11institutetext: Dipartimento di Fisica e Astronomia “Galileo Galilei”,
Università di Padova, Via Marzolo 8, 35131 Padova, Italy22institutetext: INFN,
Sezione di Padova Via Marzolo 8, 35131 Padova, Italy33institutetext: INFN,
Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano,
Italy44institutetext: Department of Physics, Faculty of Science, Chulalongkorn
University, Phayathai Road, Pathumwan, Bangkok 10330, Thailand55institutetext:
Dipartimento di Matematica, Università di Milano–Bicocca, Via Cozzi 55, 20126
Milano, Italy

# Superconformal anomalies for string defects
in six-dimensional ${\cal N}=(1,0)$ SCFTs

Fabio Apruzzi 3,4 Noppadol Mekareeya 3 Brandon Robinson 3,5 Alessandro
Tomasiello<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the anomalies of two-dimensional BPS defects in six-dimensional
${\cal N}=(1,0)$ superconformal field theories. Using a holographic
description of these defects furnished by probe D4-branes in AdS7 solutions of
ten-dimensional type IIA supergravity, we compute the two independent defect
Weyl from the on-shell action for a spherical defect and defect sphere
entanglement entropy. We find agreement between the holographic prediction for
the defect A-type anomaly coming from the defect sphere free energy and the
leading large $N$ contribution to the defect ‘t Hooft anomaly found using
anomaly inflow. We also find agreement between the holographic computation of
the expectation value of a surface operator wrapping a torus and the
supersymmetric localization computation for a circular Wilson loop in ${\cal
N}=1$ super Yang-Mills theory on $\mathds{S}^{5}$. Lastly, we holographically
compute the defect gravitational anomaly from the Wess–Zumino action of the
probe D4-brane, which provides a subleading large $N$ correction to the defect
A-type anomaly.

###### Contents

  1. 1 Introduction
  2. 2 AdS7/CFT6
  3. 3 Probe brane holography
    1. 3.1 Probe D4-brane embedding
    2. 3.2 Spherical entanglement entropy
    3. 3.3 On-shell action
    4. 3.4 Defect gravitational anomaly
  4. 4 Anomalies from Field Theory and their string theory constructions
    1. 4.1 Necklace quivers and supersymmetry localization on $\mathds{S}^{5}$
    2. 4.2 Anomaly Inflow from type IIB
  5. 5 Discussion
  6. A Coordinate Transformations
  7. B Two-dimensional quiver gauge theories

##  1 Introduction

Among the powerful holographic tools available to study gauge theories at
strong coupling, probe brane holography provides a controlled framework with
which to systematically introduce heavy charged operators Karch:2002sh  or
submanifold localized degrees of freedom DeWolfe:2001pq ; Constable:2002xt  in
a regime where the number of operators holographically sourced by the probe
brane remains parametrically small compared to the number of color degrees of
freedom. By carefully constructing the brane embedding to preserve
supersymmetry on the intersection of the curved conformal boundary of the
ambient AdS space and the brane worldvolume, probe brane holography has passed
a number of precision tests through comparison to results obtained by
supersymmetric localization Karch:2015vra ; Karch:2015kfa ; Robinson:2017sup .
On the gravity side, taking the probe limit of a brane embedding includes
taking the brane tension to be parametrically large, which to leading order
freezes out the probe brane’s gravitational degrees of freedom and suppresses
backreaction onto the background geometry. This limit greatly simplifies the
computation of key physical quantities that characterize probe brane degrees
of freedom such as the brane’s on-shell action and contributions to
Entanglement Entropy (EE), both of which will be the primary focus of our
holographic computations.

In the study of superconformal field theories (SCFTs), the $d=6$ case has a
distinguished role. It is the highest dimension in which SCFTs exist;
moreover, six-dimensional (6d) theories spawn a large variety of theories in
lower dimensions. The most famous example is the ${\mathcal{N}}=(2,0)$ theory
on M5-brane stacks Witten:1995zh , but a lot more examples exist with
${\mathcal{N}}=(1,0)$, engineered by orbifolds, M-theory walls, F-theory, IIA
brane intersections Intriligator:1997dh ; Ganor:1996mu ; Heckman:2013pva ;
Hanany:1997gh . The latter also have holographic duals, the AdS${}_{7}\times
M_{3}$ solutions in massive IIA Apruzzi:2013yva ; Apruzzi:2015wna ;
Cremonesi:2015bld .

Codimension-two defects in 6d SCFTs have played a role in their
compactifications down to four dimensions, appearing as punctures on an
internal Riemann surface; see for example Gaiotto:2009we ; Gaiotto:2009gz ;
Agarwal:2014rua ; Razamat:2022gpm . On the other hand, codimension-four
defects have been studied much less. In ${\mathcal{N}}=(2,0)$ theories they
are provided by M2-branes Alday:2009fs ; Drukker:2010jp ; Mori:2014tca . In
this paper we initiate the study of codimension-four defects in the much more
numerous ${\mathcal{N}}=(1,0)$ theories.

The generic form for the Weyl anomaly of a (super)conformal defect supported
on a two-dimensional submanifold, $\Sigma$, embedded a $d$-dimensional ambient
space is111Since our focus is on co-dimension four defects, there are only
parity even defect anomalies Jensen:2018rxu .

$\displaystyle
T^{\mu}{}_{\mu}|_{\Sigma}=\frac{1}{24\pi}(a_{\Sigma}\overline{E}_{2}+d_{1}\mathring{{\rm
II}}^{2}+d_{2}P[W]_{\Sigma})$ (1.1)

where $\overline{E}_{2}$ is the two-dimensional Euler density built from the
intrinsic metric of the defect submanifold, $\mathring{{\rm II}}^{2}$ is the
square of the trace-free second fundamental form for the embedding, and
$P[W]_{\Sigma}$ is the trace of the pullback of the ambient Weyl tensor to the
defect. The A-type Deser:1993yx  defect Weyl anomaly $a_{\Sigma}$ has been
shown to obey a weak c-theorem Jensen:2015swa ; Wang:2020xkc  and, with a
sufficient amount of supersymmetry preserved on the defect, to be related to
defect R- and gravitational anomalies Wang:2020xkc . The B-type anomalies
$d_{1}$ and $d_{2}$ are strictly non-negative in reflection positive theories
Billo:2016cpy ; Jensen:2018rxu , but do not obey any known c-theorem. However,
relevant to the cases we are interested, it is known that $d_{1}=d_{2}$ for
two-dimensional conformal defect preserving at least ${\cal N}=(2,0)$ defect
supersymmetry Bianchi:2019sxz ; Drukker:2020atp , and so in order to
characterize BPS string defects in six-dimensional ${\cal N}=(1,0)$ SCFTs, we
only need to compute $a_{\Sigma}$ and $d_{2}$. Crucial for our analysis below,
these defect anomalies are known to control two quantities that can be easily
computed using probe brane holography: the log divergent part of defect sphere
free energy, which is uniquely determined by $a_{\Sigma}$, and the universal
part of the defect sphere EE, which is fixed by a linear combination of
$a_{\Sigma}$ and $d_{2}$ Kobayashi:2018lil ; Jensen:2018rxu .

In section 2, we briefly review AdS7 solutions to type IIA SUGRA found in
Apruzzi:2013yva ; Apruzzi:2015wna ; Cremonesi:2015bld  and their holographic
dual description in terms of six-dimensional ${\cal N}=(1,0)$ SCFT at large
$N$.

In section 3, we construct solutions for embedding probe D4-brane into a
generic AdS${}_{7}\times M_{3}$ background with non-trivial Romans
mass.222Some fully back-reacted solutions for codimension-four defects exist
DHoker:2008lup ; Dibitetto:2017klx ; Lozano:2019ywa ; Lozano:2022ouq , and it
would be interesting to compare our approach to those results; the
AdS${}_{3}\times\mathds{S}^{3}$ solutions appear particularly relevant.
Further in this section, we carry out four holographic computations that
characterize the two-dimensional defect in the field theory. First, we compute
the defect contribution to the EE of a spherical region. Second, we compute
the $a_{\Sigma}$ anomaly from the free energy of a spherical defect and use
the defect EE result to obtain the B-type anomalies. In terms of the number of
probe D4-branes $q_{i}$ and partitions of D6-branes $r_{i}$ they can be
succinctly written to leading order in large $N$ as

$\displaystyle a_{\Sigma}=24(q,r),\qquad d_{1}=d_{2}=32(q,r)~{}$ (1.2)

with scalar product taken with respect to the inverse Cartan matrix of
$A_{N-1}$. Third, we holographically compute the expectation value of the
string defect operator $W$ on
$\mathds{S}^{1}_{\beta/4\pi}\times\mathds{S}^{1}$, which gives

$\displaystyle\left<W\right>=\exp\left[\beta(q,r)\right].$ (1.3)

Lastly, we compute the defect gravitational anomaly from the Wess–Zumino
action, which is shown to be subleading at large $N$ and contributes to the
$a_{\Sigma}$ anomaly

$\displaystyle a_{\Sigma}=24(q,r)+\frac{1}{2}\sum_{i}q_{i}r_{i}\,.$ (1.4)

In section 4, we compare the probe brane results to anomalies computed via
other means. In section 4.1, we compare the expectation value of the torus
Wilson surface to the dimensionally reduced circular Wilson loop obtained
purely in field theory using supersymmetric localization of ${\cal N}=1$ SYM
theory on $\mathds{S}^{5}$. Despite the holographic computation only computing
the value at large $N$ and the supersymmetric localization result being exact,
we find a precise match between the two expressions. Later, in section 4.2, we
consider the type IIB description of string defects in six-dimensional SCFTs.
By the help of these string constructions we study the 6d anomaly polynomial
inflow onto the defect. In particular, we compute the defect R-anomaly and
successfully match the leading large $N$ result for the $a_{\Sigma}$ anomaly.
As a byproduct we developed the inflow mechanism for (string) defects provided
by probe branes in IIB backgrounds engineering 6d SCFTs. This approach could
be potentially generalized to other type (non-string) of defects.

In appendix A, we explain the coordinate transformations used in section 3.1.
In appendix B, we consider a purely two-dimensional description of the
D2–D4–D6 brane system in terms of a quiver gauge theory. We demonstrate that
the quiver description correctly captures the subleading large $N$
contributions (only) to the defect anomaly, i.e. the gravitational anomaly.
Differently from the BPS strings, which become tensionless at the fixed point,
the quiver which includes the defect does not capture the full defect degrees
of freedom. We argue that the quiver only provides the intrinsic degrees of
freedom of the defect. However it does not capture the bulk-defect ones, which
are instead computed via anomaly inflow.

##  2 AdS7/CFT6

The starting point is the asymptotically AdS${}_{7}\times M_{3}$ background
solutions to type IIA SUGRA constructed in Apruzzi:2013yva ; Apruzzi:2015wna ;
Cremonesi:2015bld

$\displaystyle\begin{split}\frac{1}{\pi\sqrt{2}}ds^{2}&=8\sqrt{-\frac{\alpha}{\alpha^{\prime\prime}}}ds_{7}^{2}+\sqrt{-\frac{\alpha^{\prime\prime}}{\alpha}}\left(dz^{2}+\frac{\alpha^{2}}{{\alpha^{\prime}}^{2}-2\alpha\alpha^{\prime\prime}}d\Omega_{2}^{2}\right)\,,\\\
e^{\phi}&=3^{4}(\sqrt{2}\pi)^{\frac{5}{2}}\frac{(-\alpha/\alpha^{\prime\prime})^{\frac{3}{4}}}{\sqrt{{\alpha^{\prime}}^{2}-2\alpha\alpha^{\prime\prime}}}\,,\\\
B&=\pi\left(-z+\frac{\alpha\alpha^{\prime}}{{\alpha^{\prime}}^{2}-2\alpha\alpha^{\prime\prime}}\right)\text{vol}_{\mathds{S}^{2}}\,,\\\
F_{2}&=\left(\frac{\alpha^{\prime\prime}}{162\pi^{2}}+\frac{\pi
F_{0}\alpha\alpha^{\prime}}{{\alpha^{\prime}}^{2}-2\alpha\alpha^{\prime\prime}}\right)\text{vol}_{\mathds{S}^{2}}\,.\end{split}$
(2.1)

Here $ds_{7}^{2}$ is the line element on AdS7, $d\Omega^{2}_{2}$ is the line
element on the unit 2-sphere $\mathds{S}^{2}$, prime denotes $\partial_{z}$,
and $\alpha$ is a $C^{1}$ piecewise-cubic function on the interval $z\in[0,N]$
that satisfies $\alpha>0$, $\alpha^{\prime\prime}<0$,

$\alpha^{\prime\prime\prime}(z)=-162\pi^{3}F_{0}$ (2.2)

almost everywhere, and appropriate boundary conditions at the endpoints of the
interval. The simplest possibility is $\alpha=\alpha^{\prime\prime}=0$, which
ensures smoothness, but at either endpoint it is also possible to introduce
D6-branes (with $\alpha=0$), O8-planes ($\alpha^{\prime}=0$), O6-planes
($\alpha^{\prime\prime}=0$).

NSNS flux quantization implies $N\in{\mathbb{Z}}$, while for RR it gives

$-\frac{\alpha^{\prime\prime}(i)}{81\pi^{2}}\in{\mathbb{Z}}\,,\qquad\forall
i\in{\mathbb{Z}}\,.$ (2.3)

$F_{0}=\frac{n_{0}}{2\pi}$, $n_{0}\in{\mathbb{Z}}$ is locally constant, but
can jump across D8-branes; these are restricted to lie at integer values $\mu$
of $z$, which are in turn identified with their D6-brane charge. A cartoon of
the internal geometry is shown in Fig. 1.

The curvature and string coupling of any given solution can be rescaled and
made arbitrarily small by taking

$N\gg 1\,\qquad\mu/N\ \text{finite}\,.$ (2.4)

This is the regime relevant for holography.

The simplest solution occurs when the Romans mass $F_{0}=0$. (2.2) implies
that $\alpha^{\prime\prime}$ is constant, and (2.3) implies

$\alpha=\frac{81}{2}\pi^{2}{k}z(N-z)\,.$ (2.5)

${k}\in{\mathbb{Z}}$ is interpreted as the $F_{2}$ flux integer on the
$\mathds{S}^{2}$. There are ${k}$ D6-branes at $z=0$ and $-{k}$ anti D6-branes
at $z=N$. The resulting IIA solution is the dimensional reduction of the ${\rm
AdS}_{7}\times\mathds{S}^{4}/\mathbb{Z}_{k}$ solution of eleven-dimensional
SUGRA.

When $F_{0}\neq 0$, $\alpha^{\prime\prime}$ is piecewise linear; it can be
parameterized as

$-\frac{\alpha^{\prime\prime}(z)}{81\pi^{2}}=r_{i}+(r_{i+1}-r_{i})(z-i)$ (2.6)

on each interval $z\in[i,i+1]$. By eq. (2.3), the $r_{i}\in{\mathbb{Z}}$.

These AdS7 solutions were conjectured to arise from the near-horizon limit of
NS5–D6–D8 brane diagrams, and this led to the identification of the dual SCFT6
Gaiotto:2014lca . We now give a quick review focused on what we need in this
paper; see (Cremonesi:2015bld, , Sec. 2.1) for more details.

The brane diagram can be described as follows. There are $N$ NS5-branes along
directions $012345$, separated along direction $5$. Between the $(i-1)^{\rm
th}$ and $i^{\rm th}$ NS5-brane, $r_{i}$ D6-branes are suspended, along
directions $0123456$; crossing the latter, there are also
$f_{i}=2r_{i}-r_{i+1}-r_{i-1}$ D8-branes along $012345789$. This brane diagram
is depicted schematically in the black part of Fig. 2.

The six-dimensional field theory engineered by this diagram is a chain of
$N-1$ vector multiplets with gauge groups ${\rm U}(r_{i})$; there are
hypermultiplets in each of the bifundamental representations $\overline{\bf
r_{i}}\otimes{\bf r_{i+1}}$, and $f_{i}$ in the fundamental ${\bf r_{i}}$.
There are also $N-1$ tensor multiplets.

(2.6) tells us that the function $r(z)\equiv-\alpha^{\prime\prime}/81\pi^{2}$
gives the ranks $r_{i}$ when evaluated at $z=i$; conversely we can write

$\alpha(z)=-81\pi^{2}\frac{1}{\partial_{z}^{2}}r(z)\,.$ (2.7)

The integration constants in the double integral should be understood as being
fixed by the boundary conditions. As an example, when $F=0$ all the ranks are
equal, $r_{i}=k$, and it is easy to see that eq. (2.7) reproduces eq. (2.5).

An important piece of evidence for the AdS7/CFT6 conjecture we just reviewed
came from a holographic match of the $a$ anomaly Cremonesi:2015bld ;
Apruzzi:2017nck. At leading order in the limit eq. (2.4), the gravitational
result reads

$a_{\rm 6d,hol}=\frac{64}{189\pi^{2}}\int r\alpha\,dz=-\frac{192}{7}\int
r\frac{1}{\partial_{z}^{2}}r\,dz\,.$ (2.8)

In the second step we have used eq. (2.7). On the other hand, the leading term
in the field theory computation is given by a Green–Schwarz term and reads

$a_{\rm 6d}=\frac{192}{7}(C^{-1})^{ij}r_{i}r_{j}\,,$ (2.9)

where $C_{ij}$ is the Cartan matrix of $A_{N-1}$. $C_{ij}$ is a discrete
analogue of minus a double derivative: indeed

$C_{ij}r_{j}=-r_{i+1}+2r_{i}-r_{i-1}\sim-(\partial^{2}r)_{i}$ (2.10)

So its inverse is a discrete analogue of a double integral. Hence eq. (2.9)
matches eq. (2.8) when $N\gg 1$.

To summarize the important lesson for our present purposes, in the holographic
limit we have

$\alpha^{i}=81\pi^{2}(C^{-1})^{ij}r_{j}\,.$ (2.11)

As an example, we can again look at the $F_{0}=0$ case, where $r_{i}=k$ and
$(C^{-1})^{ij}r_{j}=\frac{1}{2}i(N-i)$, in agreement with eq. (2.5).

##  3 Probe brane holography

In this section, we construct embeddings for probe D4-branes in the
asymptotically AdS${}_{7}\times M_{3}$ geometries reviewed in section 2 that
are dual to BPS codimension-4 defects in six-dimensional $\mathcal{N}=(1,0)$
SCFTs. Using these solutions for embedded AdS3 branes, we characterize these
defects by computing their independent defect Weyl anomalies using probe brane
holography. First, we holographically compute the defect contribution to the
EE of a spherical region in the dual field theory; this furnishes for us a
specific linear combination of the two independent defect Weyl anomalies.
Second, we compute the free energy of the defect wrapping
$\mathds{S}^{2}\hookrightarrow\mathbb{R}^{6}$, which gives us the A-type
defect anomaly $a_{\Sigma}$. The value of $a_{\Sigma}$ matches the predicted
value computed from in-flow, and so, bolsters the standing of the holographic
prediction for the B-type anomaly $d_{2}$. Lastly, compute the defect
supersymmetry Casimir Energy (SCE), which is obtained holographically from the
on-shell action of a probe brane wrapping
$\mathds{S}^{1}\times\mathds{S}^{1}\hookrightarrow\mathds{S}^{1}\times\mathds{S}^{5}$.

###  3.1 Probe D4-brane embedding

Throughout this section, we will work in Euclidean signature. The probe brane
action is

$\displaystyle S_{\rm probe}=T_{4}\int_{M_{5}}d^{5}xe^{-\phi}\sqrt{\mathop{\rm
det}P[g+B]+{\cal F}}-T_{4}\int_{M_{5}}e^{{\cal F}}\wedge P[C]\,,$ (3.1)

where

$T_{4}=\frac{1}{16\pi^{4}l_{s}^{5}}\,;$ (3.2)

$M_{5}$ is the D4-brane worldvolume with coordinates $x^{a}$, $P$ denotes
pull-back onto $M_{5}$, and the worldvolume gauge field ${\cal F}=B+2\pi
l_{s}f$. Throughout, we will work with convention $l_{s}=1$.

We will consider D4-brane probes of the factorized form

$\widetilde{M}_{3}\times\mathds{S}^{2}\subset{\rm AdS}_{7}\times M_{3}\,,$
(3.3)

where $\widetilde{M}_{3}\subset{\rm AdS}_{7}$ and the $\mathds{S}^{2}$ is the
sphere appearing in eq. (2.1), whose $z$ position we will determine shortly.
Our D4-branes will carry non-zero D2-brane charge $j\in{\mathbb{Z}}$:

$f=\frac{j}{2}\text{vol}_{\mathds{S}^{2}}\,.$ (3.4)

The position $z$ of the D4 can be fixed by extremizing the action. This is
implied by the BPS condition, which in general reads
$\Gamma_{\parallel}\epsilon_{2}=\epsilon_{1}$ for a D-brane, with
$\Gamma_{\parallel}$ the product of the parallel gamma matrices and
$\epsilon_{a}$ the supersymmetry parameters preserved by the solution. In our
case, the $\epsilon_{a}$ are factorized in terms of Killing spinors $\zeta$ in
AdS7 and internal spinors $\chi_{a}$ (Apruzzi:2013yva, , (A.4)).
$\Gamma_{\parallel}$ is also factorized because of our assumption (3.3).

The internal part $\gamma_{\parallel}\chi_{1}=\chi_{2}$ of the BPS computation
can be carried out efficiently in terms of calibrations. It is in fact
identical to that for a D8/D6 bound state in (Apruzzi:2013yva, , Sec. 4.8), so
we will be brief. (This happens because intersecting or overlapping D-branes
preserve supersymmetry if the number of Neumann–Dirichlet directions is a
multiple of 4.) The relevant condition is333In the language of Apruzzi:2013yva
, the calibration is ${\rm Im}\psi^{1}_{+}$; alternatively, one can impose
that the forms $({\rm Im}\psi^{2}_{+},{\rm Re}\psi^{2}_{+},{\rm
Re}\psi^{1}_{+})$ pull back to zero.

$\displaystyle{\cal
F}=\pi\frac{\alpha\alpha^{\prime}}{{\alpha^{\prime}}^{2}-2\alpha\alpha^{\prime\prime}}\text{vol}_{\mathds{S}^{2}}.$
(3.5)

Using (2.1), (3.4), this implies

$z=j\,.$ (3.6)

As anticipated, this coincides with the BPS position for a D8-brane with D6
charge equal to $j$, which was already noticed below (2.3).

The case of a single D2 has to be treated separately; again this works in the
same way as a D6 extended along all of AdS7. The calibration analysis this
time gives $\alpha\alpha^{\prime\prime}=0$, which is only true at the extrema
of the interval, $z=0$ and $z=N$. The resulting probe D4–D2 bound state wraps
an internal $\mathds{S}^{2}$ at fixed $z=j$, which is schematically visualized
in Figures 1 and 2.

Figure 1: A cartoon of the internal three-dimensional space of the AdS7
solution. The black creases represent the D8-branes, where the metric is
continuous but not differentiable. The red locus represents a D4–D2 bound
state probe, to be discussed in the next section. Figure 2: The black part is
a schematic representation of the NS5–D6–D8 brane diagram, taken from
Cremonesi:2015bld . The nodes, horizontal lines and vertical lines represent
NS5-, D6- and D8-branes respectively. The upper part of the diagram, where the
NS5 are separated, is more useful to read the field theory. The red vertical
lines here represent additional D4-branes, which engineer line defects in the
field theory, to be discussed in the next section. The lower part, where the
NS5-branes coincide, corresponds to the conformal phase of the field theory,
and resembles more directly the gravity solution. The brane-creation effect
changes the number of D6-branes, and leads to the creation of D2-branes,
represented by red horizontal lines. In the near-horizon limit, each D8 with
D6s ending on it becomes a D8–D6 bound state, and the D4 with D2s becomes a
D4–D2 bound state.

The probe brane action then takes the form

$\displaystyle S_{\rm probe}$
$\displaystyle=T_{4}\int_{M_{5}}d^{5}xe^{3A-\phi}f_{1}^{2}f_{2}\sqrt{g_{\theta\theta}g_{\phi\phi}+{\cal
F}_{\theta\phi}^{2}}\big{|}_{z=j}$ (3.7)
$\displaystyle=\frac{64\pi}{81}T_{4}\text{vol}(\widetilde{M}_{3})~{}\sqrt{\alpha^{\prime
2}-\alpha\alpha^{\prime\prime}}\left(\left(\sqrt{-\frac{\alpha^{\prime\prime}}{\alpha}}\frac{\sqrt{2}\alpha^{2}}{\alpha^{\prime
2}-2\alpha\alpha^{\prime\prime}}\right)^{2}+\left(\frac{\alpha\alpha^{\prime}}{\alpha^{\prime
2}-2\alpha\alpha^{\prime\prime}}\right)^{2}\right)^{1/2}$
$\displaystyle=\frac{64\pi}{81}T_{4}\text{vol}(\widetilde{M}_{3})~{}\alpha^{j}\,,$

where we introduced $\alpha^{j}\equiv\alpha(j)$. In the second line, we have
used (2.1). Since the D2 charge carried by the probe D4 brane eliminated the
explicit $z$ dependence, the action completely factorized into a product of an
integral over the volume form of the $\widetilde{M}_{3}$ and an integral over
the internal $\mathds{S}^{2}$ geometry threaded by two-form flux.

In what follows we will consider a more general situation where there is a
collection of $q_{j}$ D4-branes with D2-brane equal to $j$. This will turn
(3.7) into
$\frac{64\pi}{81}T_{4}\text{vol}(\widetilde{M}_{3})~{}q_{j}\alpha^{j}$. We
will work in the probe approximation: this requires $q_{j}e^{\phi(j)}\ll 1$,
which by (2.1) is implied by $q_{j}\ll\sqrt{N}$.

It is clear from the form of the on-shell action in eq. (3.7) that we will
need to regulate divergences coming from $\text{vol}(\widetilde{M}_{3})~{}$.
To that end, we will use the holographic renormalization scheme for probe
branes developed in Karch:2005ms . As a brief review, this is done by
introducing a large radial cutoff $\Lambda$ in the background AdS7 geometry
and computing the covariant counterterms on the intersection of the radial
hypersurface and the worldvolume of the probe brane. We denote coordinates on
this intersection by $y^{a}$ and the induced metric by $\gamma_{\Lambda}$.
Since, we have no worldvolume scalars turned on in our embedding, the only
counterterm that we need to compute is the volume of the intersection

$\displaystyle S_{\rm CT}=-\frac{2\pi}{9}T_{4}\sqrt{2r_{j}\alpha^{j}}\int
d^{2}y\sqrt{\gamma_{\Lambda}}\,,$ (3.8)

where the coefficient is fixed to cancel polynomial divergences in $\Lambda$
appearing in $\text{vol}(\widetilde{M}_{3})~{}$. In the subsections below, we
denote the holographically renormalized probe brane action by

$\displaystyle S_{\rm ren}=S_{\rm probe}+S_{\rm CT}\,.$ (3.9)

###  3.2 Spherical entanglement entropy

Here we compute the contribution of the probe string defect to the spherical
entanglement entropy (EE). For simplicity, we will consider a flat embedding
of a two-dimensional defect supported on $\mathbb{R}^{1,1}$ in
$\mathbb{R}^{5,1}$, which will be taken as the conformal boundary of the AdS7
part of the background solution. Since we are working entirely within the
probe limit, we do not need to compute the backreaction and can compute the EE
by using Casini:2011kv .

First, we consider the AdS7 part of the background geometry in flat slicing

$\displaystyle
ds^{2}_{7}=\frac{1}{u^{2}}(du^{2}-dt^{2}+dx_{1}^{2}+d\xi^{2}+\xi^{2}d\Omega_{3}^{2})\,,$
(3.10)

with the probe D4-brane wrapping $\\{t,x_{1},u\\}$ and $d\Omega_{3}^{2}$ the
line element on the unit $\mathds{S}^{3}$. Following Casini:2011kv ;
Jensen:2013lxa , we map

$\displaystyle\begin{split}t&=\frac{R\sqrt{\rho^{2}-1}\sinh\tau}{\rho\cosh\upsilon+\sqrt{\rho^{2}-1}\cosh\tau},\hskip
56.9055ptu=\frac{R}{\rho\cosh\upsilon+\sqrt{\rho^{2}-1}\cosh\tau},\\\
x_{1}&=\frac{R\rho\sinh\upsilon\cos\phi}{\rho\cosh\upsilon+\sqrt{\rho^{2}-1}\cosh\tau},\hskip
56.9055pt\xi=\frac{R\rho\sinh\upsilon\sin\phi}{\rho\cosh\upsilon+\sqrt{\rho^{2}-1}\cosh\tau},\end{split}$
(3.11)

which after a Wick rotation brings us to the seven-dimensional hyperbolic
black hole Casini:2011kv

$\displaystyle
ds_{7}^{2}=\frac{d\rho^{2}}{f(\rho)}+f(\rho)d\tau^{2}+\rho^{2}d\upsilon^{2}+\rho^{2}\sinh^{2}\upsilon(d\phi^{2}+\sin^{2}\phi
d\Omega_{3}^{2})$ (3.12)

where the metric function

$\displaystyle f(\rho)=\rho^{2}-1\,,$ (3.13)

and $\tau\sim\tau+2\pi$. The BPS probe brane embedding found in section 3.1
now wraps $\\{\tau,\upsilon,\rho\\}$. The probe brane contributions to the
thermal entropy in the hyperbolic black hole background corresponds to the
holographic computation of the BPS surface defect contribution to the EE of a
spherical region in the dual field theory.

More generally, one can consider (not necessarily BPS) probe brane embeddings
in seven-dimensional black hole solutions with metric function

$\displaystyle
f(\rho)=\rho^{2}-1-\frac{\rho_{h}^{4}}{\rho^{4}}(\rho_{h}^{2}-1)$ (3.14)

with the location of the horizon determined in terms of the inverse
temperature $\beta$ by

$\displaystyle\rho_{h}=\frac{1}{3\beta}(\pi+\sqrt{\pi^{2}+6\beta^{2}}).$
(3.15)

This background is particularly useful for the holographic computation of
Rényi entropies Hung:2011nu . Moving to a branched cover of the black hole
geometry $\tau\sim\tau+\beta$ with $\beta=2\pi n$, the thermal entropy in this
background is dual to the $n^{\rm th}$-Rényi entropy

$\displaystyle S^{(n)}=\frac{2\pi n}{1-n}({\cal F}(2\pi)-{\cal F}(2\pi n)),$
(3.16)

where ${\cal F}(\beta)=\beta^{-1}S_{\rm ren}(\beta)$. Clearly from the bulk
geometry, the limit $n\to 1$ recovers the seven-dimensional hyperbolic black
hole background, and in the dual field theory $S^{(n)}|_{n\to 1}\to S_{\rm
EE}$ Hung:2011nu ; Jensen:2013lxa , which holographically

$\displaystyle S_{\rm EE}=\beta^{2}\partial_{\beta}(\beta^{-1}S_{\rm
ren}(\beta))|_{\beta\to 2\pi}\,.$ (3.17)

Computing the holographic renormalized on-shell action, we begin with

$\displaystyle\text{vol}(\widetilde{M}_{3})~{}=\int_{0}^{\beta}d\tau\
\int_{0}^{\upsilon_{c}}d\upsilon\int_{\rho_{h}}^{\Lambda}\rho
d\rho=\frac{\beta}{2}\upsilon_{c}(\Lambda^{2}-\rho_{h}^{2}).$ (3.18)

where we have introduced a large radial cutoff $\Lambda$. The $\Lambda^{2}$
divergence is regulated by eq. (3.8)

$\displaystyle S_{\rm CT}=-\frac{32\pi}{81}\beta
T_{4}\upsilon_{c}\Lambda^{2}\rho_{h}^{2}\alpha^{j}\,.$ (3.19)

Using eq. (3.11), the linear divergence in $\upsilon_{c}$ in the black hole
background is related to a log divergence at small $u=-\epsilon$

$\displaystyle\upsilon_{c}=\log\frac{2R}{\epsilon}\,,$ (3.20)

where $R$ is the radius of the boundary spherical entangling surface that
anchors the homologous Ryu–Takayanagi surface Ryu:2006bv  in the bulk. Thus,
we find that the near boundary behavior of the holographically renormalized
action is444For $\beta\neq 2\pi$, the defect embedding is expected to get
deformed; this deformation is not known analytically and would not be BPS.
However, we are evaluating the action, which on shell is stationary under
small deformations; so $\partial_{\beta}S_{\rm ren}(\beta)|_{\beta\to
2\pi}=0$, and we can use directly the expression (3.21) in (3.17)
Rodgers:2018mvq ; Kumar:2017vjv .

$\displaystyle S_{\rm
ren}=-\frac{8}{81}q_{j}\alpha^{j}T_{4}\text{vol}(\mathds{S}^{2})~{}\beta\rho_{h}^{2}\log\frac{2R}{\epsilon}+\ldots\,.$
(3.21)

Using the form of $\rho_{h}$ in eq. (3.15), we find the defect sphere EE is

$\displaystyle S_{\rm
EE}=\frac{8}{405\pi^{2}}q_{j}\alpha^{j}\log\frac{2R}{\varepsilon}+\ldots\,.$
(3.22)

In the $F_{0}=0$ case, we can evaluate the previous quantities more explicitly
using eq. (2.5). For example eq. (3.22) becomes

$\displaystyle S_{\rm
EE}=\frac{4{k}}{5}\sum_{j}q_{j}j(N-j)\log\frac{2R}{\epsilon}+\ldots.$ (3.23)

From Kobayashi:2018lil ; Jensen:2018rxu , we know that for a flat two-
dimensional conformal defect embedded in a CFT on a flat six-dimensional
background the universal part of the defect sphere EE is uniquely determined
by defect Weyl anomalies:

$\displaystyle S_{\rm
EE}=\frac{1}{3}\left(a_{\Sigma}-\frac{3}{5}d_{2}\right)\log\frac{2R}{\epsilon}$
(3.24)

Further, we know that for a two-dimensional superconformal preserving at least
${\cal N}=(2,0)$ defect supersymmetry, the two B-type defect Weyl anomaly
coefficients, $d_{1}$ and $d_{2}$, are equal Bianchi:2019sxz ; Drukker:2020atp
, and so the universal part of the defect sphere EE, when combined with an
independent computation of either $a_{\Sigma}$ or $d_{2}$, provides some
measure of characterization of the defect degrees of freedom via anomalies. In
the next subsection, we will compute the on-shell action for probe branes
wrapping $\mathds{S}^{2}$ and $\mathds{S}^{1}\times\mathds{S}^{1}$ in the
boundary geometry, which will give us independent predictions for $a_{\Sigma}$
and $d_{2}$.

In addition to the defect sphere EE, which can be found by taking $n\to 1$ in
eq. (3.16), there are other limits of $S^{(n)}$ that contain interesting
physical information (see e.g. Hung:2011nu ). By setting $\beta=2\pi n$ and
computing the renormalized on-shell action, we find that the $n^{\rm th}$
Rényi entropy is

$\displaystyle
S^{(n)}=T_{4}q_{j}\alpha^{j}\text{vol}(\mathds{S}^{2})~{}\text{vol}(\widetilde{\mathds{S}}^{2})~{}\log\frac{2R}{\epsilon}\frac{2(1-6n^{2}+\sqrt{1+24n^{2}})}{729n(1-n)}+\ldots.$
(3.25)

Taking the $n\to 0$ limit counts the number of non-zero eigenvalues of the
reduced density matrix:

$\displaystyle\lim_{n\to
0}S^{(n)}=\frac{4}{729n\pi^{2}}q_{j}\alpha^{j}\log\frac{2R}{\epsilon}+O(n^{0})\,.$
(3.26)

Taking the opposite limit, $n\to\infty$, measures the largest eigenvalue of
the reduced density matrix

$\displaystyle\lim_{n\to\infty}S^{(n)}=\frac{4}{243\pi^{2}}q_{j}\alpha^{j}\log\frac{2R}{\epsilon}+O(n^{-1})\,.$
(3.27)

###  3.3 On-shell action

In this subsection, we compute the holographically renormalized on-shell
action in eq. (3.9) starting from eq. (3.7) for the cases where the boundary
geometry of $\widetilde{M}_{3}$ is $\mathbb{R}^{2}$, $\mathds{S}^{2}$, and
$\mathds{S}^{1}\times\mathds{S}^{1}$. In the latter two cases, the
computations result in robust predictions for physical quantities that will be
compared to anomaly inflow and supersymmetric localization at large $N$ in the
following sections.

####  3.3.1 $\mathbb{R}^{2}$

Here we compute the on-shell action for a probe D4-brane embedding into eq.
(2.1) wrapping AdS${}_{3}\times\mathds{S}^{2}$ engineering a flat two-
dimensional defect supported on $\mathbb{R}^{2}$ in the dual field theory
defined on a $\mathbb{R}^{6}$ background. We take the AdS7 to be written in
flat slicing as in eq. (3.10). We fix static gauge with $\xi=0$ such that the
full ${\rm SO}(4)_{N}$ normal bundle symmetry is preserved. Computing the
pullback onto the worldvolume of the probe D4-branes, the induced metric takes
the form

$\displaystyle\frac{ds_{\Sigma}^{2}}{\pi\sqrt{2}}=f_{1}^{2}(dx_{0}^{2}+dx_{1}^{2})+f_{2}^{2}du^{2}+f_{3}^{2}d\Omega_{2}^{2}.$
(3.28)

Computing the on-shell action eq. (3.7), only finding
$\text{vol}(\widetilde{M}_{3})~{}$ remains to be done. For the embedding
described above,

$\displaystyle\text{vol}(\widetilde{M}_{3})~{}=\text{vol}(\mathbb{R}^{2})~{}\int_{-\infty}^{-\epsilon}\frac{du}{u^{3}}~{}=\frac{\text{vol}(\mathbb{R}^{2})~{}}{2\epsilon^{2}},$
(3.29)

where we have introduced a small radial cutoff $u=-\epsilon$. There are no
subleading divergences, and so the covariant counterterm action built out of
the induced volume of the cutoff slice $u=-\epsilon$ exactly cancels $S_{\rm
probe}$ leaving

$\displaystyle S_{\rm ren}=0\,,$ (3.30)

as expected.

####  3.3.2 $\mathds{S}^{2}$

Since we are studying a particularly simple D4–D2 bound state system as a
probe, we can easily consider other AdS7 and AdS3 geometries that will reveal
the anomaly. Instead of using flat slicing in Poincaré coordinates, we can
consider the background AdS7 geometry in global coordinates with radial slices
having $\mathds{H}^{5}\times\mathds{S}^{1}$ geometry using the map in appendix
A, which results in

$\displaystyle ds_{7}^{2}=d\sigma^{2}+\cosh^{2}\sigma
ds_{\mathds{H}_{5}}^{2}+\sinh^{2}\sigma d\varphi^{2}\,,$ (3.31)

where the line element on the $\mathds{H}_{5}$ is given in eq. (A.8). So
(3.31) is essentially (3.12), but with the defect now wrapping an equatorial
$\mathds{S}^{2}\hookrightarrow\mathds{S}^{4}\subset\mathds{H}_{5}$. Thus, our
D4–D2 bound state probe can be embedded in such a way that it wraps the
boundary submanifold $\psi=\pi/2$ at the boundary $\rho\to\infty$ at fixed
$\sigma$.

Computing the on-shell action in this background is straightforward:

$\displaystyle S_{\rm
probe}=\frac{16}{81}T_{4}(4\pi)^{2}\alpha^{j}\int_{0}^{\log\Lambda}d\rho\sinh^{2}\rho=\frac{32\pi^{2}}{81}T_{4}\alpha^{j}\left(\Lambda^{2}-4\log\Lambda+\ldots\right)\,.$
(3.32)

We have introduced a radial cutoff $\rho=\log\Lambda\gg 1$ and suppressed
further subleading (finite) terms. Note that in this curved space embedding,
$z=j$ still extremizes the action, and so we can use this to evaluate the
action on-shell. The counterterm action is computed from (3.8). We can tune
$c_{1}$ to remove the $\Lambda^{2}$ divergence, and so the holographically
renormalized on-shell action is

$\displaystyle S_{\rm ren}=-\frac{8}{81\pi^{2}}q_{j}\alpha^{j}\log\Lambda\,.$
(3.33)

This log divergence is physical: it holographically computes the A-type Weyl
anomaly for a defect wrapping $\mathds{S}^{2}$ inserted into the six-
dimensional ${\cal N}=(1,0)$ SCFT dual to the ten-dimensional SUGRA solution
above:

$\displaystyle\left<{\cal O}_{\mathds{S}^{2}}\right>\approx
e^{-S_{\text{ren}}}=(R\mu)^{a_{\Sigma}/3}$ (3.34)

where we have introduced the length scale $R$ on the $\mathds{S}^{2}$ along
with the RG scale $\mu$ for dimensional reasons. Equating

$\begin{split}S_{\text{ren}}=-\frac{1}{3}a_{\Sigma}\log\Lambda\end{split}$
(3.35)

and using (3.33), we find a prediction for the A-type defect Weyl anomaly to
leading order in large $N$:

$\displaystyle a_{\Sigma}=\frac{8}{27\pi^{2}}q_{j}\alpha^{j}\,.$ (3.36)

In order to compute the B-type anomaly, we insert (3.36) into (3.24) and use
(3.22) to arrive at

$\displaystyle d_{1}=d_{2}=\frac{32}{81\pi^{2}}q_{j}\alpha^{j}\,.$ (3.37)

We can express $a_{\Sigma}$ in terms of representation theoretic data, like
for the Wilson surface in the $(2,0)$ theory Estes:2018tnu ; Jensen:2018rxu .
Recalling eq. (2.11), we can rewrite

$a_{\Sigma}=24q_{i}(C^{-1})^{ij}r_{j}=24(q,r)\,,\qquad d_{1}=d_{2}=32(q,r)\,,$
(3.38)

where we defined the Killing form on the weight space:

$(\lambda,\lambda^{\prime})\equiv\lambda_{i}(C^{-1})^{ij}\lambda^{\prime}_{j}\,.$
(3.39)

As a simple example, consider the case where $F_{0}=0$. All the $r_{i}=k$, so
$r=k\rho_{W}$, where $\rho_{W}=(1,1,\ldots,1)$ is the the Dynkin label for the
$A_{N-1}$ Weyl vector. Moreover take a single probe brane in position $j$:
$q_{j}=1$, $q_{i\neq j}=0$. We obtain

$\displaystyle
a_{\Sigma}=12{k}\sum_{j}j(N-j)=24{k}(\lambda,\,\rho_{W})=\frac{3}{4}d_{2}\,$
(3.40)

where now

$\displaystyle\lambda=(\underbrace{0,\ldots,0}_{j-1},1,\underbrace{0,\ldots,0}_{N-j-1})\,.$
(3.41)

####  3.3.3 $\mathds{S}^{1}\times\mathds{S}^{1}$

Even though we already have the two independent defect Weyl anomalies in hand,
there are other defect geometries whose partition function carry non-trivial
information related to anomalies. Here we consider the probe brane geometry
dual to the Wilson surface operator wrapping
$\mathds{S}^{1}_{R_{6}}\times\mathds{S}^{1}\hookrightarrow\mathds{S}^{1}_{R_{6}}\times\mathds{S}^{5}$.
The purpose of computing the on-shell action for this probe brane
configuration is two-fold. First, upon dimensional reduction along
$\mathds{S}^{1}_{R_{6}}$, the BPS Wilson surface operator descends to the
circular Wilson loop $W$ in $\mathcal{N}=1$ SYM theory on $\mathds{S}^{5}$.
This opens up the possibility of comparing to exact results for
$\left<W\right>$ obtained in the field theory using supersymmetric
localization. Second, the partition function $Z$ of an SCFT on
$\mathds{S}^{1}_{\beta}\times\mathds{S}^{d-1}$ has long been known to be
related to the superconformal index $\mathcal{I}$ up to an exponential pre-
factor containing a term known as the supersymmetric Casimir Energy (SCE),
$Z=e^{-\beta E_{c}}{\cal I}~{}$ that can be computed in terms of various
anomalies Assel:2014paa ; Assel:2015nca ; Bobev:2015kza . In the case of a
superconformal field theory, the defect SCE can be related to Weyl anomalies
and in the case of superconformal defects, has been computed in several cases
Chalabi:2020iie . However, it only has been rigorously proven to be related to
defect anomalies for co-dimension four defects in six dimensional ${\cal
N}=(2,0)$ SCFTs Meneghelli:2022gps .

Returning the probe brane computation, the coordinate transformation to take
us to a probe brane wrapping $\mathds{S}^{1}\times\mathds{S}^{1}$ is discussed
in appendix A ending with the AdS7 metric in eq. (A.16). We then take the
periodicity around the $\mathds{S}^{1}_{R_{6}}$ to be $\tau\sim\tau+2\pi
R_{6}$. On the field theory side in the dimensional reduction along
$\mathds{S}^{1}_{R_{6}}$ to a Wilson loop wrapping a great circle on
$\mathds{S}^{5}$, the five-dimensional Yang-Mills coupling is related to the
radius of the compactified direction by

$\displaystyle R_{6}=\frac{g_{\rm YM}^{2}}{8\pi^{2}}\,.$ (3.42)

Integrating over the $\mathds{S}^{2}$ and the $\mathds{S}^{1}$s wrapped in the
boundary geometry, the on-shell action takes the form

$\displaystyle S_{\rm
probe}=\frac{128\pi^{3}R_{6}}{81}T_{4}q_{j}\alpha^{j}\int_{0}^{\log\Lambda}d\sigma\sinh
2\sigma=-\frac{64\pi^{3}R_{6}}{81}T_{4}q_{j}\alpha^{j}\left(\frac{\Lambda^{2}}{2}-1+\frac{1}{2\Lambda^{2}}+\ldots\right)$
(3.43)

Again using eq. (3.8) as the counterterm action, the $\Lambda^{2}$ divergence
is regulated, and we find the holographically renormalized on-shell action

$\displaystyle S_{\rm ren}=-\frac{64\pi^{3}R_{6}}{81}T_{4}q_{j}\alpha^{j}\,.$
(3.44)

In order to more easily compare to the field theory computation, we define
$\beta=4\pi R_{6}$, hence

$\displaystyle\left<W\right>=\exp\left[\frac{\beta}{81\pi^{2}}q_{j}\alpha^{j}\right]=\exp\left[\beta
q_{i}(C^{-1})^{ij}r_{j}\right]\,,$ (3.45)

which in the $F_{0}=0$ case gives

$\displaystyle\left<W\right>=\exp\left[\frac{\beta}{2}{k}\sum_{j}q_{j}j(N-j)\right].$
(3.46)

In the following section, we will compare eq. (3.46) to the field theory
calculation of the expectation value of a circular Wilson loop in five-
dimensional SYM theory on $\mathds{S}^{5}$ computed using supersymmetric
localization.

Due to the nature of the holographic computation, it should not be entirely
surprising that $\left<W\right>$ can be expressed in a form similar to the
defect Weyl anomalies. However, without detailed knowledge of how the defect
SCE of surface operators in six-dimensional ${\cal N}=(1,0)$ SCFTs should be
related to defect Weyl anomalies, we cannot say what linear combination of
$a_{\Sigma}$ and $d_{2}$ should be computed by $\left<W\right>$. In the case
of a Wilson surface in the six-dimensional $A_{N-1}$ $\mathcal{N}=(2,0)$
theory, however, it has been been shown using chiral algebra methods that
$d_{2}$ alone appears in the defect SCE Meneghelli:2022gps , which leads to
the conjecture that the relation to the defect Weyl anomalies in eq. (3.46) is
not just superficial and that at large $N$ the defect SCE is given by
$E_{c}=d_{2}/32$, though a proof is lacking.

###  3.4 Defect gravitational anomaly

Here, we compute the defect gravitational anomaly from the worldvolume action
of the probe brane in the AdS${}_{7}\times M_{3}$ geometry in eq. (2.1). In
the anomaly polynomial this is the coefficient of the first Pontryagin class
of the defect submanifold $\Sigma$:

$\mathcal{I}_{4}=\frac{k_{r}}{2}c_{1}(r)-\frac{k_{g}}{24}p_{1}(T)\,.$ (3.47)

$p_{1}(T\Sigma)=\frac{{\rm Tr}(R_{T}\wedge R_{T})}{8\pi^{2}}$ is related by
descent mechanism to the gravitational three-dimensional Chern–Simons form,

$p_{1}=2\pi d{\rm CS}_{3}(R_{T})\,\qquad{\rm CS}_{3}(R_{T})=\frac{1}{4\pi}{\rm
tr}\left(\Gamma_{T}\wedge d\Gamma_{T}+\frac{2}{3}\Gamma_{T}^{3}\right)\,,$
(3.48)

where
${R_{T}}^{a}{}_{b}=(d\Gamma_{T}+\Gamma_{T}^{2})^{a}{}_{b}=\frac{1}{2}R^{a}{}_{b\mu\nu}dx^{\mu}\wedge
dx^{\nu}$ is the curvature of $\Sigma$. Explicitly we have

${\rm tr}(R_{T}\wedge
R_{T})=\frac{1}{4}R^{a}{}_{b\rho\sigma}R^{b}{}_{a\mu\nu}dx^{\mu}\wedge
dx^{\nu}\wedge dx^{\rho}\wedge dx^{\sigma}$ (3.49)

The anomaly appears in the partition function as a phase,

$e^{2\pi i\mathcal{A}}=e^{ik_{g}\int_{M_{3}}\frac{\text{CS}_{3}(R_{T})}{24}}$
(3.50)

where for us $M_{3}$ will be AdS3.

In order to evaluate the gravitational anomaly, we apply the strategy adopted
in Aharony:2007dj . To start, we need to look at the worldvolume action of a
stack of $q_{i}$ D4-branes wrapping $\mathds{S}^{2}_{z=i}\subset M_{3}$ and
extending along AdS3. In particular we focus on the Wess–Zumino term that
includes the gravitational contribution,

$S_{\rm D4}^{\rm WZ}=T_{4}\int_{M_{8}}{\rm Tr}\left[\sum_{n}C_{n}\wedge
e^{\cal F}\wedge\sqrt{\widehat{A}(2\pi R_{T})}\right]_{5}\,,$ (3.51)

where the A-roof genus reads

$\widehat{A}(R)=1-\frac{1}{24}p_{1}(T\Sigma)+\ldots\,.$ (3.52)

If we ignore the term due to the non-abelian scalars, the term proportional to
$p_{1}(T\Sigma)$ reads

$S_{\rm D4}^{\rm WZ}\supset-
q_{i}\int_{M_{5}}(C_{1}-C_{-1}B_{2})\wedge\frac{1}{24}p_{1}(T\Sigma)\,,$
(3.53)

where $q_{i}$ comes from the trace of the identity matrix. In addition we also
accounted for the IIA Romans mass contribution $F_{0}$ which formally can be
written as $F_{0}=dC_{-1}$. Integrating by parts and by using (3.51), we
obtain

$S_{\rm D4}^{\rm WZ}\supset-
q_{i}\int_{\mathds{S}^{2}_{z=i}}(F_{2}-F_{0}B_{2})\int_{\text{AdS}_{3}}\frac{{\rm
CS}_{3}(R_{T})}{24\times 2\pi}\,.$ (3.54)

By inputting the eq. (2.1) and by integrating over $\mathds{S}^{2}_{z=i}$ we
get

$S_{\rm D4}^{\rm WZ}\supset-q_{i}r_{i}\int_{\text{AdS}_{3}}\frac{{\rm
CS}_{3}(R_{T})}{24}\,.$ (3.55)

We now have to compare this result with the argument in (3.50). Considering
multiple stacks of D4-branes, we obtain the gravitational anomaly

$k_{g}=-\sum_{i}q_{i}r_{i}\,.$ (3.56)

This matches the field theory computation in eq. (B.18).

Let us now briefly comment on the computation that led to eq. (3.55) with an
eye toward section 4.2. First, notice that the defect gravitational anomaly is
not directly captured by inflow. Further, it does not have the same product
structure of the bulk-defect degrees of freedom, where the metric of the
string lattice and its dual appear. A possible interpretation for this is that
the gravitational anomaly accounts for the modes living only on the defect and
not interacting with the bulk six-dimensional SCFT. The latter does not
capture the inflow contribution, but it reproduces the gravitational anomaly
computation that we just performed in gravity purely from field theory.
Finally we speculate that the gravitational anomaly computation provides the
only contribution to the subleading correction of the defect $a_{\Sigma}$
anomaly in the large $r_{i}$ and $N$ limit. In other words, the on-shell
action provides the leading contribution, whereas the WZ action gives the
subleading one. This would work exactly like in Aharony:2007dj . The total
$a_{\Sigma}$ anomaly would then read Wang:2020xkc

$a_{\Sigma}=3k_{r}-\frac{k_{g}}{2}=24q_{i}(C^{-1})^{ij}r_{j}+\frac{\sum_{i}q_{i}r_{i}}{2}\,.$
(3.57)

##  4 Anomalies from Field Theory and their string theory constructions

In this section, we test the holographic predictions provided by the probe
brane computations in the previous section in two different settings. In the
first subsection, we employ supersymmetric localization to compute the
expectation value of a circular Wilson loop operator inserted in the
antisymmetric representation of a necklace quiver gauge theory on
$\mathds{S}^{5}$. This Wilson loop operator is thought to have UV completion
to the string defect operator studied in section 3.3.3, and so we compare the
localization computation to the holographic result for $\left<W\right>$. In
the second subsection, we take a different approach. We first rely on
IIB/F-theory constructions of the 6d $\mathcal{N}=(1,0)$ SCFTs to compute
defect anomalies using inflow arguments. We then use known non-perturbative
relations between ’t Hooft anomalies and the A-type Weyl anomaly for
superconformal two-dimensional defects to compute $a_{\Sigma}$. In both cases,
we find exact agreement with the probe brane results.

###  4.1 Necklace quivers and supersymmetry localization on $\mathds{S}^{5}$

In this section, we demonstrate that the holographically renormalized on-shell
action in eq. (3.44) can be matched with the expectation value of the Wilson
loop in the five-dimensional ${\cal N}=1$ theory described by a circular
quiver with $k$ $\mathrm{SU}(N)$ gauge groups and the massless bifundamental
hypermultiplet between the two gauge groups that are next to each other. Here
the Wilson loop is taken to be in the rank-$j$ antisymmetric representation of
each gauge group. Such a quiver theory can be regarded as a Kaluza–Klein (KK)
theory whose UV completion is the six-dimensional ${\cal N}=(1,0)$ theory
living on $N$ M5-branes on the $\mathbb{C}^{2}/\mathbb{Z}_{k}$ singularity.
Just like in Sec. 2, this six-dimensional ${\cal N}=(1,0)$ theory can be
realized using the Type IIA brane system consisting of $k$ D6-branes in the
$0123456$ directions, along with $N$ NS5-branes in the $012345$ directions;
unlike in the general case, no D8-branes are necessary. The defect is regarded
as a D4-brane spanning the $01789$ directions and is located between the
$(j-1)$-th and $j$-th NS5-branes. In particular, we will show that the
expectation value of the Wilson loop in question is related to the Weyl
anomalies of the D4-brane defects in such a brane configuration.

The expectation value of the Wilson loop in the rank-$j$ antisymmetric
representation $\wedge^{j}$ of five-dimensional ${\cal N}=1$ SYM with
$\mathrm{SU}(N)$ gauge group was performed in (Mori:2014tca, , Section 2.3):

$\displaystyle\left<W_{\wedge^{j}}\right>=\exp\left(\frac{\beta}{2}j(N-j)\right)\,,\qquad\beta\equiv\frac{g_{\text{YM}}^{2}}{2\pi
r}$ (4.1)

where $r$ is the radius of the five-sphere which the Wilson loop wraps. On the
other hand, the authors of Minahan:2013jwa ; Minahan:2016xwk  also considered
the circular quiver as described above, where the Wilson loop was taken to be
in the fundamental representation $\mathbf{N}$ of one of the $\mathrm{SU}(N)$
gauge groups and in the trivial representation $\mathbf{1}$ of the others.
They found that its expectation value turns out to be equal to eq. (4.1) with
$j=1$, namely that of the Wilson loop in the fundamental representation of the
five-dimensional ${\cal N}=1$ SYM with $\mathrm{SU}(N)$ gauge group Kim:2012qf
; Minahan:2013jwa :

$\displaystyle\left<W_{(\mathbf{N},\mathbf{1},\ldots,\mathbf{1})}\right>=\exp\left(\frac{\beta}{2}(N-1)\right)\overset{N\rightarrow\infty}{\sim}\exp\left(\frac{\beta}{2}N\right)\,.$
(4.2)

where the gauge coupling of each gauge group in the circular is taken to be
equal to $g_{\text{YM}}$. Our main goal is to generalize these results, namely
to compute the expectation value $\langle
W_{(\wedge^{j},\wedge^{j},\ldots,\wedge^{j})}\rangle$ of the Wilson loop in
the rank-$j$ antisymmetric representation of every $\mathrm{SU}(N)$ gauge
group in the circular quiver.

To achieve this, we proceed as in (Minahan:2016xwk, , Section 4.3) and
(Mori:2014tca, , Section 2). According to (Minahan:2016xwk, , (3.22)), the
eigenvalue distribution of each $\mathrm{SU}(N)$ gauge group in the circular
quiver can be taken to be the same. We denote these by $\phi_{i}$, with
$i=1,\ldots,N$. The expression of the partition function then simplifies,
whereby in the large $N$ limit it takes the form

$\displaystyle
Z\sim\int\prod_{i=1}^{N}d\phi_{i}\,\exp\left[-k\frac{N^{2}}{\beta}\sum_{i=1}^{N}\phi_{i}^{2}+k\frac{N}{2}\sum_{1\leq
i\neq l\leq N}|\phi_{i}-\phi_{l}|\right]$ (4.3)

which is indeed a simple modification of the matrix model associated with
five-dimensional ${\cal N}=2$ SYM given by (Mori:2014tca, , (2.3)). The latter
can simply be obtained from eq. (4.3) by setting $k=1$. The saddle points are
the same as in (Mori:2014tca, , (2.4), (2.5)):

$\displaystyle 0$ $\displaystyle=-\frac{2N^{2}}{\beta}\phi_{i}+N\sum_{l,i\neq
l}\mathrm{sign}(\phi_{i}-\phi_{l})\,;$ (4.4) $\displaystyle\phi_{i}$
$\displaystyle=\frac{\beta}{2N}(N-2i)\,,\quad\text{for $\phi_{i}>\phi_{l}$
whenever $i<l$.}$ (4.5)

Note that, upon evaluating the exponential function in the integrand of eq.
(4.3) at the saddle points, we find that the free energy $F=-\log Z$ obtained
from above is indeed $k$ times that of the five-dimensional ${\cal N}=2$ SYM
with $\mathrm{SU}(N)$ gauge group, as pointed out in (Kallen:2012zn, ,
(3.11)), (Minahan:2013jwa, , (4.30)), and (Minahan:2016xwk, , Section 4.3).
Let us now proceed with the computation of the Wilson loop expectation value.
Similarly to (Mori:2014tca, , (2.15)), this is given by

$\displaystyle\left<W_{(\wedge^{j},\wedge^{j},\ldots,\wedge^{j})}\right>\sim\int\prod_{i=1}^{N}d\phi_{i}\,\exp\left[-k\frac{N^{2}}{\beta}\sum_{i=1}^{N}\phi_{i}^{2}+k\frac{N}{2}\sum_{1\leq
i\neq l\leq N}|\phi_{i}-\phi_{l}|+kN\sum_{i=1}^{j}\phi_{i}\right]\,.$ (4.6)

Since the derivative with respect to $\phi_{i}$ of the last term in the
exponential gives a constant $kN$ for all $i=1,\ldots,j$, the insertion of the
Wilson loop does not change the eigenvalue distribution. Evaluating the
exponential function containing the last term yields

$\displaystyle\left<W_{(\wedge^{j},\wedge^{j},\ldots,\wedge^{j})}\right>\sim\exp\left(kN\sum_{i=1}^{j}\phi_{i}\right)\Bigg{|}_{(\ref{saddles})}\sim\exp\left(\frac{\beta}{2}kj(N-j)\right)\,.$
(4.7)

The argument of the exponential function is in agreement with the holographic
prediction for the surface operator wrapping
$\mathds{S}^{1}\times\mathds{S}^{1}$ with vanishing Romans mass in eq. (3.46)
for a single defect $q_{j}=1$.

###  4.2 Anomaly Inflow from type IIB

In this subsection, we derive the $a_{\Sigma}$ anomaly on the defect via
inflow mechanism from the F-theory construction of six-dimensional
$\mathcal{N}={(1,0)}$ SCFTs Heckman:2013pva ; DelZotto:2014hpa ;
Heckman:2015bfa . These backgrounds can be described in terms of type IIB
string theory on a complex two-dimensional space with a non-trivial fibration
of the axio-dilaton. When the space is singular, IIB on the singularity
engineers the SCFT. If instead the space is resolved, which then denote by
$\mathcal{B}$, IIB provides a definition of the tensor branch effective field
theory (EFT) and corresponds to the scalars in the tensor multiplets acquiring
non-trivial vacuum expectation values. The full BPS physics of the tensor
branch EFT is determined by the geometric structure of the 2-cycles in
$\mathcal{B}$ which read

$\displaystyle\int_{\mathcal{B}}\beta_{i}\wedge\beta_{j}=A_{i}\cdot
A_{j}=\int_{A_{i}}\beta_{j}=\Omega_{ij}$ (4.8)
$\displaystyle\int_{\mathcal{B}}\beta_{i}\wedge\alpha^{j}=A_{i}\cdot
B^{j}=\int_{B^{j}}\beta_{i}=\delta_{i}^{j}\,.$

Here $\cdot$ denotes the number of intersections between cycles, weighted by
signs so as to be topologically invariant; $\beta_{i},\alpha^{i},B^{i},A_{i}$
generate the compact and non-compact cohomology and homology:

$\displaystyle\beta_{i}\in H^{2}_{\rm c}(\mathcal{B}),\quad\alpha^{i}\in
H^{2}_{\rm nc}(\mathcal{B})$ (4.9) $\displaystyle B^{i}\in H_{2}^{\rm
nc}(\mathcal{B}),\quad A_{i}\in H_{2}^{\rm c}(\mathcal{B})\,.$

The axio-dilaton depends on the coordinates of the base and it is interpreted
as the complex structure of a $\mathds{T}^{2}$ fibered over $\mathcal{B}$.
Crucially, the axio-dilaton degenerates over some loci of $\mathcal{B}$, and
the type of monodromy around these singular loci determines which D7-branes
spans the singular loci. Since we want to have supersymmetric configurations,
the singular loci themselves are the holomorphic 2-cycles of $\mathcal{B}$. In
addition, we associate a gauge algebra to a given D7-brane, which can be
trivial or non-trivial or even of exceptional type. The RR four-form potential
of type IIB, $C_{4}$, is expanded as follows, $\mathcal{B}$, and the type of
monodromy around these singular loci determines which D7-branes spans the
singular loci. Since we want to have supersymmetric configurations, the
singular loci themselves are the holomorphic 2-cycles of $\mathcal{B}$. In
addition, we associate a gauge algebra to a given D7-brane, which can be
trivial or non-trivial or even of exceptional type. The RR four-form potential
of type IIB is expanded as

$C_{4}=B_{2}^{i}\wedge\beta_{i}$ (4.10)

providing the dynamical anti-symmetric two-form fields $B_{2}^{i}$. The
bosonic content of the tensor multiplet is then completed by the real scalar
modulus corresponding to

${\rm Vol}(A_{i})=\int_{A_{i}}J=\Omega_{ij}\phi^{j},\qquad
J=\phi^{i}\beta_{i}\,,$ (4.11)

where $J$ is the Kähler form of $\mathcal{B}$. Lastly, we have D3-branes
wrapping $A_{i}$, which are electrically charged under $C_{4}$ and therefore
under the tensor multiplets. They form a lattice of BPS strings whose pairing
is given by $\Omega_{ij}$.

This lattice of string operators plays a crucial role in determining the
spectrum of defect operators. In a four-dimensional gauge theory the “defect
group” $\Lambda^{*}_{\rm root}/\Lambda_{\rm root}$ (where $\Lambda_{\rm root}$
is the root lattice of the gauge algebra and $\Lambda^{*}_{\rm root}$ is its
dual) organizes the set of charged operators that cannot be screened. On a
basic level, the dual lattice tells us about the non-trivial line defects that
the theory supports. The geometric understanding of the origin of these
lattices from the brane picture (at least for SCFTs) can be lifted to a
similar defect group for six-dimensional theories DelZotto:2015isa , where now
$\Lambda_{\rm string}$ is the charge lattice of tensionless BPS strings and
$\Lambda^{*}_{\rm string}$ is the lattice of string defects that the theory
supports. The six-dimensional defect group is then given by $\Lambda^{*}_{\rm
string}/\Lambda_{\rm string}$, which determines the strings charged under
self-dual two-forms that cannot be screened.

####  4.2.1 Anomaly Polynomial

The anomaly polynomial, $I_{8}$, capturing the perturbative ’t Hooft anomalies
for continuous symmetries of six-dimensional $\mathcal{N}={(1,0)}$ theories
has been well studied in Cordova:2015fha ; Ohmori:2014kda ;
Intriligator:2014eaa  and can be computed from the low-energy EFT on the
tensor branch. The necessary condition is that the tensor branch EFT contains
the appropriate Green–Schwarz (GS) coupling to cancel reducible gauge
anomalies. That is, $I_{8}\sim(I_{4}^{\rm gauge})^{2}$ where $I_{4}^{\rm
gauge}=1/2{\rm tr}_{\lambda}(F_{g}\wedge F_{g})$, $F_{g}$ is the field
strength of the gauge field, and $\lambda$ is a representation of the gauge
algebra. Let us define the Dirac pairing of BPS strings,

$C_{ij}=-\Omega_{ij}\,,$ (4.12)

where $C_{ij}$ is the same intersection paring introduced in (4.8). Then the
GS coupling Green:1984bx ; Sagnotti:1992qw  takes the form

$S_{\rm GS}=\int_{M_{6}}C_{ij}B_{2}^{i}\wedge I_{4}^{j}\,.$ (4.13)

$S_{\rm GS}$ couples the tensors to the gauge fields, but $I_{4}^{i}$ also
contains characteristic classes of the global symmetries like the first
Pontryagin class of the tangent bundle $p_{1}(T)$, as well as the the second
Chern class of the ${\rm SU}(2)_{R_{I}}$-symmetry bundle $c_{2}(I)$.

Let us be more specific:

$I_{4}^{i}=\frac{1}{4}{\rm Tr}(F_{i}\wedge
F_{i})+y^{i}c_{2}(I)-K^{i}p_{1}(T)\,,$ (4.14)

where ${\rm Tr}\cong{\rm tr}_{\lambda}/{\rm Ind}(\lambda)$ is the normalized
trace over the index of the representation $\lambda$, and $F_{i}$ is the field
strength of the gauge field associated to the D7-brane wrapping $A_{i}$;
keeping in mind that this can be the trivial gauge algebra as well. In
addition we have that for a one-instanton background
$\int_{\Sigma_{4}}\frac{1}{4}{\rm Tr}(F\wedge F)=1$ for a closed
$\Sigma_{4}\subset M_{6}$. $y^{i},K^{i}$ are coefficients such that also mixed
gauge–global anomalies are canceled, this is due to the fact that six-
dimensional SCFTs do not have two-form conserved currents Cordova:2020tij .
This fixes

$y_{i}=(C^{-1})^{ij}r_{j},\qquad r_{i}=h_{\mathfrak{g}_{i}}^{\vee}\,,$ (4.15)

where $h_{\mathfrak{g}_{i}}^{\vee}$ is the dual Coxeter number of the gauge
algebra $\mathfrak{g}_{i}$. In addition,

$K^{i}=(C^{-1})^{ij}(2-C_{jj})\,.$ (4.16)

The GS contribution to the anomaly polynomial then reads

$I_{\rm GS}=\frac{1}{2}C_{ij}I_{4}^{i}I_{4}^{j}\,.$ (4.17)

The origin of $I_{\rm GS}$ from the IIB background was proposed in
Ohmori:2014kda . This comes in term of a formal twelve-form which we then
integrate over the IIB base $\mathcal{B}$:

$I_{\rm GS}=-\frac{1}{2}\int_{\mathcal{B}}Z^{2}\,,\qquad dF_{5}=Z\,,$ (4.18)

where $F_{5}$ is the five-form flux, and Sadov:1996zm

$Z=\frac{1}{4}\left(c_{1}(\mathcal{B})\wedge p_{1}(T)+\sum_{i}{\rm
Tr}(F_{i}\wedge F_{i})\wedge\beta_{i}\right)\,.$ (4.19)

$c_{1}(\mathcal{B})$ is the first Chern class of the base, and we have the
relation with the anticanonical bundle of $\mathcal{B}$,

$\int_{\mathcal{B}}c_{1}(\mathcal{B})\wedge\beta_{i}=C_{ij}K^{j}=(2-C_{ii}),$
(4.20)

along with

$F_{5}=H^{i}\wedge\beta_{i}\,,\qquad dH^{i}=I_{4}^{i}\,,\qquad-
C_{ij}I^{j}=\int_{\mathcal{B}}Z\wedge\beta_{i},\qquad
Z=I_{4}^{i}\wedge\beta_{i}\,.$ (4.21)

We have assumed that $dH^{i}$ also contains a term proportional to $c_{2}(R)$,
which is necessary for the mixed anomaly cancellation. On the other hand it is
not directly visible from the IIB geometric construction because the
R-symmetry is not manifest. Plugging (4.21) in (4.18) we get (4.13).

The GS contribution can be also derived from the world-volume action of the
D7-branes where the following term should be present:

$\int_{M_{6}\times A_{i}}C_{4}\wedge I^{i}=\int_{M_{6}\times
A_{i}}B_{2}^{j}\wedge\beta_{j}\wedge I^{i}=\int_{M_{6}}C_{ij}B_{2}^{i}\wedge
I^{j}$ (4.22)

In case of D7-branes the contribution proportional to ${\rm Tr}(F_{i}\wedge
F_{i})$ and $p_{1}(T)$ come from the Wess–Zumino term,

$S_{D7}^{\rm WZ}=\mu_{7}\int_{M_{8}}{\rm Tr}\left[\sum_{n}C_{n}\wedge e^{\cal
F}\right]_{8}\,,$ (4.23)

where we ignore the dependence on the scalar fields parameterizing the
orthogonal directions.

####  4.2.2 Anomaly inflow for BPS strings and surface defects

As we anticipated, the D3-branes wrapping $A_{i}$ will provide the BPS strings
charged under $B_{2}^{i}$. On the other hand, the D3-branes wrapping non-
compact cycles $B^{i}$ realize the $\mathcal{N}=(0,4)$ surface defects in the
theory DelZotto:2015isa . Since both objects are D3-branes, they source
additional terms in $F_{5}$:

$dF_{5}=Z+Q^{i}\,(\delta^{(4)}(\mathbf{x}^{\perp})d\mathbf{x}^{\perp}+\chi_{4}(N\Sigma))\wedge\beta_{i}+q_{i}\,(\delta^{(4)}(\mathbf{x}^{\perp})d\mathbf{x}^{\perp}+\chi_{4}(N\Sigma))\wedge\alpha^{i}\equiv
Z^{\prime}$ (4.24)

where $Q^{i}$ are the BPS string charges and $q_{i}$ are the number of
defects. $\mathbf{x}^{\perp}$ are the coordinates of the space
$\mathbb{R}^{\perp}_{4}\subset M_{6}$ perpendicular to the BPS strings and
surface defects in the six-dimensional space-time. The tangent bundle $TM_{6}$
of the background splits into the tangent bundle $T\Sigma$ and the normal
bundle $N\Sigma$ (with structure group ${\rm SO}(4)$) of the defect
submanifold $\Sigma$. So, as in Shimizu:2016lbw ,

$\displaystyle\chi_{4}(N\Sigma)=c_{2}(L)-c_{2}(R)\,,$ (4.25) $\displaystyle
p_{1}(TM_{6})=p_{1}(T\Sigma)+p_{1}(N\Sigma)=p_{1}(T\Sigma)-2(c_{2}(L)+c_{2}(R))\,.$

The surface defect is a D3 probe in the geometric F-theory setup, which in a
string theory background can receive anomaly contributions from the bulk
theory by inflow. Placing the probe brane in the string theory background
produces a bulk anomaly for a given symmetry. This anomaly is then canceled by
an anomaly for the probe brane Freed:1998tg ; Kim:2012wc . We can apply the
same logic here. We need to integrate the anomaly polynomial in the presence
of the D3 brane probe source given by (4.24). We can either start from IIB or
directly from six dimensions. Only the reducible part will contribute $I_{\rm
GS}$ to the inflow:

$\int_{\mathbb{R}_{4}^{\perp}\times\mathcal{B}}I_{\rm
GS}=-\int_{\mathbb{R}_{4}^{\perp}\times\mathcal{B}}\frac{1}{2}(Z^{\prime})^{2}\supset-\int_{\mathcal{B}}I_{4}^{i}\wedge\beta_{i}\wedge(Q^{j}\beta_{j}+q_{j}\alpha^{j})=C_{ij}I_{4}^{i}Q^{j}-q_{j}\delta_{i}^{j}I^{i}_{4}\,.$
(4.26)

The first term of (4.26) would now reproduce the one for the BPS strings
computed in Shimizu:2016lbw . But let us focus on the inflow on the defect
only, neglecting the inflow contributions on the strings, setting $Q^{i}=0$.
Plugging in (4.14) and (4.15),

$\mathcal{I}_{4}(Q^{i}=0)=-q_{i}(C^{-1})^{ij}r_{j}c_{2}(I)\,.$ (4.27)

We need to convert the ${\rm SU}(2)_{R_{I}}$ into the $(0,2)$ ${\rm U}(1)_{r}$
superconformal R-symmetry, and for this we use the result of (Wang:2020xkc, ,
(4.28)),555Note that the second equality in (4.28) is slightly different from
(Wang:2020xkc, , (4.28)), since we are using half-integral charges with
respect to $R,I$. We thus remove a factor of 2 in the formula (Wang:2020xkc, ,
(4.28)).

$a_{\Sigma}=3k_{r}-\frac{k_{g}}{2},\qquad r=(R-2I)\,,$ (4.28)

recalling the definition of $k_{r}$ in (3.47).

The ${\rm SU}(2)_{R_{I}}$ vector bundle, $V_{I}$, splits in terms of its
Cartan line bundle $L_{I}$ as $V_{I}=L_{I}\oplus L_{I}^{\vee}$. This implies
that

$c_{2}(I)=-c_{1}(L_{I})^{2}=-4c_{1}(r)^{2}\,.$ (4.29)

The factor of 4 comes from (4.28); we ignored the contribution from $c_{2}(R)$
since it vanishes when $Q^{i}=0$. In the anomaly polynomial we have,

$K_{I}c_{2}(I)=-4K_{I}c_{1}(r)^{2}=\frac{k_{r}}{2}c_{1}(r)^{2},\qquad
K_{I}=-q_{i}(C^{-1})^{ij}r_{j}\,,$ (4.30)

which implies that

$\begin{split}k_{r}=-8K_{I}~{}\end{split}$ (4.31)

The inflow contribution to the surface defect $a_{\Sigma}$ anomaly of a
generic six-dimensional SCFT is then

$a_{\Sigma}=24q_{i}(C^{-1})^{ij}r_{j}\,.$ (4.32)

In the case where $C_{ij}$ is the Cartan matrix of $\mathfrak{su}(N)$ and
$r_{i}$ are the ranks of the $\mathfrak{su}(r_{i})$ gauge nodes of the six-
dimensional linear quiver in the tensor branch, the $a_{\Sigma}$ anomaly we
just obtained in (4.32) matches the gravity computation (3.38).

##  5 Discussion

In this work, we have characterized the probe limit of large ${\cal N}=(0,4)$
two-dimensional superconformal defects in six-dimensional ${\cal N}=(0,1)$
SCFTs at large $N$. We have constructed BPS solutions for embedding AdS3 probe
D4-branes in AdS${}_{7}\times M_{3}$ solutions of type IIA SUGRA with non-
trivial Romans mass Apruzzi:2013yva . For these probe brane embeddings, we
have computed the probe brane contribution to the holographic EE of a
spherical region in the dual field theory, the probe brane on-shell action for
various AdS3 boundary geometries, and the two-dimensional gravitational
anomaly coming from the probe WZ action. The defect EE and on-shell action
uniquely fix the independent defect Weyl anomalies $a_{\Sigma}$ and $d_{2}$ in
eq. (3.38) at large $N$, while the gravitational anomaly provides a subleading
correction to $a_{\Sigma}$ in eq. (3.57). These results provide novel
holographic predictions for universal quantities that are crucial for the
study of the ubiquitous string-like defects in six-dimensional SCFTs.

With our holographic predictions for defect anomalies in hand, section 4
focused on computations in field theory to verify our results by utilizing the
various string theoretic construction at our disposal. In particular on the
field theory side, we found that the expectation value of a circular Wilson
loop in ${\cal N}=1$ SYM theory on $\mathds{S}^{5}$ computed using
supersymmetric localization agreed exactly with the holographic computation of
the on-shell action of a probe D4-brane wrapping
$\mathds{S}^{1}\times\mathds{S}^{1}$ at the conformal boundary of AdS3. This
match mirrors similar computations done for Wilson surface operators in
$A_{N-1}$ six-dimensional ${\cal N}=(0,2)$ SCFTs, e.g Mori:2014tca . In string
theoretic constructions, we were able to compute defect ’t Hooft anomalies
using inflow arguments. Working in IIB, we showed that D3-branes wrapping non-
compact cycles receive a contribution from inflow to the R-anomaly in their
defect four-form anomaly polynomial that, using the non-perturbative relations
between defect ’t Hooft and Weyl anomalies in Wang:2020xkc , matches the
leading large $N$ results for $a_{\Sigma}$ obtained using probe brane
holography.

The inflow contribution provides the leading order contribution of
$a_{\Sigma}$ at large $N$ and $r_{i}$ and we argue that inflow captures the
bulk-defects modes. On the other hand the subleading contribution coming from
the Wess–Zumino action of the brane is reproduced by the gravitational anomaly
of the 2d quiver description of the system with the defect inserted, which has
been derived from the NS5–D2–D4–D6–D8 brane system. The 2d quiver
gravitational anomaly does not capture the inflow term and the anomaly inflow
does not capture the gravitational anomaly contribution to $a_{\Sigma}$. For
this reason, we argue that the 2d quiver describes the intrinsic modes of the
defect and the anomaly inflow the bulk-defect degrees of freedom. The latter
are leading whereas the former are subleading at large $N,r_{i}$.

## Acknowledgements

We thank J. Heckman and H. Shimizu for their initial involvement in this
paper, which is seeing the light after an unusually long gestation. We are
supported in part by INFN. The work of FA is supported in part by the Italian
MUR Departments of Excellence grant 2023-2027 "Quantum Frontiers”. NM is
partially supported by the MUR-PRIN grant No. 2022NY2MXY. AT is supported in
part by MUR-PRIN contract 2022YZ5BA2.

##  Appendix A Coordinate Transformations

Let’s consider the flat Euclidean geometry

$\displaystyle ds^{2}=\delta_{\mu\nu}dx^{\mu}dx^{\nu}$ (A.1)

where $\mu,\nu=1,\ldots,6$. We start from a flat defect in the $x^{5}$–$x^{6}$
plane situated at $x^{1}=x^{2}=x^{3}=x^{4}=0$. Consider the conformal
transformation $x^{\mu}\to\widetilde{x}^{\mu}(x)$ that leaves $\mathbb{R}^{6}$
invariant but maps the defect submanifold from
$\mathbb{R}^{2}\to\mathds{S}^{2}$ with radius $R$:

$\displaystyle\begin{split}\widetilde{x}^{1}&=R\frac{4|x|^{2}-R^{2}}{R^{2}+4|x|^{2}+4Rx^{1}}\,,\qquad\qquad\widetilde{x}^{a}=4R^{2}\frac{x^{a}}{R^{2}+4|x|^{2}+4Rx^{1}}\,,\end{split}$
(A.2)

where $a=2,\ldots,6$ with inverse transformation

$\displaystyle\begin{split}x^{1}&=R\frac{R^{2}-|\widetilde{x}|^{2}}{2((R-\widetilde{x}^{1})^{2}+\delta_{bc}\widetilde{x}^{b}\widetilde{x}^{c})}\,,\qquad\qquad
x^{a}=R^{2}\frac{\widetilde{x}^{a}}{(R-\widetilde{x}^{1})^{2}+\delta_{bc}\widetilde{x}^{b}\widetilde{x}^{c}}\,.\end{split}$
(A.3)

The effect of this transformation is to map eq. (A.1) to

$\displaystyle
ds^{2}=\frac{R^{4}}{((R-\widetilde{x}^{1})^{2}+\delta_{bc}\widetilde{x}^{b}\widetilde{x}^{c})^{2}}\delta_{\mu\nu}d\widetilde{x}^{\mu}d\widetilde{x}^{\nu}\,.$
(A.4)

Importantly the hyperplane $x^{1}=\ldots=x^{4}=0$ is mapped to
$\widetilde{x}^{2}=\widetilde{x}^{3}=\widetilde{x}^{4}=0$ with
$\delta_{\mu\nu}\widetilde{x}^{\mu}\widetilde{x}^{\nu}=R^{2}$.

To make the defect geometry a bit clearer, use
$(\widetilde{x}^{1})^{2}=R^{2}-(\widetilde{x}^{5})^{2}-(\widetilde{x}^{6})^{2}$
and $\widetilde{x}^{2}=\widetilde{x}^{3}=\widetilde{x}^{4}=0$ to write the
image of the line element of the defect submanifold geometry under the
conformal map as

$\displaystyle
ds^{2}|_{\Sigma}=R^{2}\frac{(R^{2}-(\widetilde{x}^{6})^{2})(d\widetilde{x}^{5})^{2}+(R^{2}-(\widetilde{x}^{5})^{2})(d\widetilde{x}^{6})^{2}+2\widetilde{x}^{5}\widetilde{x}^{6}d\widetilde{x}^{5}d\widetilde{x}^{6}}{4(R^{2}-(\widetilde{x}^{5})^{2}-(\widetilde{x}^{6})^{2})\Big{(}R-\sqrt{R^{2}-(\widetilde{x}^{5})^{2}-(\widetilde{x}^{6})^{2}}\Big{)}^{2}}\,.$
(A.5)

Then, using the coordinate transformation

$\displaystyle\widetilde{x}^{5}=R\sin\Theta\sin\Phi,\qquad\widetilde{x}^{6}=R\sin\Theta\cos\Phi$
(A.6)

we find

$\displaystyle
ds^{2}|_{\Sigma}=\frac{R^{2}}{16\sin^{4}\Theta/2}(d\Theta^{2}+\sin^{2}\Theta
d\Phi^{2})\,,$ (A.7)

where the conformal factor can be removed by a residual ${\rm SO}(2,2)$
transformation. So, the defect submanifold embedding now takes the form
$\mathds{S}^{2}\hookrightarrow\mathbb{R}^{6}$.

Since we are ultimately interested in studying the geometry of the probe brane
dual to this defect, we write the AdS7 metric as (3.31), with

$\displaystyle ds_{\mathds{H}^{5}}^{2}$ $\displaystyle=d\rho^{2}+\sinh^{2}\rho
d\Omega_{4}^{2}\,,$ $\displaystyle d\Omega_{4}^{2}$
$\displaystyle=d\zeta^{2}+\sin^{2}\zeta
d\widetilde{\Omega}_{2}^{2}+\cos^{2}\zeta d\chi^{2}\,,$ (A.8) $\displaystyle
d\widetilde{\Omega}_{2}^{2}$
$\displaystyle=d\widetilde{\theta}^{2}+\sin^{2}\widetilde{\theta}d\widetilde{\phi}^{2}\,.$

Starting from eq. (A.1), we first map to polar coordinates with

$\displaystyle\begin{split}&x^{1}=\varrho\cos\zeta\cos\chi\,,\quad
x^{2}=\varrho\cos\zeta\sin\chi\,,\\\
&x^{3}=\varrho\sin\zeta\sin\widetilde{\theta}\sin\widetilde{\phi}\,,\quad
x^{4}=\varrho\sin\zeta\sin\widetilde{\theta}\cos\widetilde{\phi}\,,\quad
x^{5}=\varrho\sin\zeta\cos\widetilde{\theta}\,,\end{split}$ (A.9)

which brings us to

$\displaystyle
ds_{6}^{2}=d\varrho^{2}+\varrho^{2}d\Omega_{4}^{2}+(dx^{6})^{2}\,.$ (A.10)

Now, we can map to $\mathds{H}^{5}\times\mathds{S}^{1}$ by

$\displaystyle\varrho=\frac{L\sinh\rho}{\cosh\rho-\cos\varphi},\qquad
x^{6}=\frac{L\sin\varphi}{\cosh\rho-\cos\varphi}\,,$ (A.11)

which gives

$\displaystyle
ds_{6}^{2}=\frac{L^{2}}{(\cosh\rho-\cos\varphi)^{2}}(d\rho^{2}+\sinh^{2}\rho\,d\Omega_{4}^{2}+d\varphi^{2})\,.$
(A.12)

Stripping off the overall conformal factor, we can extend this into the bulk
with a formula similar to (3.11):

$\begin{split}\varrho=&\frac{\cosh\sigma\sinh\rho}{\cosh\sigma\cosh\rho+\sinh\sigma\cos\varphi}\,,\qquad
x^{6}=\frac{\sinh
r\sin\varphi}{\cosh\sigma\cosh\rho+\sinh\sigma\cos\varphi}\,,\\\
z=&\frac{1}{\cosh\sigma\cosh\rho+\sinh\sigma\cos\varphi}\,.\end{split}$ (A.13)

This turns $(dz^{2}+d\varrho^{2}+\varrho^{2}d\Omega_{4}^{2}+dx_{6}^{2})/z^{2}$
into eq. (3.31). Fig. 3 illustrates this coordinate change. The defect wraps
the $\mathds{S}^{2}\hookrightarrow\mathds{H}_{5}$ at the conformal boundary of
$\mathds{H}_{5}$ at $\rho\to\infty$, $\zeta=\pi/2$. The probe brane wraps
$\\{\rho,\widetilde{\theta},\widetilde{\phi}\\}$ in the bulk AdS7 geometry and
an $\mathds{S}^{2}$ in the internal space.

Figure 3: Starting from the flat $\mathbb{R}^{6}$ slicing (horizontal blue
dotted lines) of AdS7, the map in eqs. A.9 and A.11 brings us to
$\mathds{H}_{5}\times\mathds{S}^{1}$ sliced AdS7. The conformal boundary
$\partial\text{AdS}_{7}$ in this new coordinatization can be reached by taking
either $\sigma\to\infty$ at fixed $\rho$ which lands in the bulk of
$\mathds{H}_{5}$ or $\rho\to\infty$ at fixed $\sigma$ which simultaneously
takes us to $\partial\mathds{H}_{5}\times\mathds{S}^{1}$.

We can follow a similar map that brought us to a spherical defect in
$\mathds{H}_{5}\times\mathds{S}^{1}$ to bring us to a defect wrapping
$\mathds{S}^{1}\times\mathds{S}^{1}$ in $\mathds{S}^{5}\times\mathds{S}^{1}$.
We start with eq. (A.10), and instead of eq. (A.11), we take

$\displaystyle\begin{split}\varrho=\frac{L\sin\varsigma}{\cosh\tau-\cos\varsigma},\qquad
x^{6}=\frac{L\sinh\tau}{\cosh\tau-\cos\varsigma}\,,\end{split}$ (A.14)

which brings us to

$\displaystyle
ds_{6}^{2}=\frac{L^{2}}{(\cos\varsigma-\cosh\tau)^{2}}(d\tau^{2}+d\varsigma^{2}+\sin^{2}\varsigma
d\Omega_{4}^{2})\,.$ (A.15)

The boundary metric, up to a conformal factor, is now
$\mathbb{R}\times\mathds{S}^{5}$. Compactifying the $\tau$ direction with
periodicity $\tau\sim\tau+\beta$ gives us
$\mathds{S}^{1}_{\beta}\times\mathds{S}^{5}$. Extending this geometry into the
AdS7 bulk we get

$\displaystyle
ds_{7}^{2}=d\sigma^{2}+\cosh^{2}\sigma\,d\tau^{2}+\sinh^{2}\sigma\,d\Omega_{5}^{2}\,.$
(A.16)

The probe now wraps $(\sigma,\,\tau)$ and a great circle at the
$\mathds{S}^{5}$ equator as well as the internal $\mathds{S}^{2}$.

##  Appendix B Two-dimensional quiver gauge theories

As reviewed in Section 2, the six-dimensional theory is realized on the stacks
of D6 branes ending on the NS5 branes. The defects we are considering are as
in the upper part of Fig. 2. For completeness we also include the string-like
objects realized by D2 branes ending on D4 branes. All in all we end up with
the brane intersection in (B.2). For simplicity of the analysis, we do not
consider the contribution of the D8 branes in the above configuration, but
will briefly comment about their contributions to various anomaly coefficient
at the end of this Appendix.

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}\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\pgfsys@color@rgb@stroke{0}{0.5}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0.5}{0}\pgfsys@invoke{ }\definecolor[named]{pgffillcolor}{rgb}{0,0.5,0}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}\pgfsys@moveto{99.58466pt}{-17.07182pt}\pgfsys@lineto{99.58466pt}{-17.07182pt}\pgfsys@curveto{99.58466pt}{-16.3605pt}{98.44655pt}{-16.00485pt}{98.44655pt}{-15.29353pt}\pgfsys@curveto{98.44655pt}{-14.5822pt}{99.58466pt}{-14.22655pt}{99.58466pt}{-13.51523pt}\pgfsys@lineto{99.58466pt}{17.07182pt}\pgfsys@lineto{99.58466pt}{17.07182pt}\pgfsys@stroke\pgfsys@invoke{ }\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{95.6115pt}{22.36021pt}\pgfsys@invoke{ }\hbox{{\definecolor[named]{.}{rgb}{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}\pgfsys@color@rgb@stroke{0}{0.5}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0.5}{0}\pgfsys@invoke{ }\hbox{\footnotesize{\definecolor[named]{.}{rgb}{0,0.5,0}\color[rgb]{0,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,0}$q_{N}$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{split}$ (B.1) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
D6 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | - | - | -
D2 | $\times$ | $\times$ | - | - | - | - | $\times$ | - | - | -
D4 | $\times$ | $\times$ | - | - | - | - | - | $\times$ | $\times$ | $\times$
NS5 | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | - | - | - | -
(B.2)

The two-dimensional ${\cal N}=(0,4)$ gauge theory on the D2 brane worldvolume
is described by the linear quiver in (B.3), which can be obtained in a similar
way to those in Tong:2014yna  and Haghighat:2013gba .

$\begin{split}\includegraphics[width=325.215pt]{quivworldsheet-
rev.pdf}\end{split}$ (B.3)

Let us explain the above quiver diagram as follows. First, we have the
circular (gauge) nodes in the center row, which are stacks of $Q_{i}$ D2
branes (the $Q_{0}$ and $Q_{N+1}$ nodes give rise to $U(Q_{0})$ and
$U(Q_{N+1})$ flavor symmetries respectively). The boxed nodes on the top row
are stacks of $r_{i}$ D6-branes. The boxed nodes in the bottom row are stacks
of $q_{i}$ D4-branes. The length of the row is determined by the number of NS5
branes ($N$). We take $0\leq i\leq N+1$ with $q_{0}=q_{N+1}=0$. We label each
field in the quiver as follows.

(B.4)

  * •

The ${\cal N}=(0,4)$ vector multiplet consists an ${\cal N}=(0,2)$ vector
multiplet $U^{(i)}$ and an adjoint ${\cal N}=(0,2)$ Fermi multiplet
$\Theta^{(i)}$. The vector multiplet $U^{(i)}$ also contains the ${\cal
N}=(0,2)$ field strength Fermi multiplet $\Upsilon^{(i)}$. These Fermi
multiplets are denoted by dashed green loops in the above diagram.

  * •

The solid green lines are ${\cal N}=(0,4)$ twisted hypermultiplets built out
of ${\cal N}=(0,2)$ chirals $(Y^{(i)},\,\widetilde{Y}^{(i)})$ transforming in
the adjoint representation of $U(Q_{i})$.

  * •

The horizontal lines connecting the $Q_{i}$ nodes comprise an ${\cal N}=(0,4)$
hypermultiplet in the bifundamental of $U(Q_{i})\times U(Q_{i+1})$ which in
${\cal N}=(0,2)$ language contains chirals $(X^{(i)},\widetilde{X}^{(i)})$
[solid] and Fermi multiplets $(\Gamma^{(i)},\widetilde{\Gamma}^{(i)})$
[dashed].

  * •

The red dashed diagonal lines from the top left corner to bottom right corner
are Fermi multiplets $\Psi_{i}$ charged under $U(r_{i-1})\times U(Q_{i})$. The
red dashed diagonal lines from the bottom left corner to top right corner are
Fermi multiplets $\Xi_{i}$ charged under $U(Q_{i})\times U(r_{i+1})$.

  * •

The vertical red solid lines are ${\cal N}=(0,4)$ hypermultiplets in the
bifundamental of $U(r_{i})\times U(Q_{i})$ built out of chiral multiplets
$({\cal Q}^{(i)},\widetilde{{\cal Q}}^{(i)})$.

  * •

The vertical black dashed lines are Fermi multiplets $\Sigma_{i}$ charged
under $U(r_{i})\times U(q_{i})$. If there are no D4 branes in the game
$q_{i}=0$ for all $i$, then these multiplets are absent.

  * •

The vertical blue lines are ${\cal N}=(0,4)$ hypermultiplets charged in the
bifundamental of $U(q_{i})\times U(Q_{i})$. These multiplets decompose in
${\cal N}=(0,2)$ language as a Fermi multiplet
$(\Omega^{(i)},\,\widetilde{\Omega}^{(i)})$ and chiral multiplet
$(B^{(i)},\widetilde{B}^{(i)})$.

The E-term potentials for the various Fermi multiplets and superpotential
terms can be written in a similar way as in Tong:2014yna . Explicitly, the
E-terms are

$\begin{split}\begin{tabular}[]{ll}$E_{\Theta^{(i)}}=[Y^{(i)},\widetilde{Y}^{(i)}]+\mathcal{Q}^{(i)}\widetilde{\mathcal{Q}}^{(i)}$&\quad$E_{\Gamma^{(i)}}=Y^{(i)}X^{(i)}-Y^{(i)}X^{(i-1)}$\\\
$E_{\widetilde{\Gamma}{(i)}}=-\widetilde{X}^{(i)}Y^{(i)}+\widetilde{X}^{(i-1)}Y^{(i)}$&\quad${E}_{{\Xi}^{(i)}}=X^{(i)}\mathcal{Q}^{(i+1)}$\\\
${E}_{{\Psi}^{(i)}}=\widetilde{\mathcal{Q}}^{(i)}{X}^{(i)}$&\quad$E_{\Omega^{(i)}}=Y^{(i)}B^{(i)}-Y^{(i)}B^{(i-1)}$\\\
$E_{\widetilde{\Omega}{(i)}}=-\widetilde{B}^{(i)}Y^{(i)}+\widetilde{B}^{(i-1)}Y^{(i)}$&\quad$E_{\Sigma^{(i)}}=\widetilde{\mathcal{Q}}^{(i)}B^{(i)}$\end{tabular}\end{split}$
(B.5)

and the superpotential terms are

$\begin{split}\scalebox{0.9}{ \begin{tabular}[]{ll}${\cal
W}_{\Theta^{(i)}}=\widetilde{X}^{(i)}\Theta
X^{(i)}-X^{(i-1)}\Theta\widetilde{X}^{(i-1)}$&${\cal
W}_{\Gamma^{(i)}}=\widetilde{X}^{(i)}\widetilde{Y}^{(i)}\Gamma^{(i)}-\widetilde{\Gamma}^{(i-1)}\widetilde{Y}^{(i)}X^{(i-1)}$\\\
${\cal
W}_{\widetilde{\Gamma}^{(i)}}=\widetilde{\Gamma}^{(i)}\widetilde{Y}^{(i)}X^{(i)}-\widetilde{X}^{(i-1)}\widetilde{Y}^{(i)}{\Gamma}^{(i-1)}$&${\cal
W}_{{\Xi}^{(i)}}=\Xi^{(i)}\widetilde{\mathcal{Q}}^{(i+1)}\widetilde{X}^{(i)}$\\\
${\cal
W}_{{\Psi}^{(i)}}=\Psi^{(i)}\widetilde{X}^{(i)}\mathcal{Q}^{(i)}$&${\cal
W}_{\Omega^{(i)}}=\widetilde{B}^{(i)}\widetilde{Y}^{(i)}\Omega^{(i)}-\widetilde{\Omega}^{(i-1)}\widetilde{Y}^{(i)}B^{(i-1)}$\\\
${\cal
W}_{\widetilde{\Omega}^{(i)}}=\widetilde{\Omega}^{(i)}\widetilde{Y}^{(i)}B^{(i)}-\widetilde{B}^{(i-1)}\widetilde{Y}^{(i)}{\Omega}^{(i-1)}$&${\cal
W}_{\Sigma^{(i)}}=\Sigma^{(i)}\widetilde{B}^{(i)}\mathcal{Q}^{(i)}$\,.\end{tabular}}\end{split}$
(B.6)

### Gauge Anomaly Cancellation

Let us compute the anomaly for quiver (B.3). First, the gauge anomaly of the
$U(1)_{i}$ subgroup of the $U(Q_{i})$ gauge group is given by

$\displaystyle\mathop{\rm
Tr}(\widehat{\gamma}^{3}J_{U(1)_{i}}J_{U(1)_{i}})=\sum_{j:\,\text{chiral}}{\mathfrak{q}}_{j}^{2}-\sum_{j:\,\text{Fermi}}{\mathfrak{q}}_{j}^{2}=2r_{i}-r_{i+1}-r_{i-1}$
(B.7)

where $\widehat{\gamma}^{3}$ is the two-dimensional chirality matrix. The
gauge anomaly cancellation thus determines $2r_{i}=r_{i+1}+r_{i-1}$ and is
independent of $q_{i}$.

This last point deserves a bit of comment. This result is quite different from
the gauge anomaly cancellation in Gadde:2015tra . The key difference is that
there is an additional pair of chiral multiplets $(B,\widetilde{B})$ in the
construction above that soaks up the anomaly from the
$(\Omega,\widetilde{\Omega})$ Fermi multiplets which are present in
Gadde:2015tra . In the latter case, the $n_{i}$ are fixed by gauge anomaly
cancellation to be $2r_{i}-r_{i+1}-r_{i-1}$.

The non-abelian gauge anomaly is obviously more involved but we have all the
information we need from the list of matter multiplets above:

$\displaystyle\begin{split}\mathop{\rm Tr}(\widehat{\gamma}^{3}J_{{\rm
SU}(Q_{i})}J_{{\rm SU}(Q_{i})})=&~{}(T_{{\cal Q}}(\Box)+T_{\widetilde{\cal
Q}}(\overline{\Box}))r_{i}-T_{\Xi}(\Box)r_{i+1}-T_{\Psi}(\overline{\Box})r_{i-1}\\\
&+T_{X}(\Box)Q_{i-1}+T_{\widetilde{X}}(\overline{\Box})Q_{i+1}-T_{\Gamma}(\Box)Q_{i-1}+T_{\widetilde{\Gamma}}(\overline{\Box})Q_{i+1}\\\
&+(T_{B}(\Box)+T_{\widetilde{B}}(\overline{\Box}))q_{i}-(T_{\Omega}(\Box)+T_{\widetilde{\Omega}}(\overline{\Box}))q_{i}\\\
&+T_{Y}({\bf{adj}})+T_{\widetilde{Y}}({\bf{adj}})-T_{\Upsilon}({\bf{adj}})-T_{\Theta}({\bf{adj}})\\\
=&~{}r_{i}-\frac{1}{2}(r_{i+1}+r_{i-1}),\end{split}$ (B.8)

where $T_{F}(\mathbf{R})$ denotes the index of the representation $\mathbf{R}$
of $\mathrm{SU}(Q_{i})$ in which the field $F$ transforms. We used the
convention that the indices of the fundamental and antifundamental
representations are $T(\Box)=T(\overline{\Box})=\frac{1}{2}$ and that of the
adjoint representation is $T({\bf adj})=Q_{i}$. This reproduces the abelian
gauge anomaly cancellation.

### R-symmetry

The two-dimensional ${\cal N}=(0,4)$ gauge theory in question has R-symmetry
$\mathfrak{so}(4)_{R}\cong\mathfrak{su}(2)^{+}_{R}\times\mathfrak{su}(2)^{-}_{R}$.
Let $R^{\pm}[F]$ be the charge of the field $F$ under the Cartan subalgebra of
$\mathfrak{su}(2)^{\pm}_{R}$. The $R$-charge assignment to each field is
constrained by the following conditions:

  * •

We assume the $R$-charges of the fields are independent of the superscript
$(i)$, and so we write $R^{\pm}[F^{(i)}]=R^{\pm}[F]$ for any field $F$.

  * •

Each superpotential term carries R-charge $R^{\pm}[{\cal W}]=+1$.

  * •

For each Fermi multiplet $f$, the corresponding $E$-term has charge
$R^{\pm}[E_{f}]=R^{\pm}[f]+1$.

  * •

$R^{\pm}[\Upsilon]=1$ since $\Upsilon$ is in the multiplet that contains the
field strength.

As pointed out in Tong:2014yna , the fields $\Gamma^{(i)}$ and
$\widetilde{\Gamma}^{(i)}$ are singlets under
$\mathfrak{su}(2)^{+}_{R}\times\mathfrak{su}(2)^{-}_{R}$:

$\begin{split}R^{\pm}[\Gamma]=0\,,\quad
R^{\pm}[\widetilde{\Gamma}]=0\,.\end{split}$ (B.9)

Moreover, as pointed out in Witten:1994tz ; Putrov:2015jpa  and (Tong:2014yna,
, p. 10), it is possible to have a single ${\cal N}=(0,2)$ Fermi multiplet
which is consistent with ${\cal N}=(0,4)$ supersymmetry. However, for this to
happen, the chiral fermion should be a singlet under
$\mathfrak{su}(2)^{+}_{R}\times\mathfrak{su}(2)^{-}_{R}$. Thus,

$\begin{split}R^{\pm}[\Xi]=0\,,\qquad R^{\pm}[\Psi]=0\,.\end{split}$ (B.10)

Taking these into account, we obtain the following R-charge assignment:

$\begin{split}\scalebox{0.9}{
\begin{tabular}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr&$\Upsilon$&$\Theta$&$Y$&$\widetilde{Y}$&$X$&$\widetilde{X}$&${\cal
Q}$&$\widetilde{\cal
Q}$&$B$&$\widetilde{B}$&$\Gamma$&$\widetilde{\Gamma}$&$\Omega$&$\widetilde{\Omega}$&$\Xi$&$\Psi$&$\Sigma$\\\
\hline\cr$R^{+}$&1&1&1&1&0&0&1&1&0&0&0&0&0&0&0&0&0\\\
\hline\cr$R^{-}$&1&$-1$&0&0&1&1&0&0&1&1&0&0&0&0&0&0&0\\\
\hline\cr\end{tabular}}\end{split}$ (B.11)

###  $\mathrm{SU}(2)_{D}$ flavor symmetry

Our quiver theory also has an ${\rm SU}(2)_{D}$ flavor symmetry that
transforms the ${\cal N}=(0,4)$ adjoint twisted hypermultiplet
$(Y^{(i)},\widetilde{Y}^{(i)})$ as a doublet for each $i$. We assign their
charges under the $\mathrm{SU}(2)_{D}$ Cartan subalgebra as

$\begin{split}D[Y^{(i)}]=1\,,\qquad D[\widetilde{Y}^{(i)}]=-1\,.\end{split}$
(B.12)

Since the superpotential terms have transform trivially under this symmetry,
from (B.6) we find that the consistent charge assignment is as follows:

$\begin{split}\scalebox{0.9}{
\begin{tabular}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr&$\Upsilon$&$\Theta$&$Y$&$\widetilde{Y}$&$X$&$\widetilde{X}$&${\cal
Q}$&$\widetilde{\cal
Q}$&$B$&$\widetilde{B}$&$\Gamma$&$\widetilde{\Gamma}$&$\Omega$&$\widetilde{\Omega}$&$\Xi$&$\Psi$&$\Sigma$\\\
\hline\cr$D$&0&0&1&$-1$&0&0&0&0&0&0&1&1&1&1&0&0&0\\\
\hline\cr\end{tabular}}\end{split}$ (B.13)

Note that $\Upsilon^{(i)}$ and $\Theta^{(i)}$, residing in the ${\cal
N}=(0,4)$ vector multiplet, transform trivially in the ${\rm SU}(2)_{D}$
flavor symmetry.

### Gravitational, R-symmetry and ${\rm SU}(2)_{D}$ Anomalies

The four-form anomaly polynomial of the two-dimensional quiver gauge theory
contains the following terms:

$\begin{split}\frac{1}{2}K_{R^{+}}c_{2}(R^{+})+\frac{1}{2}K_{R^{-}}c_{2}(R^{-})+\frac{1}{2}K_{D}c_{2}(D)+k_{g}p_{1}(T)\,,\end{split}$
(B.14)

where $K_{R^{\pm}}$ are R-symmetry anomalies, $K_{D}$ is the ${\rm SU}(2)_{D}$
symmetry anomaly, and $k_{g}$ is the gravitational anomaly of the quiver
theory:

$\begin{split}\begin{array}[]{lll}K_{R^{\pm}}=\mathop{\rm
Tr}(\widehat{\gamma}^{3}J_{{\rm SU}(2)^{\pm}_{R}}J_{{\rm
SU}(2)^{\pm}_{R}})\,,&\quad K_{D}=\mathop{\rm Tr}(\widehat{\gamma}^{3}J_{{\rm
SU}(2)_{D}}J_{{\rm SU}(2)_{D}})\,,&\quad k_{g}=\mathop{\rm
Tr}\,\widehat{\gamma}^{3}\\\
c_{2}(R^{\pm})=c_{2}(\mathrm{SU}(2)^{\pm}_{R})~{},&\quad
c_{2}(D)=c_{2}(\mathrm{SU}(2)_{D})~{},&\end{array}\end{split}$ (B.15)

For the R-anomalies, the left moving fermions in the Fermi multiplets $\psi$
contribute $-R[\psi]^{2}$ and the fermions in the chiral multiplet $\chi$
contribute $(R[\chi]-1)^{2}$, where $R[X]=R_{X}$ denotes either $R^{+}[X]$ or
$R^{-}[X]$. The counting results in

$\begin{split}\scalebox{0.9}{$\begin{split}K_{R}&=\sum_{i=0}^{N+1}\left[Q_{i}Q_{i+1}\left((R_{X}-1)^{2}+(R_{\widetilde{X}}-1)^{2}-R_{\Gamma}^{2}-R_{\widetilde{\Gamma}}^{2}\right)-Q_{i}r_{i+1}R_{\Xi}^{2}-Q_{i}r_{i-1}R_{\Psi}^{2}\right]\\\
&+\sum_{i=1}^{N}q_{i}\left[Q_{i}\big{(}(R_{B}-1)^{2}+(R_{\widetilde{B}}-1)^{2}-R_{\Omega}^{2}-R_{\widetilde{\Omega}}^{2}\big{)}-r_{i}R_{\Sigma}^{2}\right]\\\
&+\sum_{i=1}^{N}Q_{i}\left[Q_{i}((R_{Y}-1)^{2}+(R_{\widetilde{Y}}-1)^{2}-R_{\Upsilon}^{2}-R_{\Theta}^{2})+r_{i}(R_{\cal
Q}-1)^{2}+r_{i}(R_{\widetilde{\cal
Q}}-1)^{2}\right]\,.\end{split}$}\end{split}$ (B.16)

Explicitly, using (B.11), we have

$\begin{split}K_{R}=\begin{cases}2\sum_{i=1}^{N}r_{i}Q_{i}&\quad\text{if
$R=R^{-}$}\\\
2\sum_{i=1}^{N}(q_{i}-Q_{i})Q_{i}+2\sum_{i=0}^{N}Q_{i}Q_{i+1}&\quad\text{if
$R=R^{+}$}\,.\end{cases}\end{split}$ (B.17)

On the other hand, the gravitational anomaly can be obtained from by replacing
$(R_{F}-1)^{2}$ and $R_{F}^{2}$ in (B.16) by $1$ for every field $F$:

$\begin{split}k_{g}=-\sum_{i=1}^{N}r_{i}q_{i}+\sum_{i=1}^{N}(2r_{i}-r_{i+1}-r_{i-1})Q_{i}=-\sum_{i=1}^{N}r_{i}q_{i}\,,\end{split}$
(B.18)

where we have used the gauge anomaly cancellation to obtain the last equality.

Finally, the ${\rm SU}(2)_{D}$ anomaly can be computed by replacing
$(R_{F}-1)^{2}$ and $R_{F}^{2}$ in (B.16) by $D[F]^{2}$, where $D[F]$ is the
charge of $F$ under the Cartan subalgebra of ${\rm SU}(2)_{D}$ given by (B.16)
for every field $F$. The result is

$\begin{split}K_{D}=-K_{R^{+}}=2\sum_{i=1}^{N}(Q_{i}-q_{i})Q_{i}-2\sum_{i=0}^{N}Q_{i}Q_{i+1}\,.\end{split}$
(B.19)

To summarize, the terms (B.14) in the four-form anomaly polynomial of the two-
dimensional quiver theory can be written as

$\begin{split}&\frac{1}{2}K_{D}\left[c_{2}(D)-c_{2}(R^{+})\right]+\frac{1}{2}K_{R^{-}}c_{2}(R^{-})+k_{g}p_{1}(T)\\\
&=\Bigg{(}\sum_{i=1}^{N}(Q_{i}-q_{i})Q_{i}-\sum_{i=0}^{N}Q_{i}Q_{i+1}\Bigg{)}\left[c_{2}(D)-c_{2}(R^{+})\right]\\\
&\qquad+\Bigg{(}\sum_{i=1}^{N}r_{i}Q_{i}\Bigg{)}\,c_{2}(R^{-})+\Bigg{(}-\sum_{i=1}^{N}r_{i}q_{i}\Bigg{)}\,p_{1}(T)\,.\end{split}$
(B.20)

It is instructive to compare this result to (Shimizu:2016lbw, , (2.5)) in the
case of $q_{i}=0$. According to (Shimizu:2016lbw, , Page 7), the coefficient
of $c_{2}(R^{-})$ in (LABEL:I4explicit) coincides with that of $c_{2}(I)$ in
Shimizu:2016lbw , and so we identify our ${\rm SU}(2)_{R}^{-}$ with their
${\rm SU}(2)_{I}$. Moreover, the coefficient of
$\left[c_{2}(D)-c_{2}(R^{+})\right]$ when $q_{i}=0$ can be written as
$\frac{1}{2}\sum_{i,j=1}^{N}C_{ij}Q_{i}Q_{j}$ where $C_{ij}$ is the Cartan
matrix of the $A_{N}$ algebra. This is in agreement with the coefficient of
$c_{2}(L)-c_{2}(R)$ of (Shimizu:2016lbw, , (2.5)) when their $\eta^{ij}$ is
taken to be the Cartan matrix $C_{ij}$. We thus identify our ${\rm
SU}(2)_{D}-{\rm SU}(2)_{R}^{+}$ with their ${\rm SU}(2)_{L}-{\rm SU}(2)_{R}$.
Finally, we see that for $q_{i}=0$ the gravitation anomaly $k_{g}$ vanishes,
in agreement with the coefficient of $p_{1}(T)$ and $c_{2}(L)+c_{2}(R)$ of
(Shimizu:2016lbw, , (2.5)) when $\eta^{ij}$ is taken to be the Cartan matrix
$C_{ij}$, whose diagonal elements are precisely $2$. This is also consistent
with (B.19), namely the coefficient of $c_{2}(D)+c_{2}(R^{+})$ in our anomaly
polynomial is zero. This leads to the identification of our ${\rm
SU}(2)_{D}+{\rm SU}(2)_{R}^{+}$ with their ${\rm SU}(2)_{L}+{\rm SU}(2)_{R}$.
The R-symmetry of the two-dimensional theory in our notation is ${\rm
SU}(2)_{R}^{+}\times{\rm SU}(2)_{R}^{-}$ and in their notation is ${\rm
SU}(2)_{R}\times{\rm SU}(2)_{I}$. We therefore identify

$\begin{split}{\rm SU}(2)_{R}^{-}\equiv{\rm SU}(2)_{I}\,,\quad{\rm
SU}(2)_{R}^{+}\equiv{\rm SU}(2)_{R}\,,\quad{\rm SU}(2)_{D}\equiv{\rm
SU}(2)_{L}\,.\end{split}$ (B.21)

in our and their notations, respectively.

We can compute the defect anomaly from the formula (4.28), namely

$\begin{split}a_{\Sigma}=3k_{r}-\frac{1}{2}k_{g}~{}.\end{split}$ (B.22)

Note that the anomaly polynomial (B.14) contains the terms
$\frac{1}{2}K_{R^{+}}c_{2}(R^{+})=-\frac{1}{2}K_{R^{+}}c_{1}(r)^{2}$ and
$\frac{1}{2}K_{R^{-}}c_{2}(R^{-})=-\frac{4}{2}K_{R^{-}}c_{1}(r)^{2}=-2K_{R^{-}}c_{1}(r)^{2}$,
where we have used the fact that $c_{2}(R^{+})=-c_{1}(r)^{2}$ and
$c_{2}(R^{-})=c_{2}(I)=-4c_{1}(r)^{2}$; see (B.21), (4.28) and (4.29).
Recalling that we adopt the convention $\frac{1}{2}k_{r}c_{1}(r)^{2}$ in the
anomaly polynomial, we thus obtain

$\begin{split}k_{r}=\begin{cases}-4K_{R^{-}}&\quad\text{for\,\,
$\mathrm{SU}(2)_{R}^{-}$}\\\ -K_{R^{+}}&\quad\text{for\,\,
$\mathrm{SU}(2)_{R}^{+}$}~{}.\\\ \end{cases}\end{split}$ (B.23)

Applying the formulae (B.17) and (B.18), we obtain

$\begin{split}a_{\Sigma}=\begin{cases}-24\sum_{i=1}^{N}r_{i}Q_{i}+\frac{1}{2}\sum_{i=1}^{N}r_{i}q_{i}&\quad\text{if
$R=R^{-}$}\\\
-6\Big{(}\sum_{i=1}^{N}(q_{i}-Q_{i})Q_{i}+\sum_{i=0}^{N}Q_{i}Q_{i+1}\Big{)}+\frac{1}{2}\sum_{i=1}^{N}r_{i}q_{i}&\quad\text{if
$R=R^{+}$}\,.\end{cases}\end{split}$ (B.24)

We extract the defect anomaly by setting the BPS string charges $Q_{i}=0$.

We finally comment on the insertion of $f_{i}$ D8-branes, which span the
$012345\,789$ directions, in the brane system which engineers 6d theories
intersecting the $i$-th D6 segment. The extra contribution to 2d quiver gauge
theory is provided by D2–D8 states that are $f_{i}Q_{i}$ $(0,2)$ Fermi
multiplets.666The D4–D8 states will not contribute since they are not light 2d
degrees of freedom. In addition, the cancellation of the gauge anomalies (B.8)
and (B.7) now requires

$2r_{i}-r_{i+1}-r_{i-1}-f_{i}=0$ (B.25)

which is equivalent to the 6d irreducible gauge anomaly (proportional to ${\rm
tr}(F^{4})$ in the 6d anomaly polynomial) cancellation condition. All the
R-charge assignments are the same as $R_{\Xi(i)}$. Therefore, the
$K_{R^{\pm}}$ and $K_{D}$ will receive contributions from $f_{i}Q_{i}$ $(0,2)$
Fermi multiplets accordingly. The gravitational anomaly remains unchanged
since it explicitly depends on the gauge anomaly cancellation condition
(B.18). We conclude that the addition of D8 branes does not change the defect
anomaly contribution, but it changes the BPS strings R-symmetries anomalies.

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