Universal spindles: D2’s on $\Sigma$ and M5’s on $\Sigma\times\mathbb{H}^{3}$

Christopher<EMAIL_ADDRESS>Koen
<EMAIL_ADDRESS>
a _Department of Physics and Research Institute of Basic Science,
Kyung Hee University, Seoul 02447, Republic of Korea_

b _Institute for Theoretical Physics, Utrecht University
Princetonplein 5, 3584 CC Utrecht, The Netherlands_

Abstract

We study the uplift of a 4d black hole in 4d Einstein–Maxwell supergravity
with a spindle horizon to massive type IIA and 11d supergravity on
$S^{4}\times\mathbb{H}^{3}$. The solutions exhibit features distinct to the
uplift of the same 4d solution to 11d supergravity on an $S^{7}$. In
particular, whereas the orbifold singularities may be removed in the 11d
uplift on an $S^{7}$, they are always present in both of the classes
considered here. We compute the free energy of the solutions giving
predictions for the free energy of a class of 3d Chern–Simons theories wrapped
on a spindle.

###### Contents

  1. 1 Introduction
  2. 2 Spindle in 4d Einstein–Maxwell supergravity
    1. 2.1 Full solution
    2. 2.2 Near-horizon limit
  3. 3 Uplift to massive type IIA
    1. 3.1 Metric regularity
    2. 3.2 Flux quantization
  4. 4 M5-branes on SLAG-3 $\times$ spindle
    1. 4.1 The holographic duals of M5-branes on a SLAG-3 manifold
    2. 4.2 The Pernici–Sezgin solution and uplift
    3. 4.3 AdS${}_{2}\ltimes\Sigma\times\mathbb{H}^{3}$ solutions
  5. 5 Concluding remarks
  6. A M5-branes on a spindle

## Index

##  1 Introduction

The typical paradigm for studying SCFTs wrapped on compact manifolds considers
branes wrapped on manifolds of constant curvature Maldacena:2000mw .
Supersymmetry of the compactified SCFT is preserved by performing a partial
topological twist of the theory. Recently it has been appreciated that there
are more general setups away from constant curvature manifolds that one may
consider. D3-branes wrapping two-dimensional orbifolds were first discussed in
Ferrero:2020laf . Known as a spindle, the two-dimensional orbifold is
isomorphic to $\mathbb{WCP}^{1}_{n_{\pm}}$, i.e. a sphere with conical
deficits at both poles with orbifold weights $n_{\pm}$. This requires the
space to have a metric of non-constant curvature. Since then spindle solutions
have been found for other brane setups: D3-branes Ferrero:2020laf ;
Hosseini:2021fge ; Boido:2021szx , M2-branes Ferrero:2020twa ; Cassani:2021dwa
; Ferrero:2021ovq ; Couzens:2021rlk ; Couzens:2021cpk , M5-branes
Boido:2021szx ; Ferrero:2021wvk  and the D4-D8 brane system Faedo:2021nub ;
Giri:2021xta .

One interesting aspect of spindle solutions is the way in which supersymmetry
is preserved. Rather than a topological twist, supersymmetry is preserved via
either a _topological topological twist_ (also known as _twist_) or an _anti-
twist_ , two new mechanisms of preserving supersymmetry Ferrero:2021etw . In
both cases the R-symmetry is mixed with the azimuthal rotations of the
spindle, and the two twists can be differentiated by the amount of R-symmetry
flux threading through the spindle. As in the regular topological twist, the
amount of flux threading through the spindle equals the Euler characteristic
of the spindle for a (topological topological) twist. However, in contrast to
the the usual topological twist, the local curvature of the R-symmetry gauge
field is not equal to the curvature of the tangent bundle. The anti-twist is
even further away from the familiar topological twist: here the total
R-symmetry flux threading through the spindle is not even equal to the Euler
characteristic. To be more concrete, let us denote the total R-symmetry flux
threading through the spindle by $Q_{R}$ and the Euler-characteristic to be
$\chi(\Sigma)$. Then for such spindle solutions we have

$Q_{R}=\frac{n_{-}+\sigma
n_{+}}{n_{-}n_{+}}\,,\qquad\chi(\Sigma)=\frac{n_{-}+n_{+}}{n_{-}n_{+}}\,,$
(1.1)

with $\sigma=1$ for the twist and $\sigma=-1$ for the anti-twist.

Concurrently to the spindle story, M5-branes wrapped on another 2d space of
non-constant curvature were also studied Bah:2021hei ; Bah:2021mzw . These
spaces are topologically discs, admitting a conical deficit at the centre of
the disc and a boundary along which the solution is singular. This singularity
is related to the presence of smeared M5-branes and the disc can be identified
as a sphere with two punctures; one a maximal regular puncture giving a
conical defect, as in the spindle class of solutions, and the other an
irregular puncture leading to the singular boundary. It was shown in
Bah:2021hei ; Bah:2021mzw  that these solutions are dual to a class of
Argyres–Douglas theories. The solution can be further generalised to replace
the regular puncture of maximal type, to a regular puncture labelled by an
arbitrary Young diagram giving a partition of a flux number, Couzens:2022yjl
.333It would be interesting to understand the field theory dual of the
M5-brane on a spindle to a similar level of M5-branes where the higher amount
of supersymmetry allows for more control on the field theory side.

Disc solutions have also been found for other brane configurations since the
first discs: D3-branes in Couzens:2021rlk ; Suh:2021ifj , D4-D8 branes in
Suh:2021aik , M2-branes Couzens:2021rlk ; Suh:2021hef , M5-branes Bah:2021hei
; Bah:2021mzw ; Karndumri:2022wpu ; Couzens:2022yjl . As shown in
Couzens:2021rlk ; Couzens:2021tnv , for D3 and M2-branes respectively, though
the logic applies more widely, discs are different global completions of the
local solutions giving rise to multi-charge spindle solutions and preserve
twice the supersymmetry as the corresponding spindle solution.444One of the
disc solutions in Karndumri:2022wpu  is somewhat of an outlier here as it
preserves $\mathcal{N}=1$ only. A third, and final interesting related
solution are the defect solutions of Gutperle:2022pgw . There M5-branes wrap a
non-compact two-dimensional base giving rise to a co-dimension 2 defect in the
6d world-volume theory on a stack of M5-branes. As for the discs, this is a
different global completion of the same local solution giving rise to the
multi-charge spindle solutions.555Similar defect solutions are expected for
all the other brane configurations above where spindle solutions have been
found.  A recent further extension is to consider branes wrapping 4d orbifold
geometries, see for example Cheung:2022ilc  and Suh:2022olh  for the product
of a spindle with a Riemann surface.

Here we consider two classes of solutions obtained by uplifting the same 4d
solution of 4d $\mathcal{N}=2$ Einstein–Maxwell supergravity. The first is
rotating D2-branes in massive type IIA string theory wrapped on a spindle,
which uses the truncation of massive type IIA to 4d $\mathcal{N}=2$
Einstein–Maxwell supergravity Varela:2019vyd . Our second class exploits a
second consistent truncation to 4d $\mathcal{N}=2$ Einstein–Maxwell
supergravity on a class of geometries describing M5-branes wrapped on a SLAG
3-manifold Gauntlett:2007ma . In order to keep this paper as self-contained as
possible we review the 4d solution in detail, in particular the conical
deficits at the poles and the type of twist required to preserve
supersymmetry.

Upon uplifting the solution to massive type IIA we find that the internal 8d
geometry of the 10d solution takes the form of a
SE${}_{5}\times\text{S}^{1}_{z}$ fibration over a rectangle, while the uplift
to the SLAG geometry gives rise to a $S^{2}\times S^{1}\times
S^{1}\times\mathbb{H}^{3}$ fibration over a rectangle. In contrast to the M2
brane uplift of the same solution, we find that the internal spaces _cannot_
be made smooth, instead the conical deficits remain in the uplifted solution.
This behaviour is more reminiscent of M5-branes on a spindle Ferrero:2021wvk ,
yet the way in which supersymmetry is preserved is different. In the D2-brane
case considered here supersymmetry is preserved with an anti-twist, while the
M5-brane case uses a (topological topological) twist. Consequently, the
presence of (monopole) singularities in the uplift not a property exclusive to
either of the twists but rather a property of the manner in which one uplifts
the solution as one may somewhat surmise.

We begin by reviewing the rotating spindle solution in 4d $\mathcal{N}=2$
Einstein–Maxwell supergravity in section 2 before studying its uplift to
massive type IIA supergravity in section 3. In section 4 we uplift the
solution to 11d supergravity on $S^{4}\times\mathbb{H}^{3}$ giving rise to the
holographic duals of M5-branes wrapped on the 5d product space
$\Sigma\times\mathbb{H}^{3}$. We conclude in 5 and relegate a review of the
regularity of the M5-brane on a spindle solution to an appendix A.

##  2 Spindle in 4d Einstein–Maxwell supergravity

In this section we will consider the spindle solution of 4d $\mathcal{N}=2$
Einstein-Maxwell supergravity. This was first discussed in the context of a
spindle geometry in Ferrero:2020twa , having been originally found in
Plebanski:1976gy , and we will briefly review the analysis of the former
paper. First, we discuss the full black hole solution, and afterward we
discuss the near-horizon limit.

###  2.1 Full solution

The action for 4d $\mathcal{N}=2$ Einstein–Maxwell supergravity we will use is

$S=\frac{1}{16\pi G_{(4)}}\int\Big{(}(R+6)\star 1-F\wedge\star F\Big{)}\,,$
(2.1)

where we have normalised the cosmological constant for simplicity. A solution
to this action is666We follow the presentation of the solution in
Ferrero:2020twa  for ease of comparison with existing literature.

$\displaystyle\begin{aligned}
\mathrm{d}s^{2}&=\frac{1}{H(r,\theta)^{2}}\bigg{[}-\frac{Q(r)}{S(r,\theta)}\Big{(}\mathrm{d}t-a\sin^{2}\theta\,\mathrm{d}\phi\Big{)}^{2}+\frac{S(r,\theta)}{Q(r)}\,\mathrm{d}r^{2}\\\
&\quad\;+\frac{P(\theta)}{S(r,\theta)}\,\sin^{2}\theta\,\Big{(}a\,\mathrm{d}t-(r^{2}+a^{2})\,\mathrm{d}\phi\Big{)}^{2}+\frac{S(r,\theta)}{P(\theta)}\,\mathrm{d}\theta^{2}\bigg{]}\,,\\\\[4.0pt]
A&=-\frac{er}{S(r,\theta)}\big{(}\mathrm{d}t-a\sin^{2}\theta\mathrm{d}\phi\big{)}+\frac{g\cos\theta}{S(r,\theta)}\big{(}a\mathrm{d}t-(r^{2}+a^{2})\mathrm{d}\phi\big{)}\,,\end{aligned}$
(2.2)

where

$\displaystyle\begin{aligned} H(r,\theta)&=1-\alpha\,r\cos\theta\,,\\\
S(r,\theta)&=r^{2}+a^{2}\cos^{2}\theta\,,\\\ P(\theta)&=1-2\alpha
m\cos\theta+\Big{(}\alpha^{2}(a^{2}+e^{2}+g^{2})-a^{2}\Big{)}\cos^{2}\theta\,,\\\
Q(r)&=\Big{(}r^{2}-2mr+a^{2}+e^{2}+g^{2}\Big{)}(1-\alpha^{2}r^{2})+(a^{2}+r^{2})r^{2}\,.\end{aligned}$
(2.3)

We demand that the metric has the correct signature. This requires that the
functions $S(r,\theta)$, $Q(r)$ and $P(\theta)$ are all positive. The range of
$\theta$ is taken to be $[0,\pi]$ and it is convenient to define
$\theta_{-}=0,\,\theta_{+}=\pi$. The requirement of the correct metric
signature then implies

$\begin{split}P(\theta_{-})&=1-2\alpha
m+\Big{(}\alpha^{2}(a^{2}+e^{2}+g^{2})-a^{2}\Big{)}>0\,,\\\
P(\theta_{+})&=1+2\alpha
m+\Big{(}\alpha^{2}(a^{2}+e^{2}+g^{2})-a^{2}\Big{)}>0\,.\end{split}$ (2.4)

Following Ferrero:2020twa  we study the metric on constant $(t,r)$ slices,
expanding around $\theta_{\pm}$ the $\theta,\phi$ terms of the metric become

$\mathrm{d}s^{2}\sim\frac{1}{H(r,\theta_{\pm})^{2}}\frac{r^{2}+a^{2}}{P(\theta_{\pm})}\Big{[}\mathrm{d}\theta^{2}+(\theta-\theta_{\pm})^{2}P(\theta_{\pm})^{2}\,\mathrm{d}\phi^{2}\Big{]}\,.$
(2.5)

If the geometry is regular round two-sphere one requires that
$P(\theta_{+})^{2}=P(\theta_{-})^{2}$, however one finds

$P(\theta_{+})-P(\theta_{-})=4\alpha m\,,$ (2.6)

and therefore it is not possible to assign a period to $\phi$ such that (2.5)
is smooth at both poles when $\alpha\neq 0$, i.e. it cannot be a smooth
$S^{2}$. Instead one must contend with conical singularities at the poles. The
period of $\phi$ is then taken to be

$\Delta\phi=\frac{2\pi}{P(\theta_{+})\,n_{+}}=\frac{2\pi}{P(\theta_{-})\,n_{-}}\,,$
(2.7)

with $n_{\pm}$ positive relatively prime integers. This then defines a spindle
which is the weighted projective space $\mathbb{WCP}^{1}_{[n_{-},n_{+}]}$.

###  2.2 Near-horizon limit

If we impose extremality and take the near-horizon limit of the black hole
solution presented earlier in this section, we obtain a fibered AdS2 space,
see Ferrero:2020twa . The near-horizon geometry is given by

$\displaystyle\begin{aligned}
\mathrm{d}s^{2}_{\text{NH}}&=\sqrt{P(w)}\,\bigg{[}-r^{2}\,\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{r^{2}}+\mathrm{d}s^{2}_{\Sigma}\bigg{]}\,,\\\\[4.0pt]
\mathrm{d}s^{2}_{\Sigma}&=\frac{1}{f(w)}\,\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\,(\mathrm{d}z+jr\,\mathrm{d}t)^{2}\,,\\\\[6.0pt]
A&=h(w)\big{(}\mathrm{d}z+jr\,\mathrm{d}t\big{)}\,,\end{aligned}$ (2.8)

with777We have performed some redefinitions of the functions and coordinates
in comparison with Ferrero:2020twa  since it puts the the functions into a
more succinct form.

$\displaystyle\begin{aligned}
P(w)&=\bigg{[}(w-q)^{2}+\frac{j^{2}}{4}\bigg{]}^{2}\,,\\\\[2.0pt]
f(w)&=P(w)-\big{(}1-j^{2}\big{)}w^{2}\,,\\\\[2.0pt]
h(w)&=-\sqrt{1-j^{2}}\bigg{[}\frac{4(w-q)(w+q)-j^{2}}{4(w-q)^{2}+j^{2}}\bigg{]}\,.\end{aligned}$
(2.9)

In order to have a well-defined solution of correct signature, we need the
coordinate $w$ to be defined on an interval where both $f(w)$ and $P(w)$ are
non-negative. Note that $P(w)$ is strictly positive since it is a square, and
therefore the correct signature requires an interval between which $f(w)$ is
positive and vanishes at the end-points. This implies that $f(w)$ must have
four real roots, and the interval then lies between the two middle roots. We
denote the four roots by $w_{1},\ldots,w_{4}$, where we assign the labels in
ascending order. One can readily compute expressions for the roots, and it can
be checked that there is only one labelling consistent with $w_{1}\leq
w_{2}<w_{3}\leq w_{4}$ (we don’t allow for $w_{2}=w_{3}$ as this would not
give a finite interval for $w$). We find the roots, labeled in ascending
order, to be

$\displaystyle w_{1}$
$\displaystyle=\frac{1}{2}\bigg{(}2q-\sqrt{1-j^{2}}-\sqrt{1-2j^{2}-4q\sqrt{1-j^{2}}}\bigg{)}\,,$
(2.10) $\displaystyle w_{2}$
$\displaystyle=\frac{1}{2}\bigg{(}2q-\sqrt{1-j^{2}}+\sqrt{1-2j^{2}-4q\sqrt{1-j^{2}}}\bigg{)}\,,$
$\displaystyle w_{3}$
$\displaystyle=\frac{1}{2}\bigg{(}2q+\sqrt{1-j^{2}}-\sqrt{1-2j^{2}+4q\sqrt{1-j^{2}}}\bigg{)}\,,$
$\displaystyle w_{4}$
$\displaystyle=\frac{1}{2}\bigg{(}2q+\sqrt{1-j^{2}}+\sqrt{1-2j^{2}+4q\sqrt{1-j^{2}}}\bigg{)}\,.$

For convenience it is useful to redefine $w_{2}$ and $w_{3}$ to be $w_{-}$ and
$w_{+}$ respectively, and the interval for $w$ is then $w\in[w_{-},w_{+}]$.
Furthermore, it can be shown that zero must always lie in this interval, i.e.
$w_{-}$ and $w_{+}$ have opposite sign. For all four of the roots to be real
we must impose

$-\frac{1}{\sqrt{2}}\leq
j\leq\frac{1}{\sqrt{2}}\,,\quad\frac{2j^{2}-1}{4\sqrt{1-j^{2}}}\leq
q\leq\frac{1-2j^{2}}{4\sqrt{1-j^{2}}}\,.$ (2.11)

In figure 1 we have plotted the parameter space of the roots. The blue region
is the region with four real roots, the red is the region with two real roots
and the orange is the region with no real roots.

Figure 1: The domain of validity for the roots of the quartic $f(w)$ in $j$
(horizontal axis) and $q$ (vertical axis) parameter space. The central blue
region has four real roots, the left and right orange regions have no real
roots and the top and bottom red regions have two real roots. The boundaries
of the regions are where there are two equal roots. The boundary from the
upper half is where $w_{3}=w_{4}$ and the boundary form the bottom half is
where $w_{1}=w_{2}$.

The boundaries between the different locations are the regions with double
roots, note that there are two special points in parameter space at $j=\pm
2^{-1/2},q=0$ where there are two pairs of double roots. In figure 1 these
points are located where the three different colours meet. The double roots
leads to a degeneration of the form of a (quotient of) 2d hyperbolic space,
$\mathbb{H}^{2}$ rather than the more canonical $\mathbb{R}^{2}$ quotient
studied here. In fact these solutions are the ones considered in Klemm:2014rda
, see also Gnecchi:2013mja , where one replaces the conical defects of the
spindle with cusps. Note that two cusps are not possible for a static
geometry, in that case only one cusp is possible leading to a black bottle
geometry.

Just as for the full black hole solution, we find that the geometry cannot be
made smooth at both poles, i.e. $\Sigma$ is a spindle. The period of $z$ is

$\frac{\Delta
z}{2\pi}=\frac{2\sqrt{P(w_{\pm})}}{|f^{\prime}(w_{\pm})|\,n_{\pm}}\,,$ (2.12)

with $n_{\pm}$ positive relatively prime integers. For later it is sometimes
convenient to define the $2\pi$-periodic coordinate $\hat{z}$ via,

$\hat{z}=\frac{2\pi}{\Delta z}\,{z}\,.$ (2.13)

In order to determine via which mechanism supersymmetry is preserved we
compute both the magnetic charge and the Euler characteristic of the spindle,
and compare them. First, the total magnetic charge is computed as an integral
of the field strength

$Q_{m}=\frac{1}{2\pi}\int_{\Sigma}\mathrm{d}A=-\frac{\Delta
z}{2\pi}\frac{8\sqrt{1-j^{2}}\;w(w-q)}{j^{2}+4(w-q)^{2}}\,\bigg{|}^{w=w_{+}}_{w=w_{-}}=-\frac{\Delta
z}{2\pi}\frac{\sqrt{1-j^{2}}\;w\,P^{\prime}(w)}{2P(w)}\,\bigg{|}^{w=w_{+}}_{w=w_{-}}\,.$
(2.14)

Since $w_{\pm}$ are roots of $f(w)$, it follows that
$P(w_{\pm})=(1-j^{2})w_{\pm}^{2}$ and further that the first derivative
satisfies $P^{\prime}(w_{\pm})=f^{\prime}(w_{\pm})+2(1-j^{2})w_{\pm}$. Using
these identities, we find the expression

$Q_{m}=\frac{\Delta
z}{4\pi\sqrt{1-j^{2}}}\bigg{(}\text{sgn}(w_{+})\frac{|f^{\prime}(w_{+})|}{|w_{+}|}+\text{sgn}(w_{-})\frac{|f^{\prime}(w_{-})|}{|w_{-}|}\bigg{)}\,,$
(2.15)

for the magnetic charge of the near-horizon solution. The electric charge is
defined by

$Q_{e}=\frac{1}{2\pi}\int_{\Sigma}\star F\,,$ (2.16)

which gives

$Q_{e}=\frac{\Delta
z}{2\pi}\frac{j\sqrt{1-j^{2}}w}{\sqrt{P(w)}}\bigg{|}_{w=w_{-}}^{w=w_{+}}=j\frac{\Delta
z}{\pi}\,.$ (2.17)

The Euler characteristic can be computed as an integral of the 2d Ricci scalar
over the Riemann surface. We find

$\chi(\Sigma)=\frac{1}{4\pi}\int_{\Sigma}R_{2}\>\text{dvol}_{\Sigma}=\frac{f(w)P^{\prime}(w)-P(w)f^{\prime}(w)}{P(w)^{3/2}}\,\frac{\Delta
z}{4\pi}\,\bigg{|}^{w=w_{+}}_{w=w_{-}}\,,$ (2.18)

which, after using the properties of $f(w)$ and $P(w)$ evaluated at the roots
of $f(w)$, yields the expression

$\chi(\Sigma)=\frac{\Delta
z}{4\pi\sqrt{1-j^{2}}}\bigg{(}\frac{|f^{\prime}(w_{+})|}{|w_{+}|}+\frac{|f^{\prime}(w_{-})|}{|w_{-}|}\bigg{)}\,.$
(2.19)

Application of the expression for the period in (2.12) gives

$Q_{m}=\frac{\text{sgn}(w_{+})}{n_{+}}+\frac{\text{sgn}(w_{-})}{n_{-}}\,,\qquad\chi(\Sigma)=\frac{1}{n_{+}}+\frac{1}{n_{-}}\,.$
(2.20)

We see that the charge (2.15) and the Euler characteristic (2.19) are equal if
$w_{-}$ and $w_{+}$ have the same sign, and unequal if they have opposite
signs. We found earlier in this section that zero always lies in the interval
$[w_{-},w_{+}]$, so we always find the latter case. This implies that
supersymmetry is preserved with a mechanism known as an anti-twist, rather
than with a twist, following the language of Ferrero:2021etw ; Couzens:2021cpk
.

Before we close this section let us give the roots in terms of the orbifold
weights. To solve the period condition (2.12) we must impose

$\displaystyle
q=\frac{(1-2j^{2})(n_{-}^{2}-n_{+})^{2}}{4\sqrt{1-j^{2}}(n_{-}^{2}+n_{+}^{2})}\,.$
(2.21)

We may further eliminate $j$ in favour of the electric charge by using (2.17),

$j=\frac{n_{-}n_{+}Q_{e}}{\sqrt{2(n_{-}^{2}+n_{+}^{2}+n_{-}^{2}n_{+}^{2}Q_{e}^{2})}}\,,$
(2.22)

and the roots become

$\displaystyle w_{+}$
$\displaystyle=\frac{n_{+}^{2}+3n_{-}^{2}+n_{-}^{2}n_{+}^{2}Q_{e}^{2}-2n_{-}\sqrt{2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}}}{2\sqrt{2}\sqrt{\big{(}n_{-}^{2}+n_{+}^{2}+n_{-}^{2}n_{+}^{2}Q_{e}^{2}\big{)}\big{(}2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}\big{)}}}\,,$
(2.23) $\displaystyle w_{-}$
$\displaystyle=\frac{-(3n_{+}^{2}+n_{-}^{2}+n_{-}^{2}n_{+}^{2}Q_{e}^{2})+2n_{+}\sqrt{2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}}}{2\sqrt{2}\sqrt{\big{(}n_{-}^{2}+n_{+}^{2}+n_{-}^{2}n_{+}^{2}Q_{e}^{2}\big{)}\big{(}2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}\big{)}}}\,.$
(2.24)

##  3 Uplift to massive type IIA

In the previous section we have reviewed the rotating spindle that can be
constructed in 4d $\mathcal{N}=2$ Einstein–Maxwell supergravity. We will now
consider uplifting the solution to massive type IIA supergravity focussing on
the uplift giving rise to the holographic duals of 3d Chern–Simons SCFTs
arising from D2 branes wrapped on a spindle. Massive type IIA supergravity on
a topological six-sphere gives rise to 4d $\mathcal{N}=8$ ISO(7) supergravity.
As shown in Varela:2019vyd  there exists a consistent truncation of this
theory to 4d $\mathcal{N}=2$ Einstein–Maxwell supergravity. We may therefore
uplift the known solution reviewed in the previous section to massive type IIA
and study its properties using the uplifting formula presented in
Varela:2019vyd . In general the uplift of a solution of 4d $\mathcal{N}=2$
Einstein–Maxwell in Einstein frame is

$\displaystyle\mathrm{d}s^{2}_{10}$
$\displaystyle=L^{2}\,(2+\cos^{2}\alpha)^{1/8}\sqrt{1+\cos^{2}\alpha}\>\bigg{[}\frac{1}{3}\mathrm{d}s^{2}_{4}+\mathrm{d}s^{2}_{Y_{6}}\bigg{]}\,,$
(3.1) $\displaystyle\mathrm{d}s^{2}_{Y_{6}}$
$\displaystyle=\frac{1}{2}\,\mathrm{d}\alpha^{2}+\frac{3\sin^{2}\alpha}{4+2\cos^{2}\alpha}\,\hat{\eta}^{2}+\frac{\sin^{2}\alpha}{1+\cos^{2}\alpha}\,\mathrm{d}s^{2}_{\text{KE}_{4}}\,,$

where

$\displaystyle\hat{\eta}$
$\displaystyle=\eta+\frac{1}{3}A=\mathrm{d}\psi+\sigma+\frac{1}{3}A\,,$ (3.2)
$\displaystyle\mathrm{d}\eta$ $\displaystyle=2\,J_{\text{KE}_{4}}\,,$ (3.3)

and $A$ is the 4d gauge field. The coordinate $\alpha$ is defined on an
interval $[0,\pi]$, and the metric on $Y_{6}$ is that of a sine cone, which is
squashed due to the $\alpha$-dependent prefactors. The Kähler–Einstein metric
is normalized to satisfy $R_{mn}=6g_{mn}$ and $J_{\text{KE}_{4}}$ is the
Kähler form. Together $\eta$ and $\mathrm{d}s^{2}_{\text{KE}_{4}}$ form a
squashed 5d Sasaki–Einstein manifold.

The other fields in the uplift to massive type IIA are given by

$\displaystyle\mathrm{e}^{\Phi}$
$\displaystyle=\frac{1}{m^{4/5}}\frac{(2+\cos^{2}\alpha)^{3/4}}{1+\cos^{2}\alpha}\,,$
(3.4) $\displaystyle H_{3}$
$\displaystyle=\frac{L^{2}}{m^{2/5}}\bigg{[}\frac{2\sin^{3}\alpha}{(1+\cos^{2}\alpha)^{2}}\,J_{\text{KE}_{4}}\wedge\mathrm{d}\alpha+\frac{1}{2\sqrt{3}}\sin\alpha\,\mathrm{d}\alpha\wedge\star_{4}F\bigg{]}\,,$
(3.5) $\displaystyle F_{0}$ $\displaystyle=\frac{m}{L}\,,$ (3.6)
$\displaystyle F_{2}$
$\displaystyle=L\,m^{3/5}\bigg{[}-\frac{\sin^{2}\alpha\,\cos\alpha}{(2+\cos^{2}\alpha)(1+\cos^{2}\alpha)}\,J_{\text{KE}_{4}}-\frac{3\sin\alpha\,(2-\cos^{2}\alpha)}{2(2+\cos^{2}\alpha)^{2}}\,\mathrm{d}\alpha\wedge\hat{\eta}$
$\displaystyle\;\;\;\;+\frac{\cos\alpha}{2(2+\cos^{2}\alpha)}\,F-\frac{1}{2\sqrt{3}}\cos\alpha\,\star_{4}\\!F\bigg{]}\,,$
(3.7) $\displaystyle F_{4}$
$\displaystyle=L^{3}\,m^{1/5}\bigg{[}\frac{\sin^{4}\alpha\,(2+3\cos^{2}\alpha)}{2(1+\cos^{2}\alpha)^{2}}\,J_{\text{KE}_{4}}^{2}+\frac{3\sin^{3}\alpha\,\cos\alpha\,(4+\cos^{2}\alpha)}{2(1+\cos^{2}\alpha)(2+\cos^{2}\alpha)}\,J_{\text{KE}_{4}}\wedge\mathrm{d}\alpha\wedge\hat{\eta}$
$\displaystyle\;\;\;\;+\frac{1}{\sqrt{3}}\,\mathrm{d}\mathrm{vol}_{4}-\frac{1}{4}\sin\alpha\,\cos\alpha\,\Big{(}\frac{2\sin\alpha\,\cos\alpha}{1+\cos^{2}\alpha}\,J_{\text{KE}_{4}}+\mathrm{d}\alpha\wedge\hat{\eta}\Big{)}\wedge
F$
$\displaystyle\;\;\;\;-\frac{1}{4\sqrt{3}}\Big{(}\frac{2\sin^{2}\alpha}{1+\cos^{2}\alpha}\,J_{\text{KE}_{4}}+\frac{3\sin\alpha\,\cos\alpha}{2+\cos^{2}\alpha}\,\mathrm{d}\alpha\wedge\hat{\eta}\Big{)}\wedge\star_{4}F\bigg{]}\,.$
(3.8)

We have redefined $L$ and $m$ with respect to some of the existing literature
in order to make $L$ have length scaling dimension 1 and for $m$ to scaling
dimension 0, where necessary we will provide the dictionary.

###  3.1 Metric regularity

We now want to study the regularity of the uplifted 10d solution. The method
for analysing the regularity closely follows Ferrero:2020twa  however we will
see that the global structure here is different to that considered there. In
particular whereas the conical singularities of the spindle may be de-
singularised in the uplift of Ferrero:2020twa  it is not possible here and
mimics more the behaviour of the M5-brane on a spindle solution
Ferrero:2021wvk  and the D4-brane on a spindle solution Faedo:2021nub .

We write the 10d geometry (3.1) as

$\mathrm{d}s^{2}_{10}=L^{2}\,(2+\cos^{2}\alpha)^{1/8}\sqrt{1+\cos^{2}\alpha}\>\bigg{[}\frac{\sqrt{P(w)}}{3}\,\mathrm{d}s^{2}_{\text{AdS}_{2}}+\mathrm{d}s^{2}_{8}\bigg{]}\,,$
(3.9)

and study the regularity of the 8d metric at fixed $(t,r)$,

$\mathrm{d}s^{2}_{8}=\frac{\sqrt{P(w)}}{3f(w)}\,\mathrm{d}w^{2}+\frac{f(w)}{3\sqrt{P(w)}}\,Dz^{2}+\frac{1}{2}\,\mathrm{d}\alpha^{2}+\frac{3\sin^{2}\alpha}{4+2\cos^{2}\alpha}\,\hat{\eta}^{2}+\frac{\sin^{2}\alpha}{1+\cos^{2}\alpha}\,\mathrm{d}s^{2}_{\text{KE}_{4}}\,,$
(3.10)

where

$Dz=\mathrm{d}z+jr\mathrm{d}t\,.$ (3.11)

We find this 8d space to be an $S^{1}_{z}\times\text{SE}_{5}$ fibration over a
rectangle parametrized by the coordinates $w$ and $\alpha$. A plot of this
rectangle is given in figure 2 where we indicate the regions where we will
study regularity in detail. In the interior of the rectangle regularity is
ensured, since none of the coefficients in (3.10) vanish there.

Figure 2: A schematic plot of the rectangle over which there is a
SE${}_{5}\times S^{1}_{z}$ fibration. A monopole is located at each corner of
the rectangle and they are called region III in the main text. The red
monopoles have weight $n_{+}$ whilst the blue monopoles have weight $n_{-}$.

#### Region I: the poles of the spindle.

We will first focus on the metric away from $\alpha=0,\pi$ and study the
degeneration at the two poles of the spindle. To that end we construct the
following U$(1)$ Killing vectors, each of which has vanishing norm at one of
the poles:

$\partial_{\phi_{\pm}}=c_{\pm}\left(\partial_{z}-\frac{1}{3}h(w_{\pm})\,\partial_{\psi}\right)\,,$
(3.12)

with $c_{\pm}$ constants that we will fix momentarily. By expanding the norm
of these Killing vectors around $w_{\pm}$ we find that888Here we use the
leading order expansions $f(w)=|f^{\prime}(w_{\pm})|\,\Delta w$ and
$P(w)=(1-j^{2})w_{\pm}^{2}$ around $\Delta w=0$.

$||\partial_{\phi_{\pm}}||^{2}=\frac{c_{\pm}^{2}\,|f^{\prime}(w_{\pm})|}{3\sqrt{1-j^{2}}\,|w_{\pm}|}\,\Delta
w+\mathcal{O}(\Delta w^{2})\,,$ (3.13)

where $\Delta w=\mp(w-w_{\pm})$ is the positive distance away from the pole.
Consequently, the metric spanned by $w$ and $\phi_{\pm}$ close to $w_{\pm}$
can be written as

$\frac{\sqrt{1-j^{2}}\,|w_{\pm}|}{3|f^{\prime}(w_{\pm})|\Delta
w}\,\mathrm{d}w^{2}+\frac{c_{\pm}^{2}\,|f^{\prime}(w_{\pm})|\Delta
w}{3\sqrt{1-j^{2}}\,|w_{\pm}|}\,\mathrm{d}\phi_{\pm}^{2}=\frac{4\sqrt{1-j^{2}}\,|w_{\pm}|}{3|f^{\prime}(w_{\pm})|}\left[\mathrm{d}R^{2}+\frac{c_{\pm}^{2}\,f^{\prime}(w_{\pm})^{2}}{4(1-j^{2})w_{\pm}^{2}}\,R^{2}\mathrm{d}\phi_{\pm}^{2}\right]\,,$
(3.14)

where we have introduced the radial coordinate as $R^{2}=\Delta w$. In order
for the Killing vector $\partial_{\phi_{\pm}}$ to degenerate smoothly at the
pole, we see that we need to normalize it as

$c_{\pm}=\frac{2\sqrt{1-j^{2}}\,w_{\pm}}{f^{\prime}(w_{\pm})}\,.$ (3.15)

The coordinates $\phi_{\pm}$ now have periodicity $2\pi$ around $w=w_{\pm}$
respectively and the conical singularities are resolved. This works in an
analogous way to the discussion in Ferrero:2020twa  for the uplift of this
solution to 11d supergravity on a Sasaki–Einstein 7-manifold.

Let us study the metric close to the poles in a bit more detail. We switch to
the coordinates $\phi_{\pm}$ using the transformations

$\displaystyle z$
$\displaystyle=2\sqrt{1-j^{2}}\left(\frac{w_{+}}{f^{\prime}(w_{+})}\,\phi_{+}-\frac{w_{-}}{f^{\prime}(w_{-})}\,\phi_{-}\right)\,,$
(3.16) $\displaystyle\psi$
$\displaystyle=-\tfrac{2}{3}\sqrt{1-j^{2}}\left(\frac{w_{+}\,h(w_{+})}{f^{\prime}(w_{+})}\,\phi_{+}-\frac{w_{-}\,h(w_{-})}{f^{\prime}(w_{-})}\,\phi_{-}\right)\,.$

We may rewrite the metric in these new coordinates as

$\displaystyle\mathrm{d}s^{2}_{8}=\frac{\sqrt{P(w)}}{3f(w)}\,\mathrm{d}w^{2}+\frac{1}{2}\,\mathrm{d}\alpha^{2}+F_{+}\,D\phi_{+}^{2}+F_{-}\,D\phi_{-}^{2}+G\,D\phi_{+}D\phi_{-}+\frac{\sin^{2}\alpha}{1+\cos^{2}\alpha}\,\mathrm{d}s^{2}_{\text{KE}_{4}}\,,$
(3.17)

where

$\displaystyle
D\phi_{\pm}=\mathrm{d}\phi_{\pm}+\frac{3f^{\prime}(w_{\pm})}{2\sqrt{1-j^{2}}\,w_{\pm}(h(w_{\mp})-h(w_{\pm})}\left(\sigma+\tfrac{1}{3}jr\,h(w_{\mp})\,\mathrm{d}t\right)\,,$
(3.18)

and

$\displaystyle F_{\pm}$
$\displaystyle=\frac{2(1-j^{2})w_{\pm}^{2}\Big{(}2(2+\cos^{2}\alpha)f(w)+\sin^{2}\alpha\big{(}h(w)-h(w_{\pm})\big{)}^{2}\sqrt{P(w)}\Big{)}}{3(2+\cos^{2}\alpha)\sqrt{P(w)}f^{\prime}(w_{\pm})^{2}}\,,$
(3.19) $\displaystyle G$
$\displaystyle=\frac{4(1-j^{2})w_{+}w_{-}\Big{(}2(2+\cos^{2}\alpha)f(w)+\sin^{2}\alpha\big{(}h(w)-h(w_{+})\big{)}\big{(}h(w)-h(w_{-})\big{)}\sqrt{P(w)})\Big{)}}{3(2+\cos^{2}\alpha)\sqrt{P(w)}f^{\prime}(w_{+})f^{\prime}(w_{-})}\,.$

Note that $F_{+}(w_{+})=G(w_{+})=0$ and $F_{-}(w_{-})=G(w_{-})=0$. The
$w,\phi_{\pm}$ coordinates are topologically an $S^{3}$ at fixed $\alpha$ and
we see that we have an $S^{3}$ fibration over a Kähler–Einstein base. This is
similar to the M2-brane case.

#### Region II.

In order to study the metric close to $\alpha=0,\pi$, we simply Taylor expand
the trigonometric functions in the metric around these points. This yields

$\mathrm{d}s^{2}_{8}=\frac{\sqrt{P(w)}}{3f(w)}\,\mathrm{d}w^{2}+\frac{f(w)}{3\sqrt{P(w)}}\,Dz^{2}+\frac{1}{2}\Big{(}\mathrm{d}R^{2}+R^{2}\,\big{(}\hat{\eta}^{2}+\mathrm{d}s^{2}_{\text{KE}_{4}}\big{)}\Big{)}\,,$
(3.20)

where we use $R=\alpha,\pi-\alpha$ for the distance away from $\alpha=0,\pi$
respectively. We see that this is the metric of a 6d cone fibered over the
spindle. When KE${}_{4}=\mathbb{CP}^{2}$ this is $\mathbb{R}^{6}$ and free of
singularities, while for other Kähler–Einstein manifolds it has a singularity
at the tip of the Calabi–Yau cone. This singularity appears also in the AdS4
vacuum solution and is thus expected to remain.

#### Region III: monopoles.

In addition to the limits to the endpoints of the $w$ and $\alpha$ intervals,
we should also take the limit where both $w$ and $\alpha$ go to an endpoint
simultaneously, i.e. to a corner of the rectangle in figure 2.

We denote the endpoints of the intervals by
$\alpha\in[\alpha_{-},\alpha_{+}]=[0,\pi]$ and $w\in[w_{-},w_{+}]$. In order
to take the combined limit, we need a coordinate transformation to ‘polar
coordinates’ around the corner of the rectangle. A proper set of such
coordinates is given by $(R,\zeta)$ defined by

$\alpha=\alpha_{\pm}\mp R\cos\zeta\,,\qquad\quad
w=w_{\pm}\mp\left|\frac{3f^{\prime}(w_{\pm})}{8\sqrt{1-j^{2}}\,{w_{\pm}}}\right|R^{2}\sin^{2}\zeta\,,$
(3.21)

where the signs are chosen such that $R$ is a positive radial coordinate for
each of the four corners. The scalings and powers in (3.21) are chosen such
that we find the appropriate metric
($\mathrm{d}R^{2}+R^{2}\mathrm{d}\zeta^{2}$) in polar coordinates for large
$R$. The combined limit $\alpha\rightarrow\alpha_{\pm},w\rightarrow w_{\pm}$
is now simply $R\rightarrow 0$.

Changing to these new coordinates and expanding around $R=0$ yields the 8d
metric

$\displaystyle\mathrm{d}s^{2}_{8}=\frac{1}{2}\bigg{[}\mathrm{d}R^{2}+R^{2}\,\bigg{(}\mathrm{d}\zeta^{2}+\cos^{2}\zeta\,\Big{(}\big{(}\mathrm{d}\psi+\sigma+\tfrac{1}{3}A\big{)}^{2}+\mathrm{d}s^{2}_{\text{KE}_{4}}\Big{)}+\sin^{2}\zeta\>\frac{f^{\prime}(w_{\pm})^{2}}{4(1-j^{2})\,w_{\pm}^{2}}\,D{z}^{2}\bigg{)}\bigg{]}\,.$
(3.22)

Here the gauge field, which now appears as a fibration term in the one-form
$\hat{\eta}$ is given by $A=h(w_{\pm})\,(\mathrm{d}z+jr\,\mathrm{d}t)$. The
$\mathrm{d}z$ leg is constant so it can be absorbed by a shift in $\psi$.
Furthermore, the period of $z$ at the poles of the spindle is given by

$\Delta
z=\frac{4\pi\sqrt{1-j^{2}}\,|w_{\pm}|}{n_{\pm}|f^{\prime}(w_{\pm})|}\,.$
(3.23)

If we rescale $z$ such that it is $2\pi$ periodic, i.e. use the coordinate
$\hat{z}$ introduced in (2.13), we find the metric

$\displaystyle\mathrm{d}s^{2}_{8}=\frac{1}{2}\bigg{[}\mathrm{d}R^{2}+R^{2}\,\bigg{(}\mathrm{d}\zeta^{2}+\cos^{2}\zeta\,\Big{(}\big{(}\mathrm{d}\hat{\psi}+\sigma+\tfrac{1}{3}jr\,h(w_{\pm})\,\mathrm{d}t\big{)}^{2}+\mathrm{d}s^{2}_{\text{KE}_{4}}\Big{)}+\frac{1}{n_{\pm}^{2}}\sin^{2}\zeta\,D\hat{z}^{2}\bigg{)}\bigg{]}\,,$
(3.24)

where $\hat{\psi}$ and $\hat{z}$ are the shifted and rescaled coordinates with
correspondingly rescaled one-form $D\hat{z}$. Note that when the
Kähler–Einstein space is $\mathbb{CP}^{2}$ this is
$\mathbb{R}^{8}/\mathbb{Z}_{n_{\pm}}$. We recognize this geometry as a
monopole of weight $n_{\pm}$. The presence of these monopoles at the corners
of the $(w,\alpha)$ rectangle explains the singularities there and these
singularities cannot be resolved. This is most reminiscent of the case of
M5-branes on a spindle or disc where the uplifted theory still admits a
conical singularity Ferrero:2021wvk ; Bah:2021hei ; Bah:2021mzw . In appendix
A we have analyzed the M5-brane solution studied there in this light and found
the same type of singularity structure.

We emphasise that these monopoles and their corresponding singularities have
not been found in the uplift of this 4d setup to 11d supergravity, see
Ferrero:2020twa ; Couzens:2021rlk ; Couzens:2021cpk ; Ferrero:2021ovq .
Instead, in the M2-brane setup, the singularities of the 4d theory are removed
in the uplifted theory (under suitable conditions on the fluxes, which are the
same ones we take here). Here we find the novel feature that the singularities
cannot be removed in a different uplift of the same black hole solution.
Moreover, whereas the M5-brane solution realises supersymmetry via a twist,
here supersymmetry is realised via an anti-twist and therefore removal of
singularities is dependent on the uplift and not the particular twist used.

###  3.2 Flux quantization

We now want to quantize the fluxes of the solution. This is most simply
studied in string frame as it removes the need to take into account powers of
the dilaton. Consider first the Romans mass, it must satisfy

$F_{0}=\frac{n}{2\pi\ell_{s}}\,,\qquad n\in\mathbb{Z}\,,$ (3.25)

in order for the theory to be well-defined. Here $n$ can be interpreted as the
number of D8-branes present in the setup.

In order to work out the quantization of the remaining fluxes, we first
compute the corresponding Page fluxes. Since the Maxwell charges are
generically not-closed they do not give rise to conserved quantities. One
should instead consider Page charges since these are closed and thus
conserved. The cost of using the Page charges is that they are not gauge
invariant quantities, while the Maxwell charges are. The Page fluxes, using
polyform notation, are defined as

$\hat{f}=F\wedge\mathrm{e}^{-B_{2}}\,,$ (3.26)

where $F$ is understood to be the magnetic part of the flux999Note that in
this notation some of these fluxes $F_{p}$ obtain a minus sign with respect to
the Hodge star of the corresponding $(10-p)$ form. This is the case if $p/2$
is odd.

$F=F_{0}+F_{2}+F_{4}+F_{6}+F_{8}+F_{10}\Big{|}_{\text{magnetic}}\,,$ (3.27)

in string frame. The quantization conditions on the Page fluxes $\hat{f}_{p}$
read

$\frac{1}{(2\pi\ell_{s})^{p-1}}\int_{\Sigma_{p}}\hat{f}_{p}\,\in\,\mathbb{Z}$
(3.28)

where $\Sigma_{p}$ is an integral cycle in the compact 8d internal space. Let
us consider all possible cycles that we can construct in the geometry. There
is a single two-cycle which is simply the spindle at a fixed point on the
squashed sine cone $Y_{6}$ which we can take to be at either of the endpoints
of the $\alpha$ interval. There are no topological four-cycles in the geometry
that we can integrate the four-form flux over. There is a single six-cycle
which is given by $Y_{6}$ at a fixed point on the spindle and a single eight-
cycle given by the full internal space.

To proceed we need to choose a gauge for the $B$-field. We take

$B_{2}=F_{2}/F_{0}\,,$ (3.29)

which can be recognized as a proper choice for $B_{2}$ directly from the
Bianchi identities of massive type IIA, and can also be checked from
(3.5)-(3.7). Using this gauge choice we can compute the Page fluxes, e.g. as

$\hat{f}_{6}=-\star F_{4}-F_{4}\wedge
B_{2}+F_{2}\wedge\frac{B_{2}^{{\,2}}}{2}-F_{0}\,\frac{B_{2}^{\,3}}{6}\,,$
(3.30)

with $\star$ to be taken in the string frame metric. The quantization of this
six-form flux yields the condition

$\frac{1}{(2\pi\ell_{s})^{5}}\int_{Y_{6}}\hat{f}_{6}=\frac{1}{(2\pi\ell_{s})^{5}}\,\frac{16L^{5}}{3m^{1/5}}\,\text{Vol}_{\text{SE}_{5}}=N\,\in\,\mathbb{Z}\,.$
(3.31)

Here this $N$ can be interpreted as the number of D2-branes which give rise to
the 3d theory before being wrapped on the spindle. Together with the condition
(3.25) on $F_{0}$, this leads to the following quantization conditions on the
parameters $L$ and $m$:

$L=\frac{2^{1/6}3^{5/24}\,\pi\ell_{s}\,n^{1/24}N^{5/24}}{\text{Vol}_{\text{SE}_{5}}^{5/24}}\,,\quad\qquad
m=\frac{3^{5/24}\,n^{25/24}N^{5/24}}{2^{5/6}\,\text{Vol}_{\text{SE}_{5}}^{5/24}}\,.$
(3.32)

Note that these expressions agree with Fluder:2015eoa  for the parent theory,
after taking into account the redefinitions
$L_{\text{here}}=2^{5/16}3^{1/2}\,L_{\text{there}}$ and
$m_{\text{here}}=2^{5/16}\,(\mathrm{e}^{-5\Phi_{0}/4})_{\text{there}}$. It
turns out that the two-form Page flux vanishes. The eight-form flux
$\hat{f}_{8}$ does not vanish, but its integral over the internal space does.
The quantization conditions for these fluxes are therefore trivially satisfied
and we only have these two flux quanta to consider.

We may now compute the holographic free energy, which is simply given by

$\mathcal{F}_{1d}=\frac{1}{4G_{2}}$ (3.33)

with $G_{2}$ the 2d Newton constant, which is computed by integrating over the
volume of the internal space giving

$\frac{1}{G_{2}}=\frac{1}{G_{10}}\int_{\mathcal{M}_{8}}\text{dvol}_{8}=\frac{L^{8}\,\text{Vol}_{\text{SE}_{5}}}{3^{3/2}\,5\,\pi^{6}\ell_{s}^{8}}\,\Delta
z\,(w_{+}-w_{-})\,,$ (3.34)

where we have used that $G_{10}=2^{3}\pi^{6}\ell_{s}^{8}$. Using the
quantization of $L$ (3.32), we find

$\frac{1}{G_{2}}=\frac{2^{4/3}3^{1/6}\pi^{2}}{5\,\text{Vol}_{\text{SE}_{5}}^{2/3}}\,\Delta
z\,(w_{+}-w_{-})\,n^{1/3}\,N^{5/3}\,.$ (3.35)

The free-energy in terms of the orbifold weights and electric charge is then

$\mathcal{F}_{1d}=\frac{3^{1/6}\pi^{3}}{2^{2/3}5\text{Vol}_{\text{SE}_{5}}^{2/3}}n^{1/3}N^{5/3}\frac{\sqrt{2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}}-(n_{-}+n_{-})}{n_{-}n_{+}}\,.$
(3.36)

This may be further simplified by writing it in terms of the free-energy of
the 3d parent theory. Following the conventions in Fluder:2015eoa  the free
energy of the 3d parent theory is

$\mathcal{F}_{3d}=\frac{2^{1/3}3^{1/6}\pi^{3}}{5\text{Vol}_{\text{SE}_{5}}}n^{1/3}N^{5/3}\,,$
(3.37)

and therefore

$\mathcal{F}_{1d}=\frac{\sqrt{2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}}-(n_{-}+n_{-})}{4n_{-}n_{+}}\mathcal{F}_{3d}\,.$
(3.38)

It would be interesting to understand this from a dual field theory
computation.

##  4 M5-branes on SLAG-3 $\times$ spindle

###  4.1 The holographic duals of M5-branes on a SLAG-3 manifold

In the previous section we have studied the uplift of 4d Einstein–Maxwell to
massive type IIA using the known uplift formula. There is a further uplift
that we will consider which arises from considering M5-branes on a SLAG
3-manifold Gauntlett:2007ma . The AdS4 vacuum solution has purely magnetic
four-form flux. This may be further generalised by considering the uplift in
Larios:2019lxq , which uplifts to the most general $\mathcal{N}=2$ AdS4
solutions in 11d supergravity with mixed magnetic and electric fluxes, as
classified in Gabella:2012rc . We will consider only the uplift to the SLAG
3-manifold theories in the following. We will present the general uplift of
the solutions before reducing to considering the 3-manifold to be (a quotient
of) $\mathbb{H}^{3}$.

The metric of the seed 11d AdS4 solutions takes the form

$\displaystyle\mathrm{d}s^{2}_{11}=$
$\displaystyle\frac{1}{m\lambda}\mathrm{d}s^{2}(\text{AdS}_{4})+\mathrm{d}s^{2}(\mathcal{N}_{7})\,,$
$\displaystyle\mathrm{d}s^{2}(\mathcal{N}_{7})=$
$\displaystyle\mathrm{d}s^{2}(\mathcal{M}_{\text{SU}(2)})+\hat{w}^{2}+\frac{\lambda^{2}}{4m^{2}}\Big{(}\frac{1}{1-\lambda^{3}\rho^{2}}\mathrm{d}\rho^{2}+\rho^{2}\mathrm{d}\phi^{2}\Big{)}\,,$
(4.1)

where $\mathcal{M}_{\text{SU}(2)}$ is a four-dimensional space equipped with a
set of SU$(2)$ structure forms which we denote by $J^{a}$ and $\hat{w}$ is a
one-form. The metric is supported by the four-form flux

$G=\frac{1}{2m}\mathrm{d}\phi\wedge\mathrm{d}\bigg{[}\frac{\sqrt{1-\lambda^{3}\rho^{2}}}{\sqrt{\lambda}}J^{3}\bigg{]}\,.$
(4.2)

The solution is determined by a solution to the three differential conditions

$\displaystyle\mathrm{d}\Big{[}\frac{\sqrt{1-\lambda^{3}\rho^{2}}}{\lambda}\hat{w}\Big{]}=$
$\displaystyle\,m\lambda^{-1/2}J^{1}+\frac{\lambda^{2}\rho}{2\sqrt{1-\lambda^{3}\rho^{2}}}\hat{w}\wedge\mathrm{d}\rho\,,$
$\displaystyle\mathrm{d}\Big{[}\lambda^{-3/2}J^{3}\wedge\hat{w}-\frac{\lambda\rho}{2m\sqrt{1-\lambda^{3}\rho^{2}}}J^{2}\wedge\mathrm{d}\rho\Big{]}=$
$\displaystyle\,0\,,$ (4.3)
$\displaystyle\mathrm{d}\Big{[}J^{2}\wedge\hat{w}+\frac{\lambda^{-1/2}}{2m\rho\sqrt{1-\lambda^{3}\rho^{2}}}J^{3}\wedge\mathrm{d}\rho\Big{]}=$
$\displaystyle\,0\,.$

The KK-reduction of the solution to 4d $\mathcal{N}=2$ Einstein–Maxwell
supergravity was performed in Gauntlett:2007ma . The truncation ansatz is

$\displaystyle\mathrm{d}s^{2}_{11}=$
$\displaystyle\,\frac{1}{m\lambda}\mathrm{d}s^{2}_{4}+\mathrm{d}s^{2}(\mathcal{N}^{g}_{7})\,,$
$\displaystyle\mathrm{d}s^{2}(\mathcal{N}_{7})=$
$\displaystyle\,\mathrm{d}s^{2}(\mathcal{M}_{\text{SU}(2)})+\hat{w}^{2}+\frac{\lambda^{2}}{4m^{2}}\Big{(}\frac{1}{1-\lambda^{3}\rho^{2}}\mathrm{d}\rho^{2}+\rho^{2}\big{(}\mathrm{d}\phi+A)^{2}\Big{)}\,,$
(4.4) $\displaystyle\hat{G}=$ $\displaystyle\,G^{g}+F\wedge Y+\star_{4}F\wedge
X\,,$

where the subscript ‘$g$’ implies the gauging of the Killing vector
$\partial_{\phi}$ by the graviphoton of the 4d theory. At the level of forms
this is equivalent to the replacement
$\mathrm{d}\phi\rightarrow\mathrm{d}\phi+A$. The two two-forms $X,Y$ are given
by

$\displaystyle X=$
$\displaystyle\,-\frac{1}{2m}\Big{(}\mathrm{e}^{-\tfrac{1}{2}\lambda}J^{1}+\frac{\mathrm{e}^{2\lambda}\rho}{2\sqrt{1-\mathrm{e}^{3\lambda}\rho^{2}}}\hat{w}\wedge\mathrm{d}\rho\Big{)}\,,$
(4.5) $\displaystyle Y=$
$\displaystyle\,-\frac{1}{2m}\mathrm{e}^{-\tfrac{1}{2}\lambda}\sqrt{1-\mathrm{e}^{3\lambda}\rho^{2}}J^{3}\,.$

With the uplift formulae in hand, given a solution to (4.3) it is a simple
manner to study the uplifted solution. We will consider the uplift to 11d of a
solution originally constructed by Pernici and Sezgin in Pernici:1984nw  which
was subsequently rediscovered in Gauntlett:2000ng ; Acharya:2000mu  by
studying wrapped M5-branes. The solution is the gravity dual of a stack of
M5-branes wrapped on a quotient of 3d hyperbolic space $\mathbb{H}^{3}$. Since
the solution we will study was originally found by studying (a truncation) of
maximal 7d gauged supergravity, after the uplift we are able to interpret the
solution as the holographic dual of M5-branes wrapped on the 5d space
consisting of the direct product of a spindle and $\mathbb{H}^{3}$.

###  4.2 The Pernici–Sezgin solution and uplift

Let us first consider maximal 7d gauged supergravity. We follow the
conventions in Gauntlett:2000ng , where the solution we are interested in was
rediscovered. The Lagrangian for the bosonic fields is

$\displaystyle 2\mathcal{L}=$
$\displaystyle\,\bigg{[}R+\frac{1}{2}m^{2}(T^{2}-2T_{ij}T^{ij})-P_{\mu
ij}P^{\mu
ij}-\frac{1}{2}(\tensor{\Pi}{{}_{A}^{i}}\tensor{\Pi}{{}_{B}^{j}}F_{\mu\nu}^{AB})^{2}-m^{2}\big{(}\tensor{\Pi}{{}^{-1}_{i}{}^{A}}S_{\mu\nu\rho,A}\big{)}^{2}\bigg{]}\mathrm{d}\mathrm{vol}$
$\displaystyle-6m\delta^{AB}S_{A}\wedge
F_{B}+\sqrt{3}\epsilon_{ABCDE}\delta^{AG}S_{G}\wedge F^{BC}\wedge
F^{DE}+\frac{1}{8m}(2\Omega_{5}[A]-\Omega_{3}[A])\,,$ (4.6)

where the indices $A,B,..=1,..,5$ denote the indices of the SO$(5)$ gauge
group, while latin indices $i,j=1,..,5$ are for the SO$(5)_{c}$ local
composite gauge group and are raised with the Kronecker delta. There are 14
scalars $\tensor{\Pi}{{}_{A}^{i}}$ living in the coset
SL$(5,\mathbb{R})/$SO$(5)_{c}$ and transform in the $\boldsymbol{5}$ of
SO$(5)$ from the left and SO$(5)_{c}$ from the right. The scalar kinetic term,
$P_{\mu ij}$ is the symmetric part of

$\tensor{\Pi}{{}^{-1}_{i}{}^{A}}\big{(}\delta_{A}^{B}\partial_{\mu}+g\tensor{A}{{}_{\mu}{}_{A}^{B}}\big{)}\tensor{\Pi}{{}_{B}^{k}}\delta_{kj}\,,$
(4.7)

while the composite gauge field $Q_{\mu ij}$ is the anti-symmetric part. The
gauge fields are denoted by $\tensor{A}{{}_{A}^{B}}$ with field strength
$\tensor{F}{{}_{A}^{B}}$ and coupling constant $g=2m$. The potential terms are

$T_{ij}=\tensor{\Pi}{{}^{-1}_{i}{}^{A}}\tensor{\Pi}{{}^{-1}_{j}{}^{B}}\delta_{AB}\,,\qquad
T=\delta^{ij}T_{ij}\,.$ (4.8)

The last terms are Chern–Simons terms for the gauge fields, however they will
not play a role and so we suppress their details.

The solutions of interest are of the form AdS${}_{4}\times\mathbb{H}^{3}$. The
$SO(3)$ spin connection of $\mathbb{H}^{3}$ is identified with an SO$(3)$
subgroup of the R-symmetry via the splitting

$\text{SO}(5)\rightarrow\text{SO}(3)\times\text{SO}(2)\,,$ (4.9)

with both the scalars and gauge vectors obeying the symmetry. The solutions
here will have only a single scalar excited which allows us to write the
scalars as

$\tensor{\Pi}{{}_{A}^{i}}=\text{diag}\big{(}\mathrm{e}^{2\Phi},\mathrm{e}^{2\Phi},\mathrm{e}^{2\Phi},\mathrm{e}^{-3\Phi},\mathrm{e}^{-3\Phi}\big{)}\,.$
(4.10)

The gauge fields take a similar form, becoming only SO$(3)$ valued, i.e. only
gauge fields with the indices $1,2,3$ are turned on. Let us split the SO$(5)$
indices as $A=\\{a,\hat{a}\\}$ with $a=1,2,3$ and $\hat{a}=4,5$, then the
gauge fields are

$gA_{ab}=\omega_{ab}\,,$ (4.11)

where $\omega_{ab}$ is the spin-connection on $\mathbb{H}^{3}$. The metric is

$\mathrm{d}s^{2}_{7}=\frac{\mathrm{e}^{8\Phi}}{m^{2}}\Big{(}\mathrm{d}s^{2}(\text{AdS}_{4})+\frac{1}{2}\mathrm{d}s^{2}(\mathbb{H}^{3})\Big{)}\,,$
(4.12)

where the metric on $\mathbb{H}^{3}$ is normalised to have constant curvature
$R_{\mu\nu}=-g_{\mu\nu}$ and the scalar takes the constant value

$\mathrm{e}^{10\Phi}=2\,.$ (4.13)

This may be uplifted to 11d supergravity on an $S^{4}$ using the consistent
truncation in Pernici:1984xx ; Nastase:1999cb . The metric is given by

$\mathrm{d}s^{2}_{11}=\Delta^{-2/5}\mathrm{d}s_{7}^{2}+\frac{1}{m^{2}}\Delta^{4/5}\Big{[}\mathrm{e}^{4\Phi}DY^{a}DY^{a}+\mathrm{e}^{-6\Phi}\mathrm{d}Y^{\hat{a}}\mathrm{d}Y^{\hat{a}}\Big{]}\,,$
(4.14)

where the $Y^{A}$ are embedding coordinates for the $S^{4}$ satisfying
$Y^{a}Y^{a}+Y^{\hat{a}}Y^{\hat{a}}=1$. The gauged one-forms are defined to be

$DY^{a}=\mathrm{d}Y^{a}+2mA^{ab}Y^{b}\,,$ (4.15)

and the warp factor is

$\Delta^{-6/5}=\mathrm{e}^{-4\Phi}Y^{a}Y^{a}+\mathrm{e}^{6\Phi}Y^{\hat{a}}Y^{\hat{a}}\,.$
(4.16)

The expression for the flux will be given momentarily.

To make contact with the form of the metric in (4.1) we should introduce
explicit coordinates for the embedding coordinates $Y^{A}$. Following
Gauntlett:2006ux  we introduce the coordinates

$Y^{a}=\sqrt{1-\frac{\rho^{2}}{8}}\mu^{a}\,,\qquad
Y^{4}=\frac{1}{2\sqrt{2}}\rho\sin\phi\,,\qquad
Y^{5}=\frac{1}{2\sqrt{2}}\rho\cos\phi\,,$ (4.17)

with $\mu^{a}$ embedding coordinates for an $S^{2}$. The scalar $\Delta$ takes
the form

$\Delta^{-6/5}=\frac{\mathrm{e}^{-4\Phi}}{8}(8+\rho^{2})\,.$ (4.18)

The final metric is then

$\displaystyle\mathrm{d}s^{2}_{11}=\mathrm{e}^{-\lambda}\bigg{[}\mathrm{d}s^{2}(\text{AdS}_{4})+\frac{1}{2}\mathrm{d}s^{2}(\mathbb{H}^{3})+\frac{8-\rho^{2}}{8+\rho^{2}}D\mu^{a}D\mu^{a}+\frac{1}{2}\Big{(}\frac{\rho^{2}}{8+\rho^{2}}\mathrm{d}\phi^{2}+\frac{1}{8-\rho^{2}}\mathrm{d}\rho^{2}\Big{)}\bigg{]}\,,$
(4.19)

while the flux is given by

$G=\frac{1}{2}\mathrm{d}\phi\wedge\mathrm{d}\bigg{[}\mathrm{e}^{-\tfrac{1}{2}\lambda}\sqrt{1-\mathrm{e}^{3\lambda}\rho^{2}}J_{3}\bigg{]}\,,$
(4.20)

where we defined

$\mathrm{e}^{3\lambda}=\frac{2}{8+\rho^{2}}\,,$ (4.21)

and the two-form $J_{3}$ takes the form

$J_{3}=\frac{\mathrm{e}^{-\lambda}}{8}\epsilon^{abc}\mu^{a}\Big{(}\big{(}1-\mathrm{e}^{3\lambda}\rho^{2})D\mu^{b}\wedge
D\mu^{c}-\frac{1}{2}e_{\mathbb{H}^{3}}^{b}\wedge
e_{\mathbb{H}^{3}}^{c}\Big{)}\,.$ (4.22)

For later it is useful to note that the quantisation of the flux through the
$S^{4}$ gives the number of M5-brane

$\frac{1}{(2\pi\ell_{p})^{3}}\int_{S^{4}}G=\frac{1}{\pi
m^{3}\ell_{p}^{3}}\equiv N\in\mathbb{Z}\,,$ (4.23)

and that the free energy is

$\mathcal{F}_{3}=\frac{N^{3}}{3\pi}\text{Vol}(\mathbb{H}^{3})\,.$ (4.24)

Having given the AdS4 solution, and noted that the R-symmetry vector is
written explicitly, we can now truncate to 4d Einstein–Maxwell supergravity
using the consistent truncation given in Gauntlett:2007ma  and reviewed in the
previous section. We may further view this as a truncation of maximal 7d
gauged supergravity on $\mathbb{H}^{3}$ to 4d Einstein–Maxwell with a little
rewriting. The presence of the 4d graviphoton leads to a gauging of the
embedding coordinates $Y^{\hat{a}}$ in (4.14). That is we turn on a new
gauging

$\mathrm{d}Y^{\hat{a}}\rightarrow
DY^{\hat{a}}=\mathrm{d}Y^{\hat{a}}+A^{\hat{a}\hat{b}}Y^{\hat{b}}\,,$ (4.25)

where the only non-trivial component is

$A^{45}=A\,,$ (4.26)

with $A$ the graviphoton of the 4d theory. From the 7d truncation this amounts
to keeping the same scalars of the seed theory and modifying the gauge fields
to include an extra component in the SO$(2)$ of the SO$(5)$ splitting we
performed in (4.9). where $A$ is the graviphoton of the 4d theory. By
construction, and the results of Gauntlett:2007ma  this leads to a consistent
truncation of the 7d theory on $\mathbb{H}^{3}$ to 4d Einstein–Maxwell
supergravity.

###  4.3 AdS${}_{2}\ltimes\Sigma\times\mathbb{H}^{3}$ solutions

We may now use the consistent truncation outlined above of 7d maximal gauged
supergravity to 4d Einstein–Maxwell supergravity to uplift the spindle
solution of section 2. The 7d solution is then

$\displaystyle\mathrm{d}s^{2}=$
$\displaystyle\,\mathrm{e}^{8\Phi}\Big{(}\sqrt{P}(w)\Big{(}\mathrm{d}s^{2}(\text{AdS}_{2})+\mathrm{d}s^{2}_{\Sigma}\Big{)}+\frac{1}{2}\mathrm{d}s^{2}(\mathbb{H}^{3})\Big{)}\,,$
(4.27) $\displaystyle A^{AB}=$ $\displaystyle\,\begin{pmatrix}\omega^{ab}&0\\\
0&\mathcal{A}^{\hat{a}\hat{b}}\end{pmatrix}\,,$ (4.28)
$\displaystyle\tensor{\Pi}{{}_{A}^{i}}=$
$\displaystyle\,\text{diag}\big{(}\mathrm{e}^{2\Phi},\mathrm{e}^{2\Phi},\mathrm{e}^{2\Phi},\mathrm{e}^{-3\Phi},\mathrm{e}^{-3\Phi}\big{)}\,,$
(4.29)

where

$\mathcal{A}^{\hat{a}\hat{b}}=\epsilon^{\hat{a}\hat{b}}h(w)(\mathrm{d}z+jr\mathrm{d}t)\,,\qquad\mathrm{e}^{10\Phi}=2\,.$
(4.30)

Uplifting to 11d supergravity we find the (fibered) AdS2 metric101010In
Couzens:2020jgx  fibered AdS2 solutions of 11d supergravity with electric
four-form flux were classified. The uplift of the spindle solution we consider
here to 11d supergravity on a 7d Sasaki–Einstein manifold as studied in
Ferrero:2020twa  can be embedded into the classification. However since we
have magnetic flux here these solutions are outside the classification in
Couzens:2020jgx . It would be interesting to extend the classification
allowing for all fluxes turned on, giving a fibered AdS2 version of
Hong:2019wyi , in the same spirit that Couzens:2020jgx  gave a fibered version
of Kim:2006qu

$\displaystyle\mathrm{d}s^{2}_{11}=$
$\displaystyle\,\mathrm{e}^{-\lambda}\bigg{[}\sqrt{P(w)}\Big{(}\mathrm{d}s^{2}(\text{AdS}_{2})+\mathrm{d}s^{2}(\Sigma)\Big{)}+\frac{1}{2}(\mathbb{H}^{3})+\frac{8-\rho^{2}}{8+\rho^{2}}D\mu^{a}D\mu^{a}$
$\displaystyle\quad+\frac{1}{2}\Big{(}\frac{\rho^{2}}{8+\rho^{2}}\big{(}\mathrm{d}\phi+h(w)(\mathrm{d}z+jr\mathrm{d}t)\big{)}^{2}+\frac{1}{8-\rho^{2}}\mathrm{d}\rho^{2}\Big{)}\bigg{]}\,.$
(4.31)

The can be read off from (4.4) and (4.5) where the forms appearing in (4.5)
are

$\displaystyle J^{1}=$
$\displaystyle\frac{\sqrt{1-\mathrm{e}^{3\lambda}\rho^{2}}}{2^{5}/2\mathrm{e}^{\lambda}}D\mu^{a}\wedge
e_{\mathbb{H}^{3}}^{a}\,,\qquad\hat{w}=\frac{\mathrm{e}^{-\tfrac{1}{2}\lambda}}{2^{3/2}}\mu^{a}e^{a}_{\mathbb{H}^{3}}\,.$
(4.32)

Let us now analyse the regularity of the solution. Since we performed the
regularity analysis of the D2-brane solutions in great detail we will refer to
some of the results there and keep the regularity analysis lighter here. We
will restrict to studying the metric at a fixed point in AdS2 and ignore the
regularity conditions involving $\mathbb{H}^{3}$ since this is well
established in the literature.

We may break the regularity analysis into three regions on the boundary of a
rectangle as depicted in figure 3. The range of the coordinate $\rho$ is
$\rho\in[0,2\sqrt{2}]$, and $\rho=0$ defines region II, while its maximal
value at at $2\sqrt{2}$ defines region III. The regions I± are located at the
north and south poles of the spindle.

Figure 3: A schematic plot of the rectangle over which there is an
$S^{1}\times S^{1}\times S^{2}$ fibration. There are three distinguished
regions, plus two special points (the dots at the bottom corners) giving rise
to conical defects/monopoles. The red monopole has weight $n_{+}$ whilst the
blue monopole has weight $n_{-}$.

#### Region I

Let us consider a pole of the spindle. Similar to section 3.1 we may construct
the following suitably normalised Killing vectors with vanishing norm at one
of the poles of the spindle:

$\partial_{\phi_{\pm}}=\frac{n_{\pm}\Delta
z}{2\pi}\Big{(}\partial_{z}-h(w_{\pm})\partial_{\phi}\Big{)}\,.$ (4.33)

Away from $\rho=0$ where $\phi$ shrinks this gives rise to a smooth $S^{3}$.
To see this we should write the metric in a similar form to the one presented
in (3.17). The change of coordinates is

$\displaystyle z=\frac{n_{+}\Delta z}{2\pi}\phi_{+}+\frac{n_{-}\Delta
z}{2\pi}\phi_{-}\,,\quad\phi=-\frac{n_{+}h(w_{+})\Delta
z}{2\pi}\phi_{+}-\frac{n_{-}h(w_{-})\Delta z}{2\pi}\phi_{-}\,.$ (4.34)

Then the spindle and circle part of the metric becomes

$\displaystyle\sqrt{P}(w)\mathrm{d}s^{2}(\Sigma)+\frac{\rho^{2}}{2(8+\rho^{2})}\big{(}\mathrm{d}\phi+h(w)(\mathrm{d}z+jr\mathrm{d}t)\big{)}^{2}=$
$\displaystyle\sqrt{P(w)}{f(w)}\mathrm{d}w^{2}+F_{+}(w,\rho)D\phi_{+}^{2}+F_{-}(w,\rho)D\phi_{-}^{2}+GD\phi_{-}D\phi_{+}$
(4.35)

where

$\displaystyle F_{\pm}(w,\rho)$ $\displaystyle=\frac{n_{\pm}^{2}\Delta
z^{2}\big{(}2(8+\rho^{2})f(w)+\rho^{2}(h(w)-h(w_{\pm}))^{2}\sqrt{P(w)}\big{)}}{8\pi^{2}(8+\rho^{2})\sqrt{P(w)}}\,,$
(4.36) $\displaystyle D\phi_{\pm}$
$\displaystyle=\mathrm{d}\phi_{\pm}-u_{\pm}jr\mathrm{d}t\,,$ (4.37)
$\displaystyle u_{\pm}$ $\displaystyle=\pm\frac{2\pi h(w)}{n_{\pm}\Delta
z(h(w_{+})-h(w_{-})}\,,$ (4.38) $\displaystyle G$
$\displaystyle=\frac{n_{-}n_{+}\Delta
z^{2}\big{(}\rho^{2}\big{(}h(w)-h(w_{-})\big{)}\big{(}h(w)-h(w_{+})\big{)}\sqrt{P(w)}+2f(w)(8+\rho^{2})\big{)}}{4\pi^{2}(8+\rho^{2})\sqrt{P(w)}}\,.$
(4.39)

Note that $F_{\pm}(w_{\pm},\rho)=0$ and $G(w_{\pm})=0$. This then exhibits the
metric as a smooth $S^{3}$ as can be seen by going to the points $w=w_{\pm}$.
Note that this is only true away from $\rho=0$.

#### Region II

At $\rho=0$ the circle parametrised by $\phi$ shrinks smoothly if it has
period $2\pi$. This is of course the expected period coming from the embedding
coordinates we used above. At the corners of the rectangle in figure 3 there
are conical singularities of the same type encountered in the previous
section. One can see this by taking the double limit $w=w_{\pm}$ and $\rho=0$,
however since this proceeds in a similar fashion to the D2 brane case we will
suppress the computational details. One can understand this conceptually by
noting that the smoothing out of the spindle singularities requires another
circle so that the space becomes topologically an $S^{3}$. If this circle
shrinks somewhere the desingularisation through fibration is not possible at
this point, one is left with a copy of the spindle and thus orbifold
singularities.

#### Region III

At $\rho=2\sqrt{2}$ the $S^{2}$ shrinks smoothly. Note that this point along
the boundary is perfectly smooth, including the corners, it does not have
conical deficits there. This is because the circle fibration over the spindle
is left of finite size on this boundary. This should be contrasted with the D2
brane of the previous section where the singularities are present at all four
corners.

#### Flux quantisation

Let us now consider the quantisation of the four-form flux. The only four-
cycle is obtained from the $S^{4}$ at fixed point on the spindle and
$\mathbb{H}^{3}$. It is simple to show that

$\frac{1}{(2\pi\ell_{p})^{3}}\int_{S^{4}}G=\frac{1}{\pi
m^{3}\ell_{p}^{3}}\equiv N\in\mathbb{Z}\,,$ (4.40)

where $N$ should should be understood as the number of M5-branes being
wrapped.

#### Free Energy

Finally let us compute the free energy of the solution. As before the free
energy can be extracted from evaluating the 2d Netwon’s constant, which is
given by

$\displaystyle\mathcal{F}_{1}=\frac{1}{4G_{2}}=$
$\displaystyle\frac{1}{4G_{10}}\int_{\mathcal{M}_{9}}\mathrm{d}\mathrm{vol}_{9}=\frac{\Delta
z(w_{+}-w_{-})}{4}\mathcal{F}_{3}$
$\displaystyle=\frac{\sqrt{2(n_{-}^{2}+n_{+}^{2})+n_{-}^{2}n_{+}^{2}Q_{e}^{2}}-(n_{-}+n_{-})}{4n_{-}n_{+}}\mathcal{F}_{3}\,.$
(4.41)

##  5 Concluding remarks

In this paper we have considered the holographic duals of a large class of 3d
SCFTs wrapped on a spindle. We have exploited the consistent truncations of
massive type IIA and 11d supergravity on a SLAG manifold to 4d $\mathcal{N}=2$
Einstein–Maxwell supergravity and uplifted the known rotating
Plebanski–Demianski metric Plebanski:1976gy . We find that whereas the uplift
of the same solution to 11d supergravity on an $S^{7}$ allows for the conical
singularitites of the spindle to be resolved through fibration, this is not
possible in either of the two geometries studied here. Rather, the in the
massive type IIA case the 10d metric has four singularities consistent with
the presence of KK-monopoles and is similar to the singularities arising from
M5-branes on a spindle. On the other hand the solutions holographic dual to
M5-branes wrapped on a spindle and $\mathbb{H}^{3}$ have only two orbifold
singularities, rather than the four of the D2-brane case. One can understand
this by considering the degenerations of the internal manifold on which the
solutions have been uplifted. For odd dimensional spheres there is always a
circle which remains finite, over which we may fibre the spindle to obtain a
smooth $S^{3}$. For even dimensional spheres, there are points where all
circles shrink, and it is at these points that we are left with a copy of the
spindle and therefore orbifold singularities. Our second solution has extended
the known examples of wrapped branes on products of spindles with constant
curvature spaces, see Suh:2022olh ; Boido:2021szx  for the direct product of
spindle and a Riemann surface. It would be interesting to construct solutions
where the 5d space on which the M5-brane is wrapped is a full spindle, in the
same spirit as the 4d orbifold solutions of Cheung:2022ilc .

It is natural to interpret the conical defects of the spindle as regular
punctures of a round two-sphere. Indeed, in the case of M5-branes on a disc
this has been studied in Bah:2021hei ; Bah:2021mzw ; Couzens:2022yjl  and
matched to a dual field theory computation. The disc geometry is understood to
be a two-sphere with a regular puncture at one pole and an irregular puncture
at the other. Due to the presence of the irregular puncture the R-symmetry of
the field theory must mix with the isometries of the two-sphere, which is a
well established fact on the gravity side for spindle geometries. The disc
solutions considered in Bah:2021hei ; Bah:2021mzw  can be obtained as a
special limit of the local solution giving rise to the holographic dual of
M5-branes on a spindle Ferrero:2021wvk , see for example Couzens:2022yjl . In
Bah:2021hei ; Bah:2021mzw  it was shown that the conical defect gives rise, on
the field theory side, to a flavour symmetry described by a rectangular Young
diagram. This was further extended in Couzens:2022yjl  to allow for an
arbitrary regular puncture, therefore generalising the conical defect of the
disc. One should expect that a similar type of generalisation is possible for
spindle solutions too.

## Acknowledgments

We would like to thank Damian van de Heisteeg for discussions and previous
collaboration on related topics. CC would also like to thank Hyojoong Kim,
Nakwoo Kim, Yein Lee and Minwoo Suh for discussions and collaboration on
related work, past and present. CC is supported by the National Research
Foundation of Korea (NRF) grants 2019R1A2C2004880, 2020R1A2C1008497.

##  Appendix A M5-branes on a spindle

In the main text we have claimed that the singularity structure of the
D2-brane on a spindle solutions constructed here are more reminiscent of the
solutions for M5-branes on a spindle. To aid comparison we will review the
AdS5 solutions of 11d supergravity corresponding to M5-branes on a spindle, as
first studied in Ferrero:2021wvk . Our interest in this solution is to study
the singular behaviour of the uplifted theory and to compare with our analysis
of the solutions corresponding to D2-branes on a spindle.

The 7d metric of Ferrero:2021wvk  is

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\Big{(}wP(w)\Big{)}^{1/5}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{w}{f(w)}\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\mathrm{d}z^{2}\bigg{]}\,.$
(A.1)

The metric on AdS5 is taken to be the one with unit radius which implies that
it is Einstein satisfying $R_{\mu\nu}=-4g_{\mu\nu}$. The functions appearing
in the metric are the simple polynomials

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

The solution is supported by two real scalars and two abelian gauge fields

$A_{i}=\frac{l_{i}}{h_{i}(w)}\mathrm{d}z\,,\qquad
X_{i}(w)=\frac{\Big{(}wP(w)\Big{)}^{2/5}}{h_{i}(w)}\,,$ (A.3)

As explained in Ferrero:2021wvk  the $w$ coordinate is bounded between two
positive roots, $w_{\pm}$ of the function $f(w)$. The period of the circle
direction $z$ is fixed to be

$\frac{\Delta z}{2\pi}=\frac{2w_{\pm}^{2}}{n_{\pm}|f^{\prime}(w_{\pm})|}\,,$
(A.4)

with $n_{\pm}$ orbifold weights for the spindle at the respective poles. The
two magnetic charges are

$\displaystyle Q_{i}=\frac{1}{2\pi}\int_{\Sigma}\mathrm{d}A_{i}=\frac{\Delta
z}{2\pi}\Big{[}\frac{l_{i}}{h_{i}(w_{+})}-\frac{l_{i}}{h_{i}(w_{-})}\Big{]}\,.$
(A.5)

The solution can be uplifted to 11d supergravity on an $S^{4}$, with resultant
11d metric

$\displaystyle\mathrm{d}s^{2}_{11}=$
$\displaystyle\Omega^{1/3}\Big{(}wP(w)\Big{)}^{1/5}\bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{w}{f(w)}\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\mathrm{d}z^{2}$
$\displaystyle+\frac{1}{\Omega\big{(}wP(w)\big{)}^{1/5}}\Big{(}X_{0}^{-1}\mathrm{d}\mu_{0}^{2}+\sum_{i=1}^{2}X_{i}^{-1}\big{(}\mathrm{d}\mu_{i}^{2}+\mu_{i}^{2}(\mathrm{d}\phi_{i}+A_{i})^{2}\big{)}\Big{)}\bigg{]}\,.$
(A.6)

Here, $\mu_{I}$ are embedding coordinates for the $S^{4}$ and satisfy
$\mu_{0}^{2}+\mu_{1}^{2}+\mu_{2}^{2}=1$. The new functions appearing in the
metric are

$\displaystyle
X_{0}=X_{1}^{-2}X_{2}^{-2}\,,\qquad\Omega=\sum_{I=0}^{2}X_{I}\mu_{I}^{2}\,.$
(A.7)

It is useful to reparametrise the $\mu_{I}$ as

$\mu_{1}=\sqrt{1-\mu_{0}^{2}}\sin\theta\,,\quad\mu_{2}=\sqrt{1-\mu_{0}^{2}}\cos\theta\,.$
(A.8)

With these coordinates the metric takes the form

$\displaystyle\mathrm{d}s_{11}^{2}$
$\displaystyle=\Omega^{1/3}(wP(w))^{1/5}\Bigg{[}4\mathrm{d}s^{2}(\text{AdS}_{5})+\frac{w}{f(w)}\mathrm{d}w^{2}+\frac{f(w)}{P(w)}\mathrm{d}z^{2}$
(A.9)
$\displaystyle+\frac{1}{\Omega(wP(w))^{1/5}}\bigg{\\{}(1-\mu_{0})^{2}\bigg{[}\frac{\sin^{2}\theta}{X_{1}}D\phi_{1}^{2}+\frac{\cos^{2}\theta}{X_{2}}D\phi_{2}^{2}+\Big{(}\frac{\cos^{2}\theta}{X_{1}}+\frac{\sin^{2}\theta}{X_{2}}\Big{)}\mathrm{d}\theta^{2}\bigg{]}$
$\displaystyle+\frac{2\mu_{0}\sin\theta\cos\theta(X_{1}-X_{2})}{X_{1}X_{2}}\mathrm{d}\theta\mathrm{d}\mu_{0}+\bigg{[}X_{1}^{2}X_{2}^{2}+\frac{\mu_{0}^{2}}{1-\mu_{0}^{2}}\Big{(}\frac{\sin^{2}\theta}{X_{1}}+\frac{\cos^{2}\theta}{X_{2}}\Big{)}\bigg{]}\mathrm{d}\mu_{0}^{2}\bigg{\\}}\Bigg{]}\,,$
(A.10)

with

$D\phi_{i}=\mathrm{d}\phi_{i}+A_{i}\,,$ (A.11)

and $\Omega$ takes the form

$\Omega=\frac{\mu_{0}^{2}}{X_{1}^{2}X_{2}^{2}}+(1-\mu_{0}^{2})\Big{(}X_{1}\sin^{2}\theta+X_{2}\cos^{2}\theta\Big{)}\,.$
(A.12)

Near to $\theta=0,\pi$ an $S^{2}$ in the metric shrinks smoothly when
$\phi_{i}$ have period $2\pi$, this is of course the expected degeneration of
a three-sphere. Next consider a degeneration at the root of $f(w)$ at
$w_{\pm}$. The degenerating Killing vector with $2\pi$ period at one of the
poles of the spindle (away from $\mu_{0}^{2}=1$) is

$k_{\pm}=\pm\frac{2w_{\pm}^{2}}{|f^{\prime}(w_{\pm})|}\bigg{(}\partial_{z}-\sum_{i=1}^{2}\frac{l_{i}}{h_{i}(w_{\pm})}\partial_{\phi_{i}}\bigg{)}\,.$
(A.13)

Note that the prefactor satisfies

$\frac{2w_{\pm}^{2}}{|f^{\prime}(w_{\pm})|}=\frac{n_{\pm}\Delta z}{2\pi}\,.$
(A.14)

We have four Killing vectors which degenerate at $\theta=0,\pi$ and
$w=w_{\pm}$ and therefore they must satisfy a linear relation

$a_{+}k_{+}+a_{-}k_{-}+p_{1}\partial_{\phi_{1}}+p_{2}\partial_{\phi_{2}}=0\,.$
(A.15)

This implies three constraints on the parameters:

$n_{+}a_{+}-n_{-}a_{-}=0\,,\quad n_{+}a_{+}Q_{i}=p_{i}\,,$ (A.16)

where we have used the first condition to simplify the second and the explicit
form of the magnetic charges in (A.5). Clearly we should solve the first
condition by taking

$a_{+}=n_{-}\,,\quad a_{-}=n_{+}\,,$ (A.17)

then we find the quantisation condition

$n_{+}n_{-}Q_{i}=p_{i}\in\mathbb{Z}\,.$ (A.18)

The degeneration at $\mu_{0}=\pm 1$ leads to the shrinking of a warped $S^{3}$
as expected for the degeneration of an $S^{4}$. The manifold is thus smooth
and singularity free away from the four points $(\mu_{0},w)=(\pm 1,w_{\pm})$
which we are yet to study. Let us consider these points and for simplicity set
$X_{1}=X_{2}$.111111For $X_{1}\neq X_{2}$ the coordinate transformation is
more complicated due to the mixed term between $\mu_{0}$ and $\theta$ and the
coordinate transformation requires a coordinate transformation for the three
coordinates rather than just the two. Strictly the spindle does not exist when
$X_{1}=X_{2}$ since the required number of positive roots is not possible for
the polynomial $f(w)$. However since we are interested in the degeneration at
one end-point of the spindle, and not the full global regularity, we are
justified to consider $X_{1}=X_{2}$. We should change coordinates to

$w=w_{\pm}\mp\frac{|f^{\prime}(w_{\pm})|X_{1}(w_{\pm})}{2w_{\pm}^{9/5}}r^{2}\cos^{2}\zeta\,,\qquad\mu_{0}=\pm
1\mp r^{2}\sin^{2}\zeta\,.$ (A.19)

Then

$\displaystyle\mathrm{d}s^{2}_{6}=\frac{2X_{1}(w_{\pm})}{w_{\pm}^{4/5}}\bigg{[}\mathrm{d}r^{2}+r^{2}\Big{\\{}\mathrm{d}\zeta^{2}+\frac{1}{n_{\pm}^{2}}\cos^{2}\zeta\mathrm{d}\hat{z}^{2}+\sin^{2}\zeta\Big{(}\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi_{1}^{2}+\cos^{2}\theta\mathrm{d}\phi_{2}^{2}\Big{)}\Big{\\}}\bigg{]}\,,$
(A.20)

where we defined the $2\pi$ periodic coordinate $\hat{z}$ as

$z=(\Delta z)~{}\hat{z}\,,$ (A.21)

and we have performed a linear shift of the $\phi_{i}$ coordinates in the
final result. This is the metric on $\mathbb{R}^{6}/\mathbb{Z}_{n_{\pm}}$ and
implies the existence of monopoles located at the four points
$(\mu_{0},w)=(\pm 1,w_{\pm})$, with the orbifold weight correlated to the
$\pm$ of $w_{\pm}$. This remains when relaxing $X_{1}=X_{2}$ and thus is also
true for the spindle solution. Compare this to the D2-brane case studied in
the main text, in particular equation (3.24). Note that both have the same
phenomenon of resolving the potential line of singularities in the uplift at
the cost of singularities at isolated points instead. This should be
contrasted with the M2-brane and D3-brane solutions where the uplifted metric
can be made smooth everywhere.

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# A Durable Flash Memory Search Tree

James Clay III Kevin Wortman California State Univeristy, Fullerton
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the task of optimizing the _B-tree_ data structure, used
extensively in operating systems and databases, for sustainable usage on
multi-level flash memory. Empirical evidence shows that this new flash memory
tree, or _FM Tree,_ extends the operational lifespan of each block of flash
memory by a factor of roughly 27 to 70 times, while still supporting
logarithmic-time search tree operations.

## 1 Introduction

Flash memory has seen growing usage in recent years across all areas of
computing technology; devices utilizing flash include tablets, cell phones,
and USB drives. The lifespan of flash memories has been of serious concern
since their inception; flash memory degrades proportionally to the number of
times it is erased. While significant advances have been made to combat this
effect, flash usage continues to grow in areas where rewrite-intensive
operations are necessary. We consider the problem of extending the operational
lifetime of flash memory by avoiding erase operations, at the expense of some
additional time and space overhead.

## 2 Background

###  2.1 Flash Memory

_Multi-level flash memory_ is a form of NAND random access memory _(RAM)_
capable of storing one of $q>1$ discrete states in each flash cell. These $q$
values may interpreted as nonnegative integers in the range $[0,\,q-1].$ Cells
are aggregated into _blocks_ of fixed size. Individual multi-level flash cells
may be incremented, which is a fast and non-destructive operation. However,
idiosyncratically, bucket states can only be decreased by resetting an entire
block to state 0 _en masse._ These _resets_ or _erasures_ are costly both in
terms of time of the operation and lifetime of the device. Typical block
erases take between 1.5-2 milliseconds in comparison to seek or write times
which are in the tens or hundreds of microseconds. Each NAND flash device has
a set number of expected erasure cycles it can perform before failing. Wear
distribution can be done in multiple ways ranging from a purely round robin
approach to keeping the most actively used blocks in RAM. However recent
advances in multiple level flash memory data representation relaxes the
requirement that a block must be erased before it can be rewritten. Since
erasures are a limiting factor to both the durability and write throughput of
multi-level flash memory, we investigate approaches to avoid erasures, at the
cost of modest constant-factor expenditures of space and CPU time.

###  2.2 Tree Types

Our FM tree is an amalgam of ideas from established search tree data
structures. In this section we survey their properties. A _Bayer McCreight
B-Tree,_ henceforth denoted _B-tree_ , is a type of balanced search tree
developed for managing large blocks of data, particularly in file-systems and
databases. A B+ tree resembles a B-tree, with the addition of redundant node
links that facilitate tree traversal in common database operations. B- trees
are a relaxed version of the common B+ tree where the notoriously complex
post-deletion rebalancing operation is omitted. Perhaps surprisingly, Sen and
Tarjan showed that, despite postponing rebalance operations, B- trees still
boast asymptotically optimal update operations up to amortization, and may be
implemented simply with attractive constant factors.

###  2.3 A Durable B-tree

In creating a durable B-tree we have made a variety of changes to the
implementation of both operations on and storage of the keys within the
B-tree. These alterations reduce the maximum number of erasures per block and
the average number of erasures across all blocks.

  * •

Block erasures are performed lazily, postponing them for as long as possible.

  * •

The requirement that key/value pairs be sorted within nodes is relaxed. This
gives the insertion operation leeway to reuse key/value slots without erasing
the block, but makes searching a single node take $O(B)$ rather than $O(\log
B)$ time.

  * •

As in the B- tree, the delete operation marks unused nodes _barren_ rather
than actually splicing the node out of the tree. Barren nodes are ignored
until the tree is eventually garbage collected and rebuilt. This allows a node
to be excised by toggling flash cell(s) representing a boolean barren flag,
without performing block erasures.

## 3 Analysis

We prove that an FM tree supports the search, insert, and delete operations
all in amortized $O(\log n)$ time, matching the lower bound for amortized
search tree data structures. We show that, for any sequence of operations, an
FM tree performs strictly fewer erasures than a conventional B-tree.

## 4 Experimental Results

We present a variety of experiments performed on a Python implementation of
the FM Tree. We emulate the flash memory, FM Tree, and B-tree to run a variety
of benchmarks. Every experimental trial consists of randomly generated data
sets that are inserted into both trees. Each tree is inserted with a baseline
of 1000 elements. Following these initial insertions, 10000 randomly chosen
insertions and deletions are performed. We repeat this process a total of 4
times independently and determine the average for each data point. We then
calculate the FM Tree performance by comparing the erasures, reads and writes
between it and the B-tree. This process indicates that the FM Tree performs 27
times to 72.2 times fewer erasures. While the total read count was higher, the
total writes and erasures performed were far lower. As these are the most
expensive operations in terms of time, realistically the FM Tree would also be
far faster than the B-tree.

## 5 Conclusion

We find that the FM-Tree is a more durable, faster variant of the B-tree with
properties that make it intrinsically better for operating on flash memory. We
also show that the erasure count for the FM Tree is drastically smaller (27 to
72 times) than that of a B-tree.