA solution in eq. (18) can be asymptotically (in particular,
for small $v$) recast into the following form,

$\text{d}s^{2}=\frac{4L_{\mathds{S}^{4}}^{2}}{v^{2}}\left[\text{d}v^{2}+\alpha_{1}\text{d}s^{2}_{\text{AdS}_{3}}+\alpha_{2}\text{d}s^{2}_{\mathds{S}^{3}}\right]+L_{\mathds{S}^{4}}^{2}\left[\alpha_{3}\text{d}\phi^{2}+\alpha_{4}s^{2}_{\phi}\text{d}s^{2}_{\tilde{\mathds{S}}^{3}}\right]~{},$
(88)

for some $\mathcal{O}(v^{0})$ metric factors $\\{\alpha_{i}\\}_{i=1}^{4}$. By
imposing that the metric factor in the $v$ direction be exact in $v$, as
above, we can reconstruct the asymptotic reparametrization order by order in
$v$ and in terms of polar coordinates

$\displaystyle
r=\sqrt{z^{2}+\rho^{2}}\quad\text{and}\quad\theta=\arctan(\rho/z)$ (89)

on $\Sigma_{2}$ to be the following,

$\displaystyle\allowdisplaybreaks r(v,\phi)$
$\displaystyle=-\frac{2(\gamma+1)^{2}m_{1}}{\gamma
v^{2}}+\frac{(2\gamma+1)m_{1}\left(c_{2\phi}-3\right)}{24\gamma}+\frac{m_{2}c_{\phi}}{2m_{1}}+\Biggl{[}\frac{3\gamma^{2}m_{2}^{2}\left(7c_{2\phi}+1\right)}{4m_{1}^{3}}$
(90a)
$\displaystyle\qquad-\frac{\gamma^{2}m_{3}\left(5c_{2\phi}+3\right)}{m_{1}^{2}}-\frac{2\gamma(2\gamma+1)m_{2}c_{\phi}s_{\phi}^{2}}{m_{1}}-\frac{1}{16}(8\gamma(\gamma+1)+3)m_{1}c_{2\phi}$
$\displaystyle\qquad+\frac{37}{48}\gamma(\gamma+1)m_{1}+\frac{73m_{1}}{192}+\frac{19}{192}(2\gamma+1)^{2}m_{1}c_{4\phi}\Biggr{]}\frac{v^{2}}{48\gamma(\gamma+1)^{2}}+\mathcal{O}(v^{4}),$
$\displaystyle\theta(v,\phi)$
$\displaystyle=\phi+\frac{(2\gamma+1)m_{1}^{2}c_{\phi}+3\gamma
m_{2}}{12(\gamma+1)^{2}m_{1}^{2}}s_{\phi}v^{2}+\Biggl{[}\frac{9\gamma^{2}m_{2}^{2}s_{2\phi}}{8m_{1}^{4}}-\frac{5\gamma^{2}m_{3}s_{2\phi}}{6m_{1}^{3}}$
(90b)
$\displaystyle\qquad+\frac{c_{\phi}s_{\phi}\left(5(2\gamma+1)^{2}c_{2\phi}-24\gamma(\gamma+1)-7\right)}{48}$
$\displaystyle\qquad+\frac{\gamma(2\gamma+1)m_{2}\left(3c_{2\phi}-1\right)s_{\phi}}{12m_{1}^{2}}\Biggr{]}\frac{v^{4}}{16(\gamma+1)^{4}}+\mathcal{O}(v^{6})~{},$

where we introduced the moments

$m_{k}:=\sum_{j=1}^{2n+2}(-1)^{j}\xi_{j}^{k}~{}.$ (91)

Under this mapping, the metric takes the desired form of eq. (88) with the
following metric factors

$\displaystyle\allowdisplaybreaks\alpha_{1}(\gamma)$
$\displaystyle=1+\frac{2\gamma+3-(1+2\gamma)c_{2\phi}}{16(\gamma+1)^{2}}v^{2}+\left[9\left(4(3\gamma-13)\gamma+16\gamma^{2}\kappa-17\right)\right.$
(92a)
$\displaystyle\quad\left.-67(2\gamma+1)^{2}c_{4\phi}+12\left(8(3\gamma+1)\gamma+20\gamma^{2}\kappa+3\right)c_{2\phi}\right]\frac{v^{4}}{18432(\gamma+1)^{4}}+\mathcal{O}(v^{6})~{},$
$\displaystyle\alpha_{2}(\gamma)$
$\displaystyle=\frac{(\gamma+1)^{2}}{\gamma^{2}}+\frac{2\gamma-1-(1+2\gamma)c_{2\phi}}{16\gamma^{2}}v^{2}+\left[9\left(4(3\gamma+19)\gamma+16\gamma^{2}\kappa+47\right)\right.$
(92b)
$\displaystyle\quad\left.-67(2\gamma+1)^{2}c_{4\phi}+12\left(8(3\gamma+5)\gamma+20\gamma^{2}\kappa+19\right)c_{2\phi}\right]\frac{v^{4}}{18432\gamma^{2}(\gamma+1)^{2}}+\mathcal{O}(v^{6})~{},$
$\displaystyle\alpha_{3}(\gamma)$
$\displaystyle=1+\frac{(2\gamma+1)c_{\phi}^{2}}{4(\gamma+1)^{2}}v^{2}+\left[-12(\gamma+1)\gamma-24\gamma^{2}\kappa-9+6(2\gamma+1)^{2}c_{2\phi}\right.$
(92c)
$\displaystyle\quad\left.+7(2\gamma+1)^{2}c_{4\phi}\right]\frac{v^{4}}{768(\gamma+1)^{4}}+\mathcal{O}(v^{6})~{},$
$\displaystyle\alpha_{4}(\gamma)$
$\displaystyle=1+\frac{(2\gamma+1)(2c_{2\phi}+1)}{12(\gamma+1)^{2}}v^{2}+\left[-52(\gamma+1)\gamma-24\gamma^{2}\kappa-19\right.$
(92d)
$\displaystyle\quad\left.+79(2\gamma+1)^{2}c_{4\phi}+12\left(-2(\gamma+1)\gamma-10\gamma^{2}\kappa-3\right)c_{2\phi}\right]\frac{v^{4}}{4608(\gamma+1)^{4}}+\mathcal{O}(v^{6})~{},$

where, to the order shown, the moments $\\{m_{i}\\}$ enter the metric factors
only in the combination

$\kappa\equiv\frac{3m_{2}^{2}-4m_{1}m_{3}}{m_{1}^{4}}~{}.$ (93)

Note that $\kappa$ is invariant under coordinate transformations
$\\{\xi_{i}\mapsto\xi_{i}+\lambda\\}$, even though $m_{2}$ and $m_{3}$ are not
individually. From this mapping, we also deduce the curvature scale of the
asymptotic local $\mathds{S}^{4}$ in FG gauge,

$L_{\mathds{S}^{4}}^{3}=\frac{|1+\gamma|^{3}}{\gamma^{2}}\frac{h_{1}m_{1}}{2}~{}.$
(94)

We can now take the $\gamma\to-\infty$ limit, accompanied by appropriate
rescalings as described in section 4, to determine the asymptotic local
behavior of the metric in eq. (36). In this limit, we find that

$\frac{\kappa}{\gamma^{2}}\to\frac{12(\hat{n}_{1}^{2}-\hat{m}_{1}\hat{n}_{2})}{\hat{m}_{1}^{4}}~{},$
(95)

where we introduced the additional moments

$\hat{n}_{k}:=\sum_{j=1}^{2n+2}(-1)^{j}\nu_{j}^{k}\hat{\xi}_{j}~{}.$ (96)

Furthermore, we find that this limiting procedure trivializes the relative
warping between AdS3 and $\mathds{S}^{3}$; that is,
$\alpha_{1}(\gamma\to-\infty)=\alpha_{2}(\gamma\to-\infty)$. Overall, the
limit $\gamma\to-\infty$ produces the FG line element advertized in eq. (51),
with the following metric factors,

$\displaystyle\alpha_{1}(\gamma\to-\infty)$
$\displaystyle=1+\frac{(3+5c_{2\phi})(\hat{n}_{1}^{2}-\hat{m}_{1}\hat{n}_{2})}{32\hat{m}_{1}^{4}}v^{4}$
(97a)
$\displaystyle\quad-\frac{5(1+7c_{2\phi})(2\hat{n}_{1}^{3}-3\hat{m}_{1}\hat{n}_{1}\hat{n}_{2}+\hat{m}_{1}^{2}\hat{n}_{3})c_{\phi}}{144\hat{m}_{1}^{6}}v^{6}+\mathcal{O}(v^{8}),$
$\displaystyle\alpha_{3}(\gamma\to-\infty)$
$\displaystyle=1+\frac{3(\hat{m}_{1}\hat{n}_{2}-\hat{n}_{1}^{2})}{8\hat{m}_{1}^{4}}v^{4}+\frac{5(2\hat{n}_{1}^{3}-3\hat{m}_{1}\hat{n}_{1}\hat{n}_{2}+\hat{m}_{1}^{2}\hat{n}_{3})c_{\phi}}{12\hat{m}_{1}^{6}}v^{6}+\mathcal{O}(v^{8}),$
(97b) $\displaystyle\alpha_{4}(\gamma\to-\infty)$
$\displaystyle=1+\frac{(1+5c_{2\phi})(\hat{m}_{1}\hat{n}_{2}-\hat{n}_{1}^{2})}{16\hat{m}_{1}^{4}}v^{4}$
(97c)
$\displaystyle\quad+\frac{5(5c_{\phi}+7c_{3\phi})(2\hat{n}_{1}^{3}-3\hat{m}_{1}\hat{n}_{1}\hat{n}_{2}+\hat{m}_{1}^{2}\hat{n}_{3})}{144\hat{m}_{1}^{6}}v^{6}+\mathcal{O}(v^{8})~{},$

and with $\mathds{S}^{4}$ radius given by eq. (53).

Finally, we note that, if all points are taken to collapse to the origin,
$\nu_{j}=0$ for $j\in\\{1,2,\ldots,2n+2\\}$, the asymptotic reparametrization
in eq. (90) becomes exact,

$\displaystyle r(v,\phi)$ $\displaystyle=\frac{2\hat{m}_{1}}{v^{2}}$ (98a)
$\displaystyle\theta(v,\phi)$ $\displaystyle=\phi~{}.$ (98b)

##  Appendix B Area of the Ryu-Takayanagi hypersurface

In this appendix, we fill in the technical details of the computation of the
holographic entanglement entropy in eq. (83).

We begin by treating the integral in eq. (82). In order to exploit the
geometry of the internal space and facilitate an easier path to evaluating the
remaining integral, we first transform to polar coordinates
$\\{r,\,\theta\\}$, introduced in eq. (89), with $\theta\in[0,\pi]$. We can
then expand the summands in eq. (82) on the complete basis of $L^{2}(0,\pi)$
functions spanned by the Legendre polynomials, $P_{k}(c_{\theta})$. The
harmonic function $H$ can then be written in two equivalent representations

$H(r,\theta)=\begin{cases}\frac{h_{1}}{2}\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}\left|\nu_{j}\right|^{-3}\left(\sum_{k=0}^{\infty}r^{k}\nu_{j}^{-k}P_{k}(c_{\theta})\right)^{3},\quad
r\in[0,|\nu_{j}|),\vspace{0.2cm}\\\
\frac{h_{1}}{2}\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}r^{-3}\left(\sum_{k=0}^{\infty}r^{-k}\nu_{j}^{k}P_{k}(c_{\theta})\right)^{3},\quad
r\in(|\nu_{j}|,\Lambda_{r}(\epsilon_{v},0)],\end{cases}$ (99)

which converge when integrated in $r$ over their respective domains. We have
also introduced in the second line of eq. (99) a cutoff scale $\Lambda_{r}$ at
large $r$, which from the transformation to FG gauge in eq. (90) can be
expressed in a small $\epsilon_{v}$ expansion as

$\Lambda_{r}(\epsilon_{v},\theta)=\frac{2\hat{m}_{1}}{\epsilon_{v}^{2}}+\frac{\hat{n}_{1}c_{\theta}}{\hat{m}_{1}}+\frac{(3+5c_{2\theta})\hat{m}_{1}\hat{n}_{2}-(5+3c_{2\theta})\hat{n}_{1}^{2}}{16\hat{m}_{1}^{3}}\
\epsilon_{v}^{2}+\mathcal{O}(\epsilon_{v}^{4}).$ (100)

Thus, the remaining integral in the area functional can be partitioned into
two separate contributions

$A[\zeta_{\text{RT}}]=\frac{32\pi^{4}L_{\mathds{S}^{4}}^{9}}{\hat{m}_{1}^{3}}\log\left(\frac{2R}{\epsilon_{u}}\right)\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}\left(\mathcal{I}_{j}^{(1)}+\mathcal{I}_{j}^{(2)}\right)+\mathcal{O}(\epsilon_{u}^{2}),$
(101)

where both integrals

$\displaystyle\mathcal{I}_{j}^{(1)}$
$\displaystyle=\int_{0}^{\pi}\text{d}\theta\int_{0}^{\left|\nu_{j}\right|}\text{d}r\
r^{4}s^{3}_{\theta}\left(\frac{1}{\left|\nu_{j}\right|}\sum_{k=0}^{\infty}\left(\frac{r}{\nu_{j}}\right)^{k}P_{k}(c_{\theta})\right)^{3},$
(102a) $\displaystyle\mathcal{I}_{j}^{(2)}$
$\displaystyle=\int_{0}^{\pi}\text{d}\theta\int_{|\nu_{j}|}^{\Lambda_{r}(\epsilon_{v},\theta)}\text{d}r\
r\
s^{3}_{\theta}\left(\sum_{k=0}^{\infty}\left(\frac{\nu_{j}}{r}\right)^{k}P_{k}(c_{\theta})\right)^{3},$
(102b)

are convergent.

At first glance, eq. (102) may not seem to put us in a better position to
evaluate the integral, but we can exploit the properties of
$P_{k}(c_{\theta})$ over the interval $\theta\in[0,\pi]$. In particular, we
can make use of the orthogonality relation of the triple-product of Legendre
polynomials

$\int_{0}^{\pi}\text{d}\theta\ s_{\theta}\
P_{\ell_{1}}(c_{\theta})P_{\ell_{2}}(c_{\theta})P_{\ell_{3}}(c_{\theta})=2\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}^{2},$ (103)

where the right-hand side is the Wigner $3j$-symbol. Since this particular
$3j$-symbol has vanishing magnetic quantum numbers, if
$\ell\equiv\ell_{1}+\ell_{2}+\ell_{3}$ is even and together the $\ell_{i}$
satisfy the triangle inequality, then it can neatly be expressed as

$\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}=\frac{(-1)^{\ell/2}(\ell/2)!}{\sqrt{(\ell+1)!}}\prod_{i=1}^{3}\frac{\sqrt{(\ell-2\ell_{i})!}}{(\ell/2-\ell_{i})!}~{}.$
(104)

Otherwise, if $\ell$ is not even or the triangle inequality is violated, the
$3j$-symbol vanishes. Additionally, the following identity

$P_{\ell_{1}}P_{\ell_{2}}=\sum_{\ell_{3}=\lvert\ell_{1}-\ell_{2}\rvert}^{\ell_{1}+\ell_{2}}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}^{2}(2\ell_{3}+1)P_{\ell_{3}}~{},$ (105)

is particularly useful in the evaluation of the $\theta$-integrals in eq.
(102), where for brevity we denoted $P_{\ell}\equiv P_{\ell}(c_{\theta})$.
Eventually, after applying both eqs. (103) and (105), we find that the
$\theta$-integrals in $\mathcal{I}^{(1)}_{j}$ can be handled with the help of

$\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}=\frac{4}{3}\begin{pmatrix}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}^{2}-\frac{4}{3}\sum_{k=\lvert
2-\ell_{1}\rvert}^{2+\ell_{1}}(2k+1)\begin{pmatrix}2&\ell_{1}&k\\\
0&0&0\end{pmatrix}^{2}\begin{pmatrix}k&\ell_{2}&\ell_{3}\\\
0&0&0\end{pmatrix}^{2}.$ (106)

The integrals $\mathcal{I}_{j}^{(1)}$ and $\mathcal{I}_{j}^{(2)}$ may be
further simplified by considering the convergence properties of the sum, which
allow us to integrate each term in $r$ separately. Doing so, we rapidly see
the usefulness of the previous $\theta$-integral formulae, which appear
explicitly as

$\displaystyle\mathcal{I}_{j}^{(1)}=$
$\displaystyle\sum_{\ell_{1},\ell_{2},\ell_{3}}\frac{1}{5+\ell}\frac{\nu_{j}^{\ell+2}}{|\nu_{j}|^{\ell}}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}},$ (107a)
$\displaystyle\mathcal{I}_{j}^{(2)}=$
$\displaystyle\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\ \ell\neq
2\end{subarray}}\frac{\nu_{j}^{\ell}}{2-\ell}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}\left[\Lambda_{r}^{2-\ell}(\epsilon_{v},\theta)-\frac{\nu_{j}^{2}}{|\nu_{j}|^{\ell}}\right]$
$\displaystyle+\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\
\ell=2\end{subarray}}\nu_{j}^{2}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}\ln\left(\frac{\Lambda_{r}(\epsilon_{v},\theta)}{\lvert\nu_{j}\rvert}\right)\,,$
(107b)

where we have isolated the $\ell=2$ mode in $\mathcal{I}^{(2)}_{j}$ in order
to handle the potential log divergence as a separate case.

Starting with the sum on the second line of eq. (107b), we first expand the
integral in small $\epsilon_{v}$ using eq. (100). Using the integral formulae
for the Legendre polynomials above, we find that the leading
$\ln\left(2\hat{m}_{1}/\epsilon_{v}^{2}\right)$ divergence contains no
additional $\theta$-dependence, and so due to the constraint that $\ell=2$, is
weighted with $P_{1}^{2}+P_{2}$, and vanishes upon integration. Thus, we find
that

$\displaystyle\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\
\ell=2\end{subarray}}\nu_{j}^{2}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}\ln\left(\frac{\Lambda_{r}(\epsilon_{v},\theta)}{\lvert\nu_{j}\rvert}\right)=\mathcal{O}(\epsilon_{v}^{4})~{}.$
(108)

Hence, the only meaningful contributions to $A[\zeta_{\text{RT}}]$ from
$\mathcal{I}^{(2)}_{j}$ come from the first line in eq. (107b).

Moving on to the cutoff-dependent integrand on the first line of eq. (107b),
we can again utilize the small $\epsilon_{v}$ expansion in eq. (100). Since
the leading divergence in $\Lambda_{r}(\epsilon_{v},\theta)$ is
$\mathcal{O}(1/\epsilon_{v}^{2})$, we can neglect any integral for $\ell>2$ as
it will vanish as $\mathcal{O}(\epsilon_{v}^{2})$. Truncating to the sum to
$\ell<2$, expanding in small $\epsilon_{v}$, and evaluating the sum over $j$,
we find that the total contribution to $A[\zeta_{\text{RT}}]$ from the first
sum in eq. (107b) is

$\displaystyle\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\
\ell\neq
2\end{subarray}}\frac{\nu_{j}^{\ell}}{2-\ell}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}\Lambda_{r}^{2-\ell}(\epsilon_{v},\theta)=\frac{8\hat{m}_{1}^{3}}{3\epsilon_{v}^{4}}+\frac{2\hat{n}_{1}^{2}}{5\hat{m}_{1}}+\mathcal{O}(\epsilon_{v}^{2}).$
(109)

Finally, we treat the remaining sums in eq. (107a) and the second term on the
first line of eq. (107b) together. Firstly, we note that the integral at
$\ell=2$ in $\mathcal{I}_{j}^{(1)}$ vanishes due the integrand being of the
form $P_{1}^{2}+P_{2}$. Secondly, we make use of eq. (106) explicitly and
observe that only the $\ell=0$ term contributes. That is, if we decompose the
sum as partial sums in $\ell$,

$\displaystyle\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\ \ell\neq
2\end{subarray}}\left(\frac{1}{5+\ell}-\frac{1}{2-\ell}\right)\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}$
$\displaystyle=\sum_{\begin{subarray}{c}a=0\\\ a\neq
2\end{subarray}}^{\infty}\left(\frac{1}{5+a}-\frac{1}{2-a}\right)\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\
\ell=a\end{subarray}}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}$
$\displaystyle=-\frac{2}{5}-\frac{5}{6}\sum_{\begin{subarray}{c}\ell_{1},\ell_{2},\ell_{3}\\\
\ell=1\end{subarray}}\int_{0}^{\pi}\text{d}\theta\
s^{3}_{\theta}P_{\ell_{1}}P_{\ell_{2}}P_{\ell_{3}}+\ldots,$ (110)

we find that all the partial sums with $\ell>0$ vanish and $-2/5$ is the exact
result.

Putting all of the results above together and taking the sum over $j$, we find

$\displaystyle\sum_{j=1}^{2n+2}(-1)^{j}\hat{\xi}_{j}(\mathcal{I}_{j}^{(1)}+\mathcal{I}_{j}^{(2)})=\frac{8\hat{m}_{1}^{3}}{3\epsilon_{v}^{4}}+\frac{2}{5}\frac{\hat{n}_{1}^{2}-\hat{n}_{2}\hat{m}_{1}}{\hat{m}_{1}}+\mathcal{O}(\epsilon_{v}^{2}).$
(111)

Plugging in to eq. (101), the unregulated area of the RT hypersurface is

$\displaystyle
A[\zeta_{\text{RT}}]=\pi^{4}L_{\mathbb{S}^{4}}^{9}\log\left(\frac{2R}{\epsilon_{u}}\right)\left[\frac{256}{3}\frac{1}{\epsilon_{v}^{4}}+\frac{64}{5}\frac{\hat{n}_{1}^{2}}{\hat{m}_{1}^{4}}-\frac{64}{5}\frac{\hat{n}_{2}}{\hat{m}_{1}^{3}}+\mathcal{O}(\epsilon_{v}^{2})\right]+\mathcal{O}(\epsilon_{u}^{2}).$
(112)

It is then straightforward to see that $S_{\rm EE}$ is given by eq. (83).

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