Holographic duals of M5-branes on an irregularly punctured sphere

Christopher<EMAIL_ADDRESS>Hyojoong<EMAIL_ADDRESS>Nakwoo<EMAIL_ADDRESS>Yein<EMAIL_ADDRESS>
Department of Physics and Research Institute of Basic Science,

Kyung Hee University, Seoul 02447, Republic of Korea

Abstract

We provide explicit holographic duals of M5-branes wrapped on a sphere with
one irregular puncture and one regular puncture of arbitrary type. The
solutions generalise the solutions corresponding to M5-branes wrapped on a
disc recently constructed by Bah–Bonetti–Minasian–Nardoni by allowing for a
general choice of regular puncture. We show that the central charges, flavour
central charges and conformal dimensions of BPS operators match with a class
of Argyres–Douglas theory.

###### Contents

  1. 1 Introduction
  2. 2 M5-branes on a disc
    1. 2.1 AdS${}_{5}\times\Sigma$ solutions of 7d gauged supergravity
    2. 2.2 11d uplift and regularity
    3. 2.3 LLM reformulation
    4. 2.4 Electrostatics reformulation
    5. 2.5 Observables
  3. 3 Generalised regular puncture
    1. 3.1 Building block of the generalised regular puncture
    2. 3.2 Summing up building blocks
    3. 3.3 Observables
  4. 4 Field theory dual
    1. 4.1 Review of Argyres–Douglas theories
    2. 4.2 Observables
    3. 4.3 Setting up the holographic dictionary
    4. 4.4 Matching of observables
  5. 5 Conclusion
  6. A Special limits of the 7d metric
    1. A.1 AdS7
    2. A.2 Maldacena–Nunez solution
  7. B Anomaly inflow
    1. B.1 Constructing $E_{4}$
    2. B.2 Anomaly polynomial from anomaly inflow

## Index

##  1 Introduction

Compactifying SCFTs on compact manifolds has been a fruitful avenue for
constructing new SCFTs. Given a parent SCFT with a holographic dual, it is
natural to consider the holographic dual of the compactified theory. The
earliest examples of such dual pairs were constructed in Maldacena:2000mw ,
and studied the compactification of both 4d $\mathcal{N}=4$ super-Yang–Mills
and the 6d $\mathcal{N}=(2,0)$ theory on a Riemann surface with genus $g>1$.
Since then this avenue of research has been extended in multiple directions.
In this work we will primarily be interested in 4d $\mathcal{N}=2$ theories
that can be obtained from M-theory. Theories of class $\mathcal{S}$ are 4d
$\mathcal{N}=2$ SCFTs which arise from wrapping the 6d $\mathcal{N}=(2,0)$
theory living on M5-branes on a punctured Riemann surface with gravity duals
classified in Gaiotto:2009gz ; Gaiotto:2009we 111The type IIA picture is given
in Reid-Edwards:2010vpm ; Aharony:2012tz .. Extensions to 4d $\mathcal{N}=1$
theories from wrapped M5-branes have been studied in Gauntlett:2004zh ;
Agarwal:2014rua ; Bah:2013aha ; Bah:2012dg .

The prototypical setup for studying branes wrapped on various compact cycles
is to consider embedding the cycle in some larger geometry, for example a
Calabi–Yau or G2 manifold and taking the metric on the compact space to be the
one with constant curvature. Given that we typically view these types of
solutions as the IR fixed point of an RG flow interpolating between
dimensions, or rather the near-horizon of some black object, one can motivate
this ansatz, at least for Riemann surfaces, from known uniformization theorems
Anderson:2011cz ; Bobev:2020jlb . Under some assumptions, they state that one
can take the UV solution to have an arbitrary metric on the Riemann surface,
and along the RG flow this is washed out leaving just the constant curvature
metric at the IR fixed point. Key assumptions for these theories are that
supersymmetry is realised by a topological twist and that the metrics are
smooth.

Recently, solutions with non-constant curvature metrics have been studied in
string and M-theory, thus evading the uniformization theorems. These solutions
are known as spindles Ferrero:2020laf ; Hosseini:2021fge ; Boido:2021szx ;
Faedo:2021kur ; Cassani:2021dwa ; Ferrero:2021ovq ; Couzens:2021rlk ;
Faedo:2021nub ; Ferrero:2021etw ; Couzens:2021cpk ; Giri:2021xta  and discs
Bah:2021mzw ; Bah:2021hei ; Couzens:2021tnv ; Suh:2021ifj ; Suh:2021hef ;
Suh:2021aik ; Couzens:2021rlk ; Karndumri:2022wpu  and can be viewed as the
horizon of accelerating black objects.222For discussion on accelerating black
holes not embedded in string or M-theory see for example Anabalon:2018qfv ;
Anabalon:2018ydc ; Gregory:2017ogk  and references within. Spindles are the
orbifold $\mathbb{WCP}^{1}_{n_{-},n_{+}}$, i.e. a two-sphere with conical
deficits at both poles. Discs on the other hand have the topology of
(unsurprisingly) a disc, with an orbifold singularity at the centre and a
boundary on which the metric is locally a cylinder but singular. The two types
of solutions are intimately related and have been shown to be different global
completions of the same local solutions Couzens:2021rlk ; Couzens:2021tnv .
Apart from having metrics of non-constant curvature one of the interesting
features of these geometries is the method in which supersymmetry is
preserved. The mechanism is not the usual topological twist but instead
requires a mixing between the parent R-symmetry and the isometry direction of
the compactification space. For spindles there are two different types of
twist, the anti-twist or the topological topological twist Ferrero:2021etw ,
while for discs a different mechanism involving a holonomy for a gauge field
on the boundary of the disc allows for the preservation of supersymmetry. See
also Cheung:2022ilc ; Couzens:toappear  which consider 4d orbifolds with non-
constant curvature.

In this work we will be interested in extensions of disc solutions,
specifically for M5-branes wrapped on a disc. In Bah:2021mzw ; Bah:2021hei
(BBMN) the first disc solution was presented and a dual field theory was
proposed. The dual field theory is a 4d $\mathcal{N}=2$ SCFT of
Argyres–Douglas type and constitutes the first holographic dual for such a
theory. The dual field theory was shown to be $(I_{\hat{N},\hat{k}},Y_{l})$,
which is the theory on a stack of M5-branes of A-type wrapped on a twice
punctured sphere, with one irregular puncture of type I at one pole and a
regular puncture of a particular type at the other pole, we will review this
nomenclature in section 4. The goal of the present paper is to extend to more
general Argyres–Douglas theories. In particular we will provide holographic
duals for the SCFTs with arbitrary regular puncture and fixed type I irregular
puncture. We provide evidence for the proposal by matching various observables
on the two sides.

The outline of the paper is as follows. In section 2 we begin by reviewing the
disc solution in Bah:2021mzw ; Bah:2021hei . Our presentation uses a different
set of coordinates which are better adapted for later. We embed the solution
into the classification of $\mathcal{N}=2$ AdS5 solutions in M-theory in
Gaiotto:2009gz ; Lin:2004nb  studying the equivalent electrostatic problem.
The electrostatic problem has some new and novel features, in particular
unusual boundary conditions for the potential. In section 3 we generalise the
disc solution building on the original solution. We show that the new
solutions are well-defined supergravity solutions, performing a regularity
analysis and flux quantisation. We proceed to compute various observables of
the gravity solution with which to match to the field theory analysis in
section 4. We provide a dictionary between the field theory and gravity
solution with which we can compare the observables on the two sides and find
perfect agreement in the holographic limit. We relegate some material to two
appendices. The first shows how to obtain AdS7 and the Maldacena–Nunez
solutions from our starting point, whilst the second performs an independent
check of the observable computations by using anomaly inflow.

##  2 M5-branes on a disc

In Bah:2021hei  an AdS${}_{5}\times\Sigma$ solution in 7d gauged supergravity
was found where $\Sigma$ is topologically a disc. The boundary of the disc
corresponds to a singularity of the overall metric, while the centre of the
disc has a conical singularity. Upon uplifting the solution to 11d
supergravity on an $S^{4}$ one obtains a $1/2$ BPS AdS5 solution, and one can
give a physical interpretation of the singularities; the boundary of the disc
arises due to a stack of smeared M5-branes whereas the conical deficit is due
to the presence of a monopole. In Bah:2021hei  they conjectured that the
solution was holographically dual to an Argyres–Douglas theory with one
regular and one irregular puncture. In the following we will review the
solution, albeit from a different parametrisation, before embedding the
solution into the classification of $\mathcal{N}=2$ AdS5 solutions of 11d
supergravity constructed in Gaiotto:2009gz ; Lin:2004nb . We use the
reformulation of the solution in terms of an electrostatics problem for a
single potential, determining it for the disc solution before studying its
properties and flux quantisation within the electrostatics reformulation.

###  2.1 AdS${}_{5}\times\Sigma$ solutions of 7d gauged supergravity

In this section we will study the AdS${}_{5}\times\Sigma$ solution of 7d
U$(1)^{2}$ gauged supergravity originally found in Bah:2021hei  using the
conventions there for the gauged supergravity theory. As pointed out in
Couzens:2021tnv ; Couzens:2021rlk  one can obtain disc solutions as different
global completion of the same local solutions from which one may construct
spindle solutions. In the following we will use the parametrisation of the
spindle solutions in Ferrero:2021wvk  specialised to the disc solution.333This
solution specialised to preserve $\mathcal{N}=2$ may also be found in
Lin:2004nb . In fact in this reference a more general 7d solution is presented
up to solving a non-trivial ODE. This generalises the solution here by
including an additional scalar field. In the uplifted theory this breaks a
U$(1)$ symmetry, which will be essential for our generalisation and therefore
we stick with this less general solution. The solution is

$\displaystyle\mathrm{d}s^{2}_{7}$
$\displaystyle=\Big{(}wP(w)\Big{)}^{1/5}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{w}{f(w)}\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\mathrm{d}z^{2}\bigg{]}\,,$
$\displaystyle A_{i}$
$\displaystyle=-\frac{s_{i}}{h_{i}(w)}\mathrm{d}z\,,\qquad
X_{i}(w)=\frac{\Big{(}wP(w)\Big{)}^{2/5}}{h_{i}(w)}\,,$ (2.1)

where $\mathrm{d}s^{2}(\text{AdS}_{5})$ is the unit radius metric on AdS5,
satisfying $R_{\mu\nu}=-4g_{\mu\nu}$ and the functions take the form

$\displaystyle h_{i}(w)$ $\displaystyle=w^{2}-s_{i}\,,\qquad
P(w)=h_{1}(w)h_{2}(w)\,,\qquad f(w)=P(w)-w^{3}\,.$ (2.2)

The solution depends on two real constants, $s_{i}$. In order to obtain a disc
the functions $f(w)$ and $P(w)$ must have a common root. Given the form of the
polynomials, it is clear that this must necessarily be at $w=0$ and therefore
in order to obtain a disc we set, without loss of generality, $s_{2}=0$. In
fact this limit leads to an enhancement of supersymmetry, with the solution
preserving $\mathcal{N}=2$ supersymmetry, rather than $\mathcal{N}=1$. We will
show this later by explicitly embedding the solution into the $\mathcal{N}=2$
classification of AdS5 solutions of 11d supergravity in Gaiotto:2009gz ;
Lin:2004nb . This local metric also admits other interesting limits. One may
recover both pure AdS7 and the Maldacena–Nunez solution Maldacena:2000mw  by
taking different limits of this local metric as we show in appendix A.

Regularity in 7d

Let us begin by considering the regime of the $s_{1}$ which gives rise to a
well-defined metric. We require $f(w)$ to admit two roots, one at $0$ and the
second positive. The latter condition is necessary for the metric to have the
correct signature, since there are terms proportional to $w$ appearing in the
metric. The roots of $f(w)$ are

$w_{0}=0\,,\quad(\text{twice})\,,\qquad
w_{\pm}=\frac{1}{2}\Big{(}1\pm\sqrt{1+4s_{1}}\,\Big{)}\,.$ (2.3)

In order for there to be a real positive root and for the scalars to be
positive we require

$-\frac{1}{4}\leq s_{1}\leq 0\,.$ (2.4)

This leads to two positive roots and we take the domain of the line interval
parametrised by $w$ to be

$w\in[0,w_{-}]\,.$ (2.5)

We must now check how the metric degenerates at the end-points.

First consider the end-point at $w=w_{-}$. Since $P(w_{-})\neq 0$ we need only
consider the metric on $\Sigma$ at this end-point. Expanding the metric on
$\Sigma$ around $w=w_{-}$ we find

$\displaystyle\mathrm{d}s^{2}(\Sigma)$
$\displaystyle\simeq\frac{w_{-}}{|f^{\prime}(w_{-})|(w_{-}-w)}\mathrm{d}w^{2}+\frac{|f^{\prime}(w_{-})|}{P(w_{-})}(w_{-}-w)\mathrm{d}z^{2}$
$\displaystyle=\frac{4w_{-}}{|f^{\prime}(w_{-})|}\bigg{[}\mathrm{d}r^{2}+\frac{|f^{\prime}(w_{-})|^{2}}{4w_{-}^{4}}r^{2}\mathrm{d}z^{2}\bigg{]}\,,$
(2.6)

where $w_{-}-w=r^{2}$. Fixing the period of $z$ to be

$\frac{\Delta
z}{2\pi}=\frac{2w_{-}^{2}}{l|f^{\prime}(w_{-})|}=\frac{2}{l(w_{+}-w_{-})}\,,\qquad
l\in\mathbb{Z}^{+}$ (2.7)

the space is the orbifold $\mathbb{R}^{2}/\mathbb{Z}_{l}$.

Let us now consider the end-point at $w=0$. Expanding the 7d metric around
$w=0$ we find

$\displaystyle\mathrm{d}s^{2}_{7}$ $\displaystyle\simeq
w^{3/5}w_{-}^{1/5}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\mathrm{d}z^{2}+\frac{1}{ww_{+}w_{-}}\mathrm{d}w^{2}\bigg{]}$
$\displaystyle=r^{6/5}w_{-}^{1/5}~{}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\mathrm{d}z^{2}+\frac{4}{w_{+}w_{-}}\mathrm{d}r^{2}\bigg{]}\,,$
(2.8)

where we performed the change of coordinate $w=r^{2}$. Clearly this is
singular, as one can verify by computing the Ricci scalar, or any other
curvature invariant. In addition, the scalars also have a singular behaviour,

$X_{1}\sim\frac{r^{12/5}}{|s_{1}|^{3/5}}\,,\qquad
X_{2}\sim\frac{|s_{1}|^{2/5}}{r^{8/5}}\,.$ (2.9)

One should contrast this singular behaviour with the singular behaviour of the
4d and 5d solutions for M2-branes Couzens:2021rlk ; Suh:2021hef  and D3-branes
Suh:2021ifj ; Couzens:2021tnv  on discs. One notes that in addition to the
singular metric only a single scalar diverges in each of these cases with the
other scalars tending to zero. These scalars describe the stretching and
squashing of the sphere in the uplifted theory, when written in embedding
coordinates adapted to the U$(1)^{n}$ symmetry. In the M2-branes and D3-branes
cases the sphere diverges along one direction and shrinks in the remaining
directions. In contrast, here we have three scalars parametrising the
squashing; two of which diverge and only one vanishes. We will see later,
using the uplifted solution, that the behaviour of M5-branes on a disc is
somewhat different to that of the M2-brane and D3-brane cases.

Magnetic charge and holonomy

The metric is supported by a single magneticaly charged gauge field. The
magnetic charge is defined to be

$Q_{i}=\frac{1}{2\pi}\int_{\Sigma}F_{i}\,,$ (2.10)

which for the solution at hand is

$Q\equiv Q_{1}=-w_{-}\frac{\Delta z}{2\pi}\,,\qquad Q_{2}=0\,.$ (2.11)

Note that due to the orbifold it is not necessary that $Q$ is integer but
rather the weaker condition $l\,Q\in\mathbb{Z}$. As such let us define

$lQ=-p\,,\qquad p\in\mathbb{Z}^{+}\,.$ (2.12)

Since the disc also has a boundary one can define the holonomy of the gauge
field on the boundary. One should choose a gauge for the gauge field so that
it is globally well-defined on the disc. Since the circle shrinks at the
centre of the disc we must require that the gauge field vanishes there. This
uniquely fixes the gauge and the globally well-defined gauge field is

$A_{1}=-\bigg{(}\frac{s_{1}}{w^{2}-s_{1}}-\frac{s_{1}}{w_{-}^{2}-s_{1}}\bigg{)}\mathrm{d}z\,.$
(2.13)

The holonomy of the gauge field along the boundary is then

$\text{hol}_{\partial\Sigma}(A_{1})=\frac{1}{2\pi}\oint_{\partial\Sigma}A_{1}=w_{-}\frac{\Delta
z}{2\pi}=-Q\,,$ (2.14)

which gives minus the total magnetic charge threading through the disc.

Euler Characteristic

The final observable that we can compute is the Euler characteristic of the
disc. We have

$\displaystyle\chi(\Sigma)$
$\displaystyle=\frac{1}{4\pi}\int_{\Sigma}R\mathrm{d}\mathrm{vol}(\Sigma)+\frac{1}{2\pi}\int_{\partial\Sigma}\kappa\mathrm{d}\mathrm{vol}(\partial\Sigma)$
$\displaystyle=\frac{\Delta
z}{4\pi}\frac{w^{3/2}(3f(w)-wf^{\prime}(w))}{P(w)^{3/2}}\bigg{|}_{w=0}^{w=w_{-}}$
$\displaystyle=\frac{1}{l}\,,$ (2.15)

where in the going to the second line we have used that the geodesic curvature
of the boundary of the disc is 0, it is locally a cylinder there, and hence
the boundary contribution vanishes.

Using that $w_{+}+w_{-}=1$ we find the relation

$\chi(\Sigma)-Q=\frac{\Delta z}{4\pi}\,.$ (2.16)

This relation is a generic feature of disc geometries, with equivalent
expressions for M2-branes, D3-branes and D4-D8-brane compactifications
Couzens:2021rlk ; Couzens:2021tnv ; Suh:2021aik ; Suh:2021hef ; Suh:2021ifj .

As an aside one may express everything in terms of the orbifold weight $l$ and
the integer magnetic charge $p$ defined in (2.12). The roots in terms of these
integer parameters are

$w_{-}=\frac{p}{2(p+1)}\,,\qquad w_{+}=1-w_{-}=\frac{p+2}{2(p+1)}\,,$ (2.17)

and the period satisfies

$\gamma\equiv\frac{\Delta z}{2\pi}=\frac{2(p+1)}{l}\,.$ (2.18)

It is useful for later to introduce the $2\pi$-periodic coordinate $\hat{z}$
as

$\hat{z}=\frac{z}{\gamma}\,.$ (2.19)

Dictionary to compare with BBMN

To translate between the parametrisation given above and the solution
appearing in Bah:2021hei  one should perform the following identifications:

$\displaystyle w$
$\displaystyle=\frac{B\tilde{w}}{2\sqrt{1-\tilde{w}^{2}}}\,,\qquad\frac{\Delta
z}{4\pi}=\mathcal{C}\,,\qquad 4s_{1}=-B^{2}\,,\qquad m=1\,,\qquad
l=l_{BBMN}\,.$ (2.20)

This puts the metric into the form

$m^{2}\mathrm{d}s_{7}^{2}(BBMN)=\frac{B\tilde{w}^{3/5}}{2\sqrt{1-\tilde{w}^{2}}}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{2}{\tilde{w}h(\tilde{w})(1-\tilde{w}^{2})^{3/2}}\mathrm{d}\tilde{w}^{2}+\frac{4\mathcal{C}^{2}h(\tilde{w})}{B}\mathrm{d}\tilde{z}^{2}\bigg{]}\,,$
(2.21)

which is as given in Bah:2021hei . Note that the gauge field and scalar are
equivalent as well after the above redefinition. By using this dictionary one
finds that the regularity analysis performed above agrees with the equivalent
analysis performed in Bah:2021hei .

This concludes our review of the 7d solution and we turn our attention to the
uplift of the solution to 11d supergravity on an $S^{4}$ .

###  2.2 11d uplift and regularity

In the previous section we have studied the regularity of the 7d solution. We
have seen that the solution exhibits two distinct singular behaviours; one at
the centre of the disc and one along the boundary. In this section we will
study the 11d uplift of the solution, focussing in particular on the singular
regimes from the 7d solution.

Using the uplifting formula in Cvetic:1999xp  the metric is

$\displaystyle\mathrm{d}s^{2}_{11}$
$\displaystyle=\Omega^{1/3}\big{(}wP(w)\big{)}^{1/5}\bigg{[}\mathrm{d}s^{2}_{7}+\frac{1}{\Omega\big{(}wP(w)\big{)}^{1/5}}\Big{(}X_{0}^{-1}\mathrm{d}\mu_{0}^{2}+\sum_{i=1}^{2}X_{i}^{-1}(\mathrm{d}\mu_{i}^{2}+\mu_{i}^{2}(\mathrm{d}\phi_{i}+A_{i})^{2})\Big{)}\bigg{]}\,,$
(2.22)

with

$\displaystyle\Omega$ $\displaystyle=\sum_{I=0}^{2}X_{I}\mu_{I}^{2}\,,\qquad
X_{0}=X_{1}^{-2}X_{2}^{-2}\,,\qquad\sum_{I=0}^{2}\mu_{I}^{2}=1\,.$ (2.23)

Note that $X_{0}=X_{2}$ for the disc. Given this symmetry it is useful to
parametrise the $\mu_{I}$ as

$\mu_{0}=\sqrt{1-\mu^{2}}\cos\theta\,,\quad\mu_{2}=\sqrt{1-\mu^{2}}\sin\theta\,,\quad\mu_{1}=\mu\,,$
(2.24)

and to define

$\Omega=\frac{\tilde{\Omega}}{w^{4/5}h_{1}(w)^{3/5}}\,,\qquad\text{with}\qquad\tilde{\Omega}=w^{2}\mu^{2}+h_{1}(w)(1-\mu^{2})\,.$
(2.25)

Note that $\tilde{\Omega}$ vanishes at $(w=0,\mu=1)$ but is otherwise positive
definite. Next define

$\hat{f}(w)=(w_{-}-w)(w_{+}-w)\,,$ (2.26)

then the metric takes the form

$\displaystyle\mathrm{d}s_{11}^{2}=$
$\displaystyle~{}w^{1/3}\tilde{\Omega}^{1/3}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{1}{w\hat{f}(w)}\mathrm{d}w^{2}+\frac{\hat{f}(w)}{h_{1}(w)}\mathrm{d}z^{2}+\frac{w^{2}(1-\mu^{2})}{w\tilde{\Omega}}\mathrm{d}s^{2}(S^{2})_{(\theta,\phi_{2})}$
$\displaystyle+\frac{h_{1}(w)\mu_{1}^{2}}{w\tilde{\Omega}}D\phi_{1}^{2}+\frac{1}{w(1-\mu_{1}^{2})}\mathrm{d}\mu_{1}^{2}\bigg{]}\,,$
(2.27)

with

$h_{1}(w)=\hat{f}(w)+w\,,\qquad D\phi_{1}=\mathrm{d}\phi_{1}+A_{1}\,.$ (2.28)

Note that the singularity at $w=0$ of the 7d metric persists in the uplifted
solution. This is in contrast to the behaviour of the M2-brane and D3-brane
disc solutions discussed in Couzens:2021rlk ; Suh:2021hef  and Suh:2021ifj ;
Couzens:2021tnv  respectively, where the line $w=0$ is no longer singular only
the point $(w=0,\mu=1)$.

In order to interpret the solution it is useful to observe that it can be
written in the form of an $S^{2}\times S^{1}_{z}\times S^{1}_{\phi_{1}}$
fibration over the rectangle $[0,w_{-}]\times[0,1]$. Away from the boundary of
the rectangle the metric is smooth and the fibers are non-shrinking. Along the
boundary various fibers shrink, see figure 1.444The rectangle we take looks
somewhat different at first from the one in Bah:2021hei . To compare the two
one should take $\mu^{2}=1-\mu^{2}_{BBMN}$

Figure 1: A schematic plot of the rectangle over which the $S^{2}\times
S^{1}_{z}\times S^{1}_{\phi_{1}}$ fibers are defined. In the interior of the
rectangle all the fibers remain of finite size, whilst along the boundary
various cycles shrink. The red dot on the bottom right hand corner is the
location of a monopole of charge $l$.

Since the behaviour of the various edges of the rectangle will play a
prominent role later we will study this in detail. The results we find are in
agreement with the results in Bah:2021hei  and the reader familiar with the
analysis there may skip to the next section safely. We will first study the
degeneration along the sides away from the vertices.

$\mu=0$ degeneration

Consider first the degeneration at $\mu=0$. We can see that the $\phi_{1}$
circle shrinks smoothly giving $\mathbb{R}^{2}$ if $\phi_{1}$ has period
$2\pi$. This is of course the expected behaviour given the $S^{4}$ origin.

$\mu=1$ degeneration

At $\mu=1$ we see that the $S^{2}$ shrinks smoothly giving $\mathbb{R}^{3}$.
As before this is the expected result given the $S^{4}$ origin.

$w=w_{-}$ degeneration

To properly understand the degeneration at $w=w_{-}$ we should rewrite the
metric in the form of the $S^{1}_{z}$ circle fibered over the
$S^{1}_{\phi_{1}}$ circle. It is also convenient to perform a gauge
transformation of the gauge field whilst performing the rewriting, $\delta
A_{1}=n_{z}\mathrm{d}z$. The choice of the constant $n_{z}$ we will make is
such that the Killing spinor on the disc is independent of the angular
coordinate. This is achieved by taking $n_{z}=-\tfrac{1}{2}$. The
$S^{1}_{z}\times S^{1}_{\phi_{1}}$ part of the metric, ignoring the overall
warp-factor $w^{1/3}\tilde{\Omega}^{1/3}$, after the rewriting takes the form

$\mathrm{d}s^{2}(S^{1}_{z}\ltimes
S^{1}_{\phi_{1}})=R_{z}^{2}(\mathrm{d}z+L\mathrm{d}\phi_{1})^{2}+R_{\phi_{1}}^{2}\mathrm{d}\phi_{1}^{2}\,,$
(2.29)

with

$\displaystyle R_{z}^{2}$
$\displaystyle=\frac{\mu^{2}(w^{2}-w_{-}w_{+})^{2}+4w\hat{f}(w)\tilde{\Omega}}{4wh_{1}(w)\tilde{\Omega}}\,,$
(2.30) $\displaystyle R_{\phi_{1}}^{2}$
$\displaystyle=\frac{4\mu^{2}\hat{f}(w)h_{1}(w)}{\mu^{2}(w^{2}-w_{-}w_{+})^{2}+4w\hat{f}(w)\tilde{\Omega}}\,,$
(2.31) $\displaystyle L$
$\displaystyle=\frac{2\mu^{2}(w^{2}-w_{-}w_{+})h_{1}(w)}{\mu^{2}(w^{2}-w_{-}w_{+})^{2}+4w\hat{f}(w)\tilde{\Omega}}\,.$
(2.32)

Note that $R_{\phi_{1}}$ vanishes at both $\mu=0$ and $w=w_{-}$ whilst $R_{z}$
only vanishes at the point $(w=w_{-},\mu=0)$. Note also that the function $L$
is piecewise constant on the two-edges: $\mu=0$ and $w=w_{-}$. Along $\mu=0$
it vanishes, whilst along $w=w_{-}$ it is a non-zero constant. Since
$L\mathrm{d}\phi_{1}$ defines a connection for the fibration the physical
parameter is

$\hat{L}(w,\mu)=\frac{2\pi L(w,\mu)}{\Delta z}\,,$ (2.33)

and we find $\hat{L}(w_{-},\mu)=-l$. This signifies the presence of a monopole
at $(w=w_{-},\mu=0)$.

To see this more clearly let us take the simultaneous limit towards this
point. We change coordinates to

$\mu=r\cos^{2}\Big{(}\frac{\zeta}{2}\Big{)}\,,\qquad
w=w_{-}-\frac{w_{+}-w_{-}}{4}r^{2}\sin^{2}\Big{(}\frac{\zeta}{2}\Big{)}\,.$
(2.34)

and then take the $r\rightarrow 0$ limit. The $S^{2}$ metric remains of finite
size and the remaining 4d part of the internal space becomes

$\displaystyle\mathrm{d}s^{2}_{4}=\frac{1}{w_{-}}\bigg{[}\mathrm{d}r^{2}+\frac{r^{2}}{4}\Big{(}\frac{4}{l^{2}}\big{(}\mathrm{d}\hat{z}-\tfrac{l}{2}(1+\cos\xi)\mathrm{d}\phi_{1}\big{)}^{2}+\mathrm{d}\xi^{2}+\sin^{2}\xi\mathrm{d}\phi_{1}^{2}\Big{)}\bigg{]}\,.$
(2.35)

This is the metric on $\mathbb{R}^{4}/\mathbb{Z}_{l}$, and is due to the
presence of a monopole.

$w=0$ degeneration

The final edge of the rectangle is the one along $w=0$. Series expanding along
$w=0$ the metric reads

$\displaystyle\mathrm{d}s^{2}\simeq\,$ $\displaystyle
w^{1/3}\hat{f}(0)^{1/3}(1-\mu^{2})^{1/3}\bigg{[}\Big{(}4\mathrm{d}s^{2}(\text{AdS}_{5})+\mathrm{d}z^{2}\Big{)}+\frac{1}{w^{2/3}}\bigg{\\{}\frac{1}{1-\mu^{2}}\mathrm{d}\mu^{2}+\frac{\mu^{2}}{1-\mu^{2}}\mathrm{d}\phi_{1}^{2}$
$\displaystyle+\frac{1}{\hat{f}(0)}\Big{(}\mathrm{d}w^{2}+w^{2}\mathrm{d}s^{2}(S^{2})\Big{)}\bigg{\\}}\bigg{]}\,.$
(2.36)

Away from $\mu=1$ this is the metric on an M5-brane wrapping AdS${}_{5}\times
S^{1}_{z}$, located at the tip of $\mathbb{R}^{3}$ and smeared along the two
directions spanned by $\mu$ and $\phi_{1}$.

###  2.3 LLM reformulation

In the previous section we have studied the uplifted solution. In this section
we will extend this analysis by showing that the solution can be embedded into
the classification of $\mathcal{N}=2$ preserving AdS5 solutions of 11d
supergravity of Gaiotto:2009gz ; Lin:2004nb .555In Bah:2021hei  they showed
that the solution can indeed be embedded into the $\mathcal{N}=2$
classification. Since our conventions differ we will present the results from
scratch as they are needed as an intermediate step. Ultimately we want to
consider the equivalent electrostatic description of the problem
Gaiotto:2009gz , but for ease of exposition we will present the intermediate
steps. We will follow the conventions of Gaiotto:2009gz  in the following. Any
$\mathcal{N}=2$ AdS5 solution of 11d supergravity takes the following form:

$\displaystyle\mathrm{d}s^{2}_{11}$
$\displaystyle=\,\mathrm{e}^{2\lambda}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+y^{2}\mathrm{e}^{-6\lambda}\mathrm{d}s^{2}(S^{2})+\mathrm{d}s^{2}_{4}\bigg{]}\,,$
(2.37) $\displaystyle\mathrm{d}s^{2}_{4}$
$\displaystyle=\frac{4}{1-y\partial_{y}D}D\chi^{2}-\frac{\partial_{y}D}{y}\Big{(}\mathrm{d}y^{2}+\mathrm{e}^{D}(\mathrm{d}x_{1}^{2}+\mathrm{d}x_{2}^{2})\Big{)}\,,$
(2.38) $\displaystyle D\chi$
$\displaystyle=\mathrm{d}\chi+v_{i}\mathrm{d}x^{i}\,,\qquad
v_{i}=\frac{1}{2}\epsilon_{ij}\partial_{j}D\,,\qquad v=v_{i}\mathrm{d}x^{i}$
(2.39) $\displaystyle\mathrm{e}^{-6\lambda}$
$\displaystyle=-\frac{\partial_{y}D}{y(1-y\partial_{y}D)}\,,$ (2.40)
$\displaystyle G_{4}$
$\displaystyle=2\mathrm{d}\mathrm{vol}(S^{2})\wedge\bigg{[}D\chi\wedge\mathrm{d}(y^{3}\mathrm{e}^{-6\lambda})+y(1-y^{2}\mathrm{e}^{-6\lambda})\mathrm{d}v-\frac{1}{2}\partial_{y}\mathrm{e}^{D}\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\bigg{]}\,.$
(2.41)

As before, the metric on AdS5 is the unit radius one. The potential $D$, which
determines the full solution is a solution of the (infinite) Toda equation

$\square_{x}D+\partial_{y}^{2}\mathrm{e}^{D}=0\,.$ (2.42)

Comparing the general form of the metric with the solution (2.27) we can
immediately identify

$\displaystyle\mathrm{e}^{2\lambda}=w^{1/3}\tilde{\Omega}^{1/3}\,.$ (2.43)

After a little rewriting the metric takes the form

$\displaystyle\mathrm{d}s^{2}_{11}$
$\displaystyle=\mathrm{e}^{2\lambda}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+w^{2}(1-\mu^{2})\mathrm{e}^{-6\lambda}\mathrm{d}s^{2}(S^{2})_{(\theta,\phi_{2})}+\mathrm{d}s^{2}_{4}\bigg{]}\,,$
(2.44) $\displaystyle\mathrm{d}s^{2}_{4}$
$\displaystyle=\frac{\mathrm{d}\mu^{2}}{w(1-\mu^{2})}+\frac{\mu_{1}^{2}h_{1}(w)}{w\big{(}(1-\mu^{2})h_{1}(w)+\mu^{2}w^{2}\big{)}}(\mathrm{d}\phi_{1}+A_{1})^{2}+\frac{w}{f(w)}\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\gamma^{2}\mathrm{d}\hat{z}^{2}\,.$

From the coefficient of the two-sphere we can then identify

$y=w\sqrt{1-\mu^{2}}\,.$ (2.45)

Since the solution has an enhancement of symmetry compared to the general
classification, there is an additional U$(1)$ symmetry, we define the polar
coordinates

$x_{1}+\mathrm{i}x_{2}=r\mathrm{e}^{\mathrm{i}\beta}\,.$ (2.46)

We now want to identify the radial coordinate $r$ and potential $D$ in terms
of $w,\mu$. We find that they are given by

$\displaystyle r(w,\mu)$
$\displaystyle=H\bigg{[}\mu^{2}\big{(}w_{+}-w\big{)}^{\frac{2w_{+}}{w_{+}-w_{-}}}\big{(}w_{-}-w\big{)}^{-{\frac{2w_{-}}{w_{+}-w_{-}}}}\bigg{]}\,,$
(2.47) $\displaystyle\mathrm{e}^{D}$
$\displaystyle=\frac{w\mu}{\partial_{w}r(w,\mu)\partial_{\mu}r(w,\mu)}\,,$
(2.48)

with $H$ an arbitrary function of one variable with continuous first
derivative.666The fact that the function $H$ is undetermined is due to the
conformal symmetry of the solution in the Toda picture. We can fix the
function to be of the form $H(x)=x^{\alpha}$, with $\alpha$ a constant. There
are different choices one could make for $\alpha$, for example if one takes
$\alpha=\tfrac{1}{2}$ one finds that the potential $D$ is independent of the
coordinate $\mu$. We will instead make the seemingly crazy choice
$\alpha=-\gamma^{-1}$, see equation (2.18) for the definition of $\gamma$.
This turns out to be useful because the metric takes the canonical LLM form
upon making the change of coordinates

$\phi_{1}=-2\chi+\Big{(}2+\frac{\gamma}{2}\Big{)}\beta\,,\qquad\hat{z}=\frac{2}{\gamma}\chi-\Big{(}1+\frac{2}{\gamma}\Big{)}\beta\,,$
(2.49)

which importantly has Jacobian 1, leading to $2\pi$ periodic coordinates in
the canonical $\mathcal{N}=2$ form. Other choices of $\alpha$ do not have this
property. The potential and radial coordinate with this choice are777When
$w_{-}=w_{+}$ the centre of the disc becomes $\mathbb{H}^{2}$ rather than
$\mathbb{R}^{2}$. The radial coordinate is
$r=H[\mu^{2}\mathrm{e}^{2w_{-}/(w_{-}-w)}]\,.$ This behaviour is rather
different to the more general case that we will study here. Note that at the
$\mathbb{H}^{2}$ end-point the solution is of the form
AdS${}_{5}\times\mathbb{H}^{2}$. As we show in appendix A.2, taking the limit
carefully one obtains the Maldacena–Nunez solution.

$r=\bigg{[}\mu^{2}\big{(}w_{+}-w\big{)}^{\frac{2w_{+}}{w_{+}-w_{-}}}\big{(}w_{-}-w\big{)}^{-{\frac{2w_{-}}{w_{+}-w_{-}}}}\bigg{]}^{-\frac{1}{\gamma}}\,,\qquad\mathrm{e}^{D}=\frac{\gamma^{2}\mu^{2}(w-w_{-})(w-w_{+})}{4r(w,\mu)^{2}}\,.$
(2.50)

One can check that the potential satisfies the Toda equation (2.42) as it
should.

###  2.4 Electrostatics reformulation

In the previous section we have rewritten the solution in terms of the
classification of $\mathcal{N}=2$ AdS5 solutions of 11d supergravity,
determined by a potential satisfying the Toda equation. For solutions with two
U$(1)$-isometries, like our solution, there is a formulation one can use by
performing a Bäcklund transform Gaiotto:2009gz . Rather than being determined
by a potential satisfying the Toda equation, the solution is now determined by
a potential satisfying the 3d cylindrical Laplace equation. In terms of this
potential the problem can be interpreted as an electrostatic problem with a
linear line-charge density $\lambda$ as we will now review.

To perform the Bäcklund transform we follow the conventions in Gaiotto:2009gz
and introduce the new coordinates $\rho,\eta$ defined via

$\displaystyle r^{2}\mathrm{e}^{D}=\rho^{2}\,,\quad
y=\rho\partial_{\rho}V(\rho,\eta)\equiv\dot{V}\,,\quad\log
r=\partial_{\eta}V(\rho,\eta)\equiv V^{\prime}\,.$ (2.51)

The metric and flux after the coordinate transformation become

$\displaystyle\mathrm{d}s^{2}=$
$\displaystyle\bigg{(}\frac{\dot{V}\tilde{\Delta}}{2V^{\prime\prime}}\bigg{)}^{1/3}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{2\dot{V}V^{\prime\prime}}{\tilde{\Delta}}\mathrm{d}s^{2}(S^{2})+\frac{2V^{\prime\prime}}{\dot{V}}\Big{(}\mathrm{d}\eta^{2}+\mathrm{d}\rho^{2}+\frac{2\dot{V}}{2\dot{V}-\ddot{V}}\rho^{2}\mathrm{d}\chi^{2}\Big{)}$
$\displaystyle+\frac{2(2\dot{V}-\ddot{V})}{\dot{V}\tilde{\Delta}}\Big{(}\mathrm{d}\beta+\frac{2\dot{V}\dot{V}^{\prime}}{2\dot{V}-\ddot{V}}\mathrm{d}\chi\Big{)}^{2}\bigg{]}\,,$
(2.52) $\displaystyle C_{3}=$ $\displaystyle
2\bigg{[}-\frac{2\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}}\mathrm{d}\chi+\Big{(}\frac{\dot{V}\dot{V}^{\prime}}{\tilde{\Delta}}-\eta\Big{)}\mathrm{d}\beta\bigg{]}\wedge\mathrm{d}\mathrm{vol}(S^{2})\,,$
(2.53)

where

$\displaystyle\dot{\bullet}\equiv\rho\partial_{\rho}\,,\qquad{\bullet}^{\prime}\equiv\partial_{\eta}\,,\qquad\tilde{\Delta}=(2\dot{V}-\ddot{V})V^{\prime\prime}+(\dot{V}^{\prime})^{2}\,.$
(2.54)

The potential $V$ satisfies the 3d cylindrical Laplace equation

$\ddot{V}+\rho^{2}V^{\prime\prime}=0\,.$ (2.55)

To every potential giving rise to a sensible geometry and satisfying the
Laplace equation one can define a line-charge density

$\lambda(\eta)=y(\rho=0,\eta)\,.$ (2.56)

The benefit of the electrostatic description is that the cylindrical Laplace
equation is linear and therefore we may construct more general solutions using
superposition of known solutions.

Let us now turn our attention to obtaining the new coordinates $\rho,\eta$ and
the potential $V$ for the disc solution. The coordinate $\rho$ is simple to
extract in terms of $w,\mu$ and is given by

$\rho=\frac{\gamma\mu\sqrt{(w_{-}-w)(w_{+}-w)}}{2}\,,$ (2.57)

where we have taken the positive root without loss of generality. To compute
$\eta$ note that the integrability of the coordinate change implies the two
constraints

$\partial_{\eta}y=\mathrm{e}^{D/2}\partial_{\rho}r\,,\qquad\partial_{\rho}y=-\mathrm{e}^{D/2}\partial_{\eta}r\,,$
(2.58)

which are independent of the potential $V$. We may now solve for $\eta$ which
gives

$\eta=\frac{\gamma(1-2w)\sqrt{1-\mu^{2}}}{4}\,.$ (2.59)

This is defined up to the addition of a constant, however since this may
always be absorbed by a coordinate transformation later we set this constant
to zero. With $\rho$ and $\eta$ in hand we can now determine the potential
$V$,

$V=-\frac{1}{4}\bigg{[}\log\bigg{(}\frac{1+\sqrt{1-\mu^{2}}}{1-\sqrt{1-\mu^{2}}}\bigg{)}+\sqrt{1-\mu^{2}}\Big{(}(1-2w)\log(r)-2\Big{)}\bigg{]}\,,$
(2.60)

which is also defined up to the addition of a constant, and again this
constant is trivial. This indeed satisfies the cylindrically symmetric Laplace
equation in 3d as it should.

We have now determined both the new coordinates $\rho,\,\eta$ and the
potential $V$ for the electrostatic problem. However in our presentation above
the potential is still written in terms of the original $w,\mu$ coordinates.
To invert this we note that we may determine the $w,\mu$ coordinates in terms
of $\rho,\eta$ as

$\displaystyle w$ $\displaystyle=\frac{1}{4\gamma}\bigg{(}2\gamma-$
$\displaystyle\sqrt{2}\sqrt{\gamma^{2}(1-4w_{-}w_{+})+16(\eta^{2}+\rho^{2})+\sqrt{\big{(}16(\eta^{2}+\rho^{2})-\gamma^{2}(w_{+}-w_{-})^{2}\big{)}^{2}+64\gamma^{2}(w_{+}-w_{-})^{2}\rho^{2}}}\bigg{)}\,,$
$\displaystyle\mu^{2}$
$\displaystyle=\frac{4\rho^{2}}{\gamma^{2}(w_{-}-w)(w_{+}-w)}\,,$ (2.61)

where one should insert the expression for $w$ into $\mu^{2}$. The final
potential is

$\displaystyle-4V=\log\bigg{(}\frac{\gamma(1-2w)+4\eta}{\gamma(1-2w)-4\eta}\bigg{)}+\frac{4\eta}{\gamma}\log\Bigg{[}\frac{4\rho^{2}}{\gamma^{2}}\bigg{(}\frac{w_{+}-w}{w_{-}-w}\bigg{)}^{\frac{1}{w_{+}-w_{-}}}\Bigg{]}-\frac{8\eta}{\gamma(1-2w)}\,,$
(2.62)

and $w$ should be understood to be the function of $\rho,\eta$, depending on
the constants $p,l$ given in (2.61). It is interesting to note that the
potential can be broken into three pieces each of which are solutions of the
3d cylindrical Laplace equation on their own:

$\displaystyle-4V_{1}$
$\displaystyle=\log\bigg{(}\frac{\gamma(1-2w)+4\eta}{\gamma(1-2w)-4\eta}\bigg{)}\,,$
$\displaystyle-4V_{2}$
$\displaystyle=\frac{8\eta}{\gamma}\log\frac{2\rho}{\gamma}\,,$ (2.63)
$\displaystyle-4V_{3}$
$\displaystyle=-\frac{4\eta}{\gamma}\bigg{(}\frac{2}{1-2w}-\frac{1}{w_{+}-w_{-}}\log\frac{w_{+}-w}{w_{-}-w}\bigg{)}\,.$

The second term is the simplest, non-trivial solution to the cylindrical
Laplace equation one can construct. Note that both pure AdS7 and the
Maldacena–Nunez solution have the same form of blocks, see appendix A. Of
course this is expected given that both of these solutions can be obtained
from the same local solution considered here as we show in appendix A.

Properties of the electrostatic setup

Having reformulated the disc solution in terms of an electrostatic problem we
will now study the solution from this vantage point. The first task is to
identify the range of the $\rho$ and $\eta$ coordinates. By inserting the
boundary values of $w,\mu$ we find that the boundary of the
$(w,\mu)$-rectangle is identified with

$\mu=0\Rightarrow\rho=0\,,\quad\mu=1\Rightarrow\eta=0\,,\quad
w=w_{-}\Rightarrow\rho=0\,,\quad
w=0\Rightarrow\gamma^{2}=16\eta^{2}+\frac{4\rho^{2}}{w_{+}w_{-}}\,.$ (2.64)

We find that the ranges of $\rho$ and $\eta$ are

$\rho\in\Big{[}0,\frac{\gamma\sqrt{w_{+}w_{-}}}{2}\Big{]}\,,\qquad\eta\in\Big{[}0,\frac{\gamma}{4}\Big{]}\,.$
(2.65)

Note that the location of the smeared branes (irregular puncture) defines an
ellipse in the $(\rho,\eta)$ coordinates, see the last condition in (2.64).
The focal point of the ellipse is at

$(\rho,\eta)_{\text{Focus}}=\Big{(}0,\pm\gamma\frac{w_{+}-w_{-}}{4}\Big{)}\,,$
(2.66)

which, for the sign within the domain ($+$), is the location of the monopole!
In figure 2 we have plotted the resultant domain in the $(\rho,\eta)$
coordinates, colour coded to match the regions in figure 1 for the $(w,\mu)$
coordinates.

Figure 2: We plot the region in $(\rho,\eta)$ space where the electrostatic
problem is defined. The coloured segments correspond to the different faces of
the rectangle in $(w,\mu)$ space, see figure 1. The points are the vertices of
this rectangle in $(w,\mu)$ space. Starting with the origin and going
clockwise they are: orange=$(w_{-},1)$, magenta=$(w_{-},0)$ (monopole),
violet=$(0,0)$, cyan=$(0,1)$. The red line forming the ellipse is the location
of the smeared brane, whilst the magenta dot (lying on the $\eta$-axis at the
focus point of the ellipse) arises due to the presence of a monopole.

The prototypical example of solutions in the literature of this electrostatic
problem includes a non-compact domain, with both $\rho$ and $\eta$ non-
compact, see for example Gaiotto:2009gz ; Lin:2004nb . The typical boundary
condition imposed is that $\dot{V}$ vanishes along $\eta=0$, and we find that
this is also true of the solutions discussed here. Some solutions with $\eta$
compact have been found in the literature, see for example Lozano:2016kum ;
Aharony:2012tz ; Reid-Edwards:2010vpm , where the domain is a rectangle. The
additional boundary condition imposed in these works is that $\dot{V}$
vanishes along a line $\eta=\eta_{*}$ for some $\eta_{*}>0$. Our compact
domain is quite novel being a quarter of an ellipse. The inclusion of the
smeared brane (irregular puncture) leads to a non-trivial boundary condition
for the electrostatic problem, that is the boundary condition that $\dot{V}$
vanishes on an ellipse. This sets the disc solution apart from previous
examples in the literature.

Line Charge

Having determined the potential $V$ it is a simple matter to obtain the line
charge $\lambda(\eta)$. We find

$\lambda(\eta)=y(\rho=0,\eta)=\begin{cases}\frac{lp}{p+1}\eta&\eta\in[0,\tfrac{1}{2l}]\,,\\\
\frac{1}{2}-\frac{l}{p+1}\eta&\eta\in[\tfrac{1}{2l},\tfrac{p+1}{2l}]\,,\end{cases}$
(2.67)

where we have used (2.17) and (2.18) to express the result in terms of the
magnetic charge defined in (2.12) and the orbifold weight. Note that the
change in slope at the monopole point is $l$, the orbifold weight. The reader
familiar with the conditions in Gaiotto:2009gz  may be uneasy that $\eta$ does
not take integer values at the monopole point and that the slope is not
integer. As we will explain later this still gives rise to a well-defined
solution, in fact the constraints in Gaiotto:2009gz  are too strong and not
all constraints imposed there are needed for a well-defined solution of this
type. As we will see the non-integer slope leads to operators in the dual
field theory having non-integer scaling dimensions.

As an aside, note that there is a scaling symmetry of the solution. One may
perform the transformation

$V(\rho,\eta,p)\rightarrow M~{}V(\Xi\rho,\Xi\eta+c,p)\,,$ (2.68)

for $M$ and $\Xi$ some constants, and retain a solution to the cylindrical
Laplace equation. This transformation leads to a transformation of the line
charge as

$\lambda(\eta)\rightarrow M\lambda(\Xi\eta+c)\,,$ (2.69)

and therefore one could use this freedom to make the line charge satisfy the
conditions in Gaiotto:2009gz . Note that the parameter $c$ is precisely the
linear shift we could have performed earlier when obtaining the coordinate
$\eta$. We will refrain from performing these rescalings for the time being.

Line charge kinks analysis

Let us now substantiate our claim that the following line charge and potential
do give rise to a well-defined geometry. We will show that the monopole number
is indeed $l$ and that the flux is properly quantised. Since the analysis goes
through allowing for an arbitrary number of kinks in the line charge and we
will need this later, we perform the analysis allowing for $f$ kinks in the
line charge. We perform the analysis by assuming that the line-charge and
potential $V$ have the properties of the disc solution, in particular that
$\dot{V}$ vanishes along $\eta=0$ and along some curve generalising the
ellipse above. The final results will depend only on the implicit line-charge
and not the details of this curve. As such let us denote the locations of the
kinks of the line charge to be $n_{a}$ with $a=\\{1,..,f\\}$ and let the
changes in slope be $l_{a}$. Finally the end-point of the line charge, where
$\lambda(\eta)$ vanishes will be taken to be $n_{f+1}$.

First let us consider the expansion of the metric around a kink of the line
charge, taking this point to be $n$. Let us perform the change of coordinates

$\rho=r^{2}\sin\zeta\,,\qquad\eta=n+r^{2}\cos\zeta\,,\qquad\zeta\in[0,\pi]\,,$
(2.70)

which form semi-circles around the monopole in this 2d plane. Expanding the
terms appearing in the metric around $r=0$ we have

$\displaystyle\dot{V}$
$\displaystyle=\lambda(n)\,,\quad\ddot{V}=\frac{r^{2}\sin\zeta}{2}\delta\lambda(n)\,,\quad
V^{\prime\prime}=-\frac{1}{2r^{2}}\delta\lambda(n)\,,\quad$
$\displaystyle\dot{V}^{\prime}$
$\displaystyle=\frac{1}{2}\delta\lambda(n)(\cos\zeta-1)-\sum_{m>n}\delta\lambda(m)\,,\quad\Delta=-\frac{1}{r^{2}}\lambda(n)\delta\lambda(n)\,,$
(2.71)

where in taking the derivative of $\lambda$ around the monopole point we have

$\delta\lambda(n)=\lim_{\epsilon\rightarrow
0^{+}}\big{(}\lambda^{\prime}(n+\epsilon)-\lambda^{\prime}(n-\epsilon)\big{)}\,,$
(2.72)

and the sum is over monopoles higher along the $\eta$ axis. In this limit the
metric becomes

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\lambda(n)^{2/3}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\mathrm{d}s^{2}(S^{2})$
(2.73)
$\displaystyle-\frac{4\delta\lambda(n)}{\lambda(n)}\bigg{\\{}\mathrm{d}r^{2}+\frac{r^{2}}{4}\Big{(}\mathrm{d}\zeta^{2}+\sin^{2}\zeta\mathrm{d}\chi^{2}+4\Big{(}|\delta\lambda(n)|^{-1}\mathrm{d}\beta+\big{(}\tfrac{1}{2}(\cos\zeta-1)+\alpha\big{)}\mathrm{d}\phi\Big{)}^{2}\Big{)}\bigg{\\}}\bigg{]}\,,$

where $\alpha$ is an additive constant which can be removed by a gauge
transformation. This is then the metric on AdS${}_{5}\times
S^{2}\times\mathbb{R}^{4}/|\delta\lambda(n)|$ and implies that we should take
$\delta\lambda(n)$ to be a negative integer. This imposes that the line charge
density is convex and has integer changes in the slope at a location of the
monopole.

Flux quantisation

Next consider flux quantisation. We may rewrite the three-form potential in
the form

$C_{3}=\bigg{[}f_{\chi}(\rho,\eta)\mathrm{d}\chi+f_{\beta}(\rho,\eta)\mathrm{d}\beta\bigg{]}\wedge\mathrm{d}\mathrm{vol}(S^{2})\,,$
(2.74)

with

$f_{\chi}(\rho,\eta)=-\frac{4\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}}\,,\qquad
f_{\beta}(\rho,\eta)=2\bigg{(}\frac{\dot{V}\dot{V}^{\prime}}{\tilde{\Delta}}-\eta\bigg{)}\,.$
(2.75)

Flux quantisation imposes that

$\frac{1}{(2\pi\ell_{\text{p}})^{3}}\int_{\Sigma}G_{4}\in\mathbb{Z}\,,$ (2.76)

for all integral four-cycles $\Sigma$.

We must first identify all integral four-cycles in the geometry. There are two
types of four-cycle to consider depending on shrinking cycles in the geometry.
The first type of cycle, which we denote by $\mathcal{C}_{a}$, are constructed
by taking the cycle which shrinks along $\rho=0$ and the two-sphere which
shrinks along $\eta=0$ which is topologically a four-sphere. Pictorially they
are given by a line stretching from a point on the $\eta$ axis to the $\rho$
axis. For $f$ kinks there are $f+1$ such cycles that can be constructed by
taking the shrinking cycle to be located at some point between two kinks
$n_{a}$ (the end-points included).

The second class of four-cycle is topologically either $S^{2}\times S^{2}$ or
$S^{4}$. and is constructed by taking the shrinking $\beta$ cycle along
$\rho=0$ between two kinks (or the end-points) and the round two-sphere, and
will be denoted by $\mathcal{D}_{a}$. For the cycles involving the two end-
points the four-cycle is topologically an $S^{4}$ and otherwise is
topologically $S^{2}\times S^{2}$.

Let us first consider the four-cycles $\mathcal{C}_{a}$, and let
$a=\\{0,...,f\\}$. We must identify the exact $S^{1}$ that is shrinking along
$\rho=0$, this depends where it is located along the $\eta$-axis. We may
identify the $2\pi$ periodic coordinate which parametrises this shrinking
$S^{1}$ by finding a Killing vector with vanishing norm as $\rho\rightarrow 0$
which is normalised such that the surface gravity is $1$. The Killing vector
satisfying these properties is888When $\lambda(\eta)^{\prime}$ is integer in
all intervals along $\rho=0$ the two U$(1)$’s give rise to a regular fibration
of a $\mathbb{T}^{2}$, however if $\lambda(\eta)^{\prime}$ is instead rational
but not integer, then this gives rise to a quasi-regular fibration. As we will
see, there is no requirement from imposing regularity conditions on the metric
nor flux quantisation that constrains $\lambda(\eta)^{\prime}$ to be integer
generically, this is an assumption made in Gaiotto:2009gz . What is
constrained is the difference of $\lambda(\eta)^{\prime}$ between segments
which must be integer. As we will see the disc solution gives rise to a quasi-
regular fibration generically and $\lambda(\eta)^{\prime}$ is not integer,
though the difference between segments is. The assumption that
$\lambda(\eta)^{\prime}$ is integer is useful for writing the explicit quiver
but is not essential as we will explain at the end of this section. When
considering a line-charge which plateaus rather than vanishes one must take
the slopes to be integer however.

$\partial_{\varphi}=\partial_{\chi}-\lambda^{\prime}(\eta)\partial_{\beta}\,.$
(2.77)

Note that since $\lambda(\eta)$ is linear that this is independent of the
coordinate $\eta$, but depends on the location along the $\rho=0$ line due to
the jumps in $\lambda^{\prime}(\eta)$. We find

$\displaystyle\frac{1}{(2\pi\ell_{\text{p}})^{3}}\int_{\mathcal{C}_{a}}G_{4}$
$\displaystyle=\frac{8\pi}{(2\pi\ell_{\text{p}})^{3}}\Big{[}f_{\chi}(\rho,\eta)-\lambda^{\prime}(\eta)f_{\beta}(\rho,\eta)\Big{]}_{\eta=0}^{\rho=0}$
$\displaystyle=-\frac{16\pi^{2}}{(2\pi\ell_{\text{p}})^{3}}~{}\lambda(\eta)\Big{|}_{\text{constant
piece}}\,,$ (2.78)

where we take $\lambda$ in the interval given by the index ‘$a$’. This should
be integer for all four-cycles. Note that since $\lambda(0)=0$ the flux
through the cycle $\mathcal{C}_{0}$ is 0 always, whilst the cycle
$\mathcal{C}_{f}$ gives the total number of M5-branes wrapping the punctured
sphere.

Consider now the second type of cycle $\mathcal{D}_{a}$. Let the locations of
the monopole along the $\eta$ axis be denoted by $n_{a}$, with $n_{0}$=0, and
$n_{f+1}$ the end-point of the $\eta$-axis. We have

$\displaystyle\frac{1}{(2\pi\ell_{\text{p}})^{3}}\int_{\mathcal{D}_{a}}G_{4}$
$\displaystyle=-\frac{16\pi^{2}}{(2\pi\ell_{\text{p}})^{3}}(n_{a}-n_{a-1})\,.$
(2.79)

To simplify the results let us take the line charge to be the union of lines
of the schematic form

$\lambda=r_{a}\eta+m_{a}\,.$ (2.80)

Then the quantisation conditions in general are equivalent to

$\displaystyle r_{a-1}-r_{a}\equiv
l_{a}\in\mathbb{Z}\,,\qquad\frac{16\pi^{2}}{(2\pi\ell_{\text{p}})^{3}}m_{a}\equiv
M_{a}\in\mathbb{Z}\,,\qquad\frac{16\pi^{2}}{(2\pi\ell_{\text{p}})^{3}}n_{a}\equiv
N_{a}\in\mathbb{Z}$ (2.81)

We should understand $M_{a}$ as the number of M5-branes making the puncture
with $M_{f}$ the total number of M5-branes.

For the line charge in (2.67) we have

$n_{0}=0\,,\quad n_{1}=\frac{1}{2l}\,,\quad n_{2}=\frac{p+1}{2l}\,,\quad
r_{0}=\frac{lp}{p+1}\,,\quad r_{1}=-\frac{l}{p+1}\,,\quad m_{0}=0\,,\quad
m_{1}=\frac{1}{2}\,.$ (2.82)

Note that $r_{0}-r_{1}=l$ and is integer. The quantisation conditions are
satisfied if

$\frac{1}{\pi l\ell_{\text{p}}^{3}}=N\in\mathbb{Z}\,,$ (2.83)

and $p\in\mathbb{Z}$. Note that this is different from the quantisation
condition considered in Gaiotto:2009gz , where the $l$ is not present. We
emphasise that despite the line-charge not satisfying all the properties
required in Gaiotto:2009gz , in particular the integer slopes the solution is
globally well-defined. In Gaiotto:2009gz  for certain choices of line charge
where it plateaus one is still forced into having integer slopes, however when
the line charge does not plateau and either increases to infinity or has a
zero away from $\eta=0$ then this assumption is too restrictive. As we will
see when considering the field theory, the non-integer slope leads to
operators in the dual SCFT with fractional scaling dimensions.

Having understood the constraints for a well-defined solution let us consider
the scaling symmetry (2.68) which we may use to redefine the parameters
$n_{a}$. Under this scaling symmetry the parameters labelling the solution
transform as

$n_{a}\rightarrow\frac{n_{a}}{\Xi}\,,\quad r_{a}\rightarrow M\Xi r_{a}\,,\quad
m_{a}\rightarrow Mm_{a}\,.$ (2.84)

We want to keep the change in the slope at the kinks fixed under the symmetry
since this gives rise to the physical parameter labelling the orbifold weight.
This forces us to take $M\Xi=1$ in the following and it follows that the
scaling symmetry acts the same on $m_{a}$ and $n_{a}$. Consequently, we may
absorb this scaling by redefining the flux quanta $N$: this is to be expected
since this scaling symmetry, whilst holding the change in the slope at the
kink fixed, does not give rise to a physical parameter. We can then use the
symmetry to fix $M=\Xi^{-1}=2l$ which makes the new distinguished locations,
$n_{a}$, integer. We find the rescaled variables

$n_{0}=0\,,\quad n_{1}=1\,,\quad n_{2}=p+1\,,\quad m_{0}=0\,,\quad
m_{1}=l\,,\quad N=\frac{2}{\pi\ell_{\text{p}}^{3}}\,.$ (2.85)

We should emphasise that this rescaling has not changed the physics in any way
since it can be absorbed in the relation between $N$ and the Planck length
$\ell_{\text{p}}$. The parameter $Nm_{2}$ gives the total number of M5-branes
wrapped on the two-sphere whilst the flux $Nm_{1}$ measures the number of
M5-branes giving rise to the regular puncture at $Nn_{1}$. Note the identity

$(r_{0}-r_{1})n_{1}=-r_{1}n_{2}\,,$ (2.86)

which follows since line-charge vanishes at two points. More generally if
there are a total of $f$ kinks we have

$\sum_{a=1}^{f}N_{a}(r_{a-1}-r_{a})=-r_{f}N_{f+1}\,.$ (2.87)

Observe that the convex condition implies that each term in the sum on the
left-hand-side is positive and consistency requires each term to be integer,
thus the right-hand-side is also positive and integer. We can interpret this
constraint as a partitioning of the positive integer $-r_{f}N_{f+1}$ and will
be related to the construction of a Young diagram for the regular puncture in
section 4. In Bah:2021hei  it was shown that this particular gravity solution
is dual to a certain Argyres–Douglas theory. The goal of this paper is to
generalise this construction.

###  2.5 Observables

Having quantised the fluxes and checked regularity let us turn our attention
to the observables of the theory which we can compare to the field theory. One
such observable is the ‘$a$’ central charge, which we study in the following
section. A second is the scaling dimension of probe M2-branes, and the third
observable that we will consider is the flavour central charges of the
solution.

Central Charge

The leading order contribution to the $a$ central charge for an AdS5 solution
of the form Gauntlett:2006ai

$\mathrm{d}s^{2}_{11}=\mathrm{e}^{2A}\Big{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\mathrm{d}s^{2}(M_{6})\Big{]}\,,$
(2.88)

is

$a=\frac{2^{5}\pi^{3}}{(2\pi\ell_{\text{p}})^{9}}\int_{M_{6}}\mathrm{e}^{9A}\mathrm{d}\mathrm{vol}(M_{6})\,.$
(2.89)

From the form of the metric in electrostatic coordinates we identify

$\displaystyle\mathrm{e}^{9A}$
$\displaystyle=\Big{(}\frac{\dot{V}\tilde{\Delta}}{2V^{\prime\prime}}\Big{)}^{3/2}\,,$
(2.90) $\displaystyle\mathrm{d}\mathrm{vol}(M_{6})$
$\displaystyle=\frac{8\sqrt{2}\rho(V^{\prime\prime})^{5/2}}{\dot{V}^{1/2}\tilde{\Delta}^{3/2}}\mathrm{d}\mathrm{vol}(S^{2})\wedge\mathrm{d}\eta\wedge\mathrm{d}\rho\wedge\mathrm{d}\chi\wedge\mathrm{d}\beta\,,$
(2.91)

and therefore we have

$a=\frac{2^{7}\pi^{3}}{(2\pi\ell_{\text{p}})^{9}}\int_{M_{6}}\rho\dot{V}V^{\prime\prime}\mathrm{d}\mathrm{vol}(S^{2})\wedge\mathrm{d}\eta\wedge\mathrm{d}\rho\wedge\mathrm{d}\chi\wedge\mathrm{d}\beta\,.$
(2.92)

Using the cylindrical Laplace equation we may rewrite this as

$a=\frac{2^{10}\pi^{6}}{(2\pi\ell_{\text{p}})^{9}}\int\partial_{\rho}(\dot{V}^{2})\mathrm{d}\eta\wedge\mathrm{d}\rho\,.$
(2.93)

It follows that we can integrate over $\rho$ by defining the integration
domain to go from $\rho=0$ to the ellipse

$n^{2}(p+1)^{2}=\eta^{2}+\frac{(p+1)^{2}}{p(p+2)}\rho^{2}\,,$ (2.94)

and gives

$a=\frac{2^{10}\pi^{6}}{(2\pi\ell_{\text{p}})^{9}}\int\bigg{[}\lambda(\eta)^{2}-\dot{V}(\rho,\eta)^{2}\Big{|}_{\text{ellipse}}\bigg{]}\mathrm{d}\eta\,,$
(2.95)

where it is understood that the second term is evaluated on the ellipse by
eliminating the $\rho$ coordinate. However, the boundary condition along the
ellipse requires $\dot{V}$ to vanish and therefore the only contribution comes
from the line charge term and we have999A similar expression for the central
charge appears in Nunez:2019gbg .

$a=\frac{2^{10}\pi^{6}}{(2\pi\ell_{\text{p}})^{9}}\int\lambda(\eta)^{2}\mathrm{d}\eta\,.$
(2.96)

Note that this holds generally for a solution whose domain is fixed like the
disc solution, i.e. has a similar boundary structure to figure 2 where the
ellipse may be replaced by a more complicated curve along which $\dot{V}=0$.
Carefully performing the integral we find

$a=\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\frac{l^{2}p^{2}}{12(p+1)}\,,$
(2.97)

which upon using the quantisation condition (2.85) gives

$a=N^{3}\frac{l^{2}p^{2}}{12(p+1)}\,.$ (2.98)

This agrees with the result in Bah:2021hei  upon using the following
dictionary between our variables

$N_{BBMN}=lN\,,\quad K_{BBMN}=pN\,.$ (2.99)

Scaling dimensions of probe M2-branes

One may wrap probe M2-branes around calibrated two-cycles in the geometry
giving a BPS particle. With the metric in (2.88) the calibration condition on
a 2d submanifold $\Sigma_{2}$ reads Gauntlett:2006ai

$X\Big{|}_{\Sigma_{2}}=\mathrm{d}\mathrm{vol}_{M_{6}}(\Sigma_{2})\,,$ (2.100)

where $X$ is the calibrated two-form which can be constructed as a spinor
bilinear and will be given momentarily. The right-hand-side denotes the
restriction of the volume form on $M_{6}$ to the 2d submanifold. For such a
calibrated two-cycle the conformal dimension of the BPS particle is given by

$\Delta=\frac{4\pi}{(2\pi\ell_{\text{p}})^{3}}\int_{\Sigma_{2}}\mathrm{e}^{3A}X\,.$
(2.101)

The calibration two-form $X$ was given in Bah:2021hei ; Lin:2004nb  in terms
of the Toda frame and we refer the reader to Bah:2021hei  in particular for
further details on its construction. Following Bah:2021hei  it is convenient
to write the metric on the round two-sphere as

$\displaystyle\mathrm{d}s^{2}(S^{2})=\frac{\mathrm{d}\tau^{2}}{1-\tau^{2}}+(1-\tau^{2})\mathrm{d}\varphi^{2}\,,$
(2.102)

with $\tau\in[-1,1]$, $1$ giving the north pole and $-1$ the south pole, and
$\varphi$ is $2\pi$-periodic. Then, the calibration two-form $X$ in the Toda
frame (see (2.37)-(2.41)) is101010There are some factors of 2 and signs
different between the expression here and the one presented in Bah:2021hei
which are related to the different normalisation we employ and the different
choices of volume form.

$\displaystyle X$
$\displaystyle=y^{3}\mathrm{e}^{-9\lambda}\mathrm{d}\mathrm{vol}(S^{2})+y\mathrm{e}^{-3\lambda}(1-y^{2}\mathrm{e}^{-6\lambda})\mathrm{d}\tau\wedge
D\chi-\tau\mathrm{e}^{-3\lambda}D\chi\wedge\mathrm{d}y+\frac{\tau
y\mathrm{e}^{-9\lambda}\mathrm{e}^{D}}{1-y^{2}\mathrm{e}^{-6\lambda}}\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\,.$
(2.103)

We may transform this to the electrostatic description by the change of
coordinates

$\displaystyle y=\dot{V}\,,\quad
x_{1}+\mathrm{i}x_{2}=r\mathrm{e}^{\mathrm{i}\beta}\,,\quad\log
r=V^{\prime}\,,\quad
r^{2}\mathrm{e}^{D}=\rho^{2}\,,\quad\beta\rightarrow\beta\,,\quad\chi\rightarrow\chi+\beta\,,$
(2.104)

which gives the calibration two-form

$\displaystyle X$
$\displaystyle=2\Big{(}\frac{2V^{\prime\prime}}{\dot{V}\tilde{\Delta}}\Big{)}^{1/2}\bigg{[}\frac{\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}}\mathrm{d}\mathrm{vol}(S^{2})+\tau\Big{(}\dot{V}^{\prime}\mathrm{d}\eta+\rho^{-1}\ddot{V}\mathrm{d}\rho\Big{)}\wedge\Big{(}\mathrm{d}\chi+\frac{\dot{V}^{\prime}}{(\dot{V}^{\prime})^{2}-\ddot{V}V^{\prime\prime}}\mathrm{d}\beta\Big{)}$
(2.105)
$\displaystyle+\frac{\dot{V}}{\tilde{\Delta}}\mathrm{d}\tau\wedge\Big{(}\big{(}(\dot{V}^{\prime})^{2}-V^{\prime\prime}\ddot{V}\big{)}\mathrm{d}\chi+\dot{V}^{\prime}\mathrm{d}\beta\Big{)}+\frac{\tau\rho^{2}V^{\prime\prime}}{(\dot{V}^{\prime})^{2}-\ddot{V}V^{\prime\prime}}\big{(}V^{\prime\prime}\mathrm{d}\eta+\rho^{-1}\dot{V}^{\prime}\mathrm{d}\rho\big{)}\wedge\mathrm{d}\beta\bigg{]}\,.$

Note that this holds for any solution written in electrostatic coordinates
following the conventions used in this paper. The calibrated two-cycle that we
will consider in the following is the round two-sphere at fixed positions in
$M_{6}$. The calibration condition forces the cycle to be located at the
positions of the kink of the line charge.

Let us check the calibration condition for the two-cycles. The calibration
condition is

$2\dot{V}V^{{}^{\prime\prime}}=\tilde{\Delta}\,.$ (2.106)

This condition holds at the monopole point as one can verify from the
expansion in (2.71), in fact this is the only point that it holds true for the
potential we study. The conformal dimension of the BPS particle is then

$\displaystyle\Delta(\mathcal{O}_{1})$
$\displaystyle=\frac{4\pi}{(2\pi\ell_{\text{p}})^{3}}\int_{S^{2}}\lambda(n)\mathrm{d}\mathrm{vol}(S^{2})$
$\displaystyle=\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}\lambda(n)\,.$
(2.107)

For the solution in BBMN we find

$\Delta(\mathcal{O}_{1})=N\frac{lp}{p+1}\,,$ (2.108)

which agrees with the result in Bah:2021hei  upon using the dictionary
provided in equation (2.99).

Flavour symmetries

To each flavour symmetry we can associate a flavour central charge. One can
use anomaly inflow methods to compute a mixed U$(1)_{R}$-flavour Chern–Simons
term in AdS5. From Gaiotto:2009gz  we have that the contribution to the
flavour central charge due to the flavour group at the kink is

$k(\text{SU}(l))=\Big{(}\frac{4}{\pi\ell_{\text{p}}^{3}}\Big{)}\lambda(n)\,,$
(2.109)

which gives

$k(\text{SU}(l))=2N\frac{lp}{p+1}\,,$ (2.110)

in agreement with Bah:2021hei  after using the dictionary in equation (2.99).
As a consistency check we have obtained this result from an anomaly inflow
computation in appendix B.

##  3 Generalised regular puncture

In the previous section we have discussed how to rewrite the disc solution of
Bah:2021hei  in terms of an electrostatic problem. We have obtained a
potential $V$ which depends on two integers, $p$ and $l$. Since the
cylindrical Laplace equation is linear we may sum different potentials and
obtain a new solution. In particular we can sum up the potential of the
previous section with the different potentials depending on different
parameters $p$ and $l$. We will interpret this as giving rise to a general
Young diagram for the regular puncture. As we will show there are some
constraints that the potential blocks must satisfy in order to give rise to a
consistent solution. In the following sections we will analyse this in detail
and compute the observables of the theory considered in the previous section.

###  3.1 Building block of the generalised regular puncture

In this section we will construct the general building block form of the
potential which we will use in the remainder of the section. We saw earlier
that the potential directly following from the disc solution can be more
conveniently written by using the constant scaling symmetry of the
electrostatic description. In order to make the discussion of the
superposition of potentials simpler we will use this symmetry to construct the
building block potential. Taking the potential in (2.62) and transforming it
via

$V(\rho,\eta;\,p,l)\rightarrow
2lnV\Big{(}\frac{\rho}{2ln},\frac{\eta}{2ln};p,l\Big{)}=\mathcal{V}(\rho,\eta;p,l,n)$
(3.1)

we end up with the building block

$\displaystyle\mathcal{V}(\rho,\eta;\,p,l,n)$
$\displaystyle=\frac{l}{2}\bigg{\\{}\frac{2\eta}{(p+1)(1-2w)}-n\log\Big{[}\frac{n(p+1)(1-2w)+\eta}{n(p+1)(1-2w)-\eta}\Big{]}-\frac{2\eta}{p+1}\log\frac{\rho}{2n(p+1)}$
$\displaystyle-\eta\log\Big{[}\frac{2+p-2(p+1)w}{p-2(p+1)w}\Big{]}\bigg{\\}}$
(3.2)

where

$w(\rho,\eta;\,p,n)=\frac{1}{2}-\frac{1}{2\sqrt{2}n(p+1)}\sqrt{n^{2}+\eta^{2}+\rho^{2}+\sqrt{(\rho^{2}+\eta^{2}-n^{2})^{2}+4n^{2}\rho^{2}}}\,.$
(3.3)

Note that we have included an additional integer parameter $n$ which places
the kink at $n$ rather than at 1. This parameter is trivial in the single kink
case since it could be absorbed by redefining the flux quanta $N$, however,
with multiple kinks this is no longer true and it becomes a bonafide
parameter. The domain for the building block potential is a quarter of an
ellipse satisfying

$\rho\in[0,n\sqrt{p(p+2)}]\,,\quad\eta\in[0,n(p+1)]\,,\quad
1\geq\frac{\eta^{2}}{n^{2}(p+1)^{2}}+\frac{\rho^{2}}{n^{2}p(p+2)}\,,$ (3.4)

with focus at

$\rho=0,\,\quad\eta=n\,.$ (3.5)

It is interesting to note that the level-sets of $w$ are ellipses defined by

$w(\rho,\eta;\,p,n)=\frac{\alpha}{1+2\alpha}\,,\quad\Leftrightarrow\quad\frac{n^{2}}{(1+2\alpha)^{2}}=\frac{\eta^{2}}{(p+1)^{2}}+\frac{\rho^{2}}{(p-2\alpha)(p+2(\alpha+1))}\,,$
(3.6)

with focus at $n$. The maximal value of the level set is

$w_{\text{max}}=\frac{p}{2(p+1)}\,,$ (3.7)

which is of course the value of the root of the disc, and occurs at
$\rho=\eta=0$. In terms of $\alpha$ this corresponds to $2\alpha=p$ and
therefore we should take $-1\leq 2\alpha\leq p$.

We may rewrite the potential block $\mathcal{V}$ by first defining

$\hat{w}(\rho,\eta;\,n)=\frac{1}{\sqrt{2}n}\sqrt{n^{2}+\eta^{2}+\rho^{2}+\sqrt{(\rho^{2}+\eta^{2}-n^{2})^{2}+4n^{2}\rho^{2}}}\,,$
(3.8)

then

$\displaystyle\mathcal{V}(\rho,\eta;\,p,l,n)$
$\displaystyle=\frac{l}{2}\bigg{\\{}\frac{2\eta}{\hat{w}(\rho,\eta;n)}-\eta\log\Big{[}\frac{\hat{w}(\rho,\eta;n)+1}{\hat{w}(\rho,\eta,n)-1}\Big{]}-n\log\Big{[}\frac{n\hat{w}(\rho,\eta;n)+\eta}{n\hat{w}(\rho,\eta;n)-\eta}\Big{]}-\frac{2\eta}{p+1}\log\rho\bigg{\\}}\,,$
(3.9)

and we have removed the following term

$\delta\mathcal{V}=\frac{2\eta}{p+1}\log[2n(p+1)]\,,$ (3.10)

from the original potential in (3.2) since it is a trivial term and does not
appear in any of the functions describing the solution as it is linear in
$\eta$. Note that the potential satisfies

$\rho\partial_{\rho}\mathcal{V}=\frac{2l\eta}{p+1}\frac{w(\rho,\eta;\,p,n)}{1-2w(\rho,\eta;\,p,n)}=\frac{l\eta}{p+1}\frac{p+1-\hat{w}(\rho,\eta;n)}{\hat{w}(\rho,\eta;n)}\,,$
(3.11)

which vanishes when either $\eta=0$ or $w=0$ ($\hat{w}=p+1$). The latter has a
line of zeroes on the ellipse defined in (3.4). The line-charge for the
potential is

$\lambda(\eta)=\begin{cases}\frac{pl}{p+1}\eta\,,&0\leq\eta\leq n\,,\\\
nl-\frac{l}{p+1}\eta\,,&n\leq\eta\leq n(p+1)\,.\end{cases}$ (3.12)

###  3.2 Summing up building blocks

We now want to sum an arbitrary number of the building block potentials
$\mathcal{V}(\rho,\eta;\,p,l,n)$ with different integers $p,l,n$. We must take
the $l$’s integer and we choose to take the $n$’s integer too for later
simplicity, this can always be arranged by the quantisation condition. The
$p$’s are no longer constrained to be integer though. Let us index the
different potentials by a subscript $a$ with $a=\\{1,...,f+1\\}$, and order
the $n_{a}$ so that $0<n_{1}<n_{2}<...<n_{f}<n_{f+1}$. The largest of the
$n$’s, $n_{f+1}$ is the end-point and we understand $p_{f+1}=l_{f+1}=0$ and
$n_{0}=0$. Then, the general potential is

$V=\sum_{a=1}^{f}\mathcal{V}(\rho,\eta;\,p_{a},l_{a},n_{a})\,,$ (3.13)

and depends (naively) on $3f$ parameters in total. However, as can be seen
from the form of the potential in (3.9) the final potential actually depends
only on $2f+1$ parameters: the $f$ $l_{a}$’s, the $f$ $n_{a}$’s and the end-
point $n_{f+1}$, or equivalently the slope

$r_{f}=-\sum_{a=1}^{f}\frac{l_{a}}{p_{a}+1}\,.$ (3.14)

The resultant line charge from this general potential is

$\displaystyle\lambda(\eta)=\begin{cases}\eta\,\Big{(}r_{f}+\sum_{a=1}^{f}l_{a}\Big{)}\,,&0\leq\eta\leq
n_{1}\,,\\\ \qquad\qquad\vdots&\qquad\vdots\\\
\eta\,\Big{(}r_{f}+\sum_{a=i}^{f}l_{a}\Big{)}+\sum_{a=1}^{i-1}l_{a}n_{a}\,,&n_{i-1}\leq\eta\leq
n_{i}\,,\\\ \qquad\qquad\vdots&\qquad\vdots\\\
\eta\,r_{f}+\sum_{a=1}^{f}n_{a}l_{a}\,,&n_{f}\leq\eta\leq
n_{f+1}\,,\end{cases}$ (3.15)

where

$r_{f}=-\frac{\sum_{a=1}^{f}n_{a}l_{a}}{n_{f+1}}\,.$ (3.16)

Generically the slope $r_{f}$ is not integer, but becomes integer if
$\sum_{a=1}^{f}n_{a}l_{a}=mn_{f+1}$ for some integer $m$. The end-point
$n_{f+1}$ has been fixed by solving $\lambda(n_{f+1})=0$ and assuming that
there is no other positive 0 of the line charge. If the line charge has a zero
at a positive value smaller than $n_{f+1}$ we must cut off the line-charge
there. Note by construction that the line charge is convex and has kinks at
$n_{a}$ with change of slope $l_{a}$.111111If two of the $n_{a}$ are equal
then the change in the slope is the sum of the 2 $l$’s and there exists a
value for $p$ such that the two potentials can be described by a single
potential. The conditions arising from flux quantisation impose (this follows
straightforwardly from section 2.4 so we do not repeat the analysis)

$\displaystyle\frac{2}{\pi\ell_{\text{p}}^{3}}n_{a}=N_{a}\in\mathbb{Z}\,,\quad\frac{2}{\pi\ell_{\text{p}}^{3}}n_{a}l_{a}\in\mathbb{Z}\,,$
(3.17)

for all $a$. We may solve all these conditions by defining

$N=\frac{2}{\pi\ell_{\text{p}}^{3}}\frac{g}{\sum_{a=1}^{f}l_{a}P_{a}}\,,$
(3.18)

where

$P_{a}=\prod_{b\neq a}(p_{b}+1)\,,\quad P=\prod_{a=1}^{f}(p_{a}+1)\,,\quad
g=\text{gcd}\Big{[}P\sum_{a=1}^{f}n_{a}l_{a},\sum_{b=1}^{f}l_{b}P_{b}\Big{]}\,.$
(3.19)

We end up with $2f+1$ independent quanta, the $f$ changes of slope $l_{a}$ and
the $f+1$ positions $N_{a}$, $N_{f+1}$ included. We will relate combinations
of these quanta to the field theory parameters. The introduction of the
integer $N$ is useful for understanding the holographic limit of the solution,
however in comparing to the field theory it is more useful to work with the
integers $N_{a}$ and $N_{f+1}$.

One may wonder whether it is possible to extract out the building blocks used
to construct the potential. One can recover information about the location of
the monopoles of the building blocks and the associated change of slope,
however there is no way to recover the information about $p$. One can
construct multiple choices of building block with fixed $n_{a}$ and $l_{a}$
but varying $p_{a}$ giving rise to the same final configuration. In figure 3
we present four examples of different building blocks giving rise to the same
final theory with two kinks. It would be interesting to understand whether one
can understand this as the collision of regular punctures giving rise to the
composite regular puncture.

(a) Plot for $p_{1}=2$, $p_{2}=\frac{2}{5}$.

(b) Plot for $p_{1}=\frac{5}{3}$, $p_{2}=\frac{3}{5}$.

(c) Plot for $p_{1}=1$, $p_{2}=\frac{9}{5}$.

(d) Plot for $p_{1}=\frac{2}{3}$, $p_{2}=6$.

Figure 3: We plot various choices of the building blocks giving rise to the
same theory. The red line is the line charge of the composite potential which
is identical in all four figures with end-point $n_{3}=8$. The black line is
the line charge of the first building block giving the kink at $n_{1}=3$ with
change of slope $l_{1}=15$. The blue line is the second building block with
kink at $n_{2}=5$ and change of slope $l_{2}=7$.

Another interesting aspect of the superposition of these solutions is that one
can find solutions where the kinks and change of slope at the kinks are
identical but the location of the zero at $n_{f+1}$ is changed. This hints
that the information contained in the kinks labels a regular puncture whilst
the location of the non-trivial zero contains irregular puncture data. This
will motivate the proposal for the holographic dictionary we present in
section 4. We have plotted a few examples of this behaviour in figure 4.

Figure 4: We plot four choices of line-charge with two kinks. Each has
identical change of slopes and kink locations but with different end-point
$n_{3}$. As we will show later from the kink locations and changes of slope
one may construct a Young diagram labelling the choice of regular puncture of
the sphere. On the other hand the location $n_{f+1}$ will be data for the
irregular puncture. The solutions giving rise to the line charges plotted here
are therefore dual to theories with the same regular puncture but with a
_different_ irregular puncture.

The domain of $(\rho,\eta)$ is fixed in a similar manner to the single kink
case. We restrict to the positive quadrant in the $(\rho,\eta)$-plane. Along
$\rho=0$ the circle with coordinate $\phi$ shrinks smoothly, whilst along
$\eta=0$, $\dot{V}=0$ and the $S^{2}$ shrinks smoothly. The final bound is
obtained by solving $\dot{V}=0$ away from $\eta=0$ and is the analogue of the
ellipse in the single kink case. To understand the shape of the domain it is
useful to recall that the level sets of the function $w$ defined in (3.3) are
ellipses, with focus at $n$. We can then consider the intersection of the
various level sets of the building blocks
$\dot{\mathcal{V}}(\rho,\eta;p_{a},l_{a},n_{a})$. With the level sets defined
in (3.6), the condition $\dot{V}=0$ becomes

$\sum_{a=1}^{f}\frac{l_{a}}{p_{a}+1}\alpha_{a}=0\,.$ (3.20)

We keep fixed $p_{a},l_{a},n_{a}$ and solve this for choices of $\alpha_{a}$
subject to $-1\leq 2\alpha_{a}\leq p_{a}$. Clearly for arbitrary values of the
$\alpha_{a}$’s there may not be a solution, however for a fixed set of
$\alpha$’s with solution this gives rise to a combination of $f$ curves
(either ellipses or hyperbolas) which intersect at a point in the positive
quadrant. Plotting all possible points as we vary the possible $\alpha$’s
gives rise to a smooth curve going from $(\rho,\eta)=(0,n_{f+1})$ to a point
along the $\eta=0$ axis. This point along the $\eta=0$ axis is a solution to a
set of $f$ coupled equations and we are unable to give a closed form
expression for this point. One must solve

$\frac{n_{a}^{2}(p_{a}-2\alpha_{a})\big{(}p_{a}-2(\alpha_{a}+1)\big{)}}{(1+2\alpha_{a})^{2}}=\rho_{*}^{2}\,,\quad\forall~{}a\in[1,f]\,,$
(3.21)

subject to (3.20) for the $f-1$ remaining $\alpha_{a}$’s and finally for the
location $\rho_{*}$.

###  3.3 Observables

Central charge

We can now compute the central charge of the solution using the general form
from (2.96)

$\displaystyle a$
$\displaystyle=\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\int_{0}^{n_{f+1}}\lambda(\eta)^{2}\mathrm{d}\eta$
$\displaystyle=\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\frac{1}{12}\sum_{a=0}^{f}\frac{\lambda(n_{a+1})^{3}-\lambda(n_{a})^{3}}{r_{a}}$
(3.22)
$\displaystyle=\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\frac{1}{12}\sum_{a=0}^{f}\Big{[}r_{a}^{2}(n_{a+1}^{3}-n_{a}^{3})+3r_{a}m_{a}(n_{a+1}^{2}-n_{a}^{2})+3m_{a}^{2}(n_{a+1}-n_{a})\Big{]}\,,$

where we have used the short-hand

$m_{a}=\sum_{b=1}^{a}n_{b}l_{b}\,.$ (3.23)

It is useful to split the various terms up, we may write the first term as

$\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\sum_{a=0}^{f}r_{a}^{2}(n_{a+1}^{3}-n_{a}^{3})=\hat{N}^{2}N_{f+1}+\sum_{a=0}^{f-1}\bigg{[}\Big{(}\sum_{b=a+1}^{f}l_{b}\Big{)}^{2}(N_{a+1}^{3}-N_{a}^{3})\bigg{]}-2\frac{\hat{N}}{N_{f+1}}\sum_{a=0}^{f}N_{a}^{3}l_{a}\,,$
(3.24)

where

$\hat{N}=\sum_{a=1}^{f}N_{a}l_{a}\,,$ (3.25)

and we have used the quantisation conditions (3.17). The second term gives

$\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\sum_{a=0}^{f}r_{a}m_{a}(n_{a+1}^{2}-n_{a}^{2})=-\hat{N}^{2}N_{f+1}+\sum_{a=0}^{f-1}\big{(}N_{a+1}^{2}-N_{a}^{2}\big{)}\Big{[}\sum_{b=a+1}^{f}l_{b}\Big{]}\sum_{c=1}^{a}l_{c}N_{c}+\frac{\hat{N}}{N_{f+1}}\sum_{a=1}^{f}N_{a}^{3}l_{a}$
(3.26)

whilst the third term gives

$\Big{(}\frac{2}{\pi\ell_{\text{p}}^{3}}\Big{)}^{3}\sum_{a=0}^{f}m_{a}^{2}(n_{a+1}-n_{a})=N_{f+1}\hat{N}^{2}-N_{f}\hat{N}^{2}+\sum_{a=0}^{f-1}\big{[}\hat{N}-\sum_{b=a+1}^{f}N_{b}l_{b}\big{]}^{2}(N_{a+1}-N_{a})\,.$
(3.27)

Putting everything together we have

$\displaystyle a$
$\displaystyle=\frac{1}{12N_{f+1}}\bigg{\\{}\hat{N}^{2}N_{f+1}^{2}+\hat{N}\sum_{a=1}^{f}N_{a}^{3}l_{a}+N_{f+1}\sum_{a=0}^{f-1}\bigg{(}(N_{a+1}^{3}-N_{a}^{3})\Big{[}\sum_{b=a+1}^{f}l_{b}\Big{]}^{2}$
$\displaystyle+\Big{[}\sum_{b=a+1}^{f}l_{b}\Big{]}\Big{[}\sum_{c=1}^{a}N_{c}l_{c}\Big{]}+(N_{a+1}-N_{a})\Big{[}\sum_{b=1}^{a}N_{b}l_{b}\Big{]}^{2}\bigg{)}-N_{f}N_{f+1}\hat{N}^{2}\bigg{\\}}\,.$
(3.28)

Scaling dimensions of BPS probe M2-branes

Using the result in (2.107) the scaling dimensions of BPS probe M2-branes
located at the kinks is given by

$\Delta(\mathcal{O}_{a})=\frac{2}{\pi\ell_{\text{p}}^{3}}\lambda(n_{a})=N_{a}\Big{(}\sum_{b=a}^{f}l_{b}-\frac{\hat{N}}{N_{f+1}}\Big{)}+\sum_{b=1}^{a-1}N_{b}l_{b}\,.$
(3.29)

It is convenient to rewrite this by first defining

$A_{a}=N_{a}\sum_{b=a}^{f}l_{b}+\sum_{b=1}^{a-1}N_{b}l_{b}$ (3.30)

then

$\displaystyle\Delta(\mathcal{O}_{a})$
$\displaystyle=A_{a}\frac{N_{f+1}-\hat{N}}{N_{f+1}}+(A_{a}-N_{a})\frac{\hat{N}}{N_{f+1}}$
(3.31) $\displaystyle=A_{a}-N_{a}\frac{\hat{N}}{N_{f+1}}\,.$ (3.32)

We will see that the combination $A_{a}-N_{a}$ appears later in the field
theory section and can be obtained by studying a Young tableau.

Flavour symmetries

We may also compute the flavour central charge due to the flavour groups
arising at the location of the kinks. From (2.109) we have that the central
charge at the $a$’th kink is

$k_{F_{a}}=\frac{4}{\pi\ell_{\text{p}}^{3}}\lambda(n_{a})\,,$ (3.33)

which gives

$k_{F_{a}}=2\bigg{[}N_{a}\Big{(}\sum_{b=a}^{f}l_{b}-\frac{\hat{N}}{N_{f+1}}\Big{)}+\sum_{b=1}^{a-1}N_{b}l_{b}\bigg{]}\,,$
(3.34)

with $\hat{N}$ as given in (3.25). Note that this is twice the scaling
dimensions of the BPS operators considered above.

##  4 Field theory dual

In Bah:2021hei  the dual field theory of the disc solution that we reviewed in
section 2 was identified to be the 4d $\mathcal{N}=2$ theory

$\Big{(}I_{\hat{N},\hat{k}},\hat{Y}_{l}\Big{)}\,,$ (4.1)

with regular puncture given by the Young diagram $Y_{l}$ consisting of the
rectangle with $l$ columns and $\hat{N}/l$ rows. This class of SCFTs are of
Argyres–Douglas type and arise from the low-energy limit of $\hat{N}$
M5-branes wrapped on a sphere with irregular puncture of type
$I_{\hat{N},\hat{k}}$ and regular puncture with associated Young diagram
$Y_{l}$. Equivalently, they are the compactification of the 6d
$\mathcal{N}=(2,0)$ $A_{\hat{N}-1}$ theory on the same punctured sphere.

Motivated by this we conjecture that the solutions we constructed in section 3
are the holographic duals of the 4d $\mathcal{N}=2$ theories

$\Big{(}I_{\hat{N},\hat{k}}\,,Y\Big{)}\,,$ (4.2)

where the Young diagram $Y$ is a general partition of $\hat{N}$, not
necessarily rectangular.

To keep this paper as self-contained as possible and to clarify the notation
we will use, we first review the Argyres–Douglas theories with emphasis on the
observables that we can match with the holographic solutions in section 4.1,
the reader familiar with these theories and their notation may safely skip
this section. We then proceed to explain how to read off the holographic
dictionary between the gravity solutions in section 3, in particular from the
data contained in the line charge in equation (3.15), and the parameters of
the field theory. We show that the leading order contributions to the central
charges from gravity match the field theory results and in addition compare
the scaling dimensions of certain BPS operators.

###  4.1 Review of Argyres–Douglas theories

Argyres–Douglas theories describe a set of 4d $\mathcal{N}=2$ SCFTs which
admit fractional scaling dimensions of the Coulomb branch operators and
dimensional coupling constants. They were first found in Argyres:1995jj ;
Argyres:1995xn  as a point on the Coulomb branch of pure $\mathcal{N}=2$
SU$(3)$ gauge theory and have been extended to many different constructions
since Eguchi:1996vu ; Shapere:1999xr ; Cecotti:2010fi ; Xie:2012hs ;
Cecotti:2012jx ; Wang:2015mra . They are intrinsically strongly coupled
theories and because of the non-integer Coulomb branch operators the conformal
fixed points cannot be described by a $\mathcal{N}=2$ Lagrangian gauge
theory.121212Despite this there are 4d $\mathcal{N}=1$ Lagrangian gauge
theories that flow to Argyres Douglas theories, see for example
Maruyoshi:2016tqk ; Maruyoshi:2016aim ; Agarwal:2016pjo ; Agarwal:2017roi ;
Benvenuti:2017bpg .

One can engineer Argyres–Douglas theories using geometric engineering, in
particular there are constructions in both type IIB and in M-theory. We will
be concerned with the class of theories which can be obtained by compactifying
the 6d $\mathcal{N}=(2,0)$ theory of $\mathfrak{g}$=ADE type on a punctured
sphere. If the punctures of the sphere are all of regular type one obtains
theories of class $\mathcal{S}$, with integer scaling dimensions. If instead,
one allows for irregular singularities one can obtain Argyres–Douglas
theories, Bonelli:2011aa ; Xie:2012hs ; Wang:2015mra .

One can only obtain a 4d $\mathcal{N}=2$ SCFT with an irregular puncture by
compactifying on a sphere and not on a higher order genus Riemann surface. The
complex coordinate of the Riemann surface should transform non-trivially under
the U$(1)_{R}$ R-symmetry in the presence an irregular puncture. As such, the
puncture must be placed at a fixed point of this rotational symmetry. In the
space of Riemann surfaces, only the sphere admits a U$(1)$ action with fixed
points. It follows that we may place an irregular puncture at one of the poles
of the sphere and at most one regular puncture at the other. Any other
configurations containing an irregular puncture, whether it be a different
Riemann surface or with more than one regular puncture, will not give rise to
a $\mathcal{N}=2$ SCFT Xie:2012hs .131313Note that if there is no irregular
puncture the complex coordinate of the Riemann surface does not transform
under the U$(1)_{R}$ R-symmetry and therefore the above analysis does not
apply. One can allow for an arbitrary number of regular punctures for any
Riemann surface and obtain a theory of class $\mathcal{S}$.

The possible irregular punctures were classified in Xie:2012hs  using the
Hitchin equation. Similar to the classification of theories of type
$\mathcal{S}$ one can identify the Seiberg–Witten curve with the spectral
curve of the Hitchin system on the sphere. The Hitchin system consists of a
pair of one-forms, $(A,\phi)$ one a gauge field and the other a Higgs field,
each valued in some Lie Algebra $\mathfrak{g}$. Hitchin’s equations are
equivalent to imposing that the curvature of $\mathcal{A}=A+\mathrm{i}\phi$ is
flat. Punctures correspond to singular solutions to Hitchin’s equations, the
form of which are constrained. Let $\mathrm{z}$ be a complex coordinate on the
sphere and $\Phi(\mathrm{z})$ be the holomorphic part of the Higgs field
$\phi$. Consider the six-dimensional $A_{\hat{N}-1}$ $(2,0)$ theory, i.e. the
Hitchin system is SU$(\hat{N})$ valued. Then, for an irregular singularity at
$\mathrm{z}=0$ the Higgs field near the singularity behaves as one of the
following three forms

Type I $\displaystyle\,,\quad\text{I}_{\hat{k},\,\hat{N}}:\quad$
$\displaystyle\Phi(\mathrm{z})$
$\displaystyle=\frac{T}{\mathrm{z}^{2+r}}+\cdots\,,\quad$ $\displaystyle r=$
$\displaystyle\frac{\hat{k}}{\hat{N}}>-1\,,$ Type II
$\displaystyle\,,\quad\text{II}_{\hat{k},\,\hat{N}}:$
$\displaystyle\Phi(\mathrm{z})$
$\displaystyle=\frac{T}{\mathrm{z}^{2+r}}+\cdots\,,\quad$ $\displaystyle r=$
$\displaystyle\frac{\hat{k}}{\hat{N}-1}>-1\,,$ (4.3) Type III
$\displaystyle\,,\quad\text{III}_{Y_{l},\cdots,Y_{1}}:$
$\displaystyle\Phi(\mathrm{z})$
$\displaystyle=\frac{T_{l}}{\mathrm{z}^{l}}+\cdots+\frac{T_{1}}{\mathrm{z}}+\cdots\,,\quad$
$\displaystyle Y_{l}\subset$ $\displaystyle\,Y_{l-1}\subset\cdots\subset
Y_{1}\,.$

The final dots denote non-divergent terms and the matrices $T$ are
SU$(\hat{N})$ valued diagonal matrices. For type I and type II theories they
have $\hat{N}$ independent eigenvalues whilst for type III theories the
degeneracy of the eigenvalues is encoded in the corresponding Young tableaux
$Y_{i}$. The inclusion of a regular puncture with one of the above three
irregular punctures gives rise to a type IV theory. These theories are
labelled by the choice of irregular puncture $P$ and regular puncture $Y$, and
will be denoted $(P,Y)$. In this work we will be interested in type IV
theories where the irregular puncture is of type I. The Seiberg–Witten curve
takes the form

$0=x^{\hat{N}}+\mathrm{z}^{\hat{k}}+\sum_{(m,n)\in
S}v_{m,n}\mathrm{z}^{n}x^{m}\,,$ (4.4)

where $S$ is defined by the Newton polygon for the theory. The scaling
dimension of $x$ and $\mathrm{z}$ are fixed by taking the Seiberg–Witten
differential $x\,\mathrm{d}\mathrm{z}$ to have dimension $1$, and therefore

$[x]=\frac{\hat{k}}{\hat{k}+\hat{N}}\,,\qquad[\mathrm{z}]=\frac{\hat{N}}{\hat{k}+\hat{N}}\,.$
(4.5)

We refer the reader eager for more details on the classification of irregular
punctures and Newton polygons, after this short and simplified review, to
Xie:2012hs ; Wang:2015mra .

###  4.2 Observables

Central charges

The main observable that we wish to compare to our gravity solutions is the
leading order contribution to the ‘$a$’-central charge. There is a
“straightforward” way of obtaining the central charge from knowledge of the
central charge of the maximal puncture theory and the regular puncture theory
Giacomelli:2020ryy . We will focus on the leading order terms for large
$\hat{N},\hat{k}$ and suppress the subleading terms. As explained in
Giacomelli:2020ryy  the central charges of the $(I_{\hat{N},\hat{k}},Y)$
theory are equal to the sum of four pieces:

$\displaystyle a$
$\displaystyle=a_{Y}+\frac{\hat{N}}{\hat{N}+\hat{k}}\frac{6I_{\rho
Y}-\hat{N}(\hat{N}^{2}-1)}{12}+a_{I_{\hat{N},\hat{k}}}\,,$ (4.6)
$\displaystyle c$
$\displaystyle=c_{Y}+\frac{\hat{N}}{\hat{N}+\hat{k}}\frac{6I_{\rho
Y}-\hat{N}(\hat{N}^{2}-1)}{12}+c_{I_{\hat{N},\hat{k}}}\,.$ (4.7)

Here $a_{Y}$ and $c_{Y}$ are the standard contributions from the puncture $Y$,
$I_{\rho Y}$ is the embedding index of SU$(2)$ in SU$(\hat{N})$ associated to
the nilpotent vacuum expectation value which is turned on to deform the full
puncture to the puncture $Y$. Finally, $a_{I_{\hat{N},\hat{k}}}$ and
$c_{I_{\hat{N},\hat{k}}}$ are the central charges of the $I_{\hat{N},\hat{k}}$
theory, i.e. the theory with just the irregular puncture. Keeping only the
leading order terms the last terms contribute

$a_{I_{\hat{N},\hat{k}}}=c_{I_{\hat{N},\hat{k}}}=\frac{\hat{k}^{2}\hat{N}^{2}}{12(\hat{k}+\hat{N})}\,,$
(4.8)

to the total central charge. To understand the contributions from the regular
puncture let us set the conventions for the Young diagram. We will view the
Young diagram as a series of rectangles with height $\hat{n}_{a},\,a\in(1,f)$
and width $l_{a}$. In total the Young diagram consists of
$\hat{N}=\sum_{a=1}^{f}\hat{n}_{a}l_{a}$ boxes, thus giving a partition of
$\hat{N}$, see figure 5.

$\hat{n}_{f}$$l_{f}$$1$$l_{f}$$\ldots$$\ldots$$\hat{N}$$\vdots$$i_{f-1}$$\ldots$$\vdots$$\cdots$$\hat{n}_{a}$$l_{a}$$i_{a}$$\ldots$$\vdots$$\cdots$$i_{1}$$\ldots$$\vdots$$\hat{n}_{1}$$l_{1}$
Figure 5: Our conventions for the Young diagram and its labelling. The boxes
are labelled from left to right, bottom to top and give a partition of
$\hat{N}$. The corner boxes are distinguished and we denote them by $i_{a}$
with $a\in\\{1,..,f\\}$ and are defined in (4.17). By construction
$\hat{N}=i_{f}$.

The embedding index in these conventions is

$I_{\rho Y}=\frac{1}{6}\sum_{a=1}^{f}\hat{n}_{a}(\hat{n}_{a}^{2}-1)l_{a}\,,$
(4.9)

and this regular puncture leads to the flavour symmetry

$G_{\text{flavour}}=\text{S}\bigg{(}\prod_{a=1}^{f}\text{U}(l_{a})\bigg{)}\,.$
(4.10)

The final, as of yet unspecified, contributions are from $a_{Y}$ and $c_{Y}$.
Following the conventions in Bah:2018gwc  we have (dropping some obviously
subleading terms)

$\displaystyle a_{Y}$
$\displaystyle=\frac{1}{6}\hat{N}^{3}+\frac{1}{24}\big{(}n_{h}(Y)+5n_{v}(Y)\big{)}\,,\quad
c_{Y}=\frac{1}{6}\hat{N}^{3}+\frac{1}{12}\big{(}n_{h}(Y)+2n_{v}(Y)\big{)}\,,$
(4.11)

where $n_{h}(Y)$ and $n_{v}(Y)$ are the effective number of hypermultiplets
and vector multiplets from the regular puncture. We have

$\displaystyle n_{v}(Y)$
$\displaystyle=-\frac{1}{2}\big{(}\hat{N}^{2}-1\big{)}-\sum_{a=1}^{f}\sum_{i=\hat{n}_{a-1}+1}^{\hat{n}_{a}}\bigg{(}\sum_{b=1}^{a-1}\Big{[}(\hat{n}_{b}-\hat{n}_{b-1})\sum_{c=b}^{f}l_{c}\Big{]}+(i-\hat{n}_{a-1})\sum_{b=a}^{f}\l_{b}\bigg{)}^{2}$
(4.12) $\displaystyle n_{h}(Y)$
$\displaystyle=n_{v}(Y)-\frac{1}{2}+\frac{1}{2}\sum_{a=1}^{f}\sum_{b=1}^{a}\sum_{c=b}^{f}l_{a}(\hat{n}_{b}-\hat{n}_{b-1})l_{c}\,.$
(4.13)

Note that in the holographic limit $n_{h}(Y)=n_{v}(Y)$ since the last two
terms of (4.13) are subleading. Moreover, the first terms in (4.12) are also
subleading and can therefore be dropped too leaving only the summation term.
Note in addition that the summation term contains terms with different scaling
properties in the holographic limit.

BPS operator scaling dimensions

There are a set of distinguished BPS operators in the spectrum that we can
compute the dimensions of and match to the supergravity solution. To proceed
we label the boxes of the Young diagram, working from left to right and bottom
to top. To the $j$’th box we associate the number $s_{j}=j-\text{height}(j)$.
The $f$ boxes on the far right which have no box above them are distinguished
and we denote them by $i_{a}$. They correspond to operators associated to
monomials of the form

$\frac{\mathcal{O}_{a}}{\mathrm{z}^{s_{i_{a}}}}x^{\hat{N}-{i_{a}}}\,,$ (4.14)

in the Seiberg–Witten curve. We may compute their dimension by using the
dimension of the coordinates $x,\mathrm{z}$:

$[x]=\frac{\hat{k}}{\hat{k}+\hat{N}}\,,\quad[\mathrm{z}]=\frac{\hat{N}}{\hat{k}+\hat{N}}\,.$
(4.15)

It thus follows that the operator $\mathcal{O}_{a}$ has dimension

$\displaystyle[\mathcal{O}_{a}]$
$\displaystyle=\big{(}\hat{N}-(\hat{N}-i_{a})\big{)}[x]+s_{i_{a}}[z]$
$\displaystyle=i_{a}-\text{height}({i_{a}})\frac{\hat{N}}{\hat{k}+\hat{N}}\,.$
(4.16)

For the Young diagram pictured in figure 5 and focussing on the $a$’th
distinguished box at the $a$’th right-most corner (working form below) we can
write

$i_{a}=n_{a}\sum_{b=a}^{f}l_{b}+\sum_{b=1}^{a-1}n_{b}l_{b}\,,\quad\text{height}(i_{a})=n_{a}\,,$
(4.17)

and the conformal dimension becomes

$\Delta(\mathcal{O}_{a})=n_{a}\sum_{b=a}^{f}l_{b}+\sum_{b=1}^{a-1}n_{b}l_{b}-n_{a}\frac{\hat{N}}{\hat{k}+\hat{N}}\,.$
(4.18)

Flavour central charges

With the above operator dimensions in hand it is a simple matter to compute
the central charge of the $a$’th non-abelian gauge factor. Following the
conjecture in Xie:2013jc  the flavour central charge is twice the conformal
dimension of the operator located the corresponding right-most box of the
Young diagram. Using the results above we have

$k_{F_{a}}=2\bigg{[}n_{a}\sum_{b=a}^{f}l_{b}+\sum_{b=1}^{a-1}n_{b}l_{b}-n_{a}\frac{\hat{N}}{\hat{k}+\hat{N}}\bigg{]}\,.$
(4.19)

###  4.3 Setting up the holographic dictionary

We now want to understand how to map the gravity parameters to the field
theory parameters introduced in the previous section. One may ask whether we
can use the rules of Gaiotto:2009gz  to construct a dual quiver theory. The
short answer is that generically we cannot apply the rules to construct a
quiver. Given that the $(I_{\hat{N},\hat{k}},Y)$ theories are generically non-
Lagrangian this is somewhat reassuring since we wish to identify them as a
dual pair. To see why application of the rules in Gaiotto:2009gz  for
constructing a quiver fails one should note the non-integer slopes of the
line-charge for the solutions discussed here. In Gaiotto:2009gz  they are
taken to be integer, which is a requirement for constructing a dual quiver
since the values of the line charge at the integer values of $\eta$ become the
ranks of the gauge groups in the quiver. We emphasise that having integer
slopes for the line charge is not a requirement of a well-defined supergravity
solution, only that the change of the slopes at the kinks is integer. For
certain choices of the parameters we can make the slope $r_{f}$ integer, and
therefore all the other slopes too, which allows for the construction of a
dual quiver theory, however this is not a generic choice one can make and so
we will not focus on this possibility.

Recall that the line charge depends on $2f+1$ parameters, the $f$ changes of
slopes $l_{a}$, the $f$ kink locations $n_{a}$ and the end-point $n_{f+1}$.
The field theory we conjecture is dual also has $2f+1$ parameters: the integer
$\hat{k}$ and the $2f$ parameters describing the Young diagram giving a
partition of $\hat{N}$. We conjecture that the dictionary between the field
theory and gravity solutions is

$\hat{k}=N_{f+1}-\hat{N}\,,\quad\hat{N}=\sum_{a=1}^{f}N_{a}l_{a}\,,$ (4.20)

with the identifications

$\hat{n}_{a}=N_{a}\,,\quad l_{a}^{\text{SCFT}}=l_{a}^{\text{SUGRA}}\,,$ (4.21)

where the $N_{a}$ are defined in equation (3.17) as

$\frac{2}{\pi\ell_{s}^{3}}n_{a}=N_{a}\in\mathbb{Z}\,.$ (4.22)

Note that this reduces to the identifications made for the disc solution in
Bah:2021hei  as it should.

###  4.4 Matching of observables

We now want to match the central charge of the proposed field theory with the
gravity result in (3.22) using the above dictionary. The large $N$ limit is
obtained by taking the $\hat{n}_{a}$’s, $\hat{N}$ and $\hat{k}$ all large and
of the same order. It is useful to note that the central charge coming from
field theory may be written as

$a=\frac{1}{12N_{f+1}}\bigg{[}N_{f+1}^{2}\hat{N}^{2}+3N_{f+1}n_{h}(Y)+\hat{N}\sum_{a=1}^{f}\hat{n}_{a}^{3}l_{a}\bigg{]}\,$
(4.23)

where we have used our dictionary which gives $\hat{k}+\hat{N}=N_{f+1}$. It is
helpful to compare the $N_{f+1}$ coefficients. The first and last terms match
on the nose with the gravity expressions. The second term is not as obvious
but after a short but not particularly enlightening calculation, and removing
subleading terms one finds that this term also matches between the field
theory and gravity result. We conclude that the central charge of the gravity
solution we discuss in section 3 and the central charge of the
$(I_{\hat{N},\hat{k}},Y)$ theories match to leading order given the
holographic dictionary in section 4.3.

Similarly we may check the match for the scaling dimension of the probe BPS
M2-branes we studied in section 3.3. These BPS M2-branes were located at the
monopole positions or kinks in the line charge. In terms of the Young diagram
these should correspond to the distinguished operators at the right-most
corners. Comparing (4.18) with the gravity result in (3.32) by using the
dictionary (4.20) and (4.21) we find perfect agreement. Additionally since the
flavour central charges on both sides is given by twice the dimensions of the
corresponding state/operator they clearly match given the matching of the
conformal dimensions.

One interesting point to consider is when one can draw a dual quiver using the
prescription in Gaiotto:2009gz . As we explained earlier, their prescription
for drawing a dual quiver given a line charge, relies on the slope of the line
charge being integer. In our setup this follows if the slope $r_{f}$ in (3.16)
is integer, which imposes

$-r_{f}=\frac{\hat{N}}{\hat{k}+\hat{N}}=[\mathrm{z}]\in\mathbb{Z}\,.$ (4.24)

Thus, for these theories the scaling dimensions of the operators we consider
are integer. The non-integer slope of the line charge is therefore essential
for obtaining theories containing operators with non-integer scaling
dimensions. A final check of the proposed duality is to study the quiver
following from the prescription in Gaiotto:2009gz  when we take the slope of
the line charge to be integer. Constructing the quiver following the
prescription in Gaiotto:2009gz  one can show that the central charges match
exactly with the ones presented in (4.6)-(4.7), including subleading terms.
This gives added evidence that our conjectured duality is correct.

##  5 Conclusion

In this paper we have studied the holographic duals of the Argyres–Douglas
theories

$\Big{(}I_{\hat{N},\hat{k}}\,,Y\Big{)}\,,$ (5.1)

with $Y$ an arbitrary regular puncture generalising the work in Bah:2021hei .
We have shown that there is a perfect match between the field theory and
gravity solutions in the holographic limit. It would be interesting to go
beyond the leading order analysis conducted here and compute subleading
corrections on the gravity side, for example to study the gravitational
anomaly $a-c$ and check the matching of the central charges more generally.
Recently, 4d $\mathcal{N}=2$ SCFTs with $a=c$ have been constructed in
Kang:2021lic  with the holographic duals currently unknown. It is natural to
wonder whether using similar singular gravity solutions as those in this paper
one can construct the duals of these theories too.

One of the results of our analysis which deviates from folklore are the
necessary constraints a line charge must satisfy to give rise to a well-
defined $\mathcal{N}=2$ AdS5 solution using Gaiotto:2009gz . We have seen that
it is necessary for the integer condition of the slope to be relaxed if the
line charge does not plateau. This can be reinterpreted as the
$\mathbb{T}^{2}$ fibration of the internal manifold being quasi-regular as
opposed to regular when the slopes are all integer. This leads to M5-branes
wrapping cycles in this torus and in the quasi-regular case the dual SCFT is
non-Lagrangian and dual operators have fractional dimensions. The gravity
analysis then gives a concrete condition on when one can construct a dual
quiver and additionally by using the rules in Gaiotto:2009gz  how to do this.
It seems worthwhile to see whether relaxing this condition in previously
studied solutions in the literature gives rise to interesting solutions which
have been missed previously.

There remain a number of interesting avenues to pursue. For example it remains
to construct the holographic duals of spheres with the other types of
irregular puncture. We have constructed the holographic dual for the most
general type IV theory using a type I irregular puncture however a holographic
dual for the theories using the other irregular punctures is currently
missing. In a similar vein it would be interesting to construct the
holographic duals of the Argyres–Douglas theories constructed from the 6d
$\mathcal{N}=(2,0)$ D-series theory, one may obtain some inspiration from
Nishinaka:2012vi . Further constructing the SCFT duals for the other disc
compactifications is an outstanding problem. For the D3 discs, Couzens:2021tnv
; Suh:2021ifj  it is tempting to conjecture that this is dual to
$\mathcal{N}=4$ SYM on a punctured sphere which preserves $\mathcal{N}=(2,2)$
in 2d. Similarly the field theory dual for M2-branes on a disc Suh:2021hef ;
Couzens:2021rlk  should correspond to ABJM on a punctured sphere, and for the
D4-D8 system Suh:2021aik  to 5d $\mathcal{N}=1$ USp$(2N)$ theory on a
punctured sphere. In a similar spirit identifying the dual field theories for
compactifications on a spindle still remains an open problem.

## Acknowledgments

We would like to thank Jaewon Song, Monica Jinwoo Kang and Sakura Schäfer-
Nameki for useful discussions. This work was supported by the National
Research Foundation of Korea (NRF) grant 2019R1A2C2004880(CC, HK, NK, YL),
2020R1A2C1008497(CC, HK) and 2018R1A2A3074631(YL).

##  Appendix A Special limits of the 7d metric

In this section we will show that both the Maldacena–Nunez solution
Maldacena:2000mw  and AdS7 can be obtained from the metric in (2.1) as special
limits. We will first study the AdS7 solution before studying the limit for
the Maldacena–Nunez solution. We will focus on the uplifted 11d solution
(2.27) in the following for ease of comparison.

###  A.1 AdS7

Since the AdS7 solution should not be supported by any gauge fields we must
set them to be pure gauge. We may achieve this by setting $s_{1}=s_{2}=0$.
Note that this implies that the scalars are set to 1, which is indeed the
correct supersymmetric constant values. Changing coordinates to
$w=\cosh^{2}\zeta$, $z=2\hat{z}$ and $\mu=\sin\theta$ the metric becomes

$\mathrm{d}s^{2}_{11}=4\Big{(}\cosh^{2}\zeta\mathrm{d}s^{2}(\text{AdS}_{5})+\sinh^{2}\zeta\mathrm{d}\hat{z}^{2}+\mathrm{d}\zeta^{2}\Big{)}+\mathrm{d}\theta^{2}+\cos^{2}\theta\mathrm{d}s^{2}(S^{2})+\sin^{2}\theta\mathrm{d}\phi_{1}^{2}\,,$
(A.1)

which is the metric on AdS${}_{7}\times S^{4}$. We therefore see that the
metric in (2.27) has as special limit AdS${}_{7}\times S^{4}$. Note that the
$s_{1}=s_{2}=0$ is one of the special end-points for the range of well-defined
values for $s_{1}$ at fixed $s_{2}=0$. It is the value for which $P(w)$
obtains a 4-fold root and $f(w)$ obtains a triple root.

The potential giving the AdS7 solution in the electrostatic description is

$V=\eta\log{r}+\frac{1}{2}\cos\theta+\frac{1}{4}\log\left(\frac{1-\cos\theta}{1+\cos\theta}\right)\,,$
(A.2)

where $\eta=\frac{1}{2}\cos{\theta}\cosh{2\zeta}$,
$r=\sin{\theta}\sinh^{2}{\zeta}$. This can be obtained by taking a suitable
limit of the general potential we give in (3.2).

###  A.2 Maldacena–Nunez solution

We noted in the main text that there is a second special point in $s_{1}$
parameter space with $s_{2}=0$ fixed, namely $s_{1}=-\frac{1}{4}$. At this
point the roots of $f(w)$ are $w=0$ twice and $w=\tfrac{1}{2}$ twice. We noted
earlier that since $w=\tfrac{1}{2}$ is now a double root the metric at this
end-point looks locally like $\mathbb{H}^{2}$ rather than
$\mathbb{R}^{2}/\mathbb{Z}_{l}$ as at a generic value $-\tfrac{1}{4}<s_{1}<0$.
Indeed, as we will show momentarily by taking a suitable scaling limit to this
point in parameter space we find the Maldacena–Nunez solution.

First, let us take $\mu=\sin\theta$ and perform the rescalings141414The
coordinate names have been chosen to match the conventions in Maldacena:2000mw
and $y$ has no relation to the $y$ introduced earlier in the main text.

$l_{1}=-\frac{1}{4}\,,\quad w=\frac{\lambda}{y}+\frac{1}{2}\,,\quad
z=-\frac{x}{\lambda}\,,$ (A.3)

whilst also performing the coordinate shift

$\phi_{1}\rightarrow\phi_{1}-\frac{x}{\lambda}\,,$ (A.4)

which removes a singular term from the gauge field. Expanding around
$\lambda=0$ the metric becomes

$\displaystyle\mathrm{d}s^{2}_{11}$
$\displaystyle=\frac{1}{2}(1+\cos^{2}\theta)^{1/3}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+2\mathrm{d}\theta^{2}+\frac{2\cos^{2}\theta}{1+\cos^{2}\theta}\mathrm{d}s^{2}(S^{2})+\frac{4\sin^{2}\theta}{1+\cos^{2}\theta}(\mathrm{d}\phi_{1}+y^{-1}\mathrm{d}x)^{2}$
$\displaystyle+\frac{2}{y^{2}}\Big{(}\mathrm{d}y^{2}+\mathrm{d}x^{2}\Big{)}\bigg{]}\,,$
(A.5)

which is precisely the metric in Maldacena:2000mw , with the coordinates used
there. We therefore find that the same metric can be globally completed to
obtain three seemingly distinct solutions, disc solutions, Maldacena–Nunez
solution and pure AdS7. It is interesting to note that the Maldacena–Nunez
solution preserves supersymmetry via a topological twist whilst the disc
solution does not involve a topological twist but an altogether different
mechanism, in particular the Killing spinors of the disc are not independent
of the disc coordinates as they would be for a topological twist.

The electrostatic potential for the Maldacena–Nunez solution is

$V=\eta\log{r}+\frac{1}{2}\cos\theta+\frac{1}{4}\log\left(\frac{1-\cos\theta}{1+\cos\theta}\right)\,,$
(A.6)

where $\eta=\frac{1+y^{2}}{4y}\cos{\theta}$, $r=\frac{y-1}{y+1}$. This can
also be obtained from the general potential we provide in (3.2) after
performing the limit in (A.3).

##  Appendix B Anomaly inflow

In this section we will study the global symmetries and ‘t Hooft anomalies of
the dual SCFT by using anomaly inflow methods Freed:1998tg ; Harvey:1998bx ;
Bah:2019rgq . This section is an extension of the computations performed in
Bah:2021hei  to account for the more general flavour symmetry that our
solutions exhibit, as such we will present the bare bones computation when
there is no risk of confusion.151515We will use a different presentation in
terms of coordinates adapted to the electrostatic description in which our
solution is naturally written.  The ultimate goal of this section is to give
an independent derivation of the observables studied in the gravity theory and
to understand the breaking of a $\mathfrak{u}(1)$ isometry algebra by a
Stückelberg mechanism as observed in Bah:2021hei .

We want to understand the anomalies of the 4d SCFT living on a stack of
M5-branes compactified on a punctured sphere. The 6d internal space, $M_{6}$
arising in the holographic solution encodes the information about the global
symmetries and anomalies of the theory. Following Bah:2019rgq  one excises a
small tubular neighbourhood around the stack of M5-branes, giving the 11d
space spacetime a boundary $\partial M_{11}=M_{10}$. $M_{10}$ is a fibration
of $M_{6}$ over the 4d worldvolume $W_{4}$ on which the 4d SCFT lives. The
fibration is determined by the gauge connections for the continuous symmetries
of $M_{6}$ which descend to symmetries of the 4d SCFT. From the magnetic
source $G_{4}$ for the M5-brane stack one can define a closed, globally well-
defined four-form $E_{4}$ on $M_{10}$, with integral periods, which is
invariant under the structure group of the fibration and when restricted to
$M_{6}$ reduces to $G_{4}$. The presence of the boundary in 11d leads to the
topological terms of 11d supergravity no longer being invariant under gauge
transformations of background fields on $W_{4}$. The variation of the
Chern–Simons terms give rise to a 10-form linear in the gauge parameters
which, via the decent procedure is related to an anomaly 12-form
$\mathcal{I}_{12}$ which encodes the anomalies of the theory and takes the
form

$\mathcal{I}_{12}=-\frac{1}{(2\pi\ell_{\text{p}})^{9}}\frac{1}{3!}E_{4}\wedge
E_{4}\wedge E_{4}\,.$ (B.1)

The 6d anomaly polynomial of the 4d SCFT, in the holographic limit, is given
by

$\mathcal{I}^{SCFT}_{6}=-\int_{M_{6}}\mathcal{I}_{12}\,,$ (B.2)

with the integration over the $M_{6}$ fibers.

###  B.1 Constructing $E_{4}$

We now want to construct $E_{4}$ for the background in the main text. We will
gauge all the symmetries in $M_{6}$. As in Bah:2021hei  one finds that there
is a spontaneous breaking of the continuous symmetries of $M_{6}$ by a
Stückelberg mechanism. We will review this quickly for completeness, but refer
the reader there for further details. The four-form takes the form

$G_{4}=\mathrm{d}\mathrm{vol}(S^{2})\wedge\Big{[}\mathrm{d}f_{\chi}(\rho,\eta)\wedge\mathrm{d}\chi+\mathrm{d}f_{\beta}(\rho,\eta)\wedge\mathrm{d}\beta\Big{]}\,,$
(B.3)

with

$f_{\chi}(\rho,\eta)=-\frac{4\dot{V}^{2}V^{\prime\prime}}{\tilde{\Delta}}\,,\qquad
f_{\beta}(\rho,\eta)=2\Big{(}\frac{\dot{V}\dot{V}^{\prime}}{\tilde{\Delta}}-\eta\Big{)}\,.$
(B.4)

We may gauge the continuous isometries of $M_{6}$, in total we have the
SU$(2)$ and two U$(1)$’s to gauge. This is implemented by the replacements

$\mathrm{d}\beta\rightarrow
D\beta=\mathrm{d}\beta+A^{1}\,,\quad\mathrm{d}\chi\rightarrow
D\chi=\mathrm{d}\chi+A^{2}\,,\quad\mathrm{d}\mathrm{vol}(S^{2})\rightarrow
4\pi e_{2}\,,$ (B.5)

with $e_{2}$ the global angular form of SO$(3)$. Let us introduce the index
notation, $I=\\{1,2\\}\equiv\\{\beta,\chi\\}$ then the most general form for
$E_{4}$ is

$(2\pi\ell_{\text{p}})^{3}E_{4}=G_{4}^{\mathrm{d}\rightarrow
D}+\sum_{I=1}^{2}F^{I}\wedge\omega_{I}^{\mathrm{d}\rightarrow
D}+\sum_{I,J=1}^{2}\sigma_{(IJ)}F^{I}\wedge F^{J}\,,$ (B.6)

with $F^{I}=\mathrm{d}A^{I}$, $\omega_{I}$ two two-forms on $M_{6}$ and
$\sigma_{(IJ)}$ three real scalars on $M_{6}$. Imposing that $E_{4}$ is closed
allows us to fix the two-forms $\omega_{I}$ and scalars $\sigma_{(IJ)}$:

$\partial_{I}\lrcorner
G_{4}+\mathrm{d}\omega_{I}=0\,,\quad\partial_{(I}\lrcorner\omega_{J)}+\mathrm{d}\sigma_{(IJ)}=0\,.$
(B.7)

We have

$\partial_{I}\lrcorner G_{4}=-\mathrm{d}\Big{[}4\pi
f_{I}(\rho,\eta)e_{2}\Big{]}\,,$ (B.8)

which are both globally well-defined forms on $M_{6}$. However, only for
$I=2=\chi$ is this an exact form as is necessary to satisfy (B.7). To see this
consider the degeneration along the smeared brane where the $S^{2}$ shrinks.
For the two-form $f_{I}(\rho,\eta)e_{2}$ to be globally well-defined we
require that $f_{I}(\rho,\eta)$ vanishes there since the two-sphere shrinks
there. From (B.4) we see that on the smeared brane locus where $\dot{V}=0$
only $f_{\chi}$ vanishes there whilst $f_{\beta}$ does not. The smeared brane
source giving rise to the irregular puncture leads to the failure for
$f_{\beta}$ to vanish and thus this acts as an obstruction to constructing
$E_{4}$ as above. As explained in Bah:2021hei  this requires the introduction
of an axion which leads to the spontaneous symmetry breaking of the gauge
field $A^{\beta}$.

Following Bah:2021hei  one introduces an axion $\alpha$ with field strength
$\hat{f}$ which satisfies

$\mathrm{d}\hat{f}=\sum_{I=1}^{2}q_{I}F^{I}\,,$ (B.9)

and modifies the ansatz for $E_{4}$ to take the form

$(2\pi\ell_{\text{p}})^{3}E_{4}=G_{4}^{\mathrm{d}\rightarrow
D}+\sum_{I=1}^{2}F^{I}\wedge\omega_{I}^{\mathrm{d}\rightarrow
D}+\sum_{I,J=1}^{2}\sigma_{(IJ)}F^{I}\wedge F^{J}+\hat{f}\wedge\Lambda\,,$
(B.10)

with

$\Lambda=-4\pi\mathrm{d}f_{\beta}\wedge e_{2}\,.$ (B.11)

Closure of $E_{4}$ implies

$\partial_{I}\lrcorner
G_{4}+\mathrm{d}\omega_{I}+q_{I}\Lambda=0\,,\quad\partial_{(I}\lrcorner\omega_{J)}+\mathrm{d}\sigma_{(IJ)}=0\,.$
(B.12)

It then follows that imposing closure fixes

$\omega_{1}=0\,,\quad\omega_{2}=-4\pi
f_{\chi}e_{2}\,,\quad\sigma_{(IJ)}=0\,,\quad q_{1}=-1\,,\quad q_{2}=0\,.$
(B.13)

The Bianchi identity for the axion field strength is

$\mathrm{d}\hat{f}=-F^{1}\,,\qquad\Rightarrow\qquad\hat{f}=\mathrm{d}\alpha-A^{1}\,.$
(B.14)

As shown in Bah:2021hei  this leads to the U$(1)$ gauge field $A^{1}$ becoming
massive via a Stückelberg mechanism leading to the symmetry being
spontaneously broken.

We now want to include the contributions of the flavour symmetries. Recall
that at the locations of the kinks of the line-charge the metric is locally
the orbifold $\mathbb{R}^{4}/\mathbb{Z}_{l_{a}}$ and leads to an SU$(l_{a})$
flavour symmetry. The orbifold leads to $l_{a}-1$ resolution two-cycles on
which we may wrap the three-form potential $C_{3}$ leading to $l_{a}-1$
abelian gauge fields for each kink. Let the resolution two-cycles be denoted
by $\omega_{a,i}$ with $i\in\\{1,..,l_{a}-1\\}$ and $a\in\\{1,..,f\\}$. These
should be understood to be localised at the kink locations and therefore the
intersection of any two of these two-cycles located at different kinks is
zero. The intersection pairing then gives

$\int_{\mathbb{R}^{4}/\mathbb{Z}_{l_{a}}}\omega_{a,i}\wedge\omega_{a,j}=-C_{ij}^{\mathfrak{su}(l_{a})}\,.$
(B.15)

We can now turn on background gauge fields $\mathcal{A}_{a,i}$, with field
strength $\mathcal{F}_{a,i}$ again with $i\in\\{1,..,l_{a}-1\\}$ and
$a\in\\{1,..,f\\}$ by including the term

$\Delta
E_{4}=\sum_{a=1}^{f}\sum_{i=1}^{l_{a}-1}\omega_{a,i}\wedge\mathcal{F}_{a,i}\,.$
(B.16)

###  B.2 Anomaly polynomial from anomaly inflow

We can now insert the ansatz for $E_{4}$ into the 12-form anomaly polynomial
and integrate over $M_{6}$ to obtain the anomaly polynomial for the 4d theory.
To proceed we need a few results that may be extracted from the literature.
The Bott–Cattaneo formula 10.4310/jdg/1214460608  gives

$\int_{S^{2}}e_{2}=1\,,\quad\int_{S^{2}}e_{2}\wedge
e_{2}=0\,,\quad\int_{S^{2}}e_{2}\wedge e_{2}\wedge
e_{2}=-c_{2}(\text{SU}(2)_{R})\,,$ (B.17)

whilst the gauge field $F^{\chi}$ has bundle

$F^{\chi}=-4\pi c_{1}(\text{U}(1)_{r})\,,$ (B.18)

and

$\sum_{i,j=1}^{l_{a}-1}C_{ij}^{\mathfrak{su}(l_{a})}\mathcal{F}_{a,i}\wedge\mathcal{F}_{a,j}=2c_{2}(\text{SU}(l_{a}))\,.$
(B.19)

The final result is due to non-perturbative M2-brane states which enhance the
U$(1)^{l_{a}-1}$ symmetry to the full non-abelian symmetry SU$(l_{a})$ as
opposed to its Cartan.

Plugging all these ingredients into (B.2) we find

$\displaystyle I_{6}^{\text{SCFT}}$
$\displaystyle=-c_{1}(\text{U}(1)_{r})\wedge
c_{2}(\text{SU}(2)_{R})\frac{1}{2(\pi\ell_{\text{p}})^{3}}\int\mathrm{d}f_{\chi}^{2}\wedge\mathrm{d}f_{\beta}+\frac{4}{\pi\ell_{\text{p}}^{3}}\sum_{a=1}^{f}\lambda(n_{a})c_{1}(\text{U}(1)_{r})\wedge
c_{2}(\text{SU}(l_{a}))\,.$ (B.20)

This should be compared with the anomaly polynomial of class $\mathcal{S}$
which reads

$\displaystyle I_{6}^{\text{class}\mathcal{S}}$
$\displaystyle=(n_{v}-n_{h})\bigg{[}\frac{1}{3}c_{1}(\text{U}(1)_{r})^{3}-\frac{1}{12}c_{1}(\text{U}(1)_{r})\wedge
p_{1}(T^{4})\bigg{]}-n_{v}c_{1}(\text{U}(1)_{r})\wedge
c_{2}(\text{SU}(2)_{R})$
$\displaystyle+\sum_{\text{flavours}}k_{F}c_{1}(\text{U}(1)_{r})\wedge
c_{2}(\text{SU}(f_{a}))\,.$ (B.21)

We can immediately read off the flavour symmetry for the $a$’th kink,

$k_{F_{a}}=\frac{4}{\pi\ell_{\text{p}}^{3}}\lambda(n_{a})\,,$ (B.22)

in agreement with the result we used from Gaiotto:2009gz .

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