#
On the Class $\mathcal{S}$ Origin of Spindle Solutions

Pieter Bomans and Christopher Couzens
Mathematical Institute, University of Oxford
Andrew Wiles Building, Radcliffe Observatory Quarter
Woodstock Road, Oxford, OX2 6GG, U.K.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
0000-0002-0907-9830 0000-0001-9659-8550

###### Abstract

We analyse the backreacted geometry corresponding to a stack of M5-branes
wrapped on a spindle, with a view towards precision tests of the dual
$\mathcal{N}=1$ superconformal field theory. We carefully study the singular
loci of the uplifted geometry and show that these correspond to
$\mathbf{C}^{3}/\mathbf{Z}_{n}$ conical singularities. Therefore, these
solutions present one of the first explicit realisations of honest locally
$\mathcal{N}=1$ preserving punctures in class $\mathcal{S}$. Additionally we
study the symmetries and anomalies of the dual field theory through anomaly
inflow and compute a variety of holographic observables including dimensions
of BPS operators. This work paves the way for advancements in the study and
identification of the precise dual field theories.

###### Table of Contents

  1. 1 Introduction and summary
  2. 2 Seven-dimensional spindle solution
  3. 3 Uplift and analysis
    1. 3.1 Regularity in eleven dimensions
    2. 3.2 Flux Quantisation
  4. 4 Wrapped M2- and M5-branes
    1. 4.1 Probe M5 branes
    2. 4.2 Moduli space of probe M5-branes
    3. 4.3 M2 branes ending on M5 branes
  5. 5 Symmetries and anomalies
    1. 5.1 Construction of $E_{4}$
    2. 5.2 Flavour symmetries from punctures
    3. 5.3 Anomaly inflow
  6. A Uplift formulae
  7. B Canonical $\mathcal{N}=1$ form
  8. C From spindles to BBBW
    1. C.1 Two-sphere
    2. C.2 Torus
    3. C.3 Higher genus Riemann surface

## 1 Introduction and summary

Placing a supersymmetric field theory on a curved background generically
completely breaks supersymmetry. However, one can preserve supersymmetry by
coupling the theory to a collection of (rigid supersymmetry) background fields
[1]. Perhaps the most well known example of this mechanism is the (partial)
topological twist [2, 3, 4]. Schematically, the Killing spinor equation takes
the form

$\delta_{\epsilon}\psi_{\mu}\sim(\partial_{\mu}+\frac{1}{4}\omega_{\mu}\cdot\gamma+A_{\mu})\epsilon=0\,,$
(1.1)

and by turning on a background gauge field $A_{\mu}\propto\omega_{\mu}$, which
compensates for the curvature of spacetime, we can solve the Killing spinor
equation with a constant spinor $\epsilon$, i.e. $\partial_{\mu}\epsilon=0$.
In string or M-theory this scenario is realised by wrapping M- or D-branes on
calibrated cycles where supersymmetry is preserved by turning on the gauge
connection on the normal bundle of the cycle such that it cancels the spin
connection of the cycle. This is not the only mechanism however.

The focus of this note lies in compactifications of the six-dimensional
$\mathcal{N}=(2,0)$ theory on a complex curve $\mathcal{C}$. In this case the
curved manifold is of the form $\mathcal{M}_{4}\times\mathcal{C}$, where
$\mathcal{C}$ is a complex curve of genus $g$ possibly decorated with
punctures. The resulting four-dimensional theories are called theories of
class $\mathcal{S}$ [5] and preserve either $\mathcal{N}=1$ or $\mathcal{N}=2$
supersymmetry, depending on the choice of background flux. Even though the
resulting theories are often non-Lagrangian, the latter case is under
relatively good control and a variety of tools to study them are available,
such as the associated Hitchin system [6] or vertex operator algebras [7, 8].
Moreover, it is believed that they are classified by the type of 6d $(2,0)$
theory, the choice of Riemann surface and the additional local data specifying
the punctures.

The landscape of $\mathcal{N}=1$ class $\mathcal{S}$ theories on the other
hand remains largely unexplored. Although a variety of punctures have been
discussed in an $\mathcal{N}=1$ setting, most of them, such as the $(p,q)$
punctures discussed in [9, 10, 11], locally preserve $\mathcal{N}=2$
supersymmetry but are glued together so that globally only $\mathcal{N}=1$
supersymmetry is preserved. Alternatively, in [12], $\mathcal{N}=1$ punctures
were discussed in a probe approximation but the backreacted geometry remained
unknown. In the $\mathcal{N}=2$ setting the allowed punctures can be
understood as allowed singular boundary conditions of the associated Hitchin
system. Even though a generalised Hitchin system is believed to govern the
allowed punctures in $\mathcal{N}=1$ theories [13, 14, 15] and a variety of
special cases has been considered (see [16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26] for a necessarily incomplete set of references), a satisfactory
mathematical and physical understanding of this problem has yet to be
uncovered.

A fruitful approach to make progress in this direction is provided by
holography. In M-theory these solutions can be obtained by wrapping a stack of
M5-branes on the complex curve $\mathcal{C}$. Punctures in turn can be
described by additional flavour branes intersecting the stack of branes at a
point on $\mathcal{C}$. From a UV point of view, this setup can be described
by a stack of M5 branes wrapping a complex curve inside a Calabi-Yau three-
fold, $X$. The local geometry of $X$ is given by two line bundles over
$\mathcal{C}$, $\mathcal{L}_{1}\oplus\mathcal{L}_{2}\hookrightarrow
X\rightarrow\mathcal{C}\,$, encoding the topology of $X$. The condition to
preserve supersymmetry then reduces to the requirement that the determinant
line bundle is the canonical bundle over the curve, i.e.
$\mathcal{L}_{1}\otimes\mathcal{L}_{2}=K_{\mathcal{C}}$.111When $q_{1}$ or
$q_{2}$ vanishes the three-fold geometry simplifies to the direct product
$X=\mathbf{C}\times T^{*}\mathcal{C}$ and consequently $\mathcal{N}=2$
supersymmetry is preserved. In terms of the Chern numbers
$p_{1}=c_{1}(\mathcal{L}_{1})$ and $p_{2}=c_{1}(\mathcal{L}_{2})$ this
condition takes the form $p_{1}+p_{2}=\chi(\mathcal{C})$ where
$\chi(\mathcal{C})$ is the Euler characteristic of the curve.

From a low energy point of view, these brane setups can be represented by
backreacted $\mathop{\rm AdS}\nolimits_{5}\times\mathcal{M}_{6}$ solutions of
eleven-dimensional supergravity where $\mathcal{M}_{6}$ is an $S^{4}$
fibration over $\mathcal{C}$. Such solutions were first considered in [27] and
later extended in [23, 22] to include more general fluxes. Furthermore, in the
case of smooth Riemann surfaces, it was shown in [28, 29] that the metric on
the curve flows towards its constant curvature form along the RG flow. This is
in line with observations in ($\mathcal{N}=2$) class $\mathcal{S}$ theories
where the four-dimensional theory is seen to be independent of all the Kähler
deformations of the curve.222In addition, complex structure deformations as
well as the addition of non-trivial flat gauge fields correspond to exactly
marginal deformations of the dual SCFT.

The class $\mathcal{S}$ theories reviewed above can include additional local
contributions at marked points on the curve. The local geometry around regular
punctures in the $\mathcal{N}=2$ class $\mathcal{S}$ was studied in [5] where
they were described as a collection of singular solutions to the Toda
equation. For a specific class of punctures it was shown in [30] that such
singular solutions correspond to conical defects on the curve. When
considering curves of genus $g\geq 1$ or $g=0$ with at least three punctures
all the above arguments go through identically upon converting the usual
bundles and characteristic classes into orbibundles and orbifold
characteristic classes. Indeed, in this case supersymmetry is still preserved
by the same topological twist and the IR metric on the curve remains the
constant curvature one.

In this work, we consider a novel situation in which supersymmetry is not
preserved through the usual topological twist. Indeed, a natural way to
circumvent the results of [28] is to consider curves which do not possess a
constant curvature metric. Such cases are provided by the so-called bad
orbifolds [31] of which there exist only two cases. The spindle or the
teardrop, i.e. a sphere with one or two conical defects, denoted by ,333More
precisely, the integers $n_{\pm}$ parameterising the deficit angles
$2\pi\left(1-\frac{1}{n_{\pm}}\right)$ have to be coprime. or a disc,
$\mathbb{D}$, with one conical defect at its centre. We focus on the spindle
for the remainder of this note.

The relevant spindle solutions were constructed in maximal $\mathrm{SO}(6)$
gauged supergravity in [32] while the disc solutions were constructed in [33,
34] with a variety of generalisations considered in [35, 36, 37, 38, 39, 40,
41, 42]. For the disc solutions it is possible to preserve $\mathcal{N}=2$
supersymmetry and in this case the dual SCFTs are of Argyres-Douglas type [43,
44].444Note that for the disc, the mechanism for preserving supersymmetry is
different than that for the spindle. In this case the canonical bundle of $X$
is not trivial and hence it is not Calabi-Yau. This situation is somewhat
closer to the anti-twist scenario discussed in [45]. Since the focus of this
note is on the spindle we do not discuss this case in detail. The boundary of
the disc was understood to correspond to the irregular puncture, while the
regular puncture originates from the conical defect at the centre of the disc.
For the spindle on the other hand, there are no consistent solutions
preserving $\mathcal{N}=2$ supersymmetry and one is forced to consider
$\mathcal{N}=1$ supersymmetric solutions. In the case of the two-sphere with
up to two punctures it is understood that the dual theories do not flow to an
SCFT but instead to a topological sigma-model [46].

In both situations above, the complex curve does not allow for a constant
curvature metric, and consequently supersymmetry is realised in a distinct
way. When wrapping a stack of M5-branes on the spindle, the UV picture is
identical to the usual topological twist, however, the IR picture is rather
different. Indeed, the constraint
$p_{1}+p_{2}=\chi\left(\,{\includegraphics[height=8.0pt]{mySpindle.png}}\,\right)$
is still satisfied, but locally the spin connection is not cancelled by the
R-symmetry gauge fields, nor are the spinors constant. In other words,
supersymmetry is preserved by a topologically topological twist.

Given this novel mechanism for preserving supersymmetry a natural question is:
What is the dual field theory?. In this note we aim to make progress on this
open question by carefully investigating the eleven-dimensional solutions
corresponding to the spindle solutions, analysing their symmetries and
computing a slew of holographic observables.

##### Summary of the results

In order to extract more detailed information about the dual SCFT it proves
useful to carefully uplift the seven-dimensional solution to eleven-
dimensional supergravity. Doing so allows us to scrutinise the internal space
and give a detailed description of the geometry. Unlike the analogous M2-, D3-
solutions [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] the M5-brane
spindle solutions remain singular after uplifting them to eleven
dimensions.555The analogous D4/D8-system also admits singularities in the
uplift.

We carefully demonstrate that each orbifold singularity of the spindle results
in a pair of orbifold singularities in the uplifted geometries, locally taking
the form $\mathop{\rm
AdS}\nolimits_{5}\times\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}$ where the precise
form of the orbifold action is determined by the orbibundle data of the seven-
dimensional solution. Similar to the $\mathcal{N}=2$ class $\mathcal{S}$
story, such punctures generate additional flavour symmetries in the dual SCFT.
Indeed, this can be seen from expanding the M-theory three-form potential, or
equivalently the four-form flux, on the internal cycles of the manifold.
Analysing the crepant resolutions of such orbifold singularities, following
Ito-Reid [59], we show that there are additional localised compact cycles dual
to the exceptional divisors of the blow-up that give rise to vector fields for
the additional global symmetries. We carefully analyse the resulting flavour
symmetries for a variety of orbifold actions and describe the precise form of
the possibly non-abelian flavour symmetry as well as the flavour anomalies
$k_{\alpha\beta\gamma}$. For example, for the two infinite classes of
punctures $\mathbf{C}^{3}/\mathbf{Z}_{n}$ with $n=2k+1$ or $n=2k+2$ and
orbifold action generated by $\frac{1}{n}(1,1,n-2)$ we find that the
associated flavour symmetry of the dual SCFT is given by $\mathrm{SU}(k)^{2}$,
where each of the pair of orbifold singularities of the uplifted geometry
gives rise to one factor. This gives the first explicit holographic
description of honestly, locally, $\mathcal{N}=1$ preserving punctures and
opens the way for further explorations.

In addition to finding the global symmetry, we carefully study the anomalies
of the four-dimensional SCFT through anomaly inflow by integrating the twelve-
form characteristic class $\mathcal{I}_{12}$ [60] over the internal manifold
$\mathcal{M}_{6}$, including the effects from the two-cycles resulting from
the blow-up. Doing so, we show that in contrast to situation on the disc,
where one of the flavour symmetries is broken through a Stückelberg mechanism,
there is no obstruction to equivariantly complete the four-form flux $G_{4}$
to a $\mathrm{U}(1)^{3}$ equivariant form. Therefore, this firmly establishes
that the dual SCFT does in fact inherit the full $\mathrm{U}(1)^{3}$ global
symmetry originating from the isometries of the internal manifold.666The work
[39] gives an alternative viewpoint for the breaking of the flavour symmetry.
There it was shown that the disc solutions can be generalised to include a
scalar field which explicitly breaks the $\mathrm{U}(1)$ in the uplift.
Turning on such symmetry breaking scalar fields is not possible for the
spindle solution, and thus one expects that these are genuine flavour
symmetries of the dual field theory. Finally, we compute the conformal
dimensions of a variety of chiral primaries corresponding to M2-branes
wrapping calibrated cycles as well as study the allowed probe M5 branes
allowing us to probe a rich spectrum of information of the dual SCFT.

##### Outlook

In this work we have carefully analysed the local geometry describing the
singular loci of the uplifted solutions corresponding to M5-branes wrapped on
a spindle. We found that these singular loci describe locally $\mathcal{N}=1$
punctures and explained how to extract a variety of observables of the dual
SCFTs from such singularities.

The type of singularities we study is far from the most general class. In
order to preserve $\mathcal{N}=1$ supersymmetry such singularities are
restricted to be locally $\mathbf{C}^{3}/\Gamma$, where $\Gamma$ is a finite
subgroup of $\mathrm{SU}(3)$. Such subgroups have been classified [61, 62, 63,
64] and include more options then the abelian orbifold with
$\Gamma=\mathbf{Z}_{n}$ that arise in this work. There are more general
abelian orbifolds with $\Gamma=\mathbf{Z}_{m}\times\mathbf{Z}_{n}$ as well as
a variety of non-abelian finite subgroups $\Gamma\subset\mathrm{SU}(3)$. It
would be interesting to investigate whether we can find generalisations of the
solutions described in this work along the lines of [5, 37, 38, 40]. To make
progress in this direction it may be beneficial to embed the solutions found
here in the framework of [9] and look for a generalisation of the
$\mathcal{N}=2$ electrostatics picture allowing one to generalise the
solutions to the relevant Monge-Ampére equation.

Further, having described such general punctures on the spindle a natural
question is whether we can construct solutions describing M5-branes on higher
genus Riemann surfaces and include these more general punctures. Similar to
the $\mathcal{N}=2$ case, finding global solutions with a variety of punctures
is prohibitively hard. However, using the newly developed tools of equivariant
localisation in supergravity [65, 66, 67, 68, 69, 70] it should nonetheless be
possible to extract a wealth of information from a careful local analysis of
the relevant singularities.

Finally, in this note we studied in detail the internal geometry corresponding
to the M5 spindle solutions. We shed new light on a variety of observables and
clarified various aspects of the dual SCFTs. However, it would be very
interesting to exactly determine the dual SCFT with said properties. The
variety of punctures described holographically could have a dual description
as $\mathcal{N}=1$ quiver tails which could be added to SCFTs such as BBBW and
its generalisations providing a new larger set of $\mathcal{N}=1$ class
$\mathcal{S}$ SCFTs. We hope to come back to these question in the future.

##### Structure of this paper

The remainder of this paper is organised as follows. In Section 2, we revisit
the spindle solutions of seven-dimensional gauged supergravity and carefully
discuss the defining data of the relevant orbibundles. We proceed in Section 3
by presenting the explicit uplift of the solution to eleven-dimensions and
analyse the singular loci of the eleven-dimensional geometry. Having done so,
we continue our analysis in Section 4 and 5 where we study respectively the
holographic observables originating from wrapped branes and the global
symmetries and anomalies of the dual SCFT . In Appendices A and B we summarise
the uplift formulae and describe the eleven-dimensional solution in canonical
$\mathcal{N}=1$ coordinates. Finally, in Appendix C we show how in a set of
specific (singular) limits of the solutions presented in this note one can
recover the BBBW solutions corresponding to M5 branes wrapping a smooth
(higher-genus) Riemann surface. This allows us to better contrast the new
ingredients of the spindle field theories against the more established BBBW
solution and dual field theory.

## 2 Seven-dimensional spindle solution

The main character of this work is the spindle solution of seven-dimensional
gauged supergravity. This section serves as an introduction to the solution,
in particular the conventions we will employ in this work. This background is
a solution to the seven-dimensional $\mathrm{U}(1)^{2}\subset\mathrm{SO}(5)$
gauged supergravity BPS equations and equations of motion [71].777We use the
same conventions as in [39]. The relevant solution was described in [32, 45]
and is given as follows. The metric, gauge fields and scalar are

$\displaystyle{\rm d}s^{2}_{7}$
$\displaystyle=\,\Big{(}wH(w)\Big{)}^{1/5}\left[{\rm d}s^{2}_{\mathop{\rm
AdS}\nolimits_{5}}+\frac{w}{f(w)}{\rm
d}w^{2}+\frac{f(w)}{4H(w)}\mathfrak{r}_{z}^{2}{\rm d}z^{2}\right]\,,$ (2.1)
$\displaystyle A^{(i)}$
$\displaystyle=\,-\frac{w^{2}}{h_{i}(w)}\mathfrak{r}_{z}\,{\rm
d}z\,,\qquad\qquad X^{(i)}(w)=\frac{\left(wH(w)\right)^{2/5}}{h_{i}(w)}\,,$

where $i=1,2$ and ${\rm d}s^{2}_{\mathop{\rm AdS}\nolimits_{5}}$ is the metric
on $\mathop{\rm AdS}\nolimits_{5}$ with unit radius, satisfying
$R_{\mu\nu}=-4g_{\mu\nu}$. We explicitly include the constant
$\mathfrak{r}_{z}$ parameterizing the radius of the $z$ circle such that the
coordinate $z$ always has period $2\pi$. The functions $f$, $H$ and $h_{i}$
take the form

$h_{i}(w)=q_{i}+w^{2}\,,\qquad H(w)=h_{1}(w)h_{2}(w)\,,\qquad
f(w)=H(w)-4w^{3}\,.$ (2.2)

Figure 1: A pictorial representation of a spindle.

In this work we are interested in spindle solutions, i.e. compact global
completions of the local solution above which have the topology of a sphere
with one or two conical defects at the poles.888The local solution also
contains compact solutions with disc topology and non-compact co-dimension 2
defect solutions [72, 73, 74] as different global completions. Much of the
analysis of the probe branes and singularity structure of the solutions here
goes over for the $\mathcal{N}=1$ non-compact solutions of the latter
references.  For this reason we restrict ourselves to the portion of
$(q_{1},q_{2})$ parameter space with $q_{1}q_{2}<0$ and where the function $f$
has four distinct real roots and restrict the range of $w\in[w_{-},w_{+}]$ to
lie between the middle two roots where the function $f$ is positive. One can
slightly enlarge the truncation and for each of the two $\mathrm{U}(1)$’s add
a charged scalar which in the dual SCFT breaks the relevant $\mathrm{U}(1)$
global symmetry. However, as shown in [39] such scalars cannot be turned on
while preserving the sphere topology and we will consider them further in this
work.

We denote the two-dimensional spindle metric by

${\rm d}s^{2}_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}=\frac{w}{f(w)}{\rm
d}w^{2}+\frac{f(w)}{4H(w)}\mathfrak{r}_{z}^{2}{\rm d}z^{2}\,.$ (2.3)

Near one of the poles, i.e. near a root of $f(w)$ the two-dimensional metric
takes the form

${\rm d}s^{2}_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}\rightarrow\frac{4w_{\pm}}{|f^{\prime}(w_{\pm})|}\left({\rm
d}r^{2}+\frac{|f^{\prime}(w_{\pm})|^{2}\mathfrak{r}_{z}^{2}}{64w_{\pm}^{4}}\,r^{2}\,{\rm
d}z^{2}\right)\,,\qquad w\rightarrow w_{\pm}\,,$ (2.4)

where we changed the radial coordinate to $r^{2}=\mp(w-w_{\pm})$ at $w\simeq
w_{\pm}$ respectively. Locally, around each root we therefore recover the
metric of an $\mathbf{R}^{2}/\mathbf{Z}_{n_{\pm}}$ orbifold, provided that the
radius $\mathfrak{r}_{z}$ satisfies,

$\mathfrak{r}_{z}=\frac{8w_{\pm}^{2}}{n_{\pm}|f^{\prime}(w_{\pm})|}\,.$ (2.5)

The fluxes through the spindle are given by

$Q_{i}\equiv\frac{1}{2\pi}\int_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}F^{(i)}=-\mathfrak{r}_{z}\,\frac{w^{2}}{h_{i}(w)}\Big{|}^{w=w_{+}}_{w=w_{-}}=\frac{p_{i}}{n_{+}n_{-}}\,,\qquad\qquad
p_{i}\in\mathbf{Z}\,,$ (2.6)

in line with the quantisation conditions for $\mathrm{U}(1)$-orbibundles over
the spindle [75, 45]. The orbifold Euler characteristic of the spindle is
given by

$\chi\left(\,{\includegraphics[height=8.0pt]{mySpindle.png}}\,\right)=2-\left(1-\frac{1}{n_{-}}\right)-\left(1-\frac{1}{n_{+}}\right)=\frac{n_{-}+n_{+}}{n_{-}n_{+}}\,,$
(2.7)

while the total R-symmetry flux through the spindle is given by

$\frac{1}{2\pi}\int_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}F_{R}=-\chi\left(\,{\includegraphics[height=8.0pt]{mySpindle.png}}\,\right)\,.$
(2.8)

where $F_{R}={\rm d}A^{(1)}+{\rm d}A^{(2)}$. Hence, this solution realises the
twist scenario of [45]. Let us re-emphasise that this is not a standard
topological twist. The spindle is a so-called "bad" orbifold which does not
allow for a constant curvature metric as one would expect from a topological
twist. Similarly, the Killing spinor is not constant along the spindle. This
solution only topologically realises the topological twist. As already
mentioned, in order to find solutions with global spindle topology we need
$q_{1}q_{2}<0$ analysing the fluxes one can see that this implies that also
$p_{1}p_{2}<0$. Since from the twist condition we have
$p_{1}+p_{2}=n_{+}+n_{-}$ we have to impose either $p_{1}<0$ or
$p_{1}>n_{+}+n_{-}$.

So far we described the local form of the solution as well as the global
completion of the two-dimensional surface to a spherical topology. In the case
where  is a sphere this is a complete description and with the data provided
all $\mathrm{U}(1)$ bundles over  are uniquely determined. However, as
explained in [45], on orbifolds there is a certain ambiguity left for the
$\mathrm{U}(1)$ orbibundles. More precisely, in the neighbourhood of the
conical defects our solution is a $\mathrm{U}(1)^{2}$ orbibundle over
$\mathbf{C}/\mathbf{Z}_{n}$. On the covering space $\mathbf{C}\times
S^{1}\times S^{1}$ we can introduce complex coordinates $(x,y_{i})$ with
$|y_{i}|=1$ such that the generator of $\mathbf{Z}_{n}$ acts on $x$ as
$x\mapsto\omega x$, with $\omega=\exp\left(\frac{2\pi{\rm i}}{n}\right)$. To
fully specify the orbibundles over $\mathbf{C}/\mathbf{Z}_{n}$, we need to
pick a homomorphism $\varphi$ from the orbifold group into the fibre group,

$\varphi\quad:\quad\mathbf{Z}_{n}\rightarrow\mathrm{U}(1)^{2}\quad:\quad\omega\mapsto\varphi(\omega)=\left(\omega^{m^{(1)}},\omega^{m^{(2)}}\right)\,.$
(2.9)

This choice of homomorphism amounts to picking ‘charges’
$m^{(i)}\in\mathbf{Z}_{n}$ for the $\mathbf{Z}_{n}$ action on the fibre. The
orbibundle is then fully determined as the quotient
$(\mathbf{C}\times\mathrm{U}(1)^{2})/\mathbf{Z}_{n}$ where $\mathbf{Z}_{n}$
acts with generator $(\omega,\omega^{m^{(1)}},\omega^{m^{(2)}})$. Hence, we
need to supply the solution above with 2 pairs of integers
$(m^{(1)}_{\pm},m^{(2)}_{\pm})$ specifying the homomorphism at the two
orbifold singularities. In a patch around the singular points we can then
write the gauge fields as

$\left.A^{(i)}\right|_{w\rightarrow
w_{\pm}}=A^{(i)}_{\pm,0}+\frac{m^{(i)}_{\pm}}{n_{\pm}}{\rm d}z\,,$ (2.10)

where $A^{(i)}_{\pm,0}$ is a well-defined gauge field in the disc centred at
the conical defect, i.e. $A^{(i)}_{\pm,0}(w_{\pm})=0$. This choice fixes the
remaining gauge freedom in (2.1) and will be crucial for a proper analysis of
the singularities in eleven dimensions. Finally, in order to ensure
supersymmetry, we need to carefully analyse the regularity of the spinor at
the conical defects. As Shown in [45], in the gauge used in (2.1) the Killing
spinor can be written as

$\zeta=\begin{pmatrix}\zeta_{+}\\\
\zeta_{-}\end{pmatrix}=\mathrm{e}^{-\frac{3{\rm
i}z}{4}}\frac{y^{\frac{1}{20}}}{\sqrt{2}H^{\frac{1}{5}}}\begin{pmatrix}\left[\sqrt{H}+2y^{\frac{3}{2}}\right]^{\frac{1}{2}}\\\
\left[\sqrt{H}-2y^{\frac{3}{2}}\right]^{\frac{1}{2}}\end{pmatrix}$ (2.11)

From (2.2) we immediately see that $\zeta_{-}$ vanishes at both conical
defects, hence the focus will be on $\zeta_{+}$. To properly analyse the
regularity at the conical defect it is instructive to change gauge to (2.10)
and furthermore perform a large gauge $\mathrm{SL}(2,\mathbf{Z})$ gauge
transformation removing the fractional term. Since the spinor has charge
$q=\frac{1}{2}$ under the R-symmetry we can rewrite $\zeta_{+}$ near a conical
defect as

$\zeta_{+}=\mathrm{e}^{{\rm
i}\left(q-\frac{1+m_{\pm}^{(1)}+m_{\pm}^{(2)}}{2n_{\pm}}\right)z}f_{\pm}(r)\,.$
(2.12)

where $f_{\pm}(r)$ is a function of the radial coordinate only. Since at the
conical defect the coordinate $z$ is not well-defined, we need to impose

$q-\frac{1+m_{\pm}^{(1)}+m_{\pm}^{(2)}}{2n_{\pm}}=0\qquad\Leftrightarrow\qquad
1+m_{\pm}^{(1)}+m_{\pm}^{(2)}=n_{\pm}\,,$ (2.13)

in order for the spinor to be well-defined and the solution to preserve
supersymmetry globally. As we will see below, in the uplift this condition
precisely translates into the condition for the internal manifold to be
Calabi-Yau, as expected for a supersymmetric solution.

## 3 Uplift and analysis

A fruitful strategy to better understand the properties of these solutions, as
well as their holographic dual SCFTs, is to consider the uplift to eleven-
dimensional supergravity. The general formulae to uplift a solution of maximal
7d gauged supergravity were derived in [76, 77, 78] and can be found in
Appendix A.

To attain a more elegant presentation it is convenient to introduce the
following functions

$\displaystyle\Upsilon_{H}=\frac{\mu_{0}^{2}\,H}{w}+w\left(\mu_{2}^{2}\,h_{1}+\mu_{1}^{2}\,h_{2}\right)\,,$
(3.1)
$\displaystyle\Upsilon_{F}=\frac{\mu_{0}^{2}\,f}{4w^{3}}+\frac{1}{4w}\left(\mu_{2}^{2}\,h_{1}+\mu_{1}^{2}\,h_{2}\right)\,,$
$\displaystyle\Psi=\mu_{0}^{2}\frac{\mu_{2}^{2}-\mu_{1}^{2}}{\mu_{2}^{2}+\mu_{1}^{2}}+\frac{w^{2}}{H}\left(\mu_{2}^{2}\,h_{1}-\mu_{1}^{2}\,h_{2}\right)\,,$

where we introduced the embedding coordinates $\mu_{I}$ on the four-sphere
which satisfy $\mu_{0}^{2}+\mu_{1}^{2}+\mu_{2}^{2}=1$. A convenient choice for
our purposes is,

$\mu_{0}=\cos\xi\,,\qquad\mu_{1}=\sin\xi\cos\theta\,,\qquad\mu_{2}=\sin\xi\sin\theta\,.$
(3.2)

where $\xi\in[0,\pi]$ and $\theta\in[0,\pi/2]$. In terms of these function we
can write the eleven-dimensional metric as

$\displaystyle L^{-2}{\rm d}s_{11}^{2}=\Upsilon_{H}^{1/3}\Bigg{[}{\rm
d}s_{\mathop{\rm AdS}\nolimits_{5}}^{2}+{\rm d}s_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}^{2}+\frac{1}{\Upsilon_{H}}\Big{(}w^{2}{\rm
d}\mu_{0}^{2}$ $\displaystyle+h_{1}\,{\rm d}\mu_{1}^{2}+h_{2}\,{\rm
d}\mu_{2}^{2}$ (3.3) $\displaystyle+\mu_{1}^{2}\,h_{1}\,{\rm
D}\phi_{1}^{2}+\mu_{2}^{2}\,h_{2}\,{\rm D}\phi_{2}^{2}\Big{)}\Bigg{]}$

where we introduced the gauged one-forms

${\rm D}\phi_{i}={\rm d}\phi_{i}-A^{(i)}\,,$ (3.4)

and the coordinates $\phi_{i}$ are both $2\pi$-periodic. This metric is
supported by the four-form flux $G_{4}={\rm d}C_{3}$ with three-form
potential,

$\displaystyle L^{-3}C_{3}=$ $\displaystyle-\frac{\Psi}{2w\,\Upsilon_{H}}{\rm
D}\phi_{1}\wedge{\rm D}\phi_{2}\wedge{\rm
d}\mu_{0}-\frac{\mu_{0}\mu_{1}^{2}\mu_{2}^{2}}{(\mu_{1}^{2}+\mu_{2}^{2})}\frac{1}{w\,\Upsilon_{H}}{\rm
D}\phi_{1}\wedge{\rm D}\phi_{2}\wedge{\rm
d}\left(\log\frac{\mu_{1}}{\mu_{2}}\right)$
$\displaystyle-\mu_{0}\,\sqrt{wH}\left(\frac{q_{1}}{h_{1}^{2}}{\rm
d}\phi_{2}+\frac{q_{2}}{h_{2}^{2}}{\rm d}\phi_{1}\right)\wedge{\rm
vol}_{\hskip 1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}\,.$

These solutions are a family of $\frac{1}{4}$-BPS solutions of eleven-
dimensional supergravity and therefore should fall into the general
classification of [79]. Although conceptually straightforward, finding an
explicit map to the canonical GMSW form is very tedious. It remains a fruitful
task however since the many of the computations simplify in the classification
form. The transformation to canonical coordinates is relegated to Appendix B
to shield the reader from some eyesore.

### 3.1 Regularity in eleven dimensions

As discussed above, the seven-dimensional solution contains two conical
singularities, one at each pole of the spindle. After uplifting the solution,
let us investigate the fate of these singularities in eleven dimensions. The
eleven-dimensional metric takes the form

${\rm d}s_{11}^{2}=\mathrm{e}^{2\lambda}\left({\rm d}s_{\mathop{\rm
AdS}\nolimits_{5}}^{2}+{\rm d}s_{\mathcal{M}_{6}}^{2}\right)\,,$ (3.5)

where we can regard the six-dimensional internal space $\mathcal{M}_{6}$ as a
U$(1)^{3}$ fibration over a prism with coordinates $(w,\mu_{1},\mu_{2})$,
represented graphically in Figure 2.

$\mu_{1}^{2}$$\mu_{2}^{2}$$w$ Figure 2: A schematic representation of the
prism over which the three circles are fibred. In the interior of the prism,
all fibres remain of finite size. However, along the faces a variety of cycles
shrink as indicated on the figure. The red dots in the corners indicate the
presence of $\mathbf{R}^{6}/\mathbf{Z}_{n_{\pm}}$ singularities.

Along the diagonal face where $\mu_{0}=0$, or equivalently
$\mu_{1}^{2}+\mu^{2}_{2}=1$, and away from the faces boundary no circle
shrinks. Instead, along this face for generic $w$, the internal space looks
like $\mathbb{R}\times
S^{3}\ltimes{\includegraphics[height=8.0pt]{mySpindle.png}}$ where the $S^{3}$
is located at the equator of the $S^{4}$. This is nothing but the usual
degeneration of an $S^{4}$, written as an $S^{3}$-fibration over an interval.
The $S^{3}$ shrinks at $\mu_{1}^{2}=\mu_{2}^{2}=0$, or equivalently at
$\mu_{0}=\pm 1$. Note that this diagram only represents half of the physical
internal space, since we have $\mu_{0}\in[-1,1]$ and the diagram only
represents the positive values of $\mu_{0}$. The negative values can be
obtained by reflecting the diagram along the diagonal face. Note that this
means that also the singularities are doubled. Along the face where
$\mu_{i}=0$, the $S^{1}_{\phi_{i}}$ circle shrinks, these are just the poles
of the $S^{3}$. Both circles shrink smoothly, provided they are
$2\pi$-periodic.

The most delicate points to analyse are the simultaneous limits
$\mu_{1},\mu_{2}\rightarrow 0$ and $w\rightarrow w_{\pm}$, in figure 2 these
are the red dots.999The analysis for $\mu_{0}=\pm 1$ is identical, so below we
do not specify which one we pick. To properly do so we should be careful with
our choice of gauge and in particular keep track of large gauge
transformations. The appropriate choice for the gauge fields around the
singular points is given by (2.10) so for the analysis in this section we
substitute this expression in (3.4). To continue our analysis in the
neighbourhood of the singular points, let us redefine

$\sin\xi=\frac{2\,r\,\mathcal{C}\,w_{\pm}\cos\chi}{\sqrt{\cos^{2}\theta\,h_{1}(w_{\pm})+\sin^{2}\theta\,h_{2}(w_{\pm})}}\,,\quad
w=w_{\pm}+r^{2}\frac{\mathcal{C}^{2}\,f^{\prime}(w_{\pm})\,\sin^{2}\chi}{4w_{\pm}}\,,$
(3.6)

where $\mathcal{C}$ is an arbitrary positive constant. Notice we have
$f^{\prime}(w_{+})<0$ and $f^{\prime}(w_{-})>0$ so the change of coordinates
is well defined at both endpoints. Taking the $r\rightarrow 0$ limit (and
performing the correct large gauge transformations) we find that the metric
takes the form:

${\rm d}s^{2}_{\mathcal{M}_{6}}={\rm d}\mu_{1}^{2}+\mu_{1}^{2}\left({\rm
d}\phi_{1}+\frac{m^{(1)}_{\pm}}{n_{\pm}}{\rm d}z\right)^{2}+{\rm
d}\mu_{2}^{2}+\mu_{2}^{2}\left({\rm
d}\phi_{2}+\frac{m^{(2)}_{\pm}}{n_{\pm}}{\rm d}z\right)^{2}+{\rm
d}\mu_{3}^{2}+\frac{\mu_{3}^{2}}{n_{\pm}^{2}}{\rm d}z^{2}\,,$ (3.7)

where

$\begin{split}\mu_{1}&=\frac{r\cos\chi\sqrt{h_{1}(w_{\pm})}\cos\theta}{\sqrt{h_{1}(w_{\pm})\cos^{2}\theta+h_{2}(w_{\pm})\sin^{2}\theta}}\,,\quad\mu_{2}=\frac{r\cos\chi\sqrt{h_{2}(w_{\pm})}\sin\theta}{\sqrt{h_{1}(w_{\pm})\cos^{2}\theta+h_{2}(w_{\pm})\sin^{2}\theta}}\,,\\\
\mu_{3}&=r\sin\chi\,.\end{split}$ (3.8)

Staring at this expression for a while, one notices that this is nothing but
$\mathbf{R}^{6}/\mathbf{Z}_{n_{\pm}}$. In other words, we find that the
$\mathbf{R}^{2}/\mathbf{Z}_{n_{\pm}}$ orbifold singularities of the seven-
dimensional solution are lifted to $\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}$
singularities in eleven dimensions at the poles of the $S^{4}$, [67]. More
precisely, the action on the complex coordinates,
$(z_{i}=\mu_{i}\mathrm{e}^{{\rm i}\psi_{i}})$ is given as follows,

$\bigg{(}z_{1}\,,\,z_{2}\,,\,z_{3}\bigg{)}\rightarrow\left(\mathrm{e}^{2\pi{\rm
i}\frac{m^{(1)}_{\pm}}{n_{\pm}}}z_{1}\,,\,\mathrm{e}^{2\pi{\rm
i}\frac{m^{(2)}_{\pm}}{n_{\pm}}}z_{2}\,,\,\mathrm{e}^{2\pi{\rm
i}\frac{1}{n_{\pm}}}z_{3}\right)\,.$ (3.9)

Later we will need to know the weights of the orbifold action with respect to
the R-symmetry vector. From the orbifold action (3.9), it is simple to read
off the weights to be:

$\epsilon^{(i)}_{\pm}=\frac{1}{n_{\pm}}\left(m_{\pm}^{(1)},m_{\pm}^{(2)},1\right)\,.$
(3.10)

An alternative method to see that these are indeed the weights is to compute
the anti-symmetric matrix: $\tfrac{1}{2}({\rm d}\xi^{\\#})_{\mu\nu}$ where
$\xi^{\\#}$ is the one-form dual to the R-symmetry vector $\xi$. By writing
this in canonical anti-symmetric matrix form one can read off the weights as
the off-diagonal entries. Computing this matrix at the fixed points we find:

$\frac{1}{2}{\rm
d}\xi^{\\#}_{\mu\nu}=\frac{1}{n_{\pm}}\begin{pmatrix}0&m_{\pm}^{(1)}&0&0&0&0\\\
-m_{\pm}^{(1)}&0&0&0&0&0\\\ 0&0&0&m_{\pm}^{(2)}&0&0\\\
0&0&-m_{\pm}^{(2)}&0&0&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&-1&0\end{pmatrix}_{\mu\nu}\,,$ (3.11)

which indeed gives the same weights as in (3.10).

Summarising, we find that locally the eleven-dimensional metric takes the form

${\rm d}s_{11}^{2}={\rm d}s^{2}_{\mathop{\rm AdS}\nolimits_{5}}+{\rm
d}s_{\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}}^{2}\,.$ (3.12)

with the orbifold action given in (3.9). This is in contrast to the
$\mathcal{N}=2$ preserving class $\mathcal{S}$ punctures that locally are of
the form $\mathop{\rm AdS}\nolimits_{5}\times
S^{2}\times\mathbf{C}^{2}/\mathbf{Z}_{n}$. Hence, we find that the conical
defects of the spindle solution realise genuine locally $\mathcal{N}=1$
preserving punctures! Upon setting one of the parameters $m^{(i)}$ to zero, we
recover the more familiar locally $\mathcal{N}=2$ preserving punctures. In
this case one complex plane remains untouched by the orbifold action and the
corresponding $\mathrm{U}(1)$ is enhanced to the $\mathrm{SU}(2)$ R-symmetry.
Indeed, we see that in this case the two-dimensional orbifold acts as
$\frac{1}{n}(-1,1)$, and our punctures reduce to the well-known type $A$ du
Val singularities. I.e. in this case the local singularities are classified by
(collections of) finite subgroups of $\mathrm{SU}(2)$ while in the
$\mathcal{N}=1$ case they are classified by finite subgroups of
$\mathrm{SU}(3)$. The latter classification [80] is much richer and our
spindle solutions only realise a small subset of them given by the abelian
orbifolds of the form $\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}$. It is an
outstanding problem to find more general solutions realising punctures
corresponding to the remaining finite subgroups of $\mathrm{SU}(3)$. We hope
to come back to this problem in the near future.

### 3.2 Flux Quantisation

In order for the backgrounds described above to be consistent backgrounds of
M-theory, we impose that the $G_{4}$ flux is quantised along any four-cycle in
$\mathcal{M}_{4}$.

The first non-trivial cycle we consider, which we denote by $\mathcal{C}_{4}$,
has the topology of $S^{4}$ and can be obtained as the $S^{4}$ fibre over a
generic point on the spindle. This four-cycle can be obtained by taking a
slice of the prism perpendicular to the $w$-axis in Figure 2 and integrating
along the both ${\rm d}\phi_{i}$ circles. Using the expression for the four-
form flux above, we have the following quantisation condition

$\frac{1}{(2\pi\ell_{p})^{3}}\int_{\mathcal{C}_{4}}G_{4}=N\in\mathbf{Z}$
(3.13)

This flux can be interpreted as the number of M5-branes wrapping the spindle.
As a consequence, we find the length scale $L$ of the uplifted solution is
quantised as

$L=(\pi N)^{\frac{1}{3}}\ell_{p}\,.$ (3.14)

Next, consider the four-cycles $\mathcal{D}_{4}^{\pm}$ obtained by going to
one of the orbifold points at the poles of the spindle. The fibres over these
points take the form $S^{4}/\mathbf{Z}_{n_{\pm}}$. Similar to the above we can
consider the flux quantisation through the fibres,

$\frac{1}{(2\pi\ell_{p})^{3}}\int_{\mathcal{D}_{4}^{\pm}}G_{4}=\frac{N}{n_{\pm}}\,.$
(3.15)

The cycles $\mathcal{D}_{4}^{\pm}$ are not independent from the cycle
$\mathcal{C}_{4}$ but instead we have the homology relations

$\left[\mathcal{C}_{4}\right]=n_{\pm}\left[\mathcal{D}_{4}^{\pm}\right]\,.$
(3.16)

This agrees with the analysis of [67] using equivariant localisation.

## 4 Wrapped M2- and M5-branes

Probe M5- and M2-branes wrapping cycles of the internal manifold give rise to
an important set of protected data in the dual field theory. Probe BPS
M2-branes wrapping calibrated two-cycles correspond to chiral primaries in the
dual SCFTs [81], while the probe BPS M5-branes can be understood as
parametrising a class of supersymmetric deformations of the four-dimensional
$\mathcal{N}=1$ SCFT. In this section we study the possible embeddings of BPS
probe branes within the spindle solution. We will find that there is a richer
structure of probe branes that may be embedded in comparison with the BBBW
solutions [23] as studied in [12]. In particular, on top of the BPS M5-branes
which deform the theory by changing/adding a puncture, as in [12], there are
additional M5 branes wrapping the isometry circle of the spindle. These are
located away from the conical defects and provide a new set of codimension-1
operators in the 6d $(2,0)$ which deform the resulting four-dimensional
theory. One can attribute the presence of these additional probe M5-branes to
a larger moduli space for the probe M5-branes in the setup and therefore a
richer structure in the dual SCFT.

The analysis of [12] was performed by embedding their setup into the GMSW
framework, and exemplified with the BBBW [23] and $\mathcal{N}=1$
Maldacena–Nunez solutions [27]. After embedding our solution into this general
framework, which is detailed in Appendix B, the analysis can be repeated
verbatim. However, due to the richer setup our results include a larger set of
(new) probe branes.

### 4.1 Probe M5 branes

In order for a probe M5-brane to preserve the symmetries of our solutions it
is forced to wrap all of AdS5 together with a one-cycle in the internal space.
Let $\\{\sigma^{i}\\}$ with $i=0,\dots,4$ be the coordinates in AdS5 and let
$\tau=\sigma^{5}$ be the proper length parameter along the one-cycle within
$M_{6}$ wrapped by the probe M5-brane. The calibration condition for a probe
M5-brane fully wrapping AdS5, written using the canonical $\mathcal{N}=1$ form
(see appendix B), is

$\mathrm{e}^{-3\lambda}{\rm d}s_{4}\Big{|}_{{\rm
d}\tau}=0\,,\qquad\frac{\mathrm{e}^{-3\lambda}}{\cos\zeta}{\rm
d}y\Big{|}_{{\rm d}\tau}=0\,,\qquad\frac{\sin\zeta\cos\zeta}{3}({\rm
d}\psi+P)\Big{|}_{{\rm d}\tau}=0\,,$ (4.1)

where ${\rm d}s^{2}_{4}$ is the ($y$-dependent) Kähler metric at a fixed value
of $y$. The component $g_{\tau\tau}$ of the induced metric is

$g_{\tau\tau}=\mathrm{e}^{-6\lambda}\left[{\rm d}s_{4}\Big{|}_{{\rm
d}\tau}\right]^{2}+\frac{\mathrm{e}^{-6\lambda}}{\cos^{2}\zeta}\left[{\rm
d}y|_{{\rm d}\tau}\right]^{2}+\frac{\cos^{2}\zeta}{9}\left[({\rm
d}\psi+P)\Big{|}_{{\rm d}\tau}\right]^{2}\,,$ (4.2)

which we require to be non-zero. The only option, giving a non-trivial induced
metric $g_{\tau\tau}$, and satisfying the calibration condition is to set
$\sin\zeta=0$, which is equivalent to $y=0$. This immediately satisfies the
second condition in (4.1) since we are localised to a fixed value of $y$ as
well as the third one. It remains to satisfy the first condition in (4.1).
From the embedding of the spindle solution in the GMSW classification we have
that the coordinate $y$ is given by $y=w\cos\xi$ and therefore we have that
$y=0$ if and only if $\cos\xi=0$. This then places us at the equator of the
$S^{4}$ which has the topology of a three-sphere. The location of the probe
M5-branes is then at a point on the diagonal face of figure 2. For ease of
reference let us denote this face by $\mathcal{F}_{2}$.

As the metric on $M_{6}$ is toric, it admits a $\mathrm{U}(1)^{3}$ action
which is fibered over the prism in figure 2. The M5-brane will wrap a one-
cycle embedded in this torus and we must require that the one-cycle closes,
i.e. that it is not dense within the torus. At generic points on the face
$\mathcal{F}_{2}$ the M5-brane will then break a
$\mathrm{U}(1)^{2}\subset\mathrm{U}(1)^{3}$. The preserved symmetry is
enhanced along the boundary of $\mathcal{F}_{2}$ where an additional cycle
shrinks and therefore this symmetry is not broken by the presence of the probe
brane.

Once we have restricted to the equator of the four-sphere, it remains to
satisfy the first of calibration condition, (4.1). It is most convenient to
work in terms of a vielbein on ${\rm d}s^{2}_{4}$, which we derive in appendix
B in terms of the canonical GMSW form. These vielbein are adapted to give the
Kähler form on the base in the canonical way. The pertinent vielbein are the
ones involving the angular coordinates on the four-dimensional base, which
evaluated at $y=0$ and along the curve, are given by

$\begin{split}e^{2}\Big{|}_{y=0}&=\frac{\cos\theta\sin\theta}{\sqrt{\Theta(w,\theta)}}\left(h_{1}(w)\frac{{\rm
d}\phi_{1}}{{\rm d}\tau}-h_{2}(w)\frac{{\rm d}\phi_{2}}{{\rm
d}\tau}\right){\rm d}\tau\,,\\\
e^{4}\Big{|}_{y=0}&=-\frac{\sqrt{f(w)}}{2\sqrt{\Theta(w,\theta)}}\left(\cos^{2}\theta\frac{{\rm
d}\phi_{1}}{{\rm d}\tau}+\sin^{2}\theta\frac{{\rm d}\phi_{2}}{{\rm
d}\tau}\right){\rm d}\tau\,.\end{split}$ (4.3)

To solve (4.1) these must simultaneously vanish.

#### Generic point in $\mathcal{F}_{2}$

At a generic point $(w_{*},\theta_{*})\in\mathcal{F}_{2}$ we must impose the
two conditions

$\frac{{\rm d}\phi_{1}}{{\rm
d}\tau}=\frac{h_{2}(w_{*})}{h_{1}(w_{*})}\frac{{\rm d}\phi_{2}}{{\rm
d}\tau}\,,\qquad\frac{{\rm d}\phi_{1}}{{\rm
d}\tau}=-\tan^{2}\theta_{*}\frac{{\rm d}\phi_{2}}{{\rm d}\tau}\,.$ (4.4)

Since $h_{i}(w_{*})>0$ the only possibility is

$\frac{{\rm d}\phi_{1}}{{\rm d}\tau}=\frac{{\rm d}\phi_{2}}{{\rm d}\tau}=0\,.$
(4.5)

Hence, the probe M5-brane is located at a generic point along the $\phi_{i}$
circles. That is, at a generic point on the $S^{3}$ along the equator of the
$S^{4}$. The cycle wrapped by the probe brane is then simply the isometry
circle of the spindle,

$\text{M5}_{\text{gen}}:\leavevmode\nobreak\ \leavevmode\nobreak\
\text{AdS}_{5}\times S^{1}_{\text{spindle}}\times\\{\text{point}\in
S^{3}_{\text{equator}}\\}$ (4.6)

Since the brane is localised at points along the $\phi_{i}$ circles it follows
that the $\mathrm{U}(1)^{3}$ symmetry is broken to $\mathrm{U}(1)_{R}$ by the
probe M5-brane, with the remaining isometry dual to the R-symmetry.

#### North and south pole of $S^{3}_{\text{equator}}$

A more interesting case arises when the probe brane approaches the boundaries
of this domain, i.e. when it is located at a pole of the three-sphere.
Consider first the north pole of the $S^{3}_{\text{equator}}$ at $\mu_{1}=0$,
where the circle with coordinate $\phi_{1}$ shrinks. Note that we stay away
from $w=w_{\pm}$. In this case, the calibration conditions reduces to

$\frac{{\rm d}\phi_{2}}{{\rm d}\tau}=0\,,$ (4.7)

such that the probe M5-brane is localised at a point along the $\phi_{2}$
circle. This is just the non-shrinking circle $S^{1}_{N}$ of the north pole of
$S^{3}_{\text{equator}}$. We see that compared to the generic case there is an
enhancement of the symmetry preserved by the probe brane to
$\mathrm{U}(1)^{2}$, with the two $\mathrm{U}(1)$’s being the R-symmetry and a
flavour symmetry. The boundary at $\mu_{2}=0$ works similarly, this time with
the M5-brane localised at point along the $\phi_{1}$ circle, that is
$S^{1}_{S}$. We denote these M5-brane probes by

$\text{M}5_{w}^{N,S}=\text{AdS}_{5}\times
S^{1}_{\text{spindle}}\times\\{w\in(w_{-},w_{+})\\}\times\\{S^{3}_{N,S},\leavevmode\nobreak\
\text{point}\in S^{1}_{N,S}\\}\,.$ (4.8)

Both the $\text{M5}_{\text{gen}}$ and $\text{M}5_{w}^{N,S}$ probe M5-branes
wrap the circle of the spindle and are located at points of the equator of the
$S^{4}$. These probe M5-branes are new in comparison with the possible probe
M5-branes possible in the compactification on a Riemann surface and is an
artefact of the need to twist the R-symmetry with the isometry of the spindle.
Finally observe that there is no obstruction to moving a
$\text{M5}_{\text{gen}}$ to $\text{M}5_{w}^{N,S}$ or vice-versa.

#### Spindle fixed points

The remaining possibility is to put the probe M5-branes at the poles of the
spindle $w=w_{\pm}$. These are even more interesting and we will treat them
concurrently. At these points the $\mathrm{U}(1)$ isometry circle of the
spindle shrinks and is therefore not wrapped by the probe M5-branes. Away from
the poles of the $S^{3}_{\text{equator}}$ the embedding of the probe brane
must satisfy

$\frac{{\rm d}\phi_{1}}{{\rm
d}\tau}=\frac{h_{2}(w_{\pm})}{h_{1}(w_{\pm})}\frac{{\rm d}\phi_{2}}{{\rm
d}\tau}\,.$ (4.9)

In order for the one-cycle not to be dense, which would render it
inappropriate for the probe M5-brane to wrap, we must require that

$\frac{h_{2}(w_{\pm})}{h_{1}(w_{\pm})}\equiv\frac{\alpha_{2,\pm}}{\alpha_{1,\pm}}\in\mathbf{Q}\,,\qquad\text{where}\quad\alpha_{i,\pm}\in\mathbf{Z}\,.$
(4.10)

At the poles $w=w_{\pm}$ we have the embedding,

$\phi_{1}(\tau)=\phi_{1,*}+\alpha_{2,\pm}\tau\,,\quad\phi_{2}(\tau)=\phi_{2,*}+\alpha_{1,\pm}\tau\,,$
(4.11)

where $\phi_{i,*}$ are the initial point of the one-cycle inside
$S^{3}_{\text{equator}}$. These M5-branes therefore wrap

$\text{M}5_{\pm}^{\vec{\alpha}_{\pm}}:\mathop{\rm
AdS}\nolimits_{5}\times\left\\{S^{1}_{R}(\vec{\alpha}_{\pm})\subset
S^{3}_{\text{equator}}\right\\}\times\left\\{w=w_{\pm}\right\\}\,.$ (4.12)

For generic solutions to the calibration condition, the parameters
$\alpha_{i,\pm}$ in (4.10) are not integer and additional constraints on the
spindle parameters need to be imposed. Following [32] we may solve for the
roots and the parameters $q_{i}$ in terms of the integer magnetic charges and
orbifold weights. We have101010Note that our naming of the orbifold orders is
interchanged with that in [32].

$\begin{split}w_{\pm}&=\frac{3p_{1}p_{2}(4n_{\mp}-n_{\pm}+s)(s+p_{1}+p_{2})}{2(n_{\pm}-p_{1})(n_{\pm}-p_{2})[s+2(p_{1}+p_{2})]^{2}}\,,\\\
q_{1}&=\frac{3p_{1}p_{2}^{2}(5n_{-}-n_{+}+s)(5n_{+}-n_{-}+s)(p_{1}-2p_{2}-s)(p_{1}+p_{2}+s)^{2}}{4(n_{-}-p_{1})^{2}(n_{-}-p_{2})^{2}[s+2(p_{1}+p_{2})]^{4}}\,,\\\
q_{2}&=q_{1}\Big{|}_{p_{1}\leftrightarrow p_{2}}\,,\end{split}$ (4.13)

where

$s=\sqrt{(n_{-}+n_{+})^{2}+12(n_{-}-p_{1})(n_{+}-p_{1})}\,.$ (4.14)

For regularity one should take $n_{+}>n_{-}>0$, fix $p_{2}=n_{+}+n_{-}-p_{1}$
and take either $p_{1}<0$ or $p_{1}>n_{+}+n_{-}$. It then follows that (4.10)
is true if and only if the parameter $s$ in equation (4.14) is rational. This
also implies that such SCFTs have a rational central charge (at leading
order). Note that a similar phenomenon was found for the BBBW solutions in
[12] where the rational SCFTs allowed additional probe M5-branes at a point on
the Riemann surface, wrapping a non-trivial one-cycle on the
$S^{3}_{\text{equator}}$. The rationality condition is equivalent to

$12n_{+}n_{-}=m^{2}-(n_{+}-n_{-})^{2}+12p_{1}(n_{+}+n_{-}-p_{1})\,,$ (4.15)

with $m$ an arbitrary integer. We may solve this by defining

$n_{s}=n_{+}+n_{-}\,,\quad n_{p}=n_{+}n_{-}\,,$ (4.16)

such that

$n_{p}=(p_{1}-n_{s})p_{1}+\frac{m^{2}-n_{s}^{2}}{12}\,.$ (4.17)

In this way we see that we require $(m,n_{s})$ to satisfy the generalised
Pell’s equation:

$m^{2}-n_{s}^{2}=12\delta\,,\qquad\delta\in\mathbf{Z}\,.$ (4.18)

To simplify let

$n_{s}=12k_{1}+a\,,\qquad m=12k_{2}+b\,,\qquad\text{where}\qquad
k_{1},k_{2}\in\mathbf{Z},\qquad a,b\in\mathbf{Z}_{12}\,,$ (4.19)

then (4.18) can be solved by picking $a$ and $b$ from the sets

$\begin{split}(a,b)\in\\{0,6\\}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
(a,b)\in\\{1,5,7,11\\}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
(a,b)\in\\{2,4,8,10\\}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
(a,b)\in\\{3,9\\}\,.\end{split}$ (4.20)

One now needs to invert to solve for $n_{\pm}$. We have not found a closed
form expression for this in general however we have checked that solutions can
be found. Mathematica’s function ‘FindInstance’ almost always finds a solution
for random integer values consistent with the regularity that we have checked.

For rational spindle SCFTs, we may wrap an additional probe M5-brane along a
non-trivial cycle within $S^{3}_{\text{equator}}$, localised at the poles of
the spindle. The wrapped one-cycle depends on a pair of integers
$\alpha_{i,\pm}$ which exist for all rational SCFTs. The symmetry preserved in
this case is $\mathrm{U}(1)^{2}$, with one $\mathrm{U}(1)$ the R-symmetry and
the other a flavour symmetry.

#### Corners of $\mathcal{F}_{2}$

The final set of points on $\mathcal{F}_{2}$ that we need to discuss are the
corners. For concreteness, let us consider the corner corresponding to the
north pole of the $S^{3}_{\text{equator}}$ and the $w=w_{+}$ pole of the
spindle. At this point the calibration condition is immediately solved and the
one-cycle wrapped is given by the $S^{1}_{N}$ of the north pole. The
generalisation to the other corners is obvious hence these probe M5-branes
wrap

$\text{M5}_{\pm}^{N,S}:\text{AdS}_{5}\times
S^{1}_{N,S}\times({\includegraphics[height=8.0pt]{mySpindle.png}}^{\pm})\times(S^{3}_{\text{equator}})^{N,S}\,.$
(4.21)

These M5-branes preserve the full $\mathrm{U}(1)^{3}$ symmetry.

Consider now moving one of the $\text{M5}^{\pm}_{\alpha_{\pm}}$ towards a
neighbouring corner. For concreteness let us take the $w=w_{+}$ pole of the
spindle and move the point on the three-sphere to the north pole of
$S^{3}_{\text{equator}}$. Since the circle with coordinate $\phi_{1}$ shrinks
at this point, the M5-branes wraps the one-cycle $S^{1}_{N}$ $\alpha_{1,+}$
times. Equivalently, there are $\alpha_{1,+}$ probe M5-branes wrapping
$S^{1}_{N}$. Conversely if we have an $\text{M5}_{\pm}^{N,S}$ probe brane and
try to move it away from the corner we cannot unless we have $\alpha_{1,+}$ of
them. Doing so the $\alpha_{1,+}$ probe branes at the corner become a single
$\text{M5}^{N}_{\vec{\alpha}_{+}}$ probe brane in the away from the corner.
This is a consequence of the corner being a
$\mathbf{R}^{4}/\mathbf{Z}_{\alpha_{1,+}}$ orbifold at which we can place so-
called fractional M5-branes as we will discuss in more detail later. At the
other corners of $\mathcal{F}_{2}$ there is an analogous behaviour. This is
very similar to the BBBW solutions, [12]. There the probe M5-branes in our
language are, $\text{M5}_{-}^{N,S}$ and $\text{M5}_{-}^{\vec{\alpha}_{-}}$. We
see that the spindle leads to a doubling of these probe M5-branes with
different weights $\vec{\alpha}_{\pm}$ at the two poles of the spindle.
Finally, note that we cannot move a probe M5-brane from either of the
$w=w_{\pm}$ boundaries into the bulk to a different point on the spindle and
vice versa. This will lead to a moduli space which is not connected.

To summarise, we have found that there are two distinct types of probe
M5-branes. One class wraps the spindle circle and the other is located at the
poles of the spindle and wraps a one-cycle inside the equator of $S^{4}$. For
the M5-branes at the poles of the spindle there are additional probe M5-branes
provided that the dual SCFT is rational, which we have shown exist. These
additional M5-branes can be moved to the corners of $\mathcal{F}_{2}$ and lead
to stacks of (fractional) M5-branes localised at the corners of
$\mathcal{F}_{2}$.

### 4.2 Moduli space of probe M5-branes

Having studied the possible embeddings of probe M5-branes we turn our
attention to better understanding the moduli space of the probe M5-branes. We
mainly focus on the cases satisfying the rationality condition, since these
present the richest set of probe M5-branes. When this condition is not
satisfied part of the moduli space, coloured in blue in Figure 3, does not
exist.

$\mathcal{F}_{2}$$\text{M5}_{\rm
gen}$$\text{M5}_{w}^{N}$$\text{M5}_{w}^{S}$$\text{M5}_{-}^{\vec{\alpha}_{-}}$$\text{M5}_{+}^{\vec{\alpha}_{+}}$$\text{M5}_{+}^{N}$$\text{M5}_{-}^{N}$$\text{M5}_{+}^{S}$$\text{M5}_{-}^{S}$$\alpha_{1,\pm}$$\alpha_{2,\pm}$$w_{-}$$w_{+}$
Figure 3: The moduli space $\mathcal{F}_{2}$ of probe M5-branes in out setup.
The blue line and four black dots are disconnected from the yellow region and
green lines. Moreover, the blue lines are not accessible when the large $N$
central charge of the SCFT is not rational. When the theory is rational the
blue line and the two connected dots are topologically a spindle with weights
given by $\vec{\alpha}_{\pm}$.

Since the probe M5-branes are calibrated by the R-symmetry direction at $y=0$
the metric on the moduli space is obtained by restricting the metric on
$\mathcal{M}_{4}$ to $\mathcal{F}_{2}$. We take the $\alpha_{i,\pm}$ as
defined in (4.10) and take them to be pairwise relatively prime, i.e.
$\mathrm{gcd}(\alpha_{1,\pm},\alpha_{2,\pm})=1$. More explicitly, they are
given in terms of the $n_{\pm}$ and $p_{i}$ by

$\alpha_{1,\pm}=p_{2}\big{(}n_{-}+n_{+}+6(n_{\pm}-p_{2})-s\big{)}\,,\quad\alpha_{2,\pm}=p_{1}\big{(}n_{-}+n_{+}+6(n_{\pm}-p_{1})-s\big{)}\,,$
(4.22)

and satisfy

$\alpha_{1+}=6p_{2}(n_{+}-n_{-})+\alpha_{1-}\,,\quad\alpha_{2+}=6p_{1}(n_{+}-n_{-})+\alpha_{2-}\,.$
(4.23)

Pulling back the metric on $\mathcal{M}_{4}$ to the diagonal face
$\mathcal{F}_{2}$ in 2 we find

$\begin{split}{\rm d}s^{2}_{4}&=\Theta(w,\theta){\rm
d}\theta^{2}+\frac{\cos^{2}\theta\sin^{2}\theta}{\Theta(w,\theta)}(h_{1}(w){\rm
d}\phi_{1}-h_{2}(w){\rm d}\phi_{2})^{2}\\\
&+\frac{w^{2}\Theta(w,\theta)}{f(w)}{\rm
d}w^{2}+\frac{f(w)}{\Theta(w,\theta)}(\cos^{2}\theta{\rm
d}\phi_{1}+\sin^{2}\theta{\rm d}\phi_{2})^{2}\,.\end{split}$ (4.24)

This metric is necessarily complex and Kähler. Moreover, it describes a four-
dimensional toric orbifold. One should contrast this with the BBBW case where
the four-dimensional moduli space is a spindle fibered over a Riemann surface
[12].

To better understand the moduli space let us construct the degenerating
Killing vectors on the boundary of $\mathcal{F}_{2}$. For
$\mu_{1}=0,\mu_{2}=0$ we see that the degenerating Killing vectors, normalised
to have unit surface gravity, are respectively

$l_{1}=\partial_{\phi_{1}}\,,\quad l_{2}=\partial_{\phi_{2}}\,.$ (4.25)

The metric degenerates smoothly provided the $\phi_{i}$ have period $2\pi$
which is indeed the period inherited from the four-sphere origin. Since they
are $2\pi$-periodic we will use $\partial_{\phi_{i}}$ as a basis for the
$\mathrm{U}(1)^{2}$ toric action. In this basis we have the toric vectors:

$v_{1}=(1,0)\,,\quad v_{3}=(0,1)\,.$ (4.26)

For $w=w_{\pm}$ we see that the degenerating Killing vectors are

$k_{\pm}=\frac{1}{6n_{\mp}(n_{-}-n_{+})}\left(\alpha_{2,\pm}\partial_{\phi_{1}}+\alpha_{1,\pm}\partial_{\phi_{2}}\right)\,,$
(4.27)

where the normalisation is fixed so that the surface gravity is 1. From these
we extract out the effective toric vectors:

$v_{2}=\frac{1}{6n_{-}(n_{-}-n_{+})}(\alpha_{2,+},\alpha_{1,+})\,,\quad
v_{4}=-\frac{1}{6n_{+}(n_{-}-n_{+})}(\alpha_{2,-},\alpha_{1,-})\,.$ (4.28)

The non-integral nature of these vectors indicates the orbifolded nature of
the degeneration. One can in principle perform a rotation of the basis to
obtain an integral set of vectors [82] however we will not do this
here.111111Note that we have a genuine symplectic structure on our four-
dimensional orbifold which is not the case in [82].

It follows that at the corners of $\mathcal{F}_{2}$ the degeneration is a
cover of $\mathbf{C}^{2}/\mathbf{Z}_{\alpha_{i,\pm}}$. For example the metric
at the corner $w=w_{+}$, $\mu_{1}=0$ corner is:

${\rm d}s^{2}_{4}={\rm d}r^{2}+r^{2}\left[{\rm
d}\xi^{2}+\frac{\cos^{2}\xi}{\alpha_{1,+}^{2}}\left(\alpha_{1,+}{\rm
d}\phi_{1}-\alpha_{2,+}{\rm
d}\phi_{2}\right)^{2}+\frac{36n_{-}^{2}(n_{-}-n_{+})^{2}}{\alpha_{1,+}^{2}}\sin^{2}\xi{\rm
d}\phi_{2}^{2}\right]\,,$ (4.29)

with analogous degeneration at the other corners. The
$\mathbb{Z}_{\alpha_{i,\pm}}$, action is non-trivial and generate a finite
subgroup of $\mathrm{U}(2)$ which is not contained in $\mathrm{SU}(2)$. To see
this one can compute the weights of the Killing vector
$K=\partial_{\phi_{1}}+\partial_{\phi_{2}}$ at the fixed point at $r=0$,
resulting in

$dK|_{\mathbf{C}^{2}/\mathbf{Z}_{\alpha_{i,\pm}}}=\frac{1}{\alpha_{i,\pm}}\begin{pmatrix}0&-6n_{\mp}(n_{-}-n_{+})&0&0\\\
6n_{\mp}(n_{-}-n_{+})&0&0&0\\\ 0&0&0&\alpha_{i+1,\pm}-\alpha_{i,\pm}\\\
0&0&-\alpha_{i+1,\pm}+\alpha_{i,\pm}&0\end{pmatrix}$ (4.30)

from which the weights can immediately be extracted.

To conclude, we find that the moduli space of the probe M5-branes is a four-
dimensional toric Kähler orbifold of quadrilateral type. At the corners of the
toric polytope the degeneration is
$\mathbf{C}^{2}/\mathbf{Z}_{\alpha_{i,\pm}}$. Using the results of appendix C,
and a lot of care, one can take a limit to the moduli space of the BBBW
solutions as computed in [12].

### 4.3 M2 branes ending on M5 branes

The presence of the probe M5-branes allows for M2-branes to end on them.
Chiral BPS operators in the dual SCFT correspond to M2-branes moving along a
geodesic in AdS5 and wrapping a calibrated curve in the internal space. The
calibration two-form is [81]

$Y^{\prime}=-\sin\zeta\mathrm{e}^{-6\lambda}J+K_{1}\wedge K_{2}\,,$ (4.31)

and the calibration condition for the cycle $\mathcal{M}_{2}$ becomes

${\rm vol}(\mathcal{M}_{2})=Y^{\prime}\Big{|}_{\mathcal{M}_{2}}\,,$ (4.32)

where ${\rm vol}(\mathcal{M}_{2})$ is the volume element on $\mathcal{M}_{2}$
pulled back from $\mathcal{M}_{6}$. From the form of the metric and
calibration condition we see that the calibrated cycles a priori can fall into
two classes. They either wrap the $\mathrm{U}(1)_{R}$ direction or they do not
and are calibrated by $J$. Given a calibrated M2-brane, wrapping the cycle
$\mathcal{M}_{2}$ we may compute the conformal dimension of the dual BPS
operator via [81]

$\Delta(\mathcal{M}_{2})=\frac{N}{4\pi}\int_{\mathcal{M}_{2}}\mathrm{e}^{3\lambda}Y^{\prime}\,,$
(4.33)

where we have used (3.14) and the definition of $Y^{\prime}$ is given in
(B.16).

Consider first the embedding of an M2-brane which does not wrap the R-symmetry
direction. Then the calibration condition requires $\sin\zeta=\pm 1$, i.e. we
are at restricted to the fixed points of the $\mathrm{U}(1)_{R}$ isometry.
Since these are genuine fixed points of $\mathcal{M}_{6}$ and the entire
metric shrinks there are no non-shrinking two-cycles which the probe M2-brane
can wrap. This differs with the BBBW solutions where one could wrap the
Riemann surface and is a consequence of having a genuine $\mathcal{N}=1$
puncture rather than local $\mathcal{N}=2$ punctures.

Having concluded that the probe M2-brane must wrap the $\mathrm{U}(1)_{R}$
R-symmetry directions let us consider their embedding in more detail. There
are two options. The first consists of the M2-brane world volume wrapping the
spindle located at one of the poles of the $S^{4}$. The second option consists
of wrapping a 2-cycle with the topology of a disc, stretching from one of the
fixed points to one of the probe M5-branes at $y=0$.

The first type probe M2-brane was studied in [67] and we recover their result.
Let us parametrise the surface with the coordinates $(\tau,\sigma)$ and adapt
$\tau$ so that it parametrises the $\mathrm{U}(1)_{R}$ only. Then we have that
the brane wraps the line $(w,\xi,\theta)(\sigma)$, with end-points at
$\cos\zeta=0$. In order for the calibration condition to hold we need to make
the contribution from $J$ to $Y^{\prime}$ vanish. Clearly evaluating the
calibration form at the poles of the $S^{4}$ satisfies this demand. We denote
these cycles by $\Sigma_{N,S}$ and find that the conformal dimension of the
dual BPS operators are

$\Delta(\Sigma_{N,S})=\frac{3p_{1}p_{2}}{n_{+}n_{-}(s+2(n_{-}+n_{+}))}\,,$
(4.34)

which agrees with equation (3.58) of [67].

Next, we consider the probe M2-branes which end on a probe M5-brane. First we
consider the M2-branes which end on a $\text{M5}_{\pm}^{N,S}$, see (4.21). In
this case, the BPS condition implies that

$\sin^{4}\xi(\sigma)=C\prod_{I=1}^{4}(w(\sigma)-w_{I})^{-\tfrac{w_{I}h_{1}(w_{I})}{2f^{\prime}(w_{I})}}$
(4.35)

with the $w_{I}$ being the four roots of $f(w)$ and the constant $C$ is fixed
so that at the location of the M5-brane we have $\sin^{2}\xi=1$. The topology
of the M2-brane world volume in the internal space is that of a disc with
boundary the M5-brane. Note that we can take the centre of the disc to be at
either of the four points where $\cos^{2}\zeta=0$. The conformal dimension of
the corresponding operators when we are fixed to the pole of the spindle is
simple to compute. The two-cycle that the M2-brane wraps is then just a
linearly embedded two-sphere inside the $S^{4}$. We denote these cycles by
$\Sigma_{\pm,i}$. Despite this being a trivial cycle in homology we obtain a
non-trivial results since $Y^{\prime}$ is not closed,

$\Delta(\Sigma_{\pm,i})=\frac{3b_{i+1}}{2}N\,,$ (4.36)

where

$b_{i,\pm}=\frac{1}{2}\left(\Phi_{i}-\frac{p_{i}}{n_{+}n_{-}}\epsilon\right)\,,$
(4.37)

and121212These are given in [83] with some minor differences in naming
conventions and choice of magnetic charges. See also [67].

$\begin{split}\epsilon&=-\frac{n_{-}n_{+}(n_{-}-n_{+})(n_{-}+n_{+}+s)}{2(n_{-}-p_{1})(n_{+}-p_{1})(s+2(n_{-}+n_{+}))}\,,\\\
\Phi_{1}&=\frac{p_{1}(n_{-}+n_{+}+s)(s+6p_{1}-4(n_{-}+n_{+}))}{6(n_{-}-p_{1})(n_{+}-p_{1})(s+2(n_{+}+n_{-}))}\,\\\
\Phi_{2}&=2+\frac{\epsilon(n_{+}-n_{-})}{n_{-}n_{+}}-\Phi_{1}\,.\end{split}$
(4.38)

This form is motivated by the localisation results in [67]. Furthermore, one
can see that these reduce correctly to the results for the BBBW case in
equation (3.35) of [12] and (3.45) of [67]. For the analogous M2-branes ending
on the $\text{M5}_{\pm}^{\vec{\alpha}_{\pm}}$ there are additional factors of
$\vec{\alpha}_{\pm}$ to take into account. We were unable to find a closed
form expression for the more general M2-brane, nevertheless we have shown that
there is a rich and different structure of probe M2-branes (and therefore BPS
operators) in the SCFT arising from compactifying M5-branes on a spindle in
comparison to the compactification on a Riemann surface.

## 5 Symmetries and anomalies

After carefully scrutinising the internal geometry and identifying the precise
form of the singularities in the eleven-dimensional geometry, we proceed to
analyse the global symmetries and ’t Hooft anomalies of the dual SCFTs. The
eleven-dimensional geometry admits a $\mathfrak{u}(1)^{3}$ isometry algebra
dual to the superconformal $R$ symmetry and two additional $\mathfrak{u}(1)$
flavour symmetries of the dual SCFT. In addition, the dual SCFT enjoys flavour
symmetries associated to the $\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}$ orbifold
singularities.

The analysis of the symmetries as well as their ’t Hooft anomalies can be
performed systematically, following the anomaly inflow methods developed in
[10, 84, 85, 86, 87], building on the work of [60, 88]. This analysis employs
an auxiliary twelve-manifold $\mathcal{M}_{12}$, realised as an
$\mathcal{M}_{6}$ fibration over a closed six-manifold $\mathcal{N}_{6}$, i.e.
$\mathcal{M}_{6}\hookrightarrow\mathcal{N}_{12}\rightarrow\mathcal{N}_{6}$.
The space $\mathcal{N}_{6}$ is interpreted as a fibration over the external
spacetime on which the anomaly polynomial lives. In particular, we will obtain
the anomaly polynomial of the dual four-dimensional SCFT by integrating the
twelve-form $\mathcal{I}_{12}$ over the internal space $\mathcal{M}_{6}$.

$\mathcal{I}_{6}=\int_{\mathcal{M}_{6}}\mathcal{I}_{12}\,.$ (5.1)

Moreover, through the usual descent procedure, the six-or twelve-form anomaly
polynomial is related to a five- or eleven-form which can be identified with
the topological couplings in the relevant five- or eleven-dimensional
supergravity theory,

$\mathcal{I}_{d+2}={\rm d}\mathcal{I}^{(0)}_{d+1}\,,\qquad\qquad
S^{\rm(top)}_{d+1}=\int_{\mathcal{N}_{d}}\mathcal{I}^{(0)}_{d+1}\,.$ (5.2)

The twelve-form $\mathcal{I}_{12}$ was described in [60, 88] and is given by

$\mathcal{I}_{12}=\frac{1}{6}E_{4}\wedge E_{4}\wedge E_{4}+E_{4}\wedge
X_{8}\,,$ (5.3)

where

$X_{8}=\frac{1}{192}\left[p_{1}(T\mathcal{M}_{11})^{2}-4\,p_{2}(T\mathcal{M}_{11})\right]\,.$
(5.4)

The four-form $E_{4}$ appearing in (5.3) is a globally defined, closed four-
form on the total space $\mathcal{N}_{12}$ with integral periods on all four-
cycles in $\mathcal{N}_{12}$. When restricted to the fibre over a generic
point it reproduces the $G_{4}$ flux that supports the $\mathop{\rm
AdS}\nolimits_{5}$ solution.

### 5.1 Construction of $E_{4}$

The four-form $E_{4}$ can be constructed from the $\mathrm{U}(1)^{3}$
equivariant completion of $\widetilde{G}_{4}=(2\pi\ell_{p})^{-3}G_{4}$, where
the prefactor is added to have integer quantisation. The construction of
$E_{4}$ proceeds in two steps, first we define the $\mathfrak{u}(1)^{3}$
equivariantly closed form $V_{4}$ such that

${\rm d}_{\xi_{I}}V_{4}=\left({\rm
d}-\frac{\mathcal{F}^{I}}{2\pi}\wedge\iota_{\xi_{I}}\right)V_{4}=0\,,$ (5.5)

where we chose a basis $\xi^{I}=\\{\xi_{R},\xi_{1},\xi_{2}\\}$ for the Killing
vectors and $\mathcal{F}^{I}={\rm
d}\mathcal{A}^{I}=\\{\mathcal{F}_{R},\mathcal{F}_{1},\mathcal{F}_{2}\\}$
denote the external field strengths associated to the R-symmetry and two
flavour symmetries. The Killing vectors for the flavour symmetries can be
chosen simply as $\xi_{i}=\partial_{\phi_{i}}$. Note however that the Killing
vector dual to the R-symmetry is not simply $\partial_{z}$ but instead
$\partial_{\psi}$ where
$\psi=-\frac{3}{2}\mathfrak{r}_{z}z-\phi_{1}-\phi_{2}$. After putting the
metric in the canonical $\mathcal{N}=1$ form this can be deduced
straightforwardly by demanding that the complex three-form $\Omega$ has charge
$+1$ under the R-symmetry but is uncharged under the remaining flavour
symmetries, see appendix B.

A suitable ansatz for $V_{4}$ is given by

$V_{4}=\widetilde{G}_{4}+\sum_{I}\frac{\mathcal{F}^{I}}{2\pi}\wedge\omega_{(2)I}+\sum_{I,J}\frac{\mathcal{F}^{I}}{2\pi}\wedge\frac{\mathcal{F}^{J}}{2\pi}\,\omega_{(0)IJ}\,.$
(5.6)

Demanding equivariant closure then amounts to imposing the following equations

${\rm d}\widetilde{G}_{4}=0\,,\qquad\iota_{\xi_{I}}\widetilde{G}_{4}-{\rm
d}\omega_{(2)I}=0\,,\qquad\iota_{\xi_{I}}\widetilde{\omega}_{(2)J}-{\rm
d}\omega_{(0)IJ}=0\,,$ (5.7)

After identifying the appropriate equivariantly closed form $V_{4}$ we can
obtain an expression for $E_{4}$ by gauging $V_{4}$ with respect to the
isometries.

$E_{4}=V_{4}^{g}=\widetilde{G}_{4}^{g}+\sum_{I}\frac{\mathcal{F}^{I}}{2\pi}\wedge\omega_{(2)I}^{g}+\sum_{I,J}\frac{\mathcal{F}^{I}}{2\pi}\wedge\frac{\mathcal{F}^{J}}{2\pi}\,\omega_{(0)IJ}\,,$
(5.8)

where the superscript $g$ denotes the gauging of the internal forms
$\widetilde{G}_{4}$ and $\omega_{(2)I}$. More concretely, this gauging
operation consists of replacing in each form on $\mathcal{M}_{6}$,

${\rm d}\psi\rightarrow\mathcal{D}\psi={\rm
d}\psi-\frac{\mathcal{A}_{R}}{2\pi}\,,\qquad\qquad{\rm
d}\phi_{i}\rightarrow\mathcal{D}\phi_{i}={\rm
d}\phi_{i}-\frac{\mathcal{A}_{i}}{2\pi}\,.$ (5.9)

It is then straightforward to show that $E_{4}$ is a closed form on the
auxiliary manifold $\mathcal{N}_{12}$.

Imposing equations (5.7) uniquely fixes131313Unique up to equivariantly exact
pieces but these are irrelevant for our purposes since they do not contribute
to any observable. the internal forms $\omega_{(0)IJ}$ and $\omega_{(2)I}$ to
be,

$\displaystyle\omega_{(2)R}=$
$\displaystyle-\frac{\mathrm{e}^{3\lambda}Y^{\prime}}{3}\,,$
$\displaystyle\omega_{(0)RR}=$ $\displaystyle\frac{1}{9}y\,,$ (5.10)
$\displaystyle\omega_{(2)\phi_{1}}=$
$\displaystyle-\iota_{\phi_{1}}C_{3}+\frac{1}{9}{\rm d}y\wedge{\rm d}\psi\,,$
$\displaystyle\omega_{(0)R\phi_{1}}=$
$\displaystyle-\frac{y}{6}\frac{w}{h_{2}(w)}\,,$
$\displaystyle\omega_{(2)\phi_{2}}=$
$\displaystyle-\iota_{\phi_{2}}C_{3}+\frac{1}{9}{\rm d}y\wedge{\rm d}\psi\,,$
$\displaystyle\omega_{(0)R\phi_{2}}=$
$\displaystyle-\frac{y}{6}\frac{w}{h_{1}(w)}\,,$
$\displaystyle\omega_{(0)\phi_{1}\phi_{1}}=$ $\displaystyle 0\,,$
$\displaystyle\omega_{(0)\phi_{1}\phi_{2}}=$ $\displaystyle 0\,,$
$\displaystyle\omega_{(0)\phi_{2}\phi_{2}}=$ $\displaystyle 0\,.$

Inserting these expressions in (5.8) we have provided an explicit
representative of the globally well-defined form $E_{4}$ satisfying all the
requirements listed above. In the above expressions, the two-form $Y^{\prime}$
is the calibration form, introduced in Appendix B. Note that the zero- and
two-form part associated to the R-symmetry are identical to the expressions
found in [65, 66].

### 5.2 Flavour symmetries from punctures

In addition to the flavour symmetries discussed so far, arising from
isometries of the internal manifold, the dual SCFT can have baryonic zero-form
flavour symmetries arising from non-trivial two-cycles in the internal
geometry. Integrating the M-theory three-form on such cycles gives rise to
vector potentials in five dimensional gauged supergravity sitting in so-called
Betti multiplets. More generally, wrapping the M-theory three-form on non-
trivial $p$-cycles in the internal manifold can give rise to $(2-p)$-form
symmetries in the dual SCFT. In our setup, such additional non-trivial cycles
arise from the blow-up of the $\mathbf{C}^{3}/\mathbf{Z}_{n_{\pm}}$ orbifold
singularities.

In order to understand the resulting flavour symmetry we first will have a
look at the resolutions of generic $\mathbf{C}^{3}/\Gamma$ orbifolds, where
$\Gamma$ is a finite subgroup of $\mathrm{SU}(3)$, following the work of Ito
and Reid [59]. In our context, the requirement that
$\Gamma\subset\mathrm{SU}(3)$ can be physically motivated by the fact that
this class of orbifolds is Calabi-Yau and therefore gives rise to at least
$\mathcal{N}=1$ supersymmetry in the dual SCFT. Indeed, this requirement has
already been derived in seven dimensions when demanding the existence of a
globally well-defined Killing spinor.

Denote the conjugacy classes of $\Gamma$ by $\gamma_{i}$. All the group
elements in a given conjugacy class have the same eigenvalues $\lambda_{i}$,
which can be written as

$\lambda_{i}=\left\\{\exp\left(2\pi{\rm
i}\,\frac{a_{i}}{r_{i}}\right)\,,\,\exp\left(2\pi{\rm
i}\,\frac{b_{i}}{r_{i}}\right)\,,\,\exp\left(2\pi{\rm
i}\,\frac{c_{i}}{r_{i}}\right)\right\\}\,,$ (5.11)

where $r_{i}$ is the order of the elements in $\gamma_{i}$ and $0\leq
a_{i},b_{i},c_{i}\leq r_{i}$. Define the age of the conjugacy class as the
integer ${\rm age}(\gamma_{i})=\frac{1}{r_{i}}(a_{i}+b_{i}+c_{i})$ and denote
the set of conjugacy classes of age $m$ by $\Gamma_{m}$. Ito and Reid proved
that for each orbifold $\mathbf{C}^{3}/\Gamma$ there exists a crepant
resolutions

$\pi:\widetilde{X}\rightarrow\mathbf{C}^{3}/\mathbf{Z}_{k}\,,$ (5.12)

such that $b_{2m}(\widetilde{X})=|\Gamma_{m}|$ and $b_{3}(\widetilde{X})=0$.

Coming back to our setup, the exceptional divisors $S_{\alpha}\in
H_{4}(\widetilde{X},\mathbf{R})$ from the blow-up are dual to the compactly
supported 2-forms $\widetilde{\omega}_{(2)\alpha}\in
H^{2}_{c}(\widetilde{X},\mathbf{R})$ which can be thought of as forms
localised at the singular points in our original geometry. We can therefore
consider the following additional terms in the expansion of $E_{4}$.

$E^{\rm
punc.}_{4}=\sum_{\alpha=1}^{b_{4}(\widetilde{X})}\frac{\mathcal{F}^{\alpha}}{2\pi}\wedge\widetilde{\omega}_{(2)\alpha}\,.$
(5.13)

where $\mathcal{F}^{\alpha}={\rm d}\mathcal{A}^{\alpha}$ are external gauge
fields for the resulting flavour symmetries. In the lower dimensional gauged
supergravity these therefore give rise to massless gauge fields in the Cartan
of the gauge group, where the rank of the gauge group is given by
$r=b_{4}(\widetilde{X})$. In various cases, the gauge symmetry gets enhanced
to a non-abelian symmetry at the SCFT point.

In particular, one can wrap M2-branes on rational curves inside a compact
divisor $S_{\beta}$. These curves can be described by a geometric ruling of
the divisor by rational curves $f_{\beta}$, i.e. $\mathbf{C}\mathbf{P}^{1}$s.
In the dual QFT (in the Coulomb phase) such wrapped M2-branes give rise to
charged particles, i.e. W-bosons. The charge of the particle under the Cartan
corresponding to a divisor $S_{\alpha}$ is given by

$S_{\alpha}\cdot f_{\beta}=-C_{\alpha\beta}\,,$ (5.14)

where the left hand side denotes the intersection number in $\widetilde{X}$
and can be computed from the triple intersection numbers $S_{\beta}^{2}\cdot
S^{\alpha}$. The mass of the particle is determined by the symplectic volume
of $f_{\alpha}$ such that in the singular limit where the resolution cycles
shrink these particles become the massless W-bosons enhancing the
$\mathfrak{u}(1)^{r}$ symmetry to a possibly non-abelian $\mathfrak{g}$
symmetry. In particular, the matrix $C_{\alpha\beta}$ reproduces the entries
of Cartan matrix of the flavour symmetry of the dual SCFT. In addition to
providing a holographic description of the flavour symmetry associated to the
punctures the additional terms in the expansion of $E_{4}$ give rise to Chern-
Simons terms in the five-dimensional supergravity theory which describe the
(mixed) ’t Hooft anomalies of the dual SCFT which are therefore controlled by
the triple intersection numbers of compact divisors
$k_{\alpha\beta\gamma}=S_{\alpha}\cdot S_{\beta}\cdot S_{\gamma}$.
Furthermore, the crepant resolutions do not have odd dimensional cycles, so
they do not contribute to the one-form symmetry of the dual SCFTs.

This setup is very similar and was heavily inspired by geometric engineering
of 5d S(C)FTs in M-theory, see [89, 90, 91] and in particular [92, 93, 94, 95,
96, 97, 98, 99] for the applications with orbifolds and more general isolated
canonical singularities. However, the crucial difference is that in our case
the non-compact Calabi-Yau three-fold only gives a local description of a
globally compact three-fold. Hence, the 5d theory obtained after
compactification is a supergravity theory. For this reason we are solely
interested in the compact cycles resulting in gauge symmetries in the bulk. In
particular we ignore the effect of the additional
$b_{2}(\widetilde{X})-b_{4}(\widetilde{X})$ compact two-forms dual to non-
compact divisors which in geometric engineering encode (part of) the flavour
symmetry of the 5d S(C)FT.

So far our discussion has been fairly general and applies to any finite
subgroup of $\mathrm{SU}(3)$. The orbifolds we encounter in the uplift of our
seven-dimensional solutions however are all abelian orbifolds of the form
$\mathbf{C}^{3}/\mathbf{Z}_{n}$. From now on we specify to this case. Note
that for these cases all conjugacy classes contain a single element such that
$n=1+|\Gamma_{1}|+|\Gamma_{2}|=1+b_{2}(\widetilde{X})+b_{4}(\widetilde{X})$
and furthermore their resolution is always toric. This class of punctures is
surprisingly rich and depending on the ‘charges’ $m^{(i)}_{\pm}$ a variety of
orbifold actions is possible. For example, for $\mathbf{Z}_{6}$ the generator
can be chosen either as $\frac{1}{6}(1,1,4)$ or $\frac{1}{6}(1,2,3)$.141414We
only include those actions resulting in a three-dimensional singularity. When
the orbifold action acts trivially in one of the directions we recover the
usual $\mathcal{N}=2$ punctures. Note that in these two cases the fourth Betti
number differs so that the rank of the flavour groups is respectively $2$ or
$1$.

Before we continue let us briefly consider a few examples. A first infinite
class of examples is given by $\Gamma=\mathbf{Z}_{n}$ with generator
$\frac{1}{n}(1,1,n-2)$. For $n$ odd res. even, the resolution of this
singularity can be encoded in the toric diagrams in Figure 4. Computing the
intersection numbers corresponding to the Cartan matrix and Chern-Simons
coefficients now becomes a straightforward exercise in toric geometry. For a
review of toric geometry we refer the reader to one of the excellent textbooks
on the subject.

Up to $\mathrm{SL}(2,\mathbf{Z})$ transformations the toric diagram is
obtained from the three vertices of weighted projective space,
$\mathbf{C}\mathbf{P}^{p,q,r}$ by interpreting them as the rays of the fan of
the toric three-fold singularity.151515A set of vertices for weighted
projective space can be obtained by finding (coprime) integer solutions to the
equation $pv_{1}+qv_{2}+rv_{3}=0$. After an $\mathrm{SL}(2,\mathbf{Z})$
transformation the $\Gamma=\mathbf{Z}_{n}$ with generator
$\frac{1}{n}(1,1,n-2)$ then takes the form as in Figure 4. From this toric
diagram we can then read off the gauge symmetry in $\mathop{\rm
AdS}\nolimits_{5}$ or equivalently the flavour symmetry of the dual SCFT.
There are $k$ compact divisors, which correspond to the Cartan of the flavour
algebra. Computing the Cartan matrix from the intersection numbers we find
that the full non-abelian flavour symmetry of the dual SCFT is given by
$\mathrm{SU}(k)$ where $n=2k+1$ or $n=2k+2$. Note that a single orbifold point
in seven dimensions get lifted to a pair of orbifold singularities in eleven
dimensions so that the total global symmetry of the puncture is given by
$\mathrm{SU}(k)^{2}$.

The conical singularity of the spindle gets doubled in the internal space
which explains the doubling of the flavour symmetry. Indeed, notice that the
M2 branes wrapping $\Sigma_{\pm,i}$, extending from a singularity in the
internal space to the probe M5-brane on the diagonal face in 2, are charged
under one of these flavour symmetries. This provides evidence that indeed this
$\mathcal{N}=1$ puncture gives rise to the doubled $\mathrm{SU}(k)^{2}$
flavour symmetry.

More explicitly, in the case of odd $n=2k+1$ the compact surfaces are given by
$\mathbf{C}\mathbf{P}^{2}\cup\mathbb{F}_{3}\cup\mathbb{F}_{5}\cup\cdots\cup\mathbb{F}_{2k-1}$
and the triple intersection numbers are as follows. The triple intersection of
three non-identical compact divisors always vanishes. The intersection numbers
or two identical and one distinct neighbouring divisor are given by

$\displaystyle S_{0}^{2}\cdot S_{1}$ $\displaystyle=1\,,\quad$ $\displaystyle
S_{1}^{2}\cdot S_{0}$ $\displaystyle=-3\,,\quad$ $\displaystyle S_{1}^{2}\cdot
S_{2}$ $\displaystyle=3\,,\quad$ $\displaystyle S_{2}^{2}\cdot S_{1}$
$\displaystyle=-5\,,$ (5.15) $\displaystyle S_{2}^{2}\cdot S_{3}$
$\displaystyle=5\,,\quad$ $\displaystyle\cdots\quad$ $\displaystyle
S_{k-2}^{2}\cdot S_{k-1}$ $\displaystyle=2k-3\,,\quad$ $\displaystyle
S_{k-1}^{2}\cdot S_{k-2}$ $\displaystyle=-2k+1\,.$

Finally when the three divisors are identical, the triple intersection is
$S_{0}^{3}=9$ and $S_{i}^{3}=8$, where $i=1,\cdots,k-1$.

Figure 4: The toric diagrams for the two infinite families of $\mathbf{Z}_{n}$
singularities introduces above. The black dots indicate compact divisors,
while the yellow ones denote non-compact divisors. The red dot denotes the
compact divisor corresponding to a flavour curve in the geometric engineering
setup. The toric diagram on the left corresponds to $n=2k+1$, while the
diagram on the right corresponds to $n=2k+2$. Note that since these are toric
Calabi-Yau threefolds they can be encoded in a two-dimensional diagram where
we fixed the third coordinate of all rays to be one.

Similarly, as a last example we consider the singularity
$\Gamma=\mathbf{C}^{3}/\mathbf{Z}_{8}$ with generator
$\frac{1}{8}\left(1,3,4\right)$. The toric diagram is given in Figure 5 from
which we can read of the rank of the flavour symmetry group which in this case
is two. However, in this case we do not find a $\mathrm{SU}(3)$ flavour
symmetry but instead find that the flavour symmetry is
$\mathrm{SU}(2)\times\mathrm{SU}(2)$. Finally, in Table 1 we collect the Betti
numbers and flavour symmetries of the dual SCFTs for a variety of abelian
orbifolds of order $n<14$. 161616This table is modified from Table 5 in [97].

Figure 5: The toric diagrams for the $\mathbf{Z}_{8}$ singularity with generator $\frac{1}{8}(1,3,4)$. The black dots indicate compact divisors, while the yellow ones denote non-compact divisors. The red dots denote the compact divisors corresponding to flavour curves in the geometric engineering setup. $n$ |  Generator | rank $=\,b_{4}(\widetilde{X})$ | $\quad b_{2}(\widetilde{X})\quad$ | Flavour algebra
---|---|---|---|---
$3$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{3}(1,1,1)$ | $1$ | $1$ | —
$4$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{4}(1,1,2)$ | $1$ | $2$ | $\mathfrak{su}(2)$
$5$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{5}(1,1,3)$ | $2$ | $2$ | —
$6$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{6}(1,1,4)$ | $2$ | $4$ | $\mathfrak{su}(2)$
$6$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{6}(1,2,3)$ | $1$ | $4$ | $\mathfrak{su}(2)$
$7$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{7}(1,1,5)$ | $3$ | $3$ | —
$7$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{7}(1,2,4)$ | $3$ | $3$ | —
$8$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{8}(1,1,6)$ | $3$ | $4$ | $\mathfrak{su}(4)$
$8$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{8}(1,2,5)$ | $3$ | $4$ | —
$8$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{8}(1,3,4)$ | $2$ | $5$ | $\mathfrak{su}(2)\times\mathfrak{su}(2)$
$9$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{9}(1,1,7)$ | $4$ | $4$ | —
$9$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{9}(1,2,6)$ | $3$ | $5$ | —
$10$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{10}(1,1,8)$ | $4$ | $5$ | $\mathfrak{su}(5)$
$10$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{10}(1,2,7)$ | $4$ | $5$ | —
$10$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{10}(1,3,6)$ | $4$ | $5$ | —
$10$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{10}(1,4,5)$ | $2$ | $7$ | $\mathfrak{su}(2)\times\mathfrak{su}(2)$
$11$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{11}(1,1,9)$ | $5$ | $5$ | —
$11$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{11}(1,2,8)$ | $5$ | $5$ | —
$12$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{12}(1,1,10)$ | $5$ | $6$ | $\mathfrak{su}(6)$
$12$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{12}(1,2,9)$ | $4$ | $7$ | $\mathfrak{su}(4)\times\mathfrak{su}(2)$
$12$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{12}(1,3,8)$ | $3$ | $8$ | $\mathfrak{su}(3)\times\mathfrak{su}(2)$
$12$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{12}(1,4,7)$ | $4$ | $7$ | —
$12$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{12}(1,5,6)$ | $3$ | $8$ | $\quad\mathfrak{su}(2)\times\mathfrak{su}(2)\times\mathfrak{su}(2)\quad$
$13$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{13}(1,1,11)$ | $6$ | $6$ | —
$13$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{13}(1,2,10)$ | $6$ | $6$ | —
$13$ | $\rule[-3.87495pt]{0.0pt}{15.0694pt}\frac{1}{13}(1,3,9)$ | $6$ | $6$ | —
Table 1: List of (inequivalent) $\mathbf{C}^{3}/\mathbf{Z}_{n}$ orbifolds with
their generators and the Betti numbers of the (crepant) resolution. The last
column gives the precise form of the flavour symmetry algebra (if known) which
can be obtained using the methods of [97]. The dashes ‘—’ denote the cases
where the methods above do not predict symmetry enhancement to a non-abelian
flavour symmetry and we have
$\mathfrak{u}(1)^{\text{rank}}\subseteq\mathfrak{g}_{F}$.

The uplift of the seven-dimensional solutions shown in this paper necessarily
only provide a subset of the possible solutions in eleven dimensions.
Therefore it is reasonable to expect that after uplifting to eleven dimensions
one can significantly generalise the type of punctures. In particular, a
natural expectation is that more generic finite subgroups of $\mathrm{SU}(3)$
such as $\mathbf{C}^{3}/\mathbf{Z}_{n}\times\mathbf{Z}_{m}$ singularities or
even non-abelian $\Gamma$ can arise as $\mathcal{N}=1$ punctures in class
$\mathcal{S}$. It would be very interesting to find a local description of
such punctures by carefully analysing the associated Monge-Ampére equations
analogous to the Toda equation (or cylindrical Laplacian) in the case of
$\mathcal{N}=2$ punctures [5]. One can then use similar tools as above to
analyse the flavour symmetry originating from the puncture in this more
general setup. Note however that not all of the more general puncture are
toric. In such case more general methods, see for example [97], are needed to
resolve the singularities.

### 5.3 Anomaly inflow

Having constructed the equivariantly closed form $E_{4}$ and analysed the
additional flavour symmetries from the punctures, it is now an algorithmic
task to compute $\mathcal{I}_{12}$ and integrate along $\mathcal{M}_{6}$.
Writing $E_{4}$ as,

$E_{4}=G_{4}^{(g)}+\frac{\mathcal{F}^{a}}{2\pi}\wedge\omega_{(2)a}^{(g)}+\frac{\mathcal{F}^{a}}{2\pi}\wedge\frac{\mathcal{F}^{b}}{2\pi}\,\omega_{(0)ab}\,,$
(5.16)

where the index $a,b\in\\{I\,,\alpha\\}$ collectively run over the flavour
symmetries from isometries as well as two-cycles in the internal manifold.
Substituting this into the expression for $\mathcal{I}_{12}$ we find that the
$E_{4}^{3}$ term becomes,

$\frac{1}{6}E_{4}^{3}=\frac{\mathcal{F}^{a}}{2\pi}\wedge\frac{\mathcal{F}^{b}}{2\pi}\wedge\frac{\mathcal{F}^{c}}{2\pi}\wedge\left(\widetilde{G}_{4}\wedge\omega_{(2)a}\omega_{(0)bc}+\frac{1}{6}\omega_{(2)a}\wedge\omega_{(2)b}\wedge\omega_{(2)c}\right)+\cdots\,,$
(5.17)

where the dots denote terms that do not have the appropriate number of legs
along the internal space and hence will not contribute to the anomaly
polynomial of the resulting four-dimensional theory. Similarly, the higher
derivative contributions from $E_{4}\wedge X_{8}$ give rise to the following
contributions,

$E_{4}\wedge X_{8}=-\frac{1}{96}\frac{\mathcal{F}^{a}}{2\pi}\wedge
p_{1}(T\mathop{\rm AdS}\nolimits_{5})\wedge\omega_{(2)a}\wedge
p_{1}(T\mathcal{M}_{6})+\cdots\,,$ (5.18)

where we used the splitting principle to rewrite (5.4) in terms of the
Pontryagin classes of the internal resp. $\mathop{\rm AdS}\nolimits$ space. In
order to obtain the anomaly polynomial of the resulting four-dimensional
theory all that is left is to integrate these expressions over the internal
space $\mathcal{M}_{6}$. The resulting expression is given by

$\mathcal{I}_{6}=\frac{1}{6}k_{abc}\,c_{1}(\mathcal{F}^{a})\wedge
c_{1}(\mathcal{F}^{b})\wedge
c_{1}(\mathcal{F}^{c})-\frac{1}{24}k_{a}\,c_{1}(\mathcal{F}^{a})\wedge
p_{1}(T\mathop{\rm AdS}\nolimits_{5})\,,$ (5.19)

where $k_{abc}$ and $k_{a}$ are defined as

$\displaystyle k_{abc}=$
$\displaystyle\int_{\mathcal{M}_{6}}\left(\omega_{(2)a}\wedge\omega_{(2)b}\wedge\omega_{(2)c}+6\widetilde{G}_{4}\wedge\omega_{(2)a}\omega_{(0)bc}\right)\,,$
(5.20) $\displaystyle k_{a}=$
$\displaystyle\frac{1}{4}\int_{\mathcal{M}_{6}}\omega_{(2)a}\wedge
p_{1}(T\mathcal{M}_{6})\,,$ (5.21)

and the terms proportional to $\widetilde{G}_{4}$ vanish identically.
Comparing this with the anomaly polynomial of a generic four-dimensional
$\mathcal{N}=1$ SCFT,

$\mathcal{I}_{6}=\frac{8}{27}\left(5a-3c\right)c_{1}(\mathcal{F}_{R})^{3}+\frac{2}{3}\left(c-a\right)c_{1}(\mathcal{F}_{R})p_{1}\left(T\mathcal{M}_{4}\right)\,,$
(5.22)

the above expressions reduce to the well-known expressions for the conformal
anomalies $a$ and $c$,

$a=\frac{3}{32}\int_{\mathcal{M}_{6}}\left(3\,k_{RRR}-k_{R}\right)\,,\qquad\quad
c=\frac{1}{32}\int_{\mathcal{M}_{6}}\left(9\,k_{RRR}+5\,k_{R}\right)\,.$
(5.23)

At leading order in the large $N$ expansion we have $a\simeq c$ and it follows
from (5.10) above that our expression reduces precisely to

$a\simeq\frac{N^{3}}{128\pi^{3}}\int_{\mathcal{M}_{6}}\mathrm{e}^{9\lambda}\,{\rm
vol}_{\mathcal{M}_{6}}\,,$ (5.24)

which agrees with the expression obtained in [81]. Note that with the choice
of $E_{4}$ introduced above one obtains the integrand up to total derivatives
only. The end result after integration is obviously independent of such terms.
We can make this more explicit by explicitly performing the equivariant
integration of $E_{4}^{3}$ over $\mathcal{M}_{6}$. Indeed, in line with the
results in [65, 67] we find that the leading order central charge is given by

$a=\frac{N^{3}}{2^{5}3^{4}}\sum_{F_{0}}\frac{1}{d_{F_{0}}}\frac{y^{3}}{\epsilon_{1}\epsilon_{2}\epsilon_{3}}\,.$
(5.25)

This sum goes over the four fixed orbifold points where the weights are given
in (3.10). Hence, the resulting expression for the conformal $a$-anomaly
manifestly agrees with [32, 65, 67]. At subleading order in $N$, $a$ and $c$
typically differ. From the anomaly polynomial above we can read off that

$a-c=\frac{N}{128\pi}\int_{\mathcal{M}_{6}}\omega_{(2)R}\wedge
p_{1}(T\mathcal{M}_{6})\,,$ (5.26)

which can be computed by direct integration or more elegantly using
equivariant localisation. The flavour central charges $k_{abc}$ resulting from
the isometries of the internal manifold can be computed by direct
(equivariant) integration after substituting the relevant forms from (5.10).
The $k_{\phi_{i}\phi_{j}\phi_{k}}$ are trivially vanishing. In addition, in
line with $a$-maximisation we find that $\partial_{s}a(s)\simeq
k_{RR\phi_{i}}=0$ and $\partial^{2}_{s}a(s)\simeq k_{R\phi_{i}\phi_{j}}<0$.
The computation of the flavour central charges for the flavours corresponding
to the puncture can be computed as the triple intersection number

$k_{\alpha\beta\gamma}=S_{\alpha}\cdot S_{\beta}\cdot S_{\gamma}\,,$ (5.27)

which can be read off from the toric diagram as instructed above. Finally, we
can consider the mixed flavour central charges between the R-symmetry or
flavour symmetries from isometries and the flavour symmetries from the
puncture. Since the puncture divisors are localised at the puncture location
and provide a basis of harmonic two-forms one should expand the two form
corresponding to the R-symmetry and mesonic flavour symmetries in terms of the
localised cycles at the puncture location. After this one can compute these
mixed flavour central charges in a similar manner as a linear combination of
intersection numbers. However, we do not pursue this computation in the
present work. Similarly, we leave the computation of the $k_{a}$ open for
future research endeavours. Note however that, as can be observed from [32,
65, 67], in all anomalies and central charges related to the internal
symmetries, the dependence on $m_{\pm}^{(i)}$ can be absorbed in the fluxes
$p_{i}$. On the other hand, the flavour anomalies for the additional global
symmetries arising from the four orbifold singularities do contain dependence
on the $m_{\pm}^{(i)}$. This is natural as we expect these anomalies to
explicitly depend on the specific choice of punctures.

Acknowledgements

It is with pleasure that we thank Ibou Bah, Sakura Schafer-Nameki and James
Sparks for useful and inspiring discussions. The contributions of PB were made
possible through the support of grant No. 494786 from the Simons Foundation
(Simons Collaboration on the Non-perturbative Bootstrap) and the ERC
Consolidator Grant No. 864828, titled “Algebraic Foundations of Supersymmetric
Quantum Field Theory” (SCFTAlg). CC wishes to express gratitude for the
support received from the Mathematical Institute of the University of Oxford.

##  Appendix A Uplift formulae

The general formulae to uplift a solution of maximal 7d gauged supergravity
were derived in [76, 77, 78] which, for completeness, we restate here. The
eleven-dimensional metric is given by

$L^{-2}{\rm d}s_{11}^{2}=\Delta^{1/3}\,{\rm
d}s_{7}^{2}+\Delta^{-2/3}\left(X_{0}^{-1}\,{\rm
d}\mu_{0}^{2}+\sum_{i=1}^{2}X_{i}^{-1}\left({\rm
d}\mu_{i}^{2}+\mu_{i}^{2}\,D\phi_{i}^{2}\right)\right)\,,$ (A.1)

where we defined the function

$\Delta=\sum_{I=0}^{2}X_{I}\,\mu_{I}^{2}\,,$ (A.2)

in terms of the seven-dimensional supergravity scalars $X_{I}$ where
$X_{0}=X_{1}^{-2}X_{2}^{-2}$. The coordinates $\phi_{i}$ are $2\pi$-periodic,
while the embedding coordinates $\mu_{I}$ satisfy
$\mu_{0}^{2}+\mu_{1}^{2}+\mu_{2}^{2}=1$. A convenient choice for our purposes
is,

$\mu_{0}=\cos\xi\,,\qquad\mu_{1}=\sin\xi\cos\theta\,,\qquad\mu_{2}=\sin\xi\sin\theta\,.$
(A.3)

where $\xi\in[0,\pi]$ and $\theta\in[0,\pi/2]$. In addition, we defined the
gauged one-forms

${\rm D}\phi_{i}={\rm d}\phi_{i}-A^{(i)}\,.$ (A.4)

The eleven-dimensional three-form potential in turn can be written as,

$\displaystyle L^{-3}C_{3}=$
$\displaystyle\,\left[-\frac{1}{4\mu_{0}}\left({\rm d}\mu_{1}^{2}-{\rm
d}\mu_{2}^{2}\right)+\frac{\mu_{1}\,\mu_{2}}{\Delta\,\mu_{0}}\left(X_{2}\,\mu_{2}\,{\rm
d}\mu_{1}-X_{1}\,\mu_{2}\,{\rm d}\mu_{2}\right)\right]\wedge{\rm
D}\phi_{1}\wedge{\rm D}\phi_{2}$ (A.5)
$\displaystyle\,\,\,\,-\frac{\mu_{0}}{2}\left(F^{(1)}\wedge{\rm
D}\phi_{2}+F^{(2)}\wedge{\rm D}\phi_{1}\right)\,.$

##  Appendix B Canonical $\mathcal{N}=1$ form

In this appendix we introduce the relevant definitions and coordinate changes
in order to transform our solutions into the canonical form for
$\mathcal{N}=1$ $\mathop{\rm AdS}\nolimits_{5}$ solutions of M-theory [79],
henceforth referred to as the GMSW form. Since we know the spinors of the
seven-dimensional solution from [45] we can work directly in the six-
dimensional internal space $\mathcal{M}_{6}$ and to construct the eleven-
dimensional Killing spinor. From this one can then construct all the relevant
bilinear and reconstruct the full eleven-dimensional background.

Although conceptually the above approach is more straightforward, here we take
a slightly roundabout way to obtain the transformation into the canonical form
which turns out to be computationally easier. We use that there is a general
truncation of the GMSW classification to five-dimensional minimal gauged
supergravity as given in [81]. On the other hand, in [100] it was shown how to
perform the truncation of seven-dimensional U$(1)^{2}$ gauged supergravity on
a spindle to five-dimensional minimal gauged supergravity. By using the two
truncations from 11d, which must agree, we are able to extract out the
embedding of the spindle solution into the GMSW form.

The metric for most general $\mathcal{N}=1$ supersymmetric $\mathop{\rm
AdS}\nolimits_{5}$ solution can be written in the form

${\rm d}s_{11}^{2}=\mathrm{e}^{2\lambda}\left[{\rm d}s_{\mathop{\rm
AdS}\nolimits_{5}}^{2}+{\rm d}s_{\mathcal{M}_{6}}^{2}\right]\,,$ (B.1)

where the internal metric is

${\rm d}s_{\mathcal{M}_{6}}^{2}=\mathrm{e}^{-6\lambda}g_{ij}(x,y){\rm
d}x^{i}{\rm d}x^{j}+\mathrm{e}^{-6\lambda}\sec^{2}\zeta(x,y){\rm
d}y^{2}+\frac{1}{9}\cos^{2}\zeta(x,y)({\rm d}\psi+\rho)^{2}\,.$ (B.2)

Here $g_{ij}$ is the metric on a four-dimensional Kähler manifold (at fixed
$y$) and $\partial_{\psi}$ is a Killing vector of the metric $g$, the
functions, $\lambda$ and $\zeta$ and the four-form flux $G_{4}$. This Killing
vector realises the R-symmetry in the dual SCFT. The four-form field strength
can be written as

$G_{4}=-\left(\partial_{y}\mathrm{e}^{-6\lambda}\right){\rm
vol}_{4}-\mathrm{e}^{-9\lambda}\sec\zeta\left(\star_{4}{\rm
d}_{4}\mathrm{e}^{6\lambda}\right)\wedge
K_{1}-\frac{1}{3}\cos^{3}\zeta(\star_{4}\partial_{y}\rho)\wedge K_{2}\\\
+\mathrm{e}^{3\lambda}\left(\frac{1}{3}\cos^{2}\zeta\star_{4}{\rm
d}_{4}\rho-4\mathrm{e}^{-6\lambda}J\right)\wedge K_{1}\wedge K_{2}\,.$ (B.3)

In this expression, $\star_{4}$, ${\rm d}_{4}$ and $J$ denote the Hodge star,
exterior derivative and Kähler two-form with respect to the four-dimensional
metric $g_{ij}$. The one-forms $K_{1}$ and $K_{2}$ are defined to be

$K_{1}=\mathrm{e}^{-3\lambda}\sec\zeta{\rm d}y\,,\qquad
K_{2}=\frac{1}{3}\cos\zeta({\rm d}\psi+\rho)\,,$ (B.4)

and $2y=\sin\zeta\mathrm{e}^{3\lambda}$.

We can now uplift the solution in equation (2.1) to 11d using the results of
appendix A and then rewrite into the above form. We need to pick a gauge for
the gauge fields in the uplift and so we take the gauge in (2.1) in the
following, though any gauge transformation can be performed by

$A_{i}\rightarrow A_{i}+{\rm d}\Lambda_{i}\quad\Rightarrow\quad{\rm
d}\phi_{i}\rightarrow{\rm d}\phi_{i}-{\rm d}\Lambda_{i}\,.$ (B.5)

The choice of gauge is important in the main text when analysing the orbifold
action, however for the purposes of this appendix it is largely irrelevant.
From the form of the eleven-dimensional metric we can immediately read off the
form of the warp factor,

$\mathrm{e}^{2\lambda}=\Upsilon_{H}^{1/3}\,.$ (B.6)

Next, after writing out the metric and comparing with the truncation of [100]
we can extract out the $y$ coordinate, finding

$y=w\mu_{0}=w\cos\xi\,,$ (B.7)

and it immediately follows that the one-form dual to the R-symmetry vector is

${\rm d}\psi+\rho=-\frac{3}{2}\left(\mathfrak{r}_{z}{\rm
d}z+\frac{1}{\Upsilon_{F}}\big{(}\mu_{1}^{2}{\rm d}\phi_{1}+\mu_{2}^{2}{\rm
d}\phi_{2}\big{)}\right)\,.$ (B.8)

Moreover it is simple to show that

$\cos^{2}\zeta=\frac{1}{4w^{2}}\frac{\Upsilon_{H}}{\Upsilon_{F}}\,.$ (B.9)

Having extracted these crucial pieces we can identify the two-form $J$ to be

$\begin{split}J=e^{1}\wedge e^{2}+e^{3}\wedge e^{4}\,,\end{split}$ (B.10)

where the vielbein on the four-dimensional space are

$\displaystyle e^{1}$ $\displaystyle=\sqrt{\Theta(w,\theta)}\left(\sin\xi{\rm
d}\theta-\frac{\cos\theta\sin\theta\cos\xi(h_{1}(w)-h_{2}(w))}{\Theta(w,\theta)}{\rm
d}\xi\right)$ $\displaystyle e^{2}$
$\displaystyle=\frac{\cos\theta\sin\theta\sin\xi}{\sqrt{\Theta(w,\theta)}}\left(h_{1}(w){\rm
d}\phi_{1}-h_{2}(w){\rm d}\phi_{2}\right)\,,$ (B.11) $\displaystyle e^{3}$
$\displaystyle=\frac{\sqrt{\Theta(w,\theta)}\sin\xi\sqrt{f(w)}}{4w}\sqrt{\frac{\Upsilon_{H}}{\Upsilon_{F}}}\left(\frac{w}{f(w)}{\rm
d}w+\frac{4\cos\xi}{\Theta(w,\theta)\sin\xi}{\rm d}\xi\right)\,,$
$\displaystyle e^{4}$
$\displaystyle=-\frac{\sin\xi\sqrt{f(w)}}{w\sqrt{\Theta(w,\theta)}}\sqrt{\frac{\Upsilon_{H}}{\Upsilon_{F}}}\left(\cos^{2}\theta{\rm
d}\phi_{1}+\sin^{2}\theta{\rm d}\phi_{2}\right)\,.$

For notational brevity we defined the function

$\begin{split}\Theta(w,\theta)&=\sin^{2}\theta h_{1}(w)+\cos^{2}\theta
h_{2}(w)\,.\end{split}$ (B.12)

With the above definitions at hand we can compute everything we need. However,
note that the vielbein (B) naively seems to have legs along five directions in
the internal space. This is an artefact of our original choice of coordinates
and one can check that with this definition of the vielbein all algebraic and
differential constraints of [79] are satisfied.

In principle one should be able to find a general coordinate change to make
this explicit but in practice finding such coordinate transformations is
incredibly complicated. However, when $h_{1}(w)=h_{2}(w)$ we can make the
following change of coordinates,171717This simplification is not possible for
the spindle solution where they can never be equal, however it is possible for
other global completions.

$y=w\cos\xi\,,\quad x_{1}=\log\tan\theta\,,\quad x_{2}=W(w)+4\log\sin\xi\,,$
(B.13)

where

$W^{\prime}(w)=\frac{wh_{1}(w)}{f(w)}\,.$ (B.14)

In these coordinates the bracketed part of the two vielbeins $e^{1}$ and
$e^{3}$ become respectively ${\rm d}x_{1}$ and ${\rm d}x_{2}$. Solving (B.14)
one can find an explicit expression for $W(w)$,

$W(w)=\sum_{i=1}^{4}\frac{w_{i}(q_{1}+w_{i}^{2})}{2f^{\prime}(w_{i})}\log(w-w_{i})\,,$
(B.15)

where the sum goes over the four roots of the function $f(w)$. For $q_{1}\neq
q_{2}$ one can find an equivalent set of coordinates by expanding in
$\epsilon=q_{2}-q_{1}$ and perturbatively find a coordinate change. However,
the resulting expressions are rather messy and not very illuminating so we
refrain from stating them here. Writing the above vielbein with explicit $y$
dependence would be useful to better understand the underlying four-
dimensional metric and therefore how one may go about further generalising
these solutions.

Both in the computation of the operator dimensions form wrapped M2 branes, as
well as in the computation of the equivariant completion of the form $E_{4}$
we encountered the calibration form introduced in [81],

$Y^{\prime}=-\sin\zeta\mathrm{e}^{-6\lambda}J+K_{1}\wedge K_{2}\,.$ (B.16)

We have also stated that the correct $R$-symmetry is a linear combination of
the spindle coordinate and the $S^{4}$ coordinates. To show this it is useful
to consider the holomorphic $(2,0)$ form on the four-dimensional base. One
finds:

$\Omega=\mathrm{e}^{-{\rm
i}\left(\tfrac{3}{2}\mathfrak{r}_{z}z+\phi_{1}+\phi_{2}\right)}(e^{1}+{\rm
i}e^{2})\wedge(e^{3}+{\rm i}e^{4})\,.$ (B.17)

From GMSW we read off that the R-symmetry coordinate is given by

$-\psi=\tfrac{3}{2}\mathfrak{r}_{z}z+\phi_{1}+\phi_{2}\,,$ (B.18)

as stated in the main text.

##  Appendix C From spindles to BBBW

In this final appendix we show that the spindle solutions presented in this
note in fact include all of the solutions studied in BBBW [23, 22]. For genus
$0$ the relevant limit is rather straightforward and was already presented in
[45]. The goal of this appendix is to explicitly demonstrate that more general
limits recover the BBBW solution for arbitrary genus Riemann surfaces.

Recall that the metric on the spindle is given by

${\rm d}s^{2}_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}=\frac{w}{f(w)}{\rm
d}w^{2}+\frac{f(w)}{4H(w)}\mathfrak{r}_{z}^{2}{\rm d}z^{2}\,.$ (C.1)

We want to take various limits to recover the constant curvature
metrics:181818In the case of $\kappa=-1$ one may also consider the case where
$B(x)=x^{2}-1$. The difference is between considering global coordinates or
Poincare patch coordinates for the hyperbolic metric. Note that all that
matters in the function $B$ are the signs, one could add arbitrary constants
however they may all be removed by rescaling up to their sign.

${\rm d}s^{2}_{\Sigma_{g}}=\frac{{\rm d}x^{2}}{B(x)}+B(x){\rm
d}\phi^{2}\,,\qquad B(x)=1-\kappa x^{2}\,.$ (C.2)

To do so it turns out to be useful to first make the following redefinitions,

$w\rightarrow w_{0}+\epsilon\gamma(x+\beta)\,,\quad
z\rightarrow\frac{\phi}{\epsilon}\,,\quad q_{i}\rightarrow r_{i}+\epsilon
s_{i}\,,$ (C.3)

with $\beta,\gamma,s_{i},r_{i}$ constants and $\epsilon$ a parameter we will
eventually send to 0. Next, in order to guarantee a well-defined
$\epsilon\rightarrow 0$ limit we need to suitably tune the various parameters.
First of all, we require the function $f(w)$ to have a double root, which
fixes two of the constants to be

$r_{1}=\frac{w_{0}^{2}(w_{0}(4-w_{0})-r_{2})}{r_{2}+w_{0}^{2}}\,,\quad
s_{1}=\frac{2\beta\left[w_{0}\left(w_{0}^{3}(2-w_{0})+2r_{2}w_{0}(3-w_{0})-r_{2}^{2}\right)-2s_{2}w_{0}^{3}\right]}{(r_{2}+w_{0}^{2})^{2}}\,.$
(C.4)

while simultaneously fixing a third constant to be one of the branches of the
equation,

$r_{2}=3w_{0}-w_{0}^{2}\pm\sqrt{9w_{0}^{2}-4w_{0}^{3}}\,.$ (C.5)

Note that we are being careful with not expanding the square root term so that
we do not need to consider a further branching of solutions for whether
$w_{0}$ is positive or not.191919This turns out to be overkill, since
regularity only allows for $w_{0}>0$. However let us continue here without
using hindsight for the moment. With these conditions at hand we end up with
$f(w)$ having a double root in the limit of $\epsilon\rightarrow 0$. We will
refer to the two branches as the positive branch and negative branch depending
on the sign in the expression of $r_{2}$. Note that most parameters can be
removed by simple coordinate redefinitions, so that in the end we are naively
left with a two-parameter family of solutions, parameterised by $w_{0}$ and
$s_{2}$. However, further analysis shows that $s_{2}$ is not a physical
parameter, leading to a single parameter family of solutions. Furthermore, in
order for the solution to be physical and eventually reduce to the BBBW setup,
we must require that the scalar fields are positive definite constants. We
find:

$X_{1}=\frac{r_{2}+w_{0}^{2}}{2^{6/5}w_{0}^{7/5}}\,,\quad
X_{2}=\frac{2^{4/5}w_{0}^{8/5}}{r_{2}+w_{0}^{2}}\,,$ (C.6)

which are positive definite for both branches of solutions if and only if

$0<w_{0}<\frac{9}{4}\,.$ (C.7)

This expression for the scalars makes it clear that the two branches are in
fact related by the interchange of the $1$ and $2$ indices. We may therefore
restrict to studying only a single branch, which we choose to be the positive
branch.

We divide the remaining steps in three cases, depending on whether
$0<w_{0}<\tfrac{3}{2}$, $w_{0}=\tfrac{3}{2}$ or
$\tfrac{3}{2}<w_{0}<\tfrac{9}{4}$. For these three cases we recover the
solutions with a smooth Riemann surface of genus $g=0$, genus $g=1$ or genus
$g>1$ respectively. Let us now consider these three cases separately in more
detail.

### C.1 Two-sphere

Let us start with the range $0<w_{0}<\tfrac{3}{2}$. We parameterise $w_{0}$ as

$w_{0}=\frac{3}{2}\cos^{2}\theta_{0}\,,$ (C.8)

and fix the free (redundant) parameters $\gamma,\delta$ to be

$\gamma=\frac{s_{2}}{3\sin^{2}\theta_{0}\sqrt{6-2\cos^{2}\theta_{0}+2\sqrt{9-6\cos^{2}\theta_{0}}}}\,,\quad\delta=\frac{\cos^{2}\theta_{0}}{\gamma\mathfrak{r}_{z}\sin^{2}\theta_{0}}\,.$
(C.9)

This choice then leads to the metric

${\rm d}s^{2}_{\hskip
1.0pt{\includegraphics[height=5.0pt]{mySpindle.png}}}=\frac{1}{6\sin^{2}\theta_{0}}{\rm
d}s^{2}(S^{2})\,.$ (C.10)

We can then identify the parameter $z$ of the BBBW solution as,

$z=\frac{\sqrt{3-2\cos^{2}\theta_{0}}}{\sqrt{3}\sin^{2}\theta_{0}}\,,$ (C.11)

which indeed satisfies the $z>1$ constraint and indeed, the whole solution
reduces to the $\mathop{\rm AdS}\nolimits_{5}\times S^{2}$ solution presented
in [23, 22].

### C.2 Torus

Next, the torus can be obtained by fixing the constants

$\gamma=\frac{s_{2}}{\sqrt{2+\sqrt{3}}\sqrt{3-2w_{0}}}\,,\quad\delta=\frac{6}{\mathfrak{r}_{z}s_{2}}\sqrt{\frac{3-2w_{0}}{2-\sqrt{3}}}\,.$
(C.12)

Taking the $w_{0}\rightarrow\frac{3}{2}$ limit we precisely recover the
$\mathop{\rm AdS}\nolimits_{5}\times T^{2}$ solution of [23, 22].

### C.3 Higher genus Riemann surface

Finally, in order to obtain the BBBW solution with higher genus $g>1$ Riemann
surface we need to take $\tfrac{3}{2}<w_{0}<\tfrac{9}{4}$. Similar to the two-
sphere case, we parameterise $w_{0}$ as,

$w_{0}=\frac{3}{4}(\cos^{2}\theta_{0}+2)\,,$ (C.13)

and fix the redundant parameters to be

$\gamma=\frac{2s_{2}\sqrt{4-\cos^{2}\theta_{0}-2\sqrt{3}\sin\theta_{0}}}{3\cos^{2}\theta_{0}(2+\cos^{2}\theta_{0})}\,,\quad\delta=\frac{(2+\cos^{2}\theta_{0})}{\mathfrak{r}_{z}\cos^{2}\theta_{0}\gamma}\,.$
(C.14)

In order to complete the comparison with the solutions in BBBW we identify the
parameter $z$ with,

$z=\frac{2\sin\theta_{0}}{\sqrt{3}\cos^{2}\theta_{0}}\,.$ (C.15)

Again we see that the maps to the correct parameter range for $z$, $z>0$, and
we recover the full BBBW solution [23, 22].

Thus we have successfully shown that the local solution from which the spindle
is a global completion contains all the BBBW solutions as particular scaling
limits.

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