Branes wrapped on quadrilaterals

Federico Faedoa, Alessio Fontanarossab,c and Dario Martellib,c

aInternational Centre for Theoretical Physics Asia–Pacific,
University of Chinese Academy of Sciences, 100190 Beijing, China

bDipartimento di Matematica “Giuseppe Peano”, Università di Torino,
Via Carlo Alberto 10, 10123 Torino, Italy

cINFN, Sezione di Torino,
Via Pietro Giuria 1, 10125 Torino, Italy

We construct new families of supersymmetric
$\mathrm{AdS}_{2}\times\mathbb{M}_{4}$ solutions of $D=6$ gauged supergravity
and $\mathrm{AdS}_{3}\times\mathbb{M}_{4}$ solutions of $D=7$ gauged
supergravity, where $\mathbb{M}_{4}$ are four-dimensional toric orbifolds with
four fixed points. These are presented in a unified fashion, that highlights
the common underlying geometry of the two families. The $D=6$ solutions uplift
to massive type IIA and describe the near-horizon limit of D4-branes wrapped
on $\mathbb{M}_{4}$, while the $D=7$ solutions uplift to $D=11$ supergravity
and describe the near-horizon limit of M5-branes wrapped on $\mathbb{M}_{4}$.
We reproduce the entropy and gravitational central charge of the two families
by extremizing a function constructed gluing the orbifold gravitational blocks
proposed in [1].

###### Contents

  1. 1 Introduction
  2. 2 Four-dimensional toric orbifolds
    1. 2.1 Toric geometry ingredients
    2. 2.2 Quadrilaterals
  3. 3 Supergravity solutions
    1. 3.1 Solutions in $D$ dimensions
    2. 3.2 Equal charges solutions
    3. 3.3 Conditions of existence
  4. 4 Global structure of the solutions
    1. 4.1 Warm-up: revisiting the solutions of [1]
    2. 4.2 Degenerating Killing vectors
    3. 4.3 Complex structure and divisors
    4. 4.4 Toric data
    5. 4.5 Quantization conditions
  5. 5 Off-shell free energies and extremization
    1. 5.1 Recap of the recipe
    2. 5.2 Application to the new AdS${}_{D-4}\times\mathbb{M}_{4}$ solutions
  6. 6 Discussion
  7. A The AdS${}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$ limits
    1. A.1 The AdS${}_{2}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$ limit
    2. A.2 The AdS${}_{3}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$ limit
  8. B Uplift of the solutions
  9. C Supersymmetry of the $D=6$ solutions with equal charges
  10. D Some solutions to the Diophantine equations

##  1 Introduction

In the history of the AdS/CFT correspondence very often the construction of
explicit supergravity solutions with an anti-de Sitter (AdS) factor has
prompted the discovery of new classes of superconformal field theories (SCFTs)
and has led to tremendous progress in understanding general aspects of the
holographic dictionary. An important entry in this dictionary is the
relationship between extremization problems in SCFTs and their geometric
realization. An early example of such relationships is that between
$a$-maximization in four-dimensional SCFTs [2] and volume extremization of
Sasaki–Einstein manifolds [3, 4], followed by many other instances involving
SCFTs of different dimensions on one side and geometries of different kinds on
the other side.

The study of this feature of holography has been the subject of a conspicuous
body of work over the last twenty years and it received a boost recently,
following the uncovering of supergravity solutions comprising geometries with
orbifold singularities. While in the first examples of these constructions [5,
6, 7, 8] the main actor was the spindle, a number of supergravity solutions
based on geometries comprising four-dimensional orbifolds $\mathbb{M}_{4}$
have appeared subsequently. These include solutions describing either
D4-branes or M5-branes wrapped on $\mathbb{M}_{4}$ [9, 10, 11, 12, 1].

In this paper we will extend these constructions by presenting more general
classes of supersymmetric $\mathrm{AdS}_{2}\times\mathbb{M}_{4}$ solutions of
$D=6$ gauged supergravity and $\mathrm{AdS}_{3}\times\mathbb{M}_{4}$ solutions
of $D=7$ gauged supergravity, where $\mathbb{M}_{4}$ are four-dimensional
toric orbifolds with four fixed points for a torus action, that we refer to as
_quadrilaterals_ , because they are characterized by certain labelled two-
dimensional polytopes111We will be more precise later about how this labelled
polytope is constructed. with four edges [13, 14]. We will present the two
families of solutions in a unified fashion, exploiting the fact that the four-
dimensional factor of the geometry involves in both cases the same data,
namely a Hermitian orbifold metric on $\mathbb{M}_{4}$, together with two
gauge fields corresponding to local connections on line orbibundles, as well
as two real scalar fields, governed by very similar equations of motion and
supersymmetry conditions. In particular, we will give a unified description of
the underlying toric orbifolds in terms of their associated labelled polytopes
[15].

An important role in the stream of ideas surrounding the geometrization of the
extremization principles characterizing SCFTs has been played by the
_gravitational blocks_ [16]. These are elementary building blocks in terms of
which one can assemble various functions of the gravitational data that are
interpreted as dual to SCFT observables and whose extremization corresponds to
necessary (and, sometimes, also sufficient) conditions for the existence of
the supergravity solutions. This generally yields a powerful abstract
understanding of the geometry, and allows one to extract relevant properties
of supergravity solutions without their explicit knowledge. Orbifold
gravitational blocks for various branes wrapped on the spindle were
conjectured in [8] and, in some cases, subsequently proved using GK geometry
[17] (for M2 and D3-branes) or equivariant localization [18] (for M5-branes).

In our previous paper [1] we conjectured the form of the orbifold
gravitational blocks for D4-branes as well as M5-branes wrapped on toric
orbifolds $\mathbb{M}_{4}$ and successfully tested it against a number of
explicit supergravity solutions. Subsequently, the validity of these proposals
has been confirmed using the idea of equivariant localization. In particular,
in [19] it was shown that equivariant integration of the M5-branes anomaly
polynomial on four-dimensional toric orbifolds yields an off-shell trial
central charge of the corresponding two-dimensional SCFT that matches
precisely the gravitational block decomposition conjectured in [1]. Moreover,
in [18] and [19] has been proposed that equivariant localization plays a key
role for characterizing the geometry of supersymmetric solutions. As an
application, it has later been shown that the gravitational blocks associated
with M5 and D4-branes wrapped on four-dimensional orbifolds can be extracted
either from the equivariant volume of an associated geometry [20] or by fixed-
point evaluation222A computation of the gravitational block expression for
D4-branes wrapped on orbifolds using the approach of [18, 21] is not yet
available. of equivariant integrals of certain equivariantly closed forms
constructed with Killing spinor bilinears [21].

The results of the present paper allow us to test the validity of the orbifold
gravitational blocks and the corresponding extremization principles in the
context of some sophisticated explicit supergravity solutions. In particular,
one should bare in mind that, at present, the only extremization principle
applied to supergravity that has been rigorously proved to guarantee the
existence of solutions is that of volume extremization of Sasaki–Einstein
manifolds [4].333Analogous extremization principles have been formulated in
geometry, but so far they have not been shown to be directly relevant to
supergravity or holography. See, e.g., [22, 23, 24, 25, 26]. Thus, the
construction of explicit solutions in all other cases is a non-trivial result
and serves as further evidence of the validity of the respective extremization
principles. In general, it would be wonderful to obtain a proof that the
solution of one of these extremal problems is necessary and sufficient for the
existence of supergravity solutions.

The rest of the paper is organized as follows. In section 2 we discuss some
preliminaries about the geometry of toric orbifolds. In section 3 we present
the local form of our new solutions as a single $D$-dependent solution (with
$D=6,7$). In order to gain complete analytic control over the complicated
algebraic expressions, we will restrict to the subfamilies with equal charges.
In section 4 various global aspects are discussed, allowing us to uncover the
underlying toric orbifolds. We will show how an artful combination of analytic
and topological gadgets will allow us to extract from the solutions the data
necessary for implementing the extremization procedure. This is done in
section 5, where the off-shell free energies are computed and their
extremization is shown to reproduce the entropy and the gravitational central
charge of the respective explicit solutions. Section 6 discusses our findings
and gives some suggestions for future work. Four technical appendices
complement the paper. In appendix A we show how previous solutions can be
recovered in particular limits of the present ones. Appendix B discusses the
uplift of the solutions to $D=10$ and $D=11$ supergravities. In appendix C we
demonstrate the supersymmetry of the $D=6$ solutions by constructing explicit
Killing spinors solving the BPS conditions. In appendix D we show that the set
of solutions to some Diophantine equations that need to be satisfied to
construct the supergravity solutions is non-empty.

##  2 Four-dimensional toric orbifolds

###  2.1 Toric geometry ingredients

Before presenting the four-dimensional orbifolds $\mathbb{M}_{4}$ that will be
the main focus of this paper, we will recall some results of toric geometry
that will be used in the analysis, highlighting some aspects pertaining to
orbifolds. We start by noting that in the solutions that we will discuss the
metric on $\mathbb{M}_{4}$ solving the supersymmetry conditions will be
compatible with an integrable complex structure (thus will be Hermitian),
however we will not find a compatible Kähler structure. As emphasized in [19,
20] this fact does not prevent one to employ the formalism of symplectic toric
geometry as a tool for computing topological quantities, on which the
corresponding extremal problems, and ultimately the dual field theories, are
based. It should also be possible to reformulate our analysis entirely in
terms of complex geometry for orbifolds, and in particular to use the language
of toric stacks [27, 28, 29, 30, 31] to describe the underlying toric geometry
in rigorous terms. However, as we shall see, this will not be necessary for
the purposes of constructing the solutions and extracting the relevant data
from them.

We now give a brief review of the geometry of symplectic toric orbifolds,
referring to [19] and references therein for a more detailed discussion. Let
us consider a $2m$-dimensional symplectic toric orbifold
$(\mathbb{M}_{2m},\omega)$, equipped with an effective Hamiltonian action of
the real torus $\mathbb{T}^{m}=\mathbb{R}^{m}/2\pi\mathbb{Z}^{m}$. A
convenient set of coordinates to parameterize $\mathbb{M}_{2m}$ are the
symplectic coordinates $(y_{I},\phi_{I})$, with $I=1,\ldots,m$, where $y_{I}$
are the moment maps associated with $\mathbb{T}^{m}$ and $\phi_{I}$ are
angular coordinates on the torus, with canonical $2\pi$ periodicity. According
to the construction of [15], compact symplectic toric orbifolds are completely
characterized by the associated _labelled polytope $\mathcal{P}$_, and the
total space $\mathbb{M}_{2m}$ arises as a torus fibration over $\mathcal{P}$.
A labelled polytope is a rational simple convex polytope in $\mathbb{R}^{m}$,
obtained as the image of $\mathbb{M}_{2m}$ under the moment maps $y_{I}$,
equipped with a label attached to each of its facets. This object can be
described concisely as the subset

$\mathcal{P}=\bigcap_{a=1}^{n}\bigl{\\{}y\in\mathbb{R}^{m}\;:\;l_{a}(y)\equiv
y_{I}v_{a}^{I}-\lambda_{a}\leq 0\bigr{\\}}\,,$ (2.1)

with $\vec{v}_{a}\in\mathbb{Z}^{m}$ and $\lambda_{a}$ real constants. The
linear equations $l_{a}(y)=0$ define the facets $\mathcal{F}_{a}$ of the
polytope and we denote with $n$ their total number. In our conventions,
$\vec{v}_{a}$ are the outward-pointing normal vectors to the facets
$\mathcal{F}_{a}$. The labels $m_{a}$ are positive integers such that the
structure group of every point in the preimage of the interior of
$\mathcal{F}_{a}$ under the moment maps is $\mathbb{Z}_{m_{a}}$. In
particular, the labels $m_{a}$ enter in the above construction as a common
factor in the entries of the vectors $\vec{v}_{a}$, which can be written as
$\vec{v}_{a}=m_{a}\vec{\hat{v}}_{a}$, where $\vec{\hat{v}}_{a}$ are primitive
vectors and therefore define an ordinary fan.444Notice that we are using the
conventions of [19], which are opposite to the ones in [1]! In the context of
algebraic geometry the fan $\\{\vec{\hat{v}}_{a}\\}$ defines a toric variety,
which by construction can have orbifold singularities only in complex
codimension higher than one. In order for $\mathcal{P}$ to be simple, exactly
$m$ facets meet at each vertex $p$ and the $m$ vectors associated with these
facets form a basis for $\mathbb{R}^{m}$. These vertices correspond to fixed
points of the $\mathbb{T}^{m}$ action. Assuming that a given fixed point is
obtained as the intersection of the facets $\mathcal{F}_{a_{I}}$, with
$I=1,\ldots,m$, the order of the orbifold singularity at this point is given
by $d_{A}=\left|\det(\vec{v}_{a_{1}},\ldots,\vec{v}_{a_{m}})\right|$. Note
that in algebraic geometry, the fan defined by $\vec{v}_{a}$ (equivalently, by
$(\vec{\hat{v}}_{a},m_{a})$) does not give rise to a toric variety, but to a
generalization known as a _toric stack_ [27, 29, 31]. The data
$(\vec{\hat{v}}_{a},m_{a})$ are then referred to as a _stacky fan_ or an
associated _stacky polytope_ [30]. The toric orbifold $\mathbb{M}_{2m}$ viewed
as a torus fibration over the polytope $\mathcal{P}$ is non-degenerate in the
interior of $\mathcal{P}$. At each facet $\mathcal{F}_{a}$ a particular one-
cycle in $\mathbb{T}^{m}$ collapses, thus determining a symplectic subspace of
$\mathbb{M}_{2m}$ of (real) codimension two, which is the preimage of
$\mathcal{F}_{a}$ under the moment maps.

As we shall see, the data that we will not be able to extract directly from
the metric can be obtained exploiting the fact that there exists a complex
structure compatible with our metrics, thus the underlying spaces are actually
(compact) complex orbifolds. In particular, given a $k$-dimensional complex
orbifold $X$ (that can be either $\mathbb{M}_{4}$ or one of its divisors) from
the holomorphic $(k,0)$-form $\Omega_{X}$ one can compute the Ricci potential
$P_{X}$ through the relation

$\mathrm{d}\Omega_{X}=\mathrm{i}P_{X}\wedge\Omega_{X}\,.$ (2.2)

The Ricci form then reads $\rho_{X}=\mathrm{d}P_{X}$ and the first Chern class
of the tangent bundle of $X$ is obtained taking the explicit representative
$c_{1}(TX)=[\rho_{X}]/2\pi$. From the complex point of view, the preimage of
the $\mathcal{F}_{a}$ described above are _toric divisors_. More precisely,
these are the divisors associated with the vectors $\vec{v}_{a}$ and have to
be identified with the ramification divisors, which we will denote as $D_{a}$.
In contrast, the divisors $\hat{D}_{a}$, associated with the vectors
$\vec{\hat{v}}_{a}$, are to be identified with the branch divisors (see e.g.
[32]). Formally, the two sets of divisors are related as
$\hat{D}_{a}=m_{a}D_{a}$, which slightly more accurately may be written as
$D_{a}=\hat{D}_{a}\times\mathrm{pt}/\mathbb{Z}_{m_{a}}$, indicating that at
each point in $D_{a}$ the structure group $\mathbb{Z}_{m_{a}}$ is acting on
the complex direction $\mathbb{C}$ normal to $D_{a}$. Equivalently, this
relation may be also understood from the expression of the Poincaré dual two-
forms, given by the first Chern classes of a set of line bundles $L_{a}$.
Their explicit form is [33, 34]

$c_{1}(L_{a})=-\frac{\mathrm{i}}{2\pi}\bigl{[}\partial\bar{\partial}\log
l_{a}\bigr{]}\,,$ (2.3)

where $[\alpha]$ denotes the cohomology class of a given differential form
$\alpha$ and $\partial$, $\bar{\partial}$ are the (anti-)holomorphic
differentials. Using an explicit expression in symplectic coordinates one can
show that indeed $c_{1}(\hat{L}_{a})=m_{a}c_{1}(L_{a})$, where $\hat{L}_{a}$
are the line bundles associated with the divisors $\hat{D}_{a}$.555We note in
passing that in the framework of toric stacks, the divisors $D_{a}$ should be
understood in terms of a “root construction”, namely as the $m_{a}$-th root of
the $\hat{D}_{a}$ [28]. However, we shall refrain from making use of this
language in our subsequent discussions. One can also see that the following
$m$ relations among the Chern classes hold [19]

$\sum_{a}\vec{v}_{a}c_{1}(L_{a})=0\,,$ (2.4)

thus only $n-m$ of the line bundles are independent. These are equivalent to
the familiar $m$ linear relations among the divisors

$\sum_{a}\vec{v}_{a}D_{a}=0\,,$ (2.5)

which imply that the are only $n-m$ independent ($2m-2$)-cycles in homology.

It will also be useful to note that given a toric ramification divisor $D_{a}$
of the total space $\mathbb{M}_{2m}$, the first Chern class of the tangent
bundle $T\mathbb{M}_{2m}$ can be decomposed by means of the _adjunction
formula_ as

$c_{1}(T\mathbb{M}_{2m})\bigr{|}_{D_{a}}=c_{1}(TD_{a})+c_{1}(L_{a})\bigr{|}_{D_{a}}\,.$
(2.6)

In this formula the first Chern class of the line bundles $L_{a}$ is given by
(2.3), while $c_{1}(T\mathbb{M}_{2m})$ follows from

$c_{1}(T\mathbb{M}_{2m})=\sum_{a}c_{1}(L_{a})\,.$ (2.7)

Although we are not aware of a standard reference for this formula in the
context of orbifolds, endowing each divisor with its natural complex
structure, from the corresponding Ricci form $\rho_{D_{a}}$ we can compute
$c_{1}(TD_{a})=[\rho_{D_{a}}]/2\pi$, and then the adjunction formula (2.6) can
be verified explicitly.666We have verified this formula for toric orbifolds in
arbitrary dimension [35]. In the case of four-dimensional toric orbifolds a
proof is contained in [36]. We thank Luca Cassia for discussions on this
point.

###  2.2 Quadrilaterals

We now continue requiring that $\mathbb{M}_{4}$ has four real dimensions,
namely setting $m=2$. With this assumption, the polytope becomes planar and,
therefore, enjoys the special feature of having the same number of facets
(edges) and fixed points (vertices), that we denote with $n$. When working
with four-dimensional toric orbifolds, the intersection matrix of two toric
divisors is [19]

$D_{ab}\equiv D_{a}\cdot D_{b}=\int_{\mathbb{M}_{4}}c_{1}(L_{a})\wedge
c_{1}(L_{b})=\left\\{\begin{aligned} &\frac{1}{d_{a-1,a}}\qquad&&\text{if}\
b=a-1\,,\\\ &\frac{1}{d_{a,a+1}}\qquad&&\text{if}\ b=a+1\,,\\\
&-\frac{d_{a-1,a+1}}{d_{a-1,a}\,d_{a,a+1}}\qquad&&\text{if}\ b=a\,,\\\
&0\qquad&&\text{otherwise}\,,\end{aligned}\right.$ (2.8)

where we defined $d_{a,b}\equiv\det(\vec{v}_{a},\vec{v}_{b})$. One can verify
explicitly that

$\sum_{b}D_{ab}\,\vec{v}_{b}=0\,,$ (2.9)

which is a consequence of the cohomological relation (2.4).

It is straightforward to show that $d_{a,b}$ is invariant under
$SL(2,\mathbb{Z})$ transformations applied to all the vectors $\vec{v}_{a}$ of
the fan. This property will be particularly useful for us, because it implies
that also the intersection matrix $D_{ab}$ is not affected by such
transformations. The fan corresponding to a given toric orbifold is thus
determined only up to $SL(2,\mathbb{Z})$ transformations, and, in a similar
way, also the related polytope. In fact, this group of invariance can be
formally extended to $SL(2,\mathbb{R})$. In this case, the new fan is not
well-defined, since the vectors are not integer-valued, but the $d_{a,b}$ are
unchanged and, in particular, we still have $d_{a,b}\in\mathbb{Z}$. This
property will turn out to be crucial, as it will allow us to work with “fake”
vectors $\vec{v}_{a}$, that are not $\mathbb{Z}^{2}$-valued,777This implies
that they are expressed in a “wrong” basis for the $\mathbb{T}^{2}$ action, in
which the orbits in general do not close. in intermediate stages of the
computations.

Using the adjunction formula (2.6), we can relate the toric data, encoded in
the intersection matrix $D_{ab}$, and the complex formalism, embodied by
$c_{1}(T\mathbb{M}_{4})$ and $c_{1}(TD_{a})$, by means of the following two
relations888The second equation can be obtained from the first using (2.7)
together with $\sum_{b}D_{ab}=\int_{D_{a}}c_{1}(T\mathbb{M}_{4})$.

$\displaystyle
D_{aa}=\int_{D_{a}}c_{1}(L_{a})=\int_{D_{a}}\bigl{[}c_{1}(T\mathbb{M}_{4})-c_{1}(TD_{a})\bigr{]}\,,$
(2.10) $\displaystyle D_{a\,a-1}+D_{a\,a+1}=\sum_{b\neq
a}\int_{D_{a}}c_{1}(L_{b})=\int_{D_{a}}c_{1}(TD_{a})=\chi_{(a)}\,,$ (2.11)

where, by definition, $\chi_{(a)}$ is the orbifold Euler characteristic of the
divisor $D_{a}$.

We now specialize to the case of four fixed points, that we refer to as
quadrilaterals. Indeed, these configurations are completely characterized by a
generic polytope depicted in figure 1, which is a convex polygon with four
vertices and four edges.

$\vec{v}_{1}$$\vec{v}_{2}$$\vec{v}_{3}$$\vec{v}_{4}$ Figure 1: Fan and
polytope of a generic quadrilateral. The vectors $\vec{v}_{a}$ are _not_
primitive.

All the information on the system is encoded in the following $4\times 4$
intersection matrix (2.8)

$D_{ab}=\begin{pmatrix}\frac{d_{2,4}}{d_{4,1}\,d_{1,2}}&\frac{1}{d_{1,2}}&0&\frac{1}{d_{4,1}}\\\
\frac{1}{d_{1,2}}&-\frac{d_{1,3}}{d_{1,2}\,d_{2,3}}&\frac{1}{d_{2,3}}&0\\\
0&\frac{1}{d_{2,3}}&-\frac{d_{2,4}}{d_{2,3}\,d_{3,4}}&\frac{1}{d_{3,4}}\\\
\frac{1}{d_{4,1}}&0&\frac{1}{d_{3,4}}&\frac{d_{1,3}}{d_{3,4}\,d_{4,1}}\\\
\end{pmatrix}\,,$ (2.12)

where a vector identity implies that the $d_{a,b}\in\mathbb{Z}$ are actually
not independent, but satisfy the relation

$d_{1,2}d_{3,4}-d_{2,3}d_{4,1}=d_{1,3}d_{2,4}\,.$ (2.13)

It is convenient to parameterise the integers $d_{a,b}$ as

$d_{a,b}=m_{a}m_{b}\,\hat{d}_{a,b}\,,$ (2.14)

where
$\hat{d}_{a,b}\equiv\det(\vec{\hat{v}}_{a},\vec{\hat{v}}_{b})\in\mathbb{Z}$
and satisfy

$\hat{d}_{1,2}\hat{d}_{3,4}-\hat{d}_{2,3}\hat{d}_{4,1}=\hat{d}_{1,3}\hat{d}_{2,4}\,.$
(2.15)

Thus, a generic quadrilateral is characterised by five independent integer
paramaters $\hat{d}_{a,b}$ (subject to (2.15)) and four integer labels
$m_{a}$, which we refer to as the toric data of the orbifold. Later we will
show that the local solutions that we are about to introduce can be extended
to well-defined _global_ solutions on a quadrilateral whose toric data are
subject to some additional “Diophantine constraints” that we shall discuss.
Moreover, we will show that the magnetic fluxes of the supergravity solutions
will be written in terms of these toric data and some twisting parameters,
precisely as conjectured in [1].

##  3 Supergravity solutions

In this section we discuss solutions of specific $D=6,7$ matter-coupled gauged
supergravity theories with gauge group $U(1)^{2}$. Both of them comprise two
gauge fields $A_{1}$, $A_{2}$, a $(D-4)$-form $B$ and two real scalar fields
$\vec{\varphi}=(\varphi_{1},\varphi_{2})$.

The six-dimensional model can also be obtained as a sub-sector of an extension
of Romans $F(4)$ gauged supergravity [37], coupled to three vector multiplets
[38, 39]. The bosonic part of the action reads999Here and in what follows we
define, for any $p$-form $\omega$,
$|\omega|^{2}=\frac{1}{p!}\,\omega_{\mu_{1}\ldots\mu_{p}}\omega^{\mu_{1}\ldots\mu_{p}}$.

$\begin{split}S_{\text{6D}}&=\frac{1}{16\pi
G_{(6)}}\int\mathrm{d}^{6}x\,\sqrt{-g}\biggl{[}R-V_{6}-\frac{1}{2}|\mathrm{d}\vec{\varphi}|^{2}-\frac{1}{2}\sum_{i=1}^{2}X_{i}^{-2}|F_{i}|^{2}-\frac{1}{8}(X_{1}X_{2})^{2}|H|^{2}\\\
&-\frac{m^{2}}{4}(X_{1}X_{2})^{-1}|B|^{2}-\frac{1}{16}\frac{\varepsilon^{\mu\nu\rho\sigma\tau\lambda}}{\sqrt{-g}}B_{\mu\nu}\Bigl{(}F_{1\,\rho\sigma}F_{2\,\tau\lambda}+\frac{m^{2}}{12}B_{\rho\sigma}B_{\tau\lambda}\Bigr{)}\biggr{]}\,,\end{split}$
(3.1)

where $F_{i}=\mathrm{d}A_{i}$, $H=\mathrm{d}B$ and the scalar fields
$\vec{\varphi}$ are parameterized as

$X_{i}=\mathrm{e}^{-\vec{a}_{i}\cdot\vec{\varphi}}\qquad\text{with}\qquad\vec{a}_{1}=\bigl{(}2^{-1/2},2^{-3/2}\bigr{)}\,,\qquad\vec{a}_{2}=\bigl{(}-2^{-1/2},2^{-3/2}\bigr{)}\,.$
(3.2)

The scalar potential is

$V_{6}=m^{2}X_{0}^{2}-4g_{c}^{2}X_{1}X_{2}-4mg_{c}\,X_{0}(X_{1}+X_{2})\,,$
(3.3)

with $g_{c}$ the gauge coupling and $m$ the mass parameter, and where for
later convenience we defined $X_{0}=(X_{1}X_{2})^{-3/2}$. In what follows we
shall set $m=2g_{c}/3$.101010As pointed out in [8] and [1], this can always be
done without loss of generality when $g_{c},m>0$.

The seven-dimensional theory can be constructed as a consistent truncation
[40] of the $D=7$, maximal $SO(5)$ gauged supergravity [41]. The bosonic part
of the action is [42]

$\begin{split}S_{\text{7D}}&=\frac{1}{16\pi
G_{(7)}}\int\mathrm{d}^{7}x\,\sqrt{-g}\biggl{[}R-V_{7}-\frac{1}{2}|\mathrm{d}\vec{\varphi}|^{2}-\frac{1}{2}\sum_{i=1}^{2}X_{i}^{-2}|F_{i}|^{2}-\frac{1}{2}(X_{1}X_{2})^{2}|H|^{2}\\\
&-\frac{1}{24}\frac{\varepsilon^{\mu\nu\rho\sigma\tau\lambda\eta}}{\sqrt{-g}}B_{\mu\nu\rho}\Bigl{(}F_{1\,\sigma\tau}F_{2\,\lambda\eta}-\frac{g_{c}}{12}H_{\sigma\tau\lambda\eta}\Bigr{)}\biggr{]}\,.\end{split}$
(3.4)

Again, the field strengths are given by $F_{i}=\mathrm{d}A_{i}$ and
$H=\mathrm{d}B$, while now the scalar fields $\vec{\varphi}$ take the
parameterization

$X_{i}=\mathrm{e}^{-\vec{a}_{i}\cdot\vec{\varphi}}\qquad\text{with}\qquad\vec{a}_{1}=\bigl{(}2^{-1/2},10^{-1/2}\bigr{)}\,,\qquad\vec{a}_{2}=\bigl{(}-2^{-1/2},10^{-1/2}\bigr{)}\,.$
(3.5)

The scalar potential is

$V_{7}=\frac{g_{c}^{2}}{2}\bigl{[}X_{0}^{2}-8X_{1}X_{2}-4X_{0}(X_{1}+X_{2})\bigr{]}\,,$
(3.6)

with $g_{c}$ the gauge coupling and, in this context,
$X_{0}=(X_{1}X_{2})^{-2}$. Additionally, the following self-duality condition
must hold

$(X_{1}X_{2})^{2}\star\\!H=-g_{c}B+\frac{1}{2}A_{1}\wedge
F_{2}+\frac{1}{2}A_{2}\wedge F_{1}+\mathrm{d}\lambda\,,$ (3.7)

for some two-form $\lambda$.

###  3.1 Solutions in $D$ dimensions

We now present a family of solutions to both the $D=6$ and the $D=7$ theories
in which the spacetime has the form of a warped product
$\mathrm{AdS}_{D-4}\times\mathbb{M}_{4}$. Their local form can be obtained as
an analytic continuation of the asymptotically AdSD, non-extremal, charged and
rotating111111It is crucial that they have a single non-zero angular momentum,
so that the solutions contain a round $(D-4)$-sphere. black holes constructed
in [43], with the addition of a NUT parameter. The solutions of our interest
can then be obtained performing an analytic continuation, which results in the
presence of an $\mathrm{AdS}_{D-4}$. Specifically, these backgrounds can be
written in a unified fashion, with the line element reading

$\mathrm{d}s^{2}=(H_{1}H_{2})^{1/(D-2)}\,\biggl{(}\frac{x^{2}y^{2}}{a^{2}}\,\mathrm{d}s_{\mathrm{AdS}_{D-4}}^{2}+\mathrm{d}s_{\mathbb{M}_{4}}^{2}\biggr{)}\,,$
(3.8)

where $\mathrm{d}s_{\mathrm{AdS}_{D-4}}^{2}$ denotes the unit radius metric on
$\mathrm{AdS}_{D-4}$ and we defined the four-dimensional metric

$\begin{split}\mathrm{d}s_{\mathbb{M}_{4}}^{2}&=\frac{(H_{1}H_{2})^{-1}}{\Xi^{2}(x^{2}-y^{2})}\biggl{[}(V_{y}^{2}\Delta_{x}-V_{x}^{2}\Delta_{y})\,\mathrm{d}\psi^{2}+(\widetilde{V}_{y}^{2}\Delta_{x}-\widetilde{V}_{x}^{2}\Delta_{y})\,\mathrm{d}\phi^{2}\\\
&-\frac{4c_{1}c_{2}\tilde{c}_{1}\tilde{c}_{2}}{a^{2}}\biggl{(}\frac{N_{y}\Delta_{x}}{y^{D-5}}-\frac{N_{x}\Delta_{y}}{x^{D-5}}\biggr{)}\,\mathrm{d}\psi\,\mathrm{d}\phi\biggr{]}+\frac{y^{2}-x^{2}}{\Delta_{y}}\,\mathrm{d}y^{2}+\frac{x^{2}-y^{2}}{\Delta_{x}}\,\mathrm{d}x^{2}\,.\end{split}$
(3.9)

The scalar fields are

$X_{i}=(H_{1}H_{2})^{(D-3)/2(D-2)}H_{i}^{-1}\,,$ (3.10)

the gauge potentials are given by

$\begin{split}A_{i}&=\frac{2N_{y}s_{i}c_{i}\tilde{c}_{i}}{\Xi(x^{2}-y^{2})H_{i}y^{D-5}}\biggl{(}\frac{\tilde{c}_{1}\tilde{c}_{2}}{\tilde{c}_{i}^{2}}\,V_{x}^{(i)}\,\mathrm{d}\psi-\frac{c_{1}c_{2}}{c_{i}^{2}}\,\widetilde{V}_{x}^{(i)}\,\mathrm{d}\phi\biggr{)}\\\
&-\frac{2N_{x}s_{i}c_{i}\tilde{c}_{i}}{\Xi(x^{2}-y^{2})H_{i}x^{D-5}}\biggl{(}\frac{\tilde{c}_{1}\tilde{c}_{2}}{\tilde{c}_{i}^{2}}\,V_{y}^{(i)}\,\mathrm{d}\psi-\frac{c_{1}c_{2}}{c_{i}^{2}}\,\widetilde{V}_{y}^{(i)}\,\mathrm{d}\phi\biggr{)}\,,\end{split}$
(3.11)

and the $(D-4)$-form is

$\displaystyle B$
$\displaystyle=-\frac{4s_{1}s_{2}(N_{y}x^{3}-N_{x}y^{3})}{a^{2}(x^{2}-y^{2})}\,\mathrm{vol}(\mathrm{AdS}_{2})\qquad$
$\displaystyle(D=6)\,,$ (3.12) $\displaystyle B$
$\displaystyle=-\frac{2s_{1}s_{2}(N_{y}x^{4}-N_{x}y^{4})}{a^{3}(x^{2}-y^{2})}\,\mathrm{vol}(\mathrm{AdS}_{3})-\frac{2N_{y}gs_{1}s_{2}x}{\Xi
a^{2}y^{2}}\,\mathrm{d}\psi\wedge\mathrm{d}\phi\wedge\mathrm{d}x$
$\displaystyle-\frac{2N_{x}gs_{1}s_{2}y}{\Xi
a^{2}x^{2}}\,\mathrm{d}\psi\wedge\mathrm{d}\phi\wedge\mathrm{d}y\qquad$
$\displaystyle(D=7)\,.$

The functions are

$\displaystyle\Delta_{y}$
$\displaystyle=-y^{2}+a^{2}-\frac{2N_{y}}{y^{D-5}}+g^{2}\biggl{(}y^{2}-\frac{2N_{y}s_{1}^{2}}{y^{D-5}}\biggr{)}\biggl{(}y^{2}-\frac{2N_{y}s_{2}^{2}}{y^{D-5}}\biggr{)}-a^{2}g^{2}y^{2}-\frac{2N_{y}a^{2}g^{2}s_{1}^{2}s_{2}^{2}}{y^{D-5}}\,,$
$\displaystyle\Delta_{x}$
$\displaystyle=-x^{2}+a^{2}-\frac{2N_{x}}{x^{D-5}}+g^{2}\biggl{(}x^{2}-\frac{2N_{x}s_{1}^{2}}{x^{D-5}}\biggr{)}\biggl{(}x^{2}-\frac{2N_{x}s_{2}^{2}}{x^{D-5}}\biggr{)}-a^{2}g^{2}x^{2}-\frac{2N_{x}a^{2}g^{2}s_{1}^{2}s_{2}^{2}}{x^{D-5}}\,,$
$\displaystyle V_{y}^{2}$
$\displaystyle=V_{y}^{(1)}V_{y}^{(2)}\,,\qquad\widetilde{V}_{y}^{2}=\widetilde{V}_{y}^{(1)}\widetilde{V}_{y}^{(2)}\,,\qquad
V_{x}^{2}=V_{x}^{(1)}V_{x}^{(2)}\,,\qquad\widetilde{V}_{x}^{2}=\widetilde{V}_{x}^{(1)}\widetilde{V}_{x}^{(2)}\,,$
$\displaystyle V_{y}^{(i)}$
$\displaystyle=1-g^{2}\biggl{(}y^{2}-\frac{2N_{y}s_{i}^{2}}{y^{D-5}}\biggr{)}\,,\qquad\widetilde{V}_{y}^{(i)}=1-\frac{1}{a^{2}}\biggl{(}y^{2}-\frac{2N_{y}s_{i}^{2}}{y^{D-5}}\biggr{)}\,,$
(3.13) $\displaystyle V_{x}^{(i)}$
$\displaystyle=1-g^{2}\biggl{(}x^{2}-\frac{2N_{x}s_{i}^{2}}{x^{D-5}}\biggr{)}\,,\qquad\widetilde{V}_{x}^{(i)}=1-\frac{1}{a^{2}}\biggl{(}x^{2}-\frac{2N_{x}s_{i}^{2}}{x^{D-5}}\biggr{)}\,,$
$\displaystyle H_{i}$
$\displaystyle=1+\frac{2s_{i}^{2}}{x^{2}-y^{2}}\biggl{(}\frac{N_{y}}{y^{D-5}}-\frac{N_{x}}{x^{D-5}}\biggr{)}\,,$
$\displaystyle\Xi$ $\displaystyle=1-a^{2}g^{2}\,,\qquad
s_{i}=\sinh\delta_{i}\,,\quad
c_{i}=\cosh\delta_{i}\,,\quad\tilde{c}_{i}=\sqrt{1+a^{2}g^{2}s_{i}^{2}}\,.$

The constant $g$ is related to the gauge coupling as $g=2g_{c}/3$ in $D=6$ and
as $g=g_{c}/2$ in $D=7$. In our discussion we shall assume $g_{c}>0$ in both
theories, therefore $g>0$. The background is completely determined by a real
parameter $a$, two charges $\delta_{1,2}$ and two parameters $N_{y}$ and
$N_{x}$ (corresponding to the NUT and the mass of the Lorentzian solution).
When $x=0$ or $y=0$ a curvature singularity is encountered, therefore the
range of definition of these two coordinates will have to avoid the origin of
the real axis. Moreover, in the rest of the paper we shall consider these two
ranges to be disjoint, in order to prevent the occurrence of a singularity
when $x=y$. We refer to section 6 for a very brief discussion about the latter
case.

The equations of motion that descend from the actions (3.1) and (3.4), as well
as the self-duality condition (3.7), have been checked explicitly, with the
spacetime orientation induced by the coordinate ordering
$(\mathrm{AdS},y,\phi,x,\psi)$.

The $\mathrm{AdS}_{D-4}\times\mathbb{M}_{4}$ background is invariant under the
exchange of the coordinates $x$ and $y$, accompanied by $N_{x}\leftrightarrow
N_{y}$. Additionally, the system is also invariant under the discrete
inversion symmetry

$\begin{gathered}y\mapsto\gamma\,y\,,\qquad
x\mapsto\gamma\,x\,,\qquad\psi\mapsto\gamma\,\phi\,,\qquad\phi\mapsto\gamma\,\psi\,,\\\
a\mapsto\gamma^{2}a\,,\qquad N_{y}\mapsto\gamma^{D-1}N_{y}\,,\qquad
N_{x}\mapsto\gamma^{D-1}N_{x}\,,\qquad
s_{i}\mapsto\gamma^{-1}s_{i}\,,\end{gathered}$ (3.14)

with $\gamma=\pm 1/(ag)$, which connects solutions with positive and negative
values of $\Xi$.

To see the relation to the solutions in [43] we must perform an analytic
continuation of $\mathrm{AdS}_{D-4}$ into the sphere $S^{D-4}$ and define the
time and radial coordinates $t$ and $r$ as well as the mass parameter $M$ as

${}t=\psi\,,\qquad r=\mathrm{i}\,x\,,\qquad M=\mathrm{i}^{D-5}N_{x}\,.$ (3.15)

The result is the family of black holes of [43] for $D=6,7$, with the addition
of a NUT parameter, here denoted as $N_{y}$. In the same way it is possible to
retrieve the seven-dimensional solutions presented in [44], equivalent to the
ones of [43], but written in generalized Boyer–Lindquist coordinates.

Moreover, the solutions presented in this paper and the Kerr–NUT–AdS black
holes constructed in [45] overlap and agree in the intersecting region of
their parameters. Indeed, the uncharged subfamily of the former coincides with
the $r\to 0$ limit of the latter, when the mass parameter is set to zero and
only one angular momentum is retained. Lastly, in both six and seven
dimensions, the $\mathrm{AdS}$ vacuum solution is recovered setting
$N_{y}=N_{x}=0$, and its radius is $L_{\mathrm{AdS}}=1/g$.

In [43], the supersymmetry conditions for the non-extremal black holes
constructed therein were derived by studying the vanishing of the eigenvalues
of the Bogomolny matrix. The BPS condition obtained is $ags_{1}s_{2}=\pm 1$,
with $\delta_{1}$ and $\delta_{2}$ of the same sign, either positive or
negative.121212These combinations can be obtained considering the whole set of
eigenvalues of the Bogomolny matrix, i.e. all the possible combinations $E\pm
gJ\pm(Q_{1}+Q_{2})$, not only the one studied in [43] (see [46], setting to
zero all the angular momenta, but one). Our solution is the analytic
continuation of the black holes of [43], with the addition of NUT. Since the
mass parameter of the latter does not appear explicitly in the supersymmetry
condition, we might expect that the introduction of a NUT parameter does not
affect this constraint. Furthermore, since the BPS condition is not altered by
(3.15), we conjecture that the solution we presented is supersymmetric when
$ags_{1}s_{2}=\pm 1$, with $\delta_{1}$ and $\delta_{2}$ of the same sign.
This assumption is confirmed by the explicit computation we performed in $D=6$
when the two charges are equal ($\delta_{1}=\delta_{2}\equiv\delta$), which
shows that a non-trivial Killing spinor can be constructed if and only if
$ags^{2}=\pm 1$, with $s=\sinh\delta$ (see appendix C for all the details).
Additionally, as shown in appendix A, the supersymmetric
$\mathrm{AdS}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$ solutions
of [1, 12] and [10] can be retrieved if and only if the condition
$ags_{1}s_{2}=\pm 1$ is imposed, thus giving evidence for the validity our
hypothesis.

###  3.2 Equal charges solutions

The study of the general system presented in the previous subsection is
computationally involved due to the complicated form of the $(\psi,\phi)$ part
of the metric (3.9). For this reason, from now on we shall focus on the more
tractable subfamily of solutions with equal charges
$\delta_{1}=\delta_{2}\equiv\delta$. Despite being simpler, this system still
captures all the important features of the four-dimensional toric orbifold
$\mathbb{M}_{4}$, allowing for a detailed analysis of its structure and a
reliable test of the gravitational blocks conjecture of [1]. We are confident
that with some extra effort (and a good motivation) it should be possible to
analyse the more general solutions with unequal charges.

When the two charges are set equal, the metric on the toric orbifold (3.9)
acquires the following diagonal structure

$\begin{split}\mathrm{d}s_{\mathbb{M}_{4}}^{2}&=\frac{1}{\Xi^{2}H^{2}}\biggl{[}\frac{\Delta_{y}}{y^{2}-x^{2}}\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}^{2}+\frac{\Delta_{x}}{x^{2}-y^{2}}\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}^{2}\biggr{]}\\\
&+\frac{y^{2}-x^{2}}{\Delta_{y}}\,\mathrm{d}y^{2}+\frac{x^{2}-y^{2}}{\Delta_{x}}\,\mathrm{d}x^{2}\,,\end{split}$
(3.16)

where we defined the functions $H\equiv H_{1}=H_{2}$, $V_{\bullet}\equiv
V_{\bullet}^{(1)}=V_{\bullet}^{(2)}$ and
$\widetilde{V}_{\bullet}\equiv\widetilde{V}_{\bullet}^{(1)}=\widetilde{V}_{\bullet}^{(2)}$.
This class of solutions is characterized by having equal scalar fields and
equal gauge fields, where the latter now read

$A_{1}=A_{2}=\frac{2sc\tilde{c}}{\Xi(x^{2}-y^{2})H}\biggl{[}\frac{N_{y}}{y^{D-5}}\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}-\frac{N_{x}}{x^{D-5}}\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}\biggr{]}\,,$
(3.17)

with $s=\sinh\delta$, $c=\cosh\delta$ and
$\tilde{c}=\sqrt{1+a^{2}g^{2}s^{2}}$. For later convenience, we define the
orthonormal frame

$\displaystyle\hat{e}^{1}$
$\displaystyle=\sqrt{\frac{y^{2}-x^{2}}{\Delta_{y}}}\,\mathrm{d}y\,,\qquad$
$\displaystyle\hat{e}^{2}$
$\displaystyle=\frac{1}{\Xi\,H}\sqrt{\frac{\Delta_{y}}{y^{2}-x^{2}}}\,\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}\,,$
(3.18) $\displaystyle\hat{e}^{3}$
$\displaystyle=\sqrt{\frac{x^{2}-y^{2}}{\Delta_{x}}}\,\mathrm{d}x\,,\qquad$
$\displaystyle\hat{e}^{4}$
$\displaystyle=\frac{1}{\Xi\,H}\sqrt{\frac{\Delta_{x}}{x^{2}-y^{2}}}\,\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}\,.$

When the system is supersymmetric, additional properties arise. Assuming
$ags^{2}=-\kappa$, with $\kappa=\pm 1$, we can decompose $\Delta_{x}$ and
$\Delta_{y}$ as

$\Delta_{x}=\Delta_{x}^{+}\,\Delta_{x}^{-}\,,\qquad\Delta_{y}=\Delta_{y}^{+}\Delta_{y}^{-}\,,$
(3.19)

where

$\begin{split}\Delta_{x}^{\pm}&=g\biggl{(}x^{2}-\frac{2N_{x}s^{2}}{x^{D-5}}\biggr{)}-\kappa\,a\pm(1-\kappa\,ag)x\,,\\\
\Delta_{y}^{\pm}&=g\biggl{(}y^{2}-\frac{2N_{y}s^{2}}{y^{D-5}}\biggr{)}-\kappa\,a\pm(1-\kappa\,ag)y\,.\end{split}$
(3.20)

Considering, e.g., $\Delta_{x}$, every root of this function is necessarily a
root of either $\Delta_{x}^{+}$ or $\Delta_{x}^{-}$, therefore, given a
generic root $x_{*}$ of $\Delta_{x}$, we have

$x_{*}^{2}-\frac{2N_{x}s^{2}}{x_{*}^{D-5}}=\frac{\kappa\,a}{g}-\tau\,\frac{1-\kappa\,ag}{g}\,x_{*}\,,$
(3.21)

with $\tau=\pm$ according to the fact that $x_{*}$ is a root of
$\Delta_{x}^{\pm}$, respectively. Thanks to this relation and the
corresponding one for $y_{*}$, we can obtain simplified expressions for many
functions, when evaluated at one of the roots $x_{\pm}$ or $y_{\pm}$. In
particular, the function $H$ reads

$(x_{*}^{2}-y_{*}^{2})H(x_{*},y_{*})=-\frac{1-\kappa\,ag}{g}\,(\tau^{(x)}x_{*}-\tau^{(y)}y_{*})\,,$
(3.22)

while the derivatives of $\Delta_{x}$ and $\Delta_{y}$ become, collectively,

$\Delta_{z}^{\prime}(z_{*})=-2\tau(1-\kappa\,ag)\bigl{[}(D-3)gz_{*}^{2}+\tau(D-4)(1-\kappa\,ag)z_{*}-\kappa(D-5)a\bigr{]}\,,$
(3.23)

where $z$ can be either $x$ or $y$.

For the sake of simplicity, from now on we shall restrict the supersymmetry
condition to $ags^{2}=+1$, discarding the case $-1$ and with no assumption on
the sign of $\delta$. The discussion could be carried out in the other case
with minor modifications.

###  3.3 Conditions of existence

We will now begin investigating global aspects of the solutions, starting from
some basic necessary conditions of existence. In particular, in order for the
solutions to well-defined we require that all the quantities are real-valued,
the signature of the metric is Lorentzian and the scalar fields are positive.
These requirements are met if and only if

$\frac{\Delta_{x}}{x^{2}-y^{2}}>0\,,\qquad\frac{\Delta_{y}}{y^{2}-x^{2}}>0\,,\qquad
H>0$ (3.24)

in two closed intervals $[x_{-},x_{+}]$ and $[y_{-},y_{+}]$ not intersecting
and not containing the curvature singularities in $x=0$ and $y=0$. Without
loss of generality we restrict to positive values of $x$ and $y$. This can
always be done in $D=7$, while in $D=6$ a change in the sign of $N_{x}$ or
$N_{y}$ may be needed.131313In the latter case, the two-form $B$ changes its
sign, but the original form can always be restored changing the sign of the
$\mathrm{AdS}_{2}$ time coordinate. Taking advantage of the exchange symmetry
between $x$ and $y$ we furthermore assume $x>y$, and the first two conditions
in (3.24) boil down to $\Delta_{x}>0$ and $\Delta_{y}<0$. Having restricted to
$x,y>0$, we can analyse the loci of the roots and the positivity of
$\Delta_{x}$ and $\Delta_{y}$ also studying the polynomials $P_{x}\equiv
x^{2(D-5)}\Delta_{x}$ and $P_{y}\equiv y^{2(D-5)}\Delta_{y}$. The positivity
of $H$ can be examined considering the function $P_{H}\equiv(x^{2}-y^{2})H$
because $x>y$.

For the rest of this subsection, we shall assume the background is
supersymmetric, i.e. $ags^{2}=1$. This relation implies that $a>0$, being $g$
positive. By means of the decompositions (3.19), we can write

$P_{x}=P_{N_{x}}^{+}(x)\,P_{N_{x}}^{-}(x)\,,\qquad
P_{y}=P_{N_{y}}^{+}(y)\,P_{N_{y}}^{-}(y)\,,$ (3.25)

where we defined the polynomials

$P_{M}^{\pm}(z)=\left\\{\begin{aligned}
&gz^{3}\pm(1+ag)z^{2}+az-\frac{2M}{a}\qquad&(D&=6)\,,\\\
&gz^{4}\pm(1+ag)z^{3}+az^{2}-\frac{2M}{a}\qquad&(D&=7)\,,\end{aligned}\right.$
(3.26)

in which $z$ can be either $x$ or $y$ and $M$ is $N_{x}$ or $N_{y}$
accordingly. Notice that, e.g., $P_{N_{x}}^{\pm}(x)=x^{D-5}\Delta_{x}^{\pm}$.

As it can be shown, $P_{x}>0$ in $x=0$ and $P_{x}\to+\infty$ when $x$ goes to
infinity. Since we need $P_{x}$ to be positive in a closed interval in the
positive $x$-axis, this polynomial must have at least four positive roots. By
Descartes’ rule of signs, a necessary condition for this to happen is that
$N_{x}>0$. In general, when $M>0$, $P_{M}^{+}(z)$ has one positive root, while
$P_{M}^{-}(z)$ can have one or three of them. In the desired configuration
$P_{N_{x}}^{-}(x)$ must have three positive roots, so to bring the total
amount to four.

Applying Descartes’ rule, we see that, in $D=6$, $P_{N_{x}}^{-}(x)$ has no
negative root, hence all the existing ones are positive. In six dimensions
$P_{N_{x}}^{-}(x)$ is a cubic polynomial, which admits three distinct real
roots if and only if its discriminant is positive. Restricting to $N_{x}>0$,
this condition is met when

$0<N_{x}<a^{3}\,\frac{-2q^{3}+3q^{2}+3q-2+2(q^{2}-q+1)^{3/2}}{54q^{2}}\,,$
(3.27)

where we defined $q=ag$.

In seven dimensions $P^{-}_{N_{x}}(x)$ has one negative root, therefore the
existence of four roots would ensure that three of them are positive. In this
case $P^{-}_{N_{x}}(x)$ is a quartic polynomial and its discriminant is
positive if and only if

$0<N_{x}<\frac{a^{3}}{g}\,\frac{-27q^{4}+36q^{3}-2q^{2}+36q-27q+(1+q)(9q^{2}-14q+9q)^{3/2}}{1024q^{2}}\,,$
(3.28)

where, again, we restricted to $N_{x}>0$ and introduced the constant $q=ag$.
In the case of a quartic polynomial this alone does not guarantee the
existence of four distinct real roots. We need the additional conditions

$-3q^{2}+2q-3q<0\,,\qquad-3q^{4}+4q^{3}-2(1+64N_{x})q^{2}+4q-3<0\,.$ (3.29)

The first one is always satisfied for every real $q$, while the second one
gives

$N_{x}>-\frac{(1-q)^{2}(3+2q+3q^{2})}{128q^{2}}\,.$ (3.30)

Since the right-hand side of the above inequality is never positive, the
condition (3.30) is automatically implied by the constraint $N_{x}>0$.

Considering $a$ and $N_{x}$ such that $P_{x}$ has four distinct real roots,
the structure of $P_{N_{x}}^{\pm}(x)$ is depicted in figure 2. Specifically,
$x_{-}=x_{2}$, the smallest root of $P_{N_{x}}^{-}(x)$, and $x_{+}=x_{3}$, the
middle root of $P_{N_{x}}^{-}(x)$, since in this interval $P_{N_{x}}^{\pm}(x)$
are both positive and so is their product $P_{x}$. The location of the two
functions follows from the fact that $P_{M}^{+}(z)>P_{M}^{-}(z)$ where $z>0$
and that they are equal in $z=0$. Because $x_{\pm}$ are both roots of
$P_{N_{x}}^{-}(x)$, we have $\tau^{(x_{\pm})}=-1$.

$x$$x_{1}$$x_{2}$$x_{3}$$x_{4}$$-\dfrac{2N_{x}}{a}$$P_{N_{x}}^{+}$$P_{N_{x}}^{-}$
Figure 2: Example of graphs of $P_{N_{x}}^{\pm}(x)$ in the configuration in
which $P_{x}$ admits four positive roots.

We now move to the analysis of $P_{y}$, which has to be negative in a closed
positive interval $[y_{-},y_{+}]$ such that $y_{+}<x_{-}$. First of all, we
notice from their definition in (3.26) that $P_{N_{x}}^{+}(x)$ and
$P_{N_{y}}^{+}(y)$ have the same shape, but are shifted vertically; similarly
happens for $P_{N_{x}}^{-}(x)$ and $P_{N_{y}}^{-}(y)$. The strategy is to
start from the configuration $N_{y}=N_{x}$, in which $P_{N_{y}}^{\pm}(y)$ and
$P_{N_{x}}^{\pm}(x)$ are coincident, and to deform it, studying how the system
modifies. In the initial case, $P_{y}$ is negative in the intervals
$[y_{1},y_{2}]$ and $[y_{3},y_{4}]$, with $y_{i}=x_{i}$. Because
$[y_{1},y_{2}]$ lies to the left of $[x_{-},x_{+}]$, it will be our candidate.
As $N_{y}$ increases, $P_{N_{y}}^{-}(y)$ moves below $P_{N_{x}}^{-}(x)$ and
the root $y_{2}$ is shifted to the right of $x_{2}$, yielding
$y_{+}=y_{2}>x_{-}$. When, moreover, $N_{y}$ exceeds a specific value, $y_{2}$
and $y_{3}$ disappear, giving the identification $y_{+}=y_{4}>x_{+}$. Both
cases are not allowed. On the contrary, when $N_{y}$ decreases, both
$P_{N_{y}}^{\pm}(y)$ move up and the roots $y_{1}$ and $y_{2}$ are shifted to
the left of $x_{1}$ and $x_{2}$, respectively, resulting in the desired closed
interval with $y_{-}=y_{1}$ and $y_{+}=y_{2}$. However, when $N_{y}$ becomes
negative, the roots $y_{1}$ and $y_{2}$ disappear, while $y_{3}$ and $y_{4}$,
if still existing, are brought closer to one another. In this case, the only
possible interval has $y_{-}=y_{3}>x_{+}$ and is, thus, not acceptable.
Ultimately, the roots of $P_{y}$ meet all the requirements for the metric to
have a correct signature if and only if141414In the special configuration
$N_{y}=N_{x}$ we have $y_{2}=x_{2}$, hence $y_{+}=x_{-}$.

$0<N_{y}<N_{x}\,,$ (3.31)

with $N_{x}$ satisfying (3.27), in $D=6$, or (3.28), in $D=7$. In this case
$y_{-}=y_{1}$, the unique root of $P_{N_{y}}^{+}(y)$, and $y_{+}=y_{2}$, the
smallest root of $P_{N_{y}}^{-}(y)$. These results yield $\tau^{(y_{-})}=+1$
and $\tau^{(y_{+})}=-1$.

The last condition we examine is the positivity of $H$ or, equivalently, of
$P_{H}$. It can be shown that, in the range of definition of the coordinates,
$\partial_{x}P_{H}>0$ and $\partial_{y}P_{H}<0$, therefore the minimum value
is reached in $x=x_{-}$ and $y=y_{+}$. When supersymmetry is imposed, we can
make use of equation (3.22), with $\kappa=-1$ and $\tau^{(x_{-})}=-1$, because
$x_{-}$ is a root of $P_{N_{x}}^{-}(x)$, hence of $\Delta_{x}^{-}$. Since
$x>y$, it follows that $P_{H}(x_{-},y_{+})>0$, hence $P_{H}$ is positive in
the region of interest.

##  4 Global structure of the solutions

We will now analyse global aspects of the solutions, uncovering the geometry
of the underlying four-dimensional toric orbifolds $\mathbb{M}_{4}$. This will
be done combining the information obtained from the Killing vectors of the
metrics, the compatible complex structure, as well as the flux quantization
conditions, ensuring that the gauge fields are well-defined connections on
orbifold line bundles. Although this is a standard practice, we will find that
the analysis for our new solutions is unusually sophisticated, due to the lack
of a fibration structure (present in the solutions of [10, 12, 1]). Below we
will concentrate on the global regularity of the solutions in $D=6,7$
dimensions. Although in principle one should perform this analysis in ten or
eleven dimensions, the regularity of the metrics in string/M-theory will
follow from the regularity of the lower-dimensional metrics together with the
regularity of the scalars and the correct quantization conditions for the
gauge fields. The explicit uplifted solutions are discussed in appendix B,
where we also address the quantization conditions of all the fluxes and give
the relevant gravitational observables.

###  4.1 Warm-up: revisiting the solutions of [1]

In order to illustrate our strategy for analysing the solutions presented
here, we will first apply the same logic to the solutions in [1]. The geometry
of the toric orbifolds associated with those solutions has some additional
structure that makes the analysis considerably simpler.

The four-dimensional orbifold $\mathbb{M}_{4}$ of [1], which describes the
fibration of a spindle over another spindle
$\mathbbl{\Sigma}_{1}(x,\psi)\ltimes\mathbbl{\Sigma}_{2}(y,z)$, is locally
described by the metric

$\begin{split}\mathrm{d}s_{\mathbb{M}_{4}}^{2}=\frac{x^{2}}{q}\,\mathrm{d}x^{2}+\frac{q}{4x^{2}}\,\mathrm{d}\psi^{2}+\frac{y^{2}}{F}\,\mathrm{d}y^{2}+\frac{F}{h_{1}h_{2}}\Bigl{(}\mathrm{d}z-\frac{1}{2m}\Bigl{(}1-\frac{\mathtt{a}}{x}\Bigr{)}\,\mathrm{d}\psi\Bigr{)}^{2}\,.\end{split}$
(4.1)

The coordinates $x$ and $y$ range in $[x_{-},x_{+}]$ and $[y_{-},y_{+}]$,
respectively, where $q(x_{\pm})=0$ and $F(y_{\pm})=0$. The singularities of
these spindles are encoded by the following quantization conditions

$\displaystyle\frac{q^{\prime}(x_{-})}{4x_{-}^{2}}\,\Delta\psi$
$\displaystyle=\frac{2\pi}{m_{-}}\,,\qquad$
$\displaystyle-\frac{q^{\prime}(x_{+})}{4x_{+}^{2}}\,\Delta\psi$
$\displaystyle=\frac{2\pi}{m_{+}}\,,$ (4.2)
$\displaystyle\frac{gF^{\prime}(y_{-})}{3y_{-}^{3}}\,\Delta z$
$\displaystyle=\frac{2\pi}{n_{-}}\,,\qquad$
$\displaystyle-\frac{gF^{\prime}(y_{+})}{3y_{+}^{3}}\,\Delta z$
$\displaystyle=\frac{2\pi}{n_{+}}\,,$

where the minus signs are due to
$\operatorname{sign}(q^{\prime}(x_{+}))=\operatorname{sign}(F^{\prime}(y_{+}))=-1$.
The fibration is required to be well-defined in the orbifold sense, that is

$\frac{1}{2\pi}\int_{\mathbbl{\Sigma}_{1}}\mathrm{d}\eta=\frac{t}{m_{+}m_{-}}\,,\qquad\eta=\frac{2\pi}{\Delta
z}\Bigl{(}\mathrm{d}z-\frac{1}{2m}\Bigl{(}1-\frac{\mathtt{a}}{x}\Bigr{)}\,\mathrm{d}\psi\Bigr{)}\,,\qquad
t\in\mathbb{Z}\,,$ (4.3)

and we assume, for the moment, that $t$ is coprime to $m_{\pm}$. In this way
$\eta$ is a connection on $\mathcal{O}(t)$, and at a fixed value of $y$ the
metric (4.1) describes a lens space151515Notice that here $t$ is negative.
$L(-t,1)=S^{3}/\mathbb{Z}_{-t}$ with base $\mathbbl{\Sigma}_{1}$. This metric
possesses four Killing vectors $\xi_{a}$ which degenerates on the loci

$\mathcal{L}_{1}=\\{y=y_{-}\\}\,,\qquad\mathcal{L}_{2}=\\{x=x_{-}\\}\,,\qquad\mathcal{L}_{3}=\\{y=y_{+}\\}\,,\qquad\mathcal{L}_{4}=\\{x=x_{+}\\}\,.$
(4.4)

To focus on these $\mathcal{L}_{a}$, we define adapted coordinates

$\tilde{\psi}=m_{+}\phi_{-}-m_{-}\phi_{+}\,,\quad\tilde{z}=-m_{+}\frac{\mathtt{a}-x_{+}}{2mx_{+}}\frac{\Delta\psi}{\Delta
z}\,\phi_{-}+m_{-}\frac{\mathtt{a}-x_{-}}{2mx_{-}}\frac{\Delta\psi}{\Delta
z}\,\phi_{+}\,,\quad\frac{(2\pi)^{2}}{-t}=\Delta\phi_{-}\Delta\phi_{+}\,,$
(4.5)

where $\tilde{\psi}=\frac{2\pi}{\Delta\psi}\psi$ and
$\tilde{z}=\frac{2\pi}{\Delta z}z$ are $2\pi$-periodic and the total metric,
restricted to $\mathcal{L}_{a}$, reads

$\begin{split}\mathrm{d}s_{(1)}^{2}&=\mathrm{d}s_{(3)}^{2}=\frac{x^{2}}{q}\,\mathrm{d}x^{2}+\frac{q}{4x^{2}}\,\mathrm{d}\psi^{2}\,,\\\
\mathrm{d}s_{(2)}^{2}&=\frac{y^{2}}{F}\,\mathrm{d}y^{2}+\frac{t^{2}}{m_{-}^{2}}\frac{F}{h_{1}h_{2}}\biggl{(}\frac{\Delta
z}{2\pi}\biggr{)}^{2}\mathrm{d}\phi_{-}^{2}\,,\\\
\mathrm{d}s_{(4)}^{2}&=\frac{y^{2}}{F}\,\mathrm{d}y^{2}+\frac{t^{2}}{m_{+}^{2}}\frac{F}{h_{1}h_{2}}\biggl{(}\frac{\Delta
z}{2\pi}\biggr{)}^{2}\mathrm{d}\phi_{+}^{2}\,.\end{split}$ (4.6)

Looking, e.g., at $\mathrm{d}s_{(2)}^{2}$, we see that it describes
$\mathbbl{\Sigma}_{2}/\mathbb{Z}_{m_{-}}$ if we require
$(\Delta\phi_{-})^{(2)}=-2\pi/t$, where $(\Delta\phi_{-})^{(2)}$ is the
periodicity of $\phi_{-}$ when referred to $\mathcal{L}_{2}$. From (4.5), we
have then $(\Delta\phi_{+})^{(2)}=2\pi$. Similarly, the line element
$\mathrm{d}s_{(4)}^{2}$ describes $\mathbbl{\Sigma}_{2}/\mathbb{Z}_{m_{+}}$
when $(\Delta\phi_{+})^{(4)}=-2\pi/t$ and $(\Delta\phi_{-})^{(4)}=2\pi$. The
reason for which the periodicities seem not to be globally defined and “jump”
to different values on the loci $\mathcal{L}_{2}$ and $\mathcal{L}_{4}$, is
that the quotients must be done locally in patches, and then suitably glued.
Thus we should have defined local coordinates $\phi_{\pm}^{(2,4)}$ on a patch
that contains $\mathcal{L}^{(2,4)}$, for which $(\Delta\phi_{+})^{(2)}$ is
more precisely $\Delta(\phi_{+}^{(2)})$. Eventually, we will write
$\Delta\phi_{+}^{(2)}$ when it is clear what we mean. This will happen again
in the following, when we will use our procedure to extract the labels
associated with the new solutions.

From the degenerating Killing vectors on $\mathcal{L}_{a}$ we extract the
$\mathbb{Z}^{2}$-valued vectors [1]

$\vec{w}_{1}=(n_{-},0)\,,\qquad\vec{w}_{2}=(t\,a_{+},m_{-})\,,\qquad\vec{w}_{3}=(-n_{+},0)\,,\qquad\vec{w}_{4}=(t\,a_{-},-m_{+})\,,$
(4.7)

and an associated labelled polytope (or labelled fan), depicted in figure
3(a).

$p_{3}$ $p_{4}$ $p_{1}$ $p_{2}$
$n_{+}$$n_{-}$$\vec{\hat{w}}_{3}$$\vec{\hat{w}}_{4}$$\vec{\hat{w}}_{1}$$\vec{\hat{w}}_{2}$
(a) $\gcd(t,m_{\pm})=1$.

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$n_{-}$$m_{-}$$n_{+}$$m_{+}$$\vec{\hat{n}}_{1}$$\vec{\hat{n}}_{2}$$\vec{\hat{n}}_{3}$$\vec{\hat{n}}_{4}$
(b) $t=k_{+}k_{-}\overline{t}$.

Figure 3: Polytope of the $\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$
orbifold corresponding to the vectors (4.7) and (4.14), respectively.

Here $a_{\pm}\in\mathbb{Z}$ are chosen such that
$a_{+}\,m_{+}+a_{-}\,m_{-}=1$, which always exist by Bézout’s lemma for
coprime $m_{\pm}$. The set of labels derived by (4.7) is
$m_{a}=(n_{-},1,n_{+},1)$. To understand better why this is the case and the
reason for which $m_{\pm}$ are not labels, we look directly at the metric
(4.1). The four loci (4.4), with metrics (4.6), correspond to the divisors
$\hat{D}_{a}$ and topologically are

$\hat{D}_{1}=\mathbbl{\Sigma}_{1}\,,\qquad\hat{D}_{2}=\frac{\mathbbl{\Sigma}_{2}}{\mathbb{Z}_{m_{-}}}\,,\qquad\hat{D}_{3}=\mathbbl{\Sigma}_{1}\,,\qquad\hat{D}_{4}=\frac{\mathbbl{\Sigma}_{2}}{\mathbb{Z}_{m_{+}}}\,,$
(4.8)

where $\hat{D}_{1,3}$ are copies of the base $\mathbbl{\Sigma}_{1}$ at the
north and south poles of the fibre $\mathbbl{\Sigma}_{2}$, respectively. On
the other hand, at the poles of $\mathbbl{\Sigma}_{1}$, the four-dimensional
orbifold is locally modelled by
$(\mathbb{C}\times\mathbbl{\Sigma}_{2})/\mathbb{Z}_{m_{\mp}}$, and as a
consequence $\hat{D}_{2,4}$ are global quotients of the fibre
$\mathbbl{\Sigma}_{2}$. Notice that, even if
$\hat{D}_{2,4}=\mathbbl{\Sigma}_{2}/\mathbb{Z}_{m_{\mp}}$, there are no
orbifold singularities at generic points (i.e. different from the north and
south poles) on $\hat{D}_{2,4}$ and thus the labels are $m_{2,4}=1$. These are
then divisors associated with an ordinary fan, with primitive vectors
$\vec{\hat{w}}_{2,4}=\vec{w}_{2,4}$. The situation is different on
$\mathcal{L}_{1,3}$. Zooming in for example near to $y=y_{-}$, the metric at
leading order in $R^{2}=\frac{4y_{-}^{2}}{F^{\prime}(y_{-})}(y-y_{-})$
reads161616See section 4.4 of [34] for an analogous computation.

$\mathrm{d}s_{\mathbb{M}_{4}}^{2}\underset{y_{-}}{\simeq}\frac{x^{2}}{q}\,\mathrm{d}x^{2}+\frac{q}{4x^{2}}\,\mathrm{d}\psi^{2}+\,\mathrm{d}R^{2}+\frac{R^{2}}{n_{-}}\Bigl{(}\mathrm{d}\tilde{z}-\frac{1}{2m}\Bigl{(}1-\frac{\mathtt{a}}{x}\Bigr{)}\frac{1}{\Delta
z}\,\mathrm{d}\psi\Bigr{)}^{2}\,.$ (4.9)

When we set $y=y_{-}$, the metric reduces to $\mathrm{d}s^{2}_{(1)}$ in (4.6),
which is $\hat{D}_{1}=\mathbbl{\Sigma}_{1}$. However, at each point $x_{-}\leq
x\leq x_{+}$, from (4.9) we see that there is a normal conical singularity
giving, locally, a copy of
$\mathbbl{\Sigma}_{1}\ltimes(\mathbb{C}/\mathbb{Z}_{n_{-}})$. As a
consequence, $n_{-}$ is a label with associated divisor
$D_{1}=\hat{D}_{1}\times\mathrm{pt}/\mathbb{Z}_{n_{-}}$. The metric (4.9) can
be compared with its counterpart when approaching the locus $\mathcal{L}_{2}$.
In this case, we zoom in near to $x=x_{-}$ defining the new coordinate $R$
such that $x=x_{-}+\frac{q^{\prime}(x_{-})}{4x_{-}^{2}}R^{2}$. In the adapted
coordinates (4.5), the metric on $\mathbb{M}_{4}$ at leading order in $R^{2}$
reads

$\mathrm{d}s_{\mathbb{M}_{4}}^{2}\underset{x_{-}}{\simeq}\mathrm{d}R^{2}+R^{2}(\mathrm{d}\phi_{+}^{(2)}+c(y)\,\mathrm{d}\phi_{-}^{(2)})^{2}+\frac{y^{2}}{F}\,\mathrm{d}y^{2}+\frac{t^{2}}{m_{-}^{2}}\frac{F}{h_{1}h_{2}}\biggl{(}\frac{\Delta
z}{2\pi}\biggr{)}^{2}(\mathrm{d}\phi_{-}^{(2)})^{2}\,,$ (4.10)

with $c(y)$ a regular function on $[y_{-},y_{+}]$ such that
$c(y_{\pm})=-m_{+}/m_{-}$. Assuming $(\Delta\phi_{+})^{(2)}=2\pi$ as before,
the first contribution to the total metric is the smooth complex plane in
polar coordinates, therefore no singularity along the divisor $D_{2}$ is
present. Setting $R=0$, which accounts in moving exactly on the divisor, the
line element reduces to the second line of (4.6), which, as already discussed,
corresponds to the metric on $\mathbbl{\Sigma}_{2}/\mathbb{Z}_{m_{-}}$, once
the correct periodicity for $\phi_{-}^{(2)}$ is considered. An analogous
reasoning can be applied to $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$, showing
that there is a $\mathbb{Z}_{n_{+}}$ acting on $\hat{D}_{3}$, but no
singularity along $D_{4}$.

We now consider the case when $t$ and $m_{\pm}$ have common factors, plugging
a small gap in the discussion [1]. On general grounds, it is known that in
this case the fibration does not smooth the orbifold points on the base [6].
Then, the four-dimensional orbifold will be characterized by four different
labels with $m_{2}=\gcd(t,m_{-})$ and $m_{4}=\gcd(t,m_{+})$, as we shall
understand in a moment. To see explicitly the difference with the previous
case, we can zoom in near $y=y_{\pm}$ in (4.10), obtaining

$\displaystyle\mathrm{d}s_{\mathbb{M}_{4}}^{2}$
$\displaystyle\underset{x_{-},y_{-}}{\simeq}\mathrm{d}R^{2}+R^{2}\Bigl{(}\mathrm{d}\phi_{+}^{(2)}-\frac{m_{+}}{m_{-}}\
\mathrm{d}\phi_{-}^{(2)}\Bigr{)}^{2}+\mathrm{d}P^{2}+\Bigl{(}\frac{t}{m_{-}n_{-}}\Bigr{)}^{2}P^{2}(\mathrm{d}\phi_{-}^{(2)})^{2}\,,$
(4.11) $\displaystyle\mathrm{d}s_{\mathbb{M}_{4}}^{2}$
$\displaystyle\underset{x_{-},y_{+}}{\simeq}\mathrm{d}R^{2}+R^{2}\Bigl{(}\mathrm{d}\phi_{+}^{(2)}-\frac{m_{+}}{m_{-}}\
\mathrm{d}\phi_{-}^{(2)}\Bigr{)}^{2}+\mathrm{d}P^{2}+\Bigl{(}\frac{t}{m_{-}n_{+}}\Bigr{)}^{2}P^{2}(\mathrm{d}\phi_{-}^{(2)})^{2}\,.$

We can now take $t=k_{+}\overline{t}_{+}=k_{-}\overline{t}_{-}$ and
$m_{\pm}=k_{\pm}\overline{m}_{\pm}$, which implies $t=k_{+}k_{-}\overline{t}$,
such that $\gcd(t,m_{\pm})=k_{\pm}$. The requirement
$\Delta{\phi}_{-}^{(2)}=2\pi/(-k_{+}\overline{t})$ translates into

$\displaystyle\frac{2\pi}{-k_{+}\overline{t}}\,\Delta\phi_{+}^{(2)}=\Delta{\phi}_{-}^{(2)}\Delta\phi_{+}^{(2)}=\frac{(2\pi)^{2}}{-t}\quad\implies\quad\Delta\phi_{+}^{(2)}=\frac{2\pi}{k_{-}}\,.$
(4.12)

Then the space is
$(\mathbbl{\Sigma}_{2}/\mathbb{Z}_{\overline{m}_{-}})\ltimes(\mathbb{C}/\mathbb{Z}_{k_{-}})$
and $k_{-}$ acts as a label on the transverse space to the divisor.
Subsequently, there exists an associated divisor $D_{2}$ with
$\hat{D}_{2}=\mathbbl{\Sigma}_{2}/\mathbb{Z}_{\overline{m}_{-}}$. This can
also be seen from the vectors (4.7), which become

$\vec{w}_{1}=n_{-}(1,0)\,,\quad\vec{w}_{2}=k_{-}(a_{+}k_{+}\overline{t},\overline{m}_{-})\,,\quad\vec{w}_{3}=n_{+}(-1,0)\,,\quad\vec{w}_{4}=k_{+}(a_{-}k_{-}\overline{t},-\overline{m}_{+})\,,$
(4.13)

and the labels are $m_{a}=(n_{-},k_{-},n_{+},k_{+})$, as stated before. The
case $\overline{m}_{\pm}=1$ (that is $t=m_{+}m_{-}\overline{t}$) is
particularly familiar, in that the connection is on
$\mathcal{O}(m_{+}m_{-}\overline{t})$ and at a fixed value of $y$ the metric
(4.1) describes a branched lens space $L_{m_{\pm}}(-\overline{t},1)$ [47].
Using an $SL(2,\mathbb{Z})$ transformation, the vectors (4.13) can then be
rotated to

$\vec{n}_{1}=n_{-}(1,0)\,,\qquad\vec{n}_{2}=m_{-}(\overline{t},1)\,,\qquad\vec{n}_{3}=n_{+}(-1,0)\,,\qquad\vec{n}_{4}=m_{+}(0,-1)\,,$
(4.14)

which are in $\mathbb{Z}^{2}$, and the labels are simply
$m_{a}=(n_{-},m_{-},n_{+},m_{+})$. The associated labelled polytope is
sketched in figure 3(b), and is clearly a labelled (or, stacky) version of the
polytope associated with the Hirzebruch surfaces $\mathbb{F}_{-\overline{t}}$.

To summarize, in this simple example we can see that the presence of a
fibration can result in different types of polytopes, associated with
different $\mathbb{M}_{4}$, according to the values of $\gcd(t,m_{\pm})$. When
$t$ is relatively coprime with $m_{\pm}$, the integers $m_{\pm}$
characterizing the base spindle are not labels, but act as quotients on the
fixed points; vice versa, if $t$ has common factors with one or both
$m_{\pm}$, the resulting space contains orbifold points also on the base, with
more than two labels associated with it. In our more general solutions (3.16),
all the hatted divisors will be global quotients of some spindle, but with
also labels attached.

###  4.2 Degenerating Killing vectors

The four-dimensional orbifold $\mathbb{M}_{4}$ possesses a $U(1)^{2}$ isometry
associated with the two Killing vectors $\partial_{\psi}$ and
$\partial_{\phi}$ of the metric (3.16), therefore it is natural to study this
system within the framework of toric geometry. For the solutions in [1] we
showed that the toric data characterizing a given orbifold as well as the
fluxes, could be derived from the analysis of degenerate Killing vectors of
the metric. For our new solutions we will need to complement the study of the
degenerating Killing vectors with some additional information, in order to
extract all the data required to implement the extremization procedure.

The metric (3.16) has in total four degenerate Killing vectors

$\displaystyle\mathtt{j}_{\pm}$
$\displaystyle=\frac{2a^{2}}{\Delta_{x}^{\prime}(x_{\pm})}\bigl{[}\widetilde{V}_{x}(x_{\pm})\,\partial_{\psi}+V_{x}(x_{\pm})\,\partial_{\phi}\bigr{]}\equiv
J^{(\psi)}_{\pm}\partial_{\psi}+J^{(\phi)}_{\pm}\partial_{\phi}\,,$ (4.15)
$\displaystyle\mathtt{k}_{\pm}$
$\displaystyle=\frac{2a^{2}}{\Delta_{y}^{\prime}(y_{\pm})}\bigl{[}\widetilde{V}_{y}(y_{\pm})\,\partial_{\psi}+V_{y}(y_{\pm})\,\partial_{\phi}\bigr{]}\equiv
K^{(\psi)}_{\pm}\partial_{\psi}+K^{(\phi)}_{\pm}\partial_{\phi}\,,$ (4.16)

whose norm vanishes at $x=x_{\pm}$ and $y=y_{\pm}$, respectively. All of them
are normalized so to have unitary surface gravity. In order to present the
results in a uniform way we introduce the four loci

$\hat{D}_{1}=\\{x=x_{-}\\}\,,\qquad\hat{D}_{2}=\\{y=y_{-}\\}\,,\qquad\hat{D}_{3}=\\{x=x_{+}\\}\,,\qquad\hat{D}_{4}=\\{y=y_{+}\\}\,,$
(4.17)

and define the four vectors

$\xi_{1}=\mathtt{j}_{-}\,,\qquad\xi_{2}=\mathtt{k}_{-}\,,\qquad\xi_{3}=\mathtt{j}_{+}\,,\qquad\xi_{4}=\mathtt{k}_{+}\,,$
(4.18)

so that $\xi_{a}$ is the Killing vector that degenerates at $\hat{D}_{a}$.

For later convenience, we define the two alternative sets of coordinates
$\psi_{\pm}$ and $\chi_{\pm}$ such that
$\mathtt{j}_{\pm}=\varkappa_{J}\,\partial_{\psi_{\pm}}$ or
$\mathtt{k}_{\pm}=\varkappa_{K}\,\partial_{\chi_{\pm}}$, with
$\varkappa_{J}=\pm$ and $\varkappa_{K}=\pm$. These coordinates are adapted to
the direction generated by the corresponding Killing vector $\xi_{a}$ and, for
this reason, they will play a fundamental role when zooming in on the
different loci $\hat{D}_{a}$. They are defined by

$\displaystyle\psi$
$\displaystyle=\varkappa_{J}\sum_{\sigma=\pm}J_{\sigma}^{(\psi)}\psi_{\sigma}\,,\qquad$
$\displaystyle\phi$
$\displaystyle=\varkappa_{J}\sum_{\sigma=\pm}J_{\sigma}^{(\phi)}\psi_{\sigma}\,,$
(4.19)

or, alternatively, by

$\displaystyle\psi$
$\displaystyle=\varkappa_{K}\sum_{\sigma=\pm}K_{\sigma}^{(\psi)}\chi_{\sigma}\,,\qquad$
$\displaystyle\phi$
$\displaystyle=\varkappa_{K}\sum_{\sigma=\pm}K_{\sigma}^{(\phi)}\chi_{\sigma}\,.$
(4.20)

Here $\psi_{\sigma}$ should be intended as well-defined coordinates on a patch
which contains $\hat{D}_{1}$ or $\hat{D}_{3}$, and subsequently we will denote
them as $\psi_{\sigma}^{(1)}$ or $\psi_{\sigma}^{(3)}$ with periodicities
$\Delta\psi^{(1)}$ and $\Delta\psi^{(3)}$, as explained in the previous
section. The same applies also to $\chi_{\sigma}$, which will be written as
$\chi_{\sigma}^{(2)}$ and $\chi_{\sigma}^{(4)}$ on $\hat{D}_{2}$ and
$\hat{D}_{4}$. The reason for the introduction of $\varkappa_{J}$ and
$\varkappa_{K}$ and the prescription for their choice will be explained
shortly. The Jacobian matrices of these two transformations have determinant

$\det(\mathbf{J})=J_{+}^{(\psi)}J_{-}^{(\phi)}-J_{-}^{(\psi)}J_{+}^{(\phi)},\quad\;\;\;\det(\mathbf{K})=K_{+}^{(\psi)}K_{-}^{(\phi)}-K_{-}^{(\psi)}K_{+}^{(\phi)},$
(4.21)

and the periodicities of the old and new coordinates are connected by

$\displaystyle\Delta\psi\,\Delta\phi$
$\displaystyle=\bigl{|}\det(\mathbf{J})\bigr{|}\,\Delta\psi_{+}^{(1)}\,\Delta\psi_{-}^{(1)}=\bigl{|}\det(\mathbf{J})\bigr{|}\,\Delta\psi_{+}^{(3)}\,\Delta\psi_{-}^{(3)}\,,$
(4.22) $\displaystyle\Delta\psi\,\Delta\phi$
$\displaystyle=\bigl{|}\det(\mathbf{K})\bigr{|}\,\Delta\chi_{+}^{(2)}\,\Delta\chi_{-}^{(2)}=\bigl{|}\det(\mathbf{K})\bigr{|}\,\Delta\chi_{+}^{(4)}\,\Delta\chi_{-}^{(4)}\,.$

The signs $\varkappa_{J}$ and $\varkappa_{K}$ are given by

$\begin{split}\varkappa_{J}&=\operatorname{sign}\bigl{[}\Xi\,(x^{2}-y^{2})\det(\mathbf{J})\bigr{]}=\operatorname{sign}(x^{2}-y^{2})\,,\\\
\varkappa_{K}&=-\operatorname{sign}\bigl{[}\Xi\,(x^{2}-y^{2})\det(\mathbf{K})\bigr{]}=-\operatorname{sign}(x^{2}-y^{2})\,,\end{split}$
(4.23)

where, in the last steps, we restricted to $x,y>0$ and imposed the signature
conditions $N_{x},N_{y}>0$; in this case, both $\det(\mathbf{J})$ and
$\det(\mathbf{K})$ have the same sign of $\Xi$. By means of (4.23), all the
coefficients of the orthonormal frame (3.18) are positive when restricted to
$\hat{D}_{1}$ and $\hat{D}_{3}$ or to $\hat{D}_{2}$ and $\hat{D}_{4}$. Indeed,
on these loci we have

$\begin{split}\Xi^{-1}\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}\Bigr{|}_{x=x_{\mp}}&=\pm\varkappa_{J}\,\frac{\Delta_{x}^{\prime}(x_{\mp})\det(\mathbf{J})}{2\Xi
a^{2}}\,\mathrm{d}\psi_{\pm}=\frac{|\Delta_{x}^{\prime}(x_{\mp})\det(\mathbf{J})|}{2|\Xi|a^{2}}\,\mathrm{d}\psi_{\pm}\,,\\\
\Xi^{-1}\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}\Bigr{|}_{y=y_{\mp}}&=\pm\varkappa_{K}\,\frac{\Delta_{y}^{\prime}(y_{\mp})\det(\mathbf{K})}{2\Xi
a^{2}}\,\mathrm{d}\chi_{\pm}=\frac{|\Delta_{y}^{\prime}(y_{\mp})\det(\mathbf{K})|}{2|\Xi|a^{2}}\,\mathrm{d}\chi_{\pm}\,,\end{split}$
(4.24)

where we placed the absolute values because the conditions for a well-defined
metric imply
$\operatorname{sign}(x^{2}-y^{2})=\pm\operatorname{sign}[\Delta_{x}^{\prime}(x_{\mp})]=\mp\operatorname{sign}[\Delta_{y}^{\prime}(y_{\mp})]$.

###  4.3 Complex structure and divisors

A valuable insight into the orbifold structure of $\mathbb{M}_{4}$ can be
gained studying the four loci $\hat{D}_{a}$ in (4.17), viewing them as
_divisors_ from the point of view of the complex geometry associated with
$\mathbb{M}_{4}$. Let us first show that indeed there exists an integrable
complex structure compatible with our metrics (3.16). This can be done
defining the holomorphic $(2,0)$-form

$\Omega=(\hat{e}^{1}+\mathrm{i}\,\hat{e}^{2})\wedge(\hat{e}^{3}+\mathrm{i}\,\hat{e}^{4})\equiv\Omega^{(12)}\wedge\Omega^{(34)}\,,$
(4.25)

which, by construction, is compatible with the metrics (3.16). The fact that
the associated complex structure is integrable follows from the relation

$\mathrm{d}\Omega=\mathrm{i}P_{\rho}\wedge\Omega\,,$ (4.26)

where $P_{\rho}$ is the Ricci form potential $P_{\rho}$, which reads

$P_{\rho}=\frac{H\,\partial_{y}\bigl{(}H^{-2}\Delta_{y}\bigr{)}}{2\Xi(x^{2}-y^{2})}\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}-\frac{H\,\partial_{x}\bigl{(}H^{-2}\Delta_{x}\bigr{)}}{2\Xi(x^{2}-y^{2})}\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}\,,$
(4.27)

thus establishing that $\mathbb{M}_{4}$ are _complex orbifolds_. Although in
principle we could introduce complex coordinates, this is technically
involved, and ultimately not necessary. We will then continue with the natural
coordinates in which the metric (3.16) was originally presented, where the
$(x,y)$ coordinates play the role of “moment maps”, despite the lack of a
compatible symplectic structure. The Ricci form $\rho=\mathrm{d}P_{\rho}$
defines the first Chern class of the tangent bundle of $\mathbb{M}_{4}$ as
$\rho=2\pi c_{1}(T\mathbb{M}_{4})$, and will be used in the analysis later.

It is then clear that the four (real) codimension two loci $\hat{D}_{a}$ are
divisors in $\mathbb{M}_{4}$ and we will now describe these in more detail. We
first zoom in on $\hat{D}_{1}$, determined by the condition $x=x_{-}$. To this
end, we define the coordinate $\zeta$ such that $x=x_{-}+\lambda\zeta$ and
take the $\lambda\to 0$ limit. Reaching the point $x=x_{-}$ the whole space
$\mathbb{M}_{4}$ degenerates to a two-dimensional orbifold, whose metric,
expressed in terms of the coordinates $\psi_{\pm}^{(1)}$, is

$\mathrm{d}s_{(1)}^{2}=\frac{y^{2}-x_{-}^{2}}{\Delta_{y}}\,\mathrm{d}y^{2}+\frac{\Delta_{x}^{\prime}(x_{-})^{2}\det(\mathbf{J})^{2}}{4\Xi^{2}a^{4}H(x_{-},y)^{2}}\frac{\Delta_{y}}{y^{2}-x_{-}^{2}}\,(\mathrm{d}\psi_{+}^{(1)})^{2}\,.$
(4.28)

$\hat{D}_{1}$ is a complex orbifold of (complex) dimension one, therefore it
is natural to define the holomorphic $(1,0)$-form

$\Omega_{(1)}=\sqrt{\frac{y^{2}-x_{-}^{2}}{\Delta_{y}}}\,\mathrm{d}y+\mathrm{i}\,\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{2|\Xi|a^{2}H(x_{-},y)}\sqrt{\frac{\Delta_{y}}{y^{2}-x_{-}^{2}}}\,\mathrm{d}\psi_{+}^{(1)}\,.$
(4.29)

Notice that, defined $\Omega^{(12)}\equiv\hat{e}^{1}+\mathrm{i}\,\hat{e}^{2}$,
with $\hat{e}^{1}$ and $\hat{e}^{2}$ given in (3.18), $\Omega_{(1)}$ is simply
$\Omega^{(12)}$ restricted to $\hat{D}_{1}$. It can be checked that
$\mathrm{d}\Omega_{(1)}=\mathrm{i}P_{(1)}\wedge\Omega_{(1)}$, where, by
definition, $P_{(1)}$ is the Ricci form potential, whose expression is given
by

$P_{(1)}=\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{4|\Xi|a^{2}}\biggl{[}\frac{\Delta_{y}^{\prime}}{(x_{-}^{2}-y^{2})H(x_{-},y)}-\frac{\Delta_{y}\,\partial_{y}\bigl{(}(x_{-}^{2}-y^{2})H(x_{-},y)^{2}\bigr{)}}{(x_{-}^{2}-y^{2})^{2}H(x_{-},y)^{3}}\biggr{]}\mathrm{d}\psi_{+}^{(1)}\,.$
(4.30)

This potential allows us to determine the Ricci form of $\hat{D}_{1}$, namely
$\rho_{(1)}=\mathrm{d}P_{(1)}$, and, in turn, the first Chern class of its
tangent bundle $c_{1}(T\hat{D}_{1})=\rho_{(1)}/2\pi$. Denoting, in general,
with $\hat{\chi}_{(a)}$ the orbifold Euler characteristic of $\hat{D}_{a}$,
this can be determined by the integral

$\hat{\chi}_{(a)}=\frac{1}{2\pi}\int_{\hat{D}_{a}}\rho_{(a)}=\int_{\hat{D}_{a}}c_{1}(T\hat{D}_{a})\,.$
(4.31)

In the case at hand we have

$\hat{\chi}_{(1)}=\frac{\Delta\psi_{+}^{(1)}}{2\pi}\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{4|\Xi|a^{2}}\biggl{[}\frac{\Delta_{y}^{\prime}(y_{+})}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}+\frac{-\Delta_{y}^{\prime}(y_{-})}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]}\,.$
(4.32)

Endowed with metric (4.28), $\hat{D}_{1}$ is a compact surface parameterized
by the periodic azimuthal coordinate $\psi_{+}^{(1)}$ and the compact “polar”
coordinate $y$, with $y\in[y_{-},y_{+}]$. As $y$ approaches one of the
endpoints of this interval, say $y_{\alpha}$, the line element becomes

$\mathrm{d}s_{(1)}^{2}\simeq\mathrm{d}\varrho^{2}+\varrho^{2}\,\frac{\Delta_{x}^{\prime}(x_{-})^{2}\Delta_{y}^{\prime}(y_{\alpha})^{2}\det(\mathbf{J})^{2}}{16\Xi^{2}a^{4}(x_{-}^{2}-y_{\alpha}^{2})^{2}H(x_{-},y_{\alpha})^{2}}\,(\mathrm{d}\psi_{+}^{(1)})^{2}\,,$
(4.33)

where we defined $\varrho^{2}=|y-y_{\alpha}|$. In order to have a smooth
orbifold metric on $\hat{D}_{1}$ we must impose the following conditions at
the north ($y_{-}$) and south ($y_{+}$) poles, respectively,

$\displaystyle\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{4|\Xi|a^{2}}\frac{\Delta_{y}^{\prime}(y_{-})}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\,\Delta\psi_{+}^{(1)}$
$\displaystyle=-\frac{2\pi}{\mathtt{m}_{-}^{(1)}}\,,\qquad\quad$
$\displaystyle\mathtt{m}_{-}^{(1)}\in\mathbb{N}\,,$ (4.34)
$\displaystyle\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{4|\Xi|a^{2}}\frac{\Delta_{y}^{\prime}(y_{+})}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}\,\Delta\psi_{+}^{(1)}$
$\displaystyle=\frac{2\pi}{\mathtt{m}_{+}^{(1)}}\,,\qquad\quad$
$\displaystyle\mathtt{m}_{+}^{(1)}\in\mathbb{N}\,,$

where the minus in the first relation is due to the fact that
$\Delta_{y}^{\prime}(y_{-})/(x_{-}^{2}-y_{-}^{2})<0$. These constraints ensure
that the divisor in question has the orbifold structure of a spindle,
specifically
$\hat{D}_{1}=\mathbb{WCP}^{1}_{[\mathtt{m}_{-}^{(1)},\mathtt{m}_{+}^{(1)}]}$,171717Here,
$\mathtt{m}_{+}^{(1)}$ and $\mathtt{m}_{-}^{(1)}$ must be natural numbers, but
they need not be coprime. In order to be more precise we should write
$\hat{D}_{1}=\mathbb{WCP}^{1}_{[\bar{\mathtt{m}}_{-}^{(1)},\bar{\mathtt{m}}_{+}^{(1)}]}/\mathbb{Z}_{\mathtt{m}_{0}^{(1)}}$,
where $\mathtt{m}_{0}^{(1)}=\gcd(\mathtt{m}_{-}^{(1)},\mathtt{m}_{+}^{(1)})$
and
$\bar{\mathtt{m}}_{\pm}^{(1)}=\mathtt{m}_{\pm}^{(1)}/\mathtt{m}_{0}^{(1)}$.
and in terms of the $\mathtt{m}_{\pm}^{(1)}$ its Euler characteristic (4.32)
can be written as

$\hat{\chi}_{(1)}=\frac{1}{\mathtt{m}_{-}^{(1)}}+\frac{1}{\mathtt{m}_{+}^{(1)}}\,.$
(4.35)

A similar analysis can be carried out for the remaining divisors. Introducing
the one-form $\Omega^{(34)}\equiv\hat{e}^{3}+\mathrm{i}\,\hat{e}^{4}$, we
define the holomorphic $(1,0)$-forms
$\Omega_{(2)}=\Omega^{(34)}|_{\hat{D}_{2}}$,
$\Omega_{(3)}=\Omega^{(12)}|_{\hat{D}_{3}}$ and
$\Omega_{(4)}=\Omega^{(34)}|_{\hat{D}_{4}}$. Then, we compute the
corresponding Ricci forms $\rho_{(a)}$ and the Euler characteristic of each
divisor

$\displaystyle\hat{\chi}_{(2)}$
$\displaystyle=\frac{\Delta\chi_{+}^{(2)}}{2\pi}\frac{|\Delta_{y}^{\prime}(y_{-})\det(\mathbf{K})|}{4|\Xi|a^{2}}\biggl{[}\frac{-\Delta_{x}^{\prime}(x_{+})}{(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}+\frac{\Delta_{x}^{\prime}(x_{-})}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]}\,,$
(4.36) $\displaystyle\hat{\chi}_{(3)}$
$\displaystyle=\frac{\Delta\psi_{-}^{(3)}}{2\pi}\frac{|\Delta_{x}^{\prime}(x_{+})\det(\mathbf{J})|}{4|\Xi|a^{2}}\biggl{[}\frac{\Delta_{y}^{\prime}(y_{+})}{(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}+\frac{-\Delta_{y}^{\prime}(y_{-})}{(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}\biggr{]}\,,$
(4.37) $\displaystyle\hat{\chi}_{(4)}$
$\displaystyle=\frac{\Delta\chi_{-}^{(4)}}{2\pi}\frac{|\Delta_{y}^{\prime}(y_{+})\det(\mathbf{K})|}{4|\Xi|a^{2}}\biggl{[}\frac{-\Delta_{x}^{\prime}(x_{+})}{(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}+\frac{\Delta_{x}^{\prime}(x_{-})}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}\biggr{]}\,.$
(4.38)

Restricting to each single divisor and zooming in on the associated “poles”,
it is possible to impose suitable quantization conditions in order to give to
the divisor the structure of a spindle. Specifically, we will have

$\hat{\chi}_{(a)}=\frac{1}{\mathtt{m}^{(a)}_{-}}+\frac{1}{\mathtt{m}^{(a)}_{+}}\,,$
(4.39)

where $\mathtt{m}^{(a)}_{\pm}$ are eight integer parameters. We will see
shortly that these parameters will be entirely determined in terms of the
toric data.

###  4.4 Toric data

As we discussed in section 2, given a toric orbifold $\mathbb{M}_{4}$, its
toric data are encoded in a fan with labels, i.e. a set of ordered non-
primitive vectors $\vec{V}_{a}$ with integer-valued components. These data,
however, are basis-dependent and therefore not unique. For quadrilaterals a
set of “gauge invariant” toric data is given by the $\hat{d}_{a,b}$ and the
labels $m_{a}$. The vectors $\vec{V}_{a}$ can be extracted from the degenerate
Killing vectors $\xi_{a}$ associated with the metric on $\mathbb{M}_{4}$ once
a basis of an effective two-torus action $\\{E_{1},E_{2}\\}$ is fixed, namely

$\xi_{a}=\vec{V}_{a}\cdot(E_{1},E_{2})\,.$ (4.40)

We will postpone determining such a basis $\\{E_{1},E_{2}\\}$ and instead work
in the “reference” basis $\\{e_{1},e_{2}\\}$, in which we will be able to
determine all the parameters of the solutions in terms of the gauge invariant
toric data. Specifically, we introduce the $2\pi$-periodic coordinates
$\nu_{1}=\frac{2\pi}{\Delta\psi}\psi$, $\nu_{2}=\frac{2\pi}{\Delta\phi}\phi$
and consider the basis
$\\{e_{1},e_{2}\\}=\\{\partial_{\nu_{1}},\partial_{\nu_{2}}\\}$. From the
Killing vectors (4.15) and (4.16) we can derive the vectors

$\displaystyle\vec{v}_{1}$
$\displaystyle=\Bigl{(}\frac{2\pi}{\Delta\psi}\,J^{(\psi)}_{-},\frac{2\pi}{\Delta\phi}\,J^{(\phi)}_{-}\Bigr{)}\,,\qquad$
$\displaystyle\vec{v}_{2}$
$\displaystyle=\Bigl{(}\frac{2\pi}{\Delta\psi}\,K^{(\psi)}_{-},\frac{2\pi}{\Delta\phi}\,K^{(\phi)}_{-}\Bigr{)}\,,$
(4.41) $\displaystyle\vec{v}_{3}$
$\displaystyle=\Bigl{(}\frac{2\pi}{\Delta\psi}\,J^{(\psi)}_{+},\frac{2\pi}{\Delta\phi}\,J^{(\phi)}_{+}\Bigr{)}\,,\qquad$
$\displaystyle\vec{v}_{4}$
$\displaystyle=\Bigl{(}\frac{2\pi}{\Delta\psi}\,K^{(\psi)}_{+},\frac{2\pi}{\Delta\phi}\,K^{(\phi)}_{+}\Bigr{)}\,.$

Since we will not be able to determine the periodicities $\Delta\psi$ and
$\Delta\phi$ separately, but only in the combination $\Delta\psi\,\Delta\phi$,
we actually do not know the explicit form of the vectors in (4.41), thus in
particular we do not know whether they belong to $\mathbb{Z}^{2}$. We earlier
referred to these vectors as providing a “fake” fan. Nevertheless, the proper
basis will be related to $\\{e_{1},e_{2}\\}$ through an $SL(2,\mathbb{R})$
rotation $S$, acting on the vectors and the basis as
$V_{a}^{I}=S_{IJ}v_{a}^{J}$ and $E_{I}=S_{JI}^{-1}e_{J}$, respectively. Since
$\det(\vec{v}_{a},\vec{v}_{b})$, and in turn the intersection matrix, is
invariant under these transformations we can still use the vectors (4.41) to
compute the matrix $D_{ab}=D_{ab}[\vec{v}_{a}]$ and trust the results
obtained. There is only one caveat: the vectors (4.41) must be dual to a
convex polytope and, in our conventions, must be ordered counter-clockwise,
which imply $\det(\vec{v}_{a},\vec{v}_{a+1})>0$ for any $a$. If this condition
is not met, the vectors can be ordered in the correct way by means of a
reflection about a line in the $\mathbb{Z}^{2}$ plane, which can be realized,
e.g., swapping the two components of each vector. This transformation accounts
in exchanging $\partial_{\nu_{1}}$ and $\partial_{\nu_{2}}$ and the two bases
$\\{\partial_{\nu_{2}},\partial_{\nu_{1}}\\}$ and $\\{E_{1},E_{2}\\}$ will now
be related through a matrix with determinant equal to $-1$. As a consequence,
the intersection matrices computed starting from the two sets of vectors
extracted from the two aforementioned bases will have opposite sign. In order
to keep track of this fact we introduce the sign $\varkappa_{D}=\pm$, telling
whether the vectors are ordered as in (4.41) ($+$) or with the components
swapped ($-$). $\varkappa_{D}$ will multiply every determinant computed from
the vectors (4.41) and its value will be fixed shortly.

The orbifold information of $\mathbb{M}_{4}$ are encoded in the determinants
of the vectors $\vec{v}_{a}$, defined before as
$d_{a,b}=\det(\vec{v}_{a},\vec{v}_{b})$. Indeed $d_{a,a+1}$ gives the order of
the quotient singularity at the intersection of the adjacent divisors $D_{a}$
and $D_{a+1}$, namely $\mathbb{C}^{2}/\mathbb{Z}_{d_{a,a+1}}$. Using the set
of vectors (4.41) we compute

$\displaystyle\begin{split}d_{1,2}&=-\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}{\Delta_{x}^{\prime}(x_{-})\Delta_{y}^{\prime}(y_{-})}\,,\\\
d_{2,3}&=\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}{\Delta_{x}^{\prime}(x_{+})\Delta_{y}^{\prime}(y_{-})}\,,\\\
d_{3,4}&=-\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}{\Delta_{x}^{\prime}(x_{+})\Delta_{y}^{\prime}(y_{+})}\,,\\\
d_{4,1}&=\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}{\Delta_{x}^{\prime}(x_{-})\Delta_{y}^{\prime}(y_{+})}\,,\end{split}$
(4.42)

where we made use of the identity

$J_{\sigma_{1}}^{(\psi)}K_{\sigma_{2}}^{(\phi)}-K_{\sigma_{2}}^{(\psi)}J_{\sigma_{1}}^{(\phi)}=-\frac{4\Xi
a^{2}(x_{\sigma_{1}}^{2}-y_{\sigma_{2}}^{2})H(x_{\sigma_{1}},y_{\sigma_{2}})}{\Delta_{x}^{\prime}(x_{\sigma_{1}})\Delta_{y}^{\prime}(y_{\sigma_{2}})}\,,\qquad\sigma_{1},\sigma_{2}=\pm\,.$
(4.43)

Imposing the condition $\det(\vec{v}_{a},\vec{v}_{a+1})>0$ we can fix the
value of $\varkappa_{D}$, obtaining181818In order to do so we must use the
relations between the signs of $\Delta_{x}^{\prime}(x_{\pm})$ and
$\Delta_{y}^{\prime}(y_{\pm})$.

$\varkappa_{D}=\operatorname{sign}\bigl{[}\Xi(x^{2}-y^{2})\bigr{]}\,.$ (4.44)

Recalling the definition of $\varkappa_{J}$ and $\varkappa_{K}$ in (4.23), we
then have

$\varkappa_{D}\varkappa_{J}=\operatorname{sign}\bigl{[}\det(\mathbf{J})\bigr{]}\,,\qquad\varkappa_{D}\varkappa_{K}=-\operatorname{sign}\bigl{[}\det(\mathbf{K})\bigr{]}\,.$
(4.45)

Following the notation of [1], we also define

$t_{J}\equiv
d_{1,3}=-\varkappa_{D}\,\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\det(\mathbf{J})\,,\qquad
t_{K}\equiv
d_{2,4}=-\varkappa_{D}\,\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\det(\mathbf{K})\,.$
(4.46)

The four equations (4.42) together with the two equations (4.46) provide the
“quantization conditions” for the parameters of the solution, in terms of the
gauge invariant toric data and we postpone their analysis to subsection 4.5.
Note that the six $d_{a,b}$ defined above automatically satisfy the relation
(2.13).

The intersection matrix describing a given set of toric divisors was defined
in (2.8). In our construction, with a bit of computation the diagonal terms
can be cast in the form

$\displaystyle D_{11}$
$\displaystyle=-\varkappa_{D}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\frac{\Delta_{x}^{\prime}(x_{-})^{2}}{4\Xi
a^{2}}\biggl{[}\frac{1}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}-\frac{1}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]}\,,$
(4.47) $\displaystyle D_{22}$
$\displaystyle=-\varkappa_{D}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\frac{\Delta_{y}^{\prime}(y_{-})^{2}}{4\Xi
a^{2}}\biggl{[}\frac{1}{(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}-\frac{1}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]}\,,$
(4.48) $\displaystyle D_{33}$
$\displaystyle=\varkappa_{D}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\frac{\Delta_{x}^{\prime}(x_{+})^{2}}{4\Xi
a^{2}}\biggl{[}\frac{1}{(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}-\frac{1}{(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}\biggr{]}\,,$
(4.49) $\displaystyle D_{44}$
$\displaystyle=\varkappa_{D}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\frac{\Delta_{y}^{\prime}(y_{+})^{2}}{4\Xi
a^{2}}\biggl{[}\frac{1}{(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}-\frac{1}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}\biggr{]}\,,$
(4.50)

whereas the off-diagonal terms are simply ($D_{ab}=D_{ba}$)

$D_{12}=\frac{1}{d_{1,2}}\,,\qquad D_{23}=\frac{1}{d_{2,3}}\,,\qquad
D_{34}=\frac{1}{d_{3,4}}\,,\qquad D_{41}=\frac{1}{d_{4,1}}\,,$ (4.51)

with $d_{a,a+1}$ given in (4.42).

The toric orbifold $\mathbb{M}_{4}$ is complex, therefore the intersection
matrix can also be computed from its complex structure making use of formulas
(2.10) and (2.11). In particular, the diagonal terms are given by

$D_{aa}=\frac{1}{m_{a}}\int_{\hat{D}_{a}}\bigl{[}c_{1}(T\mathbb{M}_{4})-c_{1}(T\hat{D}_{a})\bigr{]}\,,$
(4.52)

where the integration should be performed over the ramification divisors,
linked to the branch divisors (4.17) through
$D_{a}=\hat{D}_{a}\times\mathrm{pt}/\mathbb{Z}_{m_{a}}$, from which it follows
that $\int_{D_{a}}\\!\Lambda=\frac{1}{m_{a}}\int_{\hat{D}_{a}}\\!\Lambda$ for
any two-form $\Lambda$. Here, we also made use of the property
$c_{1}(TD_{a})=c_{1}(T\hat{D}_{a})$, valid for any toric divisor. This
relation follows from the fact that the first Chern class of a tangent bundle
is computed from the curvature of the underlying base space (a local quantity)
and is, therefore, insensitive to the (global) action of the cyclic group
$\mathbb{Z}_{m_{a}}$. While $c_{1}(T\hat{D}_{a})$ can be obtained from the
Ricci form $\rho_{(a)}$ of $\hat{D}_{a}$, as explained in subsection 4.3,
$c_{1}(T\mathbb{M}_{4})$ is obtained from $c_{1}(T\mathbb{M}_{4})=\rho/2\pi$,
where $P_{\rho}$ is given in (4.27). Taking, as an example, the divisor
$\hat{D}_{1}$, an explicit computation gives

$\begin{split}\rho\bigr{|}_{\hat{D}_{1}}-\rho_{(1)}&=\varkappa_{J}\frac{\Delta_{x}^{\prime}(x_{-})}{\Delta_{x}^{\prime}(x_{+})}\,\partial_{y}\biggl{[}\frac{(x_{+}^{2}-y^{2})H(x_{+},y)}{(x_{-}^{2}-y^{2})H(x_{-},y)}\biggr{]}\mathrm{d}y\wedge\mathrm{d}\psi_{+}^{(1)}\\\
&-\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{2|\Xi|a^{2}}\,\partial_{y}\biggl{[}\frac{y\,\Delta_{y}}{(x_{-}^{2}-y^{2})^{2}H(x_{-},y)}\biggr{]}\mathrm{d}y\wedge\mathrm{d}\psi_{+}^{(1)}\,,\end{split}$
(4.53)

hence, performing the integration,

$\displaystyle D_{11}$
$\displaystyle=\varkappa_{J}\frac{1}{m_{1}}\frac{\Delta\psi_{+}^{(1)}}{2\pi}\frac{\Delta_{x}^{\prime}(x_{-})}{\Delta_{x}^{\prime}(x_{+})}\biggl{[}\frac{(x_{+}^{2}-y_{+}^{2})H(x_{+},y_{+})}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}-\frac{(x_{+}^{2}-y_{-}^{2})H(x_{+},y_{-})}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]}$
(4.54)
$\displaystyle=-\varkappa_{J}\frac{1}{m_{1}}\frac{\Delta\psi_{+}^{(1)}}{2\pi}\frac{\Delta_{x}^{\prime}(x_{-})^{2}\det(\mathbf{J})}{4\Xi
a^{2}}\biggl{[}\frac{1}{(x_{-}^{2}-y_{+}^{2})H(x_{-},y_{+})}-\frac{1}{(x_{-}^{2}-y_{-}^{2})H(x_{-},y_{-})}\biggr{]},$

where, in the last line, we used the identity

$\begin{split}&\frac{(x_{\pm}^{2}-y_{+}^{2})H(x_{\pm},y_{+})}{(x_{\mp}^{2}-y_{+}^{2})H(x_{\mp},y_{+})}-\frac{(x_{\pm}^{2}-y_{-}^{2})H(x_{\pm},y_{-})}{(x_{\mp}^{2}-y_{-}^{2})H(x_{\mp},y_{-})}=\\\
&\qquad\qquad=\mp\frac{\det(\mathbf{J})}{4\Xi
a^{2}}\biggl{[}\frac{\Delta_{x}^{\prime}(x_{-})\Delta_{x}^{\prime}(x_{+})}{(x_{\mp}^{2}-y_{+}^{2})H(x_{\mp},y_{+})}-\frac{\Delta_{x}^{\prime}(x_{-})\Delta_{x}^{\prime}(x_{+})}{(x_{\mp}^{2}-y_{-}^{2})H(x_{\mp},y_{-})}\biggr{]}\,.\end{split}$
(4.55)

We can now compare the expressions for $D_{11}$ presented in (4.47) and
(4.54), thus obtaining

$\varkappa_{D}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}=\varkappa_{J}\frac{1}{m_{1}}\frac{\Delta\psi_{+}^{(1)}}{2\pi}\det(\mathbf{J})\,.$
(4.56)

By means of (4.22) this relation becomes (see also (4.45))

$\varkappa_{D}\frac{\Delta\psi_{+}^{(1)}}{2\pi}\frac{\Delta\psi_{-}^{(1)}}{2\pi}\bigl{|}\det(\mathbf{J})\bigr{|}=\varkappa_{J}\frac{1}{m_{1}}\frac{\Delta\psi_{+}^{(1)}}{2\pi}\det(\mathbf{J})\qquad\implies\qquad\frac{\Delta\psi_{-}^{(1)}}{2\pi}=\frac{1}{m_{1}}\,,$
(4.57)

Similarly, the inspection of the components $D_{22}$, $D_{33}$ and $D_{44}$
gives, respectively,

$\frac{\Delta\chi_{-}^{(2)}}{2\pi}=\frac{1}{m_{2}}\,,\qquad\quad\frac{\Delta\psi_{+}^{(3)}}{2\pi}=\frac{1}{m_{3}}\,,\qquad\quad\frac{\Delta\chi_{+}^{(4)}}{2\pi}=\frac{1}{m_{4}}\,.$
(4.58)

We note that these formulas correlate the label $m_{a}$ associated with a
given divisor $\hat{D}_{a}$ and the periodicity of the null coordinate
generated precisely by $\xi_{a}$ on the same divisor. As we will see below,
they will be crucial for determining the fluxes in terms of the toric data,
and ultimately for implementing the extremization procedure of section 5.

Before computing the off-diagonal terms of the intersection matrix, we combine
equations (4.34) and (4.56). Comparing the results with (4.42), these two
remarkable relations follow:

$d_{1,2}=m_{1}\,\mathtt{m}_{-}^{(1)}\,,\qquad
d_{4,1}=m_{1}\,\mathtt{m}_{+}^{(1)}\,,$ (4.59)

which can also be expressed as $\mathtt{m}_{-}^{(1)}=m_{2}\,\hat{d}_{1,2}$ and
$\mathtt{m}_{+}^{(1)}=m_{4}\,\hat{d}_{1,4}$. Similar formulas can be proven,
which connect the integers $d_{a,a+1}$ with the label and the order of the
orbifold singularities of the remaining divisors, namely

$\displaystyle d_{2,3}=m_{2}\,\mathtt{m}_{+}^{(2)}\,,\qquad
d_{1,2}=m_{2}\,\mathtt{m}_{-}^{(2)}\,,$ (4.60) $\displaystyle
d_{3,4}=m_{3}\,\mathtt{m}_{+}^{(3)}\,,\qquad
d_{2,3}=m_{3}\,\mathtt{m}_{-}^{(3)}\,,$ $\displaystyle
d_{4,1}=m_{4}\,\mathtt{m}_{-}^{(4)}\,,\qquad
d_{3,4}=m_{4}\,\mathtt{m}_{+}^{(4)}\,.$

Going back to the intersection matrix, we can rewrite relation (2.11) as

$D_{a\,a-1}+D_{a\,a+1}=\frac{1}{m_{a}}\int_{\hat{D}_{a}}c_{1}(T\hat{D}_{a})=\frac{1}{m_{a}}\,\hat{\chi}_{(a)}\,.$
(4.61)

It is now straightforward to compare the expressions obtained from the toric
geometry and from the complex structure. Indeed, if we take as an example
$a=1$, the left-hand side can be read from (4.51), while $\hat{\chi}_{(1)}$ is
given in (4.39), and the equality is satisfied by means of (4.59). In the same
way it is possible to prove the consistency of (4.61) for the other divisors.

Let us now return to the issue of finding a basis for an effective torus
action. With the information that we have obtained so far, it is possible to
show that the following $SL(2,\mathbb{R})$ matrix

$S=\begin{pmatrix}\frac{2\pi}{\Delta\phi}(a_{+}K^{(\phi)}_{+}-a_{-}K^{(\phi)}_{-})&-\frac{2\pi}{\Delta\psi}(a_{+}K^{(\psi)}_{+}-a_{-}K^{(\psi)}_{-})\\\
-\frac{2\pi}{\Delta\phi}\frac{J^{(\phi)}_{-}}{m_{1}}&\frac{2\pi}{\Delta\psi}\frac{J^{(\psi)}_{-}}{m_{1}}\end{pmatrix}\,,$
(4.62)

acting on the “fake” vectors (4.41) transforms them in the
$\mathbb{Z}^{2}$-valued set

$\displaystyle\vec{V}_{1}$ $\displaystyle=(m_{1},0)\,,\qquad$
$\displaystyle\vec{V}_{2}$ $\displaystyle=(a_{+}d_{2,4},d_{1,2}/m_{1})\,,$
(4.63) $\displaystyle\vec{V}_{3}$
$\displaystyle=(a_{-}d_{2,3}+a_{+}d_{3,4},d_{1,3}/m_{1})\,,\qquad$
$\displaystyle\vec{V}_{4}$ $\displaystyle=(a_{-}d_{2,4},-d_{4,1}/m_{1})\,,$

where $a_{\pm}\in\mathbb{Z}$ are integers such that191919Here, for simplicity
assumed that gcd$(\mathtt{m}_{+}^{(1)},\mathtt{m}^{(1)}_{-})=1$ and that
$\varkappa_{D}=1$.

$a_{-}\mathtt{m}_{-}^{(1)}+a_{+}\mathtt{m}_{+}^{(1)}=-1\,,$ (4.64)

which always exist by Bèzout’s lemma. The resulting basis reads

$\begin{split}E_{1}&=\frac{1}{m_{1}}\bigl{(}J^{(\psi)}_{-}\partial_{\psi}+J^{(\phi)}_{-}\partial_{\phi}\bigr{)}\,,\\\
E_{2}&=\bigl{(}a_{+}K^{(\psi)}_{+}-a_{-}K^{(\psi)}_{-}\bigr{)}\,\partial_{\psi}+\bigl{(}a_{+}K^{(\phi)}_{+}-a_{-}K^{(\phi)}_{-}\bigr{)}\,\partial_{\phi}\,,\end{split}$
(4.65)

and the fact that the $\vec{V}_{a}$ are $\mathbb{Z}^{2}$-valued indicates that
this is a basis for an effectively acting torus. Notice that, remarkably,
although the basis $\\{e_{1},e_{2}\\}$ and the vectors in (4.41) depended on
the periodicities $\Delta\psi$ and $\Delta\phi$ separately (that we do not
know), the new basis $\\{E_{1},E_{2}\\}$ and the new vectors $\vec{V}_{a}$
will be explicitly determined in terms of the gauge invariant toric data, once
we complete the analysis of the quantization conditions, to which we now turn.

###  4.5 Quantization conditions

The analysis of the toric data of $\mathbb{M}_{4}$ performed in the previous
subsection shows clearly that this space is a well-defined orbifold only if
the $d_{a,b}$ computed explicitly in (4.42) and (4.46) are integers. In this
subsection we will show how to solve these “quantization” conditions by
writing all the parameters of the solution, namely $a$, $N_{x}$, $N_{y}$ and
$(\Delta\psi\,\Delta\phi)$,202020In the supersymmetric case, the one we shall
focus on, the charge $\delta$ is fixed in terms of $a$ by the supersymmetry
conditions $ags^{2}=-\kappa$. in terms of the six integers $d_{a,b}$.

Instead of tackling directly the aforementioned quantization conditions, we
follow an alternative path and begin studying the total magnetic fluxes across
the divisors. They are defined as

$\hat{\mathfrak{q}}_{i}^{a}\equiv\frac{g_{c}}{2\pi}\int_{\hat{D}_{a}}F_{i}\,,$
(4.66)

where $g_{c}$ is the gauge coupling constant and the field strengths $F_{i}$
can be conveniently written as

$F_{i}=\frac{c\tilde{c}}{\Xi\,sH^{2}}\bigl{[}\partial_{x}H\,\mathrm{d}x\wedge\bigl{(}V_{y}\,\mathrm{d}\psi-\widetilde{V}_{y}\,\mathrm{d}\phi\bigr{)}+\partial_{y}H\,\mathrm{d}y\wedge\bigl{(}V_{x}\,\mathrm{d}\psi-\widetilde{V}_{x}\,\mathrm{d}\phi\bigr{)}\bigr{]}\,.$
(4.67)

We start considering, as an example, the first divisor $\hat{D}_{1}$, whose
associated magnetic fluxes are

$\hat{\mathfrak{q}}_{1}^{1}=\hat{\mathfrak{q}}_{2}^{1}=-\frac{\Delta\psi_{+}^{(1)}}{2\pi}\,\frac{g_{c}c\tilde{c}}{s}\frac{|\Delta_{x}^{\prime}(x_{-})\det(\mathbf{J})|}{2|\Xi|a^{2}}\biggl{[}\frac{1}{H(x_{-},y_{+})}-\frac{1}{H(x_{-},y_{-})}\biggr{]}\,.$
(4.68)

We now introduce the magnetic fluxes across the ramification divisors $D_{a}$

$\mathfrak{q}_{i}^{a}\equiv\frac{\hat{\mathfrak{q}}_{i}^{a}}{m_{a}}=\frac{g_{c}}{2\pi}\int_{D_{a}}F_{i}\,,$
(4.69)

where, we recall, $m_{a}\int_{D_{a}}\\!\Lambda=\int_{\hat{D}_{a}}\\!\Lambda$
for any two-form $\Lambda$. By means of (4.57) and (4.22), the $R$-symmetry
flux $\mathfrak{q}_{R}^{a}=\mathfrak{q}_{1}^{a}+\mathfrak{q}_{2}^{a}$ across
$D_{1}$ reads

$\mathfrak{q}_{R}^{1}=-\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g_{c}c\tilde{c}}{s}\frac{|\Delta_{x}^{\prime}(x_{-})|}{|\Xi|a^{2}}\biggl{[}\frac{1}{H(x_{-},y_{+})}-\frac{1}{H(x_{-},y_{-})}\biggr{]}\,.$
(4.70)

When supersymmetry is imposed ($ags^{2}=-\kappa$), one can take advantage of
the relations presented at the end of subsection 3.2 and, employing (3.22),
the expression of $\mathfrak{q}_{R}^{1}$ boils down to212121Here, we took
$\delta>0$; for negative values the sign is flipped and similarly happens for
the other fluxes.

$\mathfrak{q}_{R}^{1}=\varkappa_{D}\,\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g\,g_{c}\Delta_{x}^{\prime}(x_{-})}{\Xi
a^{2}}\bigl{(}\tau^{(y_{+})}y_{+}-\tau^{(y_{-})}y_{-}\bigr{)}\,.$ (4.71)

Exploiting the simplifications brought by supersymmetry, it can be shown that

$\mathfrak{q}_{R}^{1}=-\frac{\tau^{(y_{+})}}{d_{4,1}}-\frac{\tau^{(y_{-})}}{d_{1,2}}-\tau^{(x_{-})}\,\frac{t_{K}}{d_{4,1}d_{1,2}}\,.$
(4.72)

where, crucially, we used the relation (4.57), obtained using the adjunction
formula. This relation implies that, once the quantization conditions (4.42)
and (4.46) are satisfied, the $R$-symmetry flux $\mathfrak{q}_{R}^{1}$ is
automatically quantized and its expression is given in terms of integers.

In a similar way we can compute the total magnetic fluxes across the remaining
divisors

$\begin{split}\mathfrak{q}_{R}^{2}&=-\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g_{c}c\tilde{c}}{s}\frac{|\Delta_{y}^{\prime}(y_{-})|}{|\Xi|a^{2}}\biggl{[}\frac{1}{H(x_{+},y_{-})}-\frac{1}{H(x_{-},y_{-})}\biggr{]}\,,\\\
\mathfrak{q}_{R}^{3}&=-\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g_{c}c\tilde{c}}{s}\frac{|\Delta_{x}^{\prime}(x_{+})|}{|\Xi|a^{2}}\biggl{[}\frac{1}{H(x_{+},y_{+})}-\frac{1}{H(x_{+},y_{-})}\biggr{]}\,,\\\
\mathfrak{q}_{R}^{4}&=-\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g_{c}c\tilde{c}}{s}\frac{|\Delta_{y}^{\prime}(y_{+})|}{|\Xi|a^{2}}\biggl{[}\frac{1}{H(x_{+},y_{+})}-\frac{1}{H(x_{-},y_{+})}\biggr{]}\,.\end{split}$
(4.73)

Imposing supersymmetry, the $R$-symmetry fluxes take the form

$\begin{split}\mathfrak{q}_{R}^{2}&=-\varkappa_{D}\,\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g\,g_{c}\Delta_{y}^{\prime}(y_{-})}{\Xi
a^{2}}\bigl{(}\tau^{(x_{+})}x_{+}-\tau^{(x_{-})}x_{-}\bigr{)}\,,\\\
\mathfrak{q}_{R}^{3}&=-\varkappa_{D}\,\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g\,g_{c}\Delta_{x}^{\prime}(x_{+})}{\Xi
a^{2}}\bigl{(}\tau^{(y_{+})}y_{+}-\tau^{(y_{-})}y_{-}\bigr{)}\,,\\\
\mathfrak{q}_{R}^{4}&=\varkappa_{D}\,\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\,\frac{g\,g_{c}\Delta_{y}^{\prime}(y_{+})}{\Xi
a^{2}}\bigl{(}\tau^{(x_{+})}x_{+}-\tau^{(x_{-})}x_{-}\bigr{)}\,,\end{split}$
(4.74)

and the following quantization formulas can be proven

$\begin{split}\mathfrak{q}_{R}^{2}&=-\frac{\tau^{(x_{-})}}{d_{1,2}}-\frac{\tau^{(x_{+})}}{d_{2,3}}+\tau^{(y_{-})}\,\frac{t_{J}}{d_{1,2}d_{2,3}}\,,\\\
\mathfrak{q}_{R}^{3}&=-\frac{\tau^{(y_{-})}}{d_{2,3}}-\frac{\tau^{(y_{+})}}{d_{3,4}}+\tau^{(x_{+})}\,\frac{t_{K}}{d_{2,3}d_{3,4}}\,,\\\
\mathfrak{q}_{R}^{4}&=-\frac{\tau^{(x_{+})}}{d_{3,4}}-\frac{\tau^{(x_{-})}}{d_{4,1}}-\tau^{(y_{+})}\,\frac{t_{J}}{d_{3,4}d_{4,1}}\,.\end{split}$
(4.75)

Remarkably, the above conditions take exactly the form

$\mathfrak{q}_{R}^{a}=\sum_{b}D_{ab}\,\sigma^{b}\,,$ (4.76)

with the $\sigma^{a}$ being signs, i.e. $\sigma^{a}=\pm 1$, as conjectured in
[1]. These are solving the topological constraints
$\sum_{a}\mathfrak{q}_{R}^{a}\vec{V}_{a}=0$, with the $\sigma^{a}$ specifying
the type of supersymmetry-preserving twist.

In addition, we note this couple of useful relations

$\frac{\mathfrak{q}_{R}^{1}}{\mathfrak{q}_{R}^{3}}=-\frac{\Delta_{x}^{\prime}(x_{-})}{\Delta_{x}^{\prime}(x_{+})}\,,\qquad\frac{\mathfrak{q}_{R}^{2}}{\mathfrak{q}_{R}^{4}}=-\frac{\Delta_{y}^{\prime}(y_{-})}{\Delta_{y}^{\prime}(y_{+})}\,.$
(4.77)

Coming back to the quantization conditions for the metric parameters, the
scheme for their resolution consists in expressing $a$, $N_{x}$, $N_{y}$ and
the product $(\Delta\psi\,\Delta\phi)$ in terms of the roots $x_{\pm}$ and
$y_{\pm}$, and, subsequently, in writing the latter in terms of the integers
$d_{a,b}$ and of the quantized fluxes $\mathfrak{q}_{R}^{a}$. To avoid clumsy
notation, until the end of this subsection we shall consider $0<y<x$. As
explained in subsection 3.3, in this case the roots are organized such that
$\Delta_{x}^{-}(x_{+})=0=\Delta_{x}^{-}(x_{-})$ and
$\Delta_{y}^{-}(y_{+})=0=\Delta_{y}^{+}(y_{-})$, which amounts in taking
$\tau^{(x_{+})}=\tau^{(x_{-})}=\tau^{(y_{+})}=-1$ and $\tau^{(y_{-})}=+1$
where needed. With this choice, all the $R$-symmetry fluxes are negative, a
fact that can be easily seen trading the $\varkappa_{D}$ for the absolute
values. Combining appropriately the pair of equations
$\Delta_{x}^{-}(x_{\pm})=0$, from (3.20) we obtain

$\begin{split}a&=-\kappa\,\frac{x_{+}^{D-4}(1-gx_{+})-x_{-}^{D-4}(1-gx_{-})}{x_{+}^{D-5}(1-gx_{+})-x_{-}^{D-5}(1-gx_{-})}\,,\\\
N_{x}&=(-\kappa\,a)\,\frac{x_{+}^{D-5}x_{-}^{D-5}(x_{+}-x_{-})(1-gx_{+})(1-gx_{-})}{2[x_{+}^{D-5}(1-gx_{+})-x_{-}^{D-5}(1-gx_{-})]}\,,\end{split}$
(4.78)

whereas applying the same method to $\Delta_{y}^{\pm}$ we are able to isolate
$a$ and $N_{y}$ in terms of $y_{\pm}$

$\begin{split}a&=-\kappa\,\frac{y_{+}^{D-4}(1-gy_{+})+y_{-}^{D-4}(1+gy_{-})}{y_{+}^{D-5}(1-gy_{+})-y_{-}^{D-5}(1+gy_{-})}\,,\\\
N_{y}&=(-\kappa\,a)\,\frac{y_{+}^{D-5}y_{-}^{D-5}(y_{+}+y_{-})(1-gy_{+})(1+gy_{-})}{2[y_{+}^{D-5}(1-gy_{+})-y_{-}^{D-5}(1+gy_{-})]}\,.\end{split}$
(4.79)

Consider, now, conditions (4.42). Taking the ratios first/fourth and
second/third and imposing supersymmetry, we obtain the two equations

$\frac{d_{1,2}\,\mathfrak{q}_{R}^{2}}{d_{4,1}\,\mathfrak{q}_{R}^{4}}=\frac{x_{-}+y_{-}}{x_{-}-y_{+}}\,,\qquad\frac{d_{2,3}\,\mathfrak{q}_{R}^{2}}{d_{3,4}\,\mathfrak{q}_{R}^{4}}=\frac{x_{+}+y_{-}}{x_{+}-y_{+}}\,,$
(4.80)

which are solved by

$x_{-}=\frac{d_{1,2}\mathfrak{q}_{R}^{2}\,y_{+}+d_{4,1}\mathfrak{q}_{R}^{4}\,y_{-}}{d_{1,2}\mathfrak{q}_{R}^{2}-d_{4,1}\mathfrak{q}_{R}^{4}}\,,\qquad
x_{+}=\frac{d_{2,3}\mathfrak{q}_{R}^{2}\,y_{+}+d_{3,4}\mathfrak{q}_{R}^{4}\,y_{-}}{d_{2,3}\mathfrak{q}_{R}^{2}-d_{3,4}\mathfrak{q}_{R}^{4}}\,.$
(4.81)

All the other possible ratios of (4.42) and (4.46) are identically satisfied
by (4.81) making use of (4.72), (4.75) and (4.77). Now that we have the roots
$x_{\pm}$ as functions of $y_{\pm}$, we can express all the physical
parameters in terms of the latter. In order to proceed, we write down the
expressions of $t_{J}$ and $t_{K}$ in the supersymmetric case, which simplify
to

$\begin{split}t_{J}&=\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(1-\kappa\,ag)}{g}\frac{x_{+}-x_{-}}{\Delta_{x}^{\prime}(x_{+})\Delta_{x}^{\prime}(x_{-})}\,,\\\
t_{K}&=\varkappa_{D}\frac{2\pi}{\Delta\psi}\frac{2\pi}{\Delta\phi}\frac{4\Xi
a^{2}(1-\kappa\,ag)}{g}\frac{y_{+}+y_{-}}{\Delta_{y}^{\prime}(y_{+})\Delta_{y}^{\prime}(y_{-})}\,.\end{split}$
(4.82)

Notice that, when $0<y<x$, both $t_{J}$ and $t_{K}$ are negative. Having the
expressions of $t_{J}$ and $t_{K}$, we can immediately derive the product
$\Delta\psi\,\Delta\phi$ in terms of the roots $y_{\pm}$ and of the other
integers characterizing the system

$\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}=-\varkappa_{D}\,\frac{a^{2}(1+\kappa\,ag)}{4g\,g_{c}^{2}(y_{+}+y_{-})^{3}}\frac{(t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3})^{2}}{t_{K}\mathfrak{q}_{R}^{2}\mathfrak{q}_{R}^{4}}\,.$
(4.83)

The next step is to determine the roots $y_{\pm}$ in terms of the integers
$d_{a,b}$. Combining in an appropriate way the fluxes $\mathfrak{q}_{R}^{a}$
with $t_{J}$, we obtain

$\begin{split}\Delta_{y}^{\prime}(y_{-})&=-\frac{4\mathfrak{q}_{R}^{2}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}\,g_{c}(1-\kappa\,ag)(y_{+}+y_{-})^{2}\,,\\\
\Delta_{y}^{\prime}(y_{+})&=\frac{4\mathfrak{q}_{R}^{4}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}\,g_{c}(1-\kappa\,ag)(y_{+}+y_{-})^{2}\,,\end{split}$
(4.84)

which is a pair of equations containing only $y_{\pm}$, once $a$ is
substituted as in (4.79).

In order to solve this system of equations we need to specify the number of
spacetime dimensions. Starting with $D=6$, we substitute in (4.84) the
expression of $\Delta_{y}^{\prime}$ given in (3.23) and parameterize the roots
of $\Delta_{y}$ as $y_{\pm}=w(1\pm\mathtt{x})$. When choosing the integers
that describe $\mathbb{M}_{4}$ we shall need to impose $w>0$ and
$0<\mathtt{x}<1$ in order to have $0<y_{-}<y_{+}$. Solving the resulting
equations for $w$ and $\mathtt{x}$ we obtain

$w=-\frac{3(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})\mathtt{x}-\mathtt{x}^{3}\pm\sqrt{3-6(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})-3(1-3(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})^{2})\mathtt{x}^{2}+\mathtt{x}^{4}}}{g[3+\mathtt{x}^{4}-6(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})(1+\mathtt{x}^{2})]}\,,$
(4.85)

where we defined
$\mathfrak{Q}_{+}=\frac{\mathfrak{q}_{R}^{4}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}$
and
$\mathfrak{Q}_{-}=\frac{\mathfrak{q}_{R}^{2}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}$
and $\mathtt{x}$ is solution of the cubic equation

$\mathtt{x}^{3}-3(\mathfrak{Q}_{+}-\mathfrak{Q}_{-})\mathtt{x}^{2}-3[1-2(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})]\mathtt{x}+3(\mathfrak{Q}_{+}-\mathfrak{Q}_{-})=0\,.$
(4.86)

The two signs in (4.85) generate the two sets of physical parameters connected
by the inversion symmetry (3.14) and, in particular, the plus yields the
configuration with $\Xi<0$. This last statement can be proven noting that
$\Xi=(1-\kappa\,ag)(1+\kappa\,ag)$ and rearranging the second term as

$1+\kappa\,ag=\mp\frac{2gw\sqrt{3-6(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})-3(1-3(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})^{2})\mathtt{x}^{2}+\mathtt{x}^{4}}}{1-\mathtt{x}^{2}}\,.$
(4.87)

The first factor of $\Xi$ is positive due to supersymmetry, while the second
one contributes to the sign and tells whether $\Xi$ is positive or negative.
The last thing to do is to equate the expression for $a$ obtained by means of
(4.78) and (4.79). This gives the following Diophantine equation for the
integers $d_{a,b}$

$t_{K}\mathfrak{q}_{R}^{2}\mathfrak{q}_{R}^{4}-3(\mathfrak{q}_{R}^{1}-\mathfrak{q}_{R}^{3})=0\,,$
(4.88)

whose analysis is postponed to appendix D. The origin of this constraint can
be traced back in the specific structure of our solution. Indeed, the
quantization conditions involve the four parameters $a$, $N_{x}$, $N_{y}$ and
$(\Delta\psi\,\Delta\phi)$, but, as we discussed, the integers $d_{a,b}$
characterizing the system are five. This means that there must be a redundancy
among them and equation (4.88) precisely resolves it. Considering a system
with different charges $\delta_{i}$ would increase by one the number of
parameters, but would also bring an additional quantum number into play, thus
not changing the balance. It is possible that the constraint (4.88) would be
eliminated if a more general solution could be constructed, but we will not
pursue this here. Similar unnatural Diophantine constraints have been observed
in previous solutions associated with orbifolds [10, 12, 1].

We now focus on $D=7$ and solve system (4.84) following the same method
adopted in the six-dimensional case. Parameterizing again the two roots as
$y_{\pm}=w(1\pm\mathtt{x})$, we obtain

$\displaystyle w$
$\displaystyle=-\frac{4(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})\mathtt{x}\pm\sqrt{2}\sqrt{[1-2(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})][1+(1-4(\mathfrak{Q}_{+}+\mathfrak{Q}_{-}))\mathtt{x}^{2}]}}{2g[1+\mathtt{x}^{2}-2(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})(1+3\mathtt{x}^{2})]}\,,$
(4.89) $\displaystyle\mathtt{x}$
$\displaystyle=\frac{\mathfrak{Q}_{+}-\mathfrak{Q}_{-}}{1-3(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})}\,,$
(4.90)

where $\mathfrak{Q}_{\pm}$ are defined as in the previous case. As before, the
two signs in (4.89) correspond to the two configurations related by (3.14) and
the plus sign gives $\Xi<0$. This can be seen considering the analogous
expressions of (4.87), but in $D=7$,

$1+\kappa\,ag=\mp\,2\sqrt{2}\,gw\sqrt{[1-2(\mathfrak{Q}_{+}+\mathfrak{Q}_{-})][1+(1-4(\mathfrak{Q}_{+}+\mathfrak{Q}_{-}))\mathtt{x}^{2}]}\,.$
(4.91)

Imposing the consistency condition that comes equating (4.78) with (4.79) we
obtain the following Diophantine equation

$\begin{split}&2\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}t_{J}\bigl{[}(\mathfrak{q}_{R}^{1})^{3}d_{1,2}d_{4,1}(d_{1,2}-d_{4,1})-(\mathfrak{q}_{R}^{3})^{3}d_{2,3}d_{3,4}(d_{2,3}-d_{3,4})\bigr{]}\\\
&\qquad+\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}t_{J}\bigl{[}8\bigl{(}(\mathfrak{q}_{R}^{1})^{2}d_{1,2}d_{4,1}-(\mathfrak{q}_{R}^{3})^{2}d_{2,3}d_{3,4}\bigr{)}-\bigl{(}(\mathfrak{q}_{R}^{1})^{4}d_{1,2}^{2}d_{4,1}^{2}-(\mathfrak{q}_{R}^{3})^{4}d_{2,3}^{2}d_{3,4}^{2}\bigr{)}\bigr{]}\\\
&\qquad+4(\mathfrak{q}_{R}^{1})^{3}d_{1,2}d_{4,1}\bigl{(}\mathfrak{q}_{R}^{2}d_{4,1}-\mathfrak{q}_{R}^{4}d_{1,2}\bigr{)}-4(\mathfrak{q}_{R}^{3})^{3}d_{2,3}d_{3,4}\bigl{(}\mathfrak{q}_{R}^{2}d_{3,4}-\mathfrak{q}_{R}^{4}d_{2,3}\bigr{)}\\\
&\qquad+3(\mathfrak{q}_{R}^{2}+\mathfrak{q}_{R}^{4})\bigl{[}(\mathfrak{q}_{R}^{1})^{4}d_{1,2}^{2}d_{4,1}^{2}-(\mathfrak{q}_{R}^{3})^{4}d_{2,3}^{2}d_{3,4}^{2}\bigr{]}=0\,.\end{split}$
(4.92)

Although this is surprisingly much more complicated than the analogous
constraint (4.88) in $D=6$, similar comments apply. We refer to appendix D for
its resolution.

We close this subsection with a brief recap of the procedure that, starting
from the set of integers characterizing a given solution, allows to
reconstruct all its parameters. The first step is to choose four positive
integers $d_{a,a+1}$ and two integers $d_{1,3}=t_{J}$ and $d_{2,4}=t_{K}$ that
satisfy constraint (2.13) and the Diophantine equation (4.88), if $D=6$, or
(4.92), for $D=7$. In six dimensions we compute the auxiliary quantities
$\mathtt{x}$ and $w$ using (4.86) and (4.85), while for the seven-dimensional
solution the corresponding relations are (4.90) and (4.89). In both cases,
$\mathfrak{Q}_{+}=\frac{\mathfrak{q}_{R}^{4}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}$,
$\mathfrak{Q}_{-}=\frac{\mathfrak{q}_{R}^{2}}{t_{J}\mathfrak{q}_{R}^{1}\mathfrak{q}_{R}^{3}}$
and the charges $\mathfrak{q}_{R}^{a}$ are given by (4.72) and (4.75), where,
for the system in question, $\tau^{(x_{+})}=\tau^{(x_{-})}=\tau^{(y_{+})}=-1$
and $\tau^{(y_{-})}=+1$. The endpoints of the range of the coordinate $y$ read
$y_{\pm}=w(1\pm\mathtt{x})$, while $x_{\pm}$ can be obtained from (4.81).
Lastly, the parameters $a$, $N_{x}$ and $N_{y}$ follow from (4.78) and (4.79),
while the product $\Delta\psi\,\Delta\phi$ in given by (4.83), where the sign
$\varkappa_{D}$ is fixed so to have a positive result.

##  5 Off-shell free energies and extremization

In this section we present the off-shell free energy proposed in [1], whose
construction is based on the idea of gluing “gravitational blocks”, introduced
in [16]. This recipe extends the results of [8] to four-dimensional toric
orbifolds with an arbitrary number of fixed points. For $D=7$, the field
theory counterpart of this construction was given in [19], integrating on
$\mathbb{M}_{4}$ the anomaly polynomial of the six-dimensional SCFTs
associated with M5-branes. In $D=6$ a solid proof is still lacking,
nevertheless the gravitational block conjectured in [1] was retrieved in [20]
through the integration of the equivariant volume of the associated geometry,
with the addition of higher times. Below we will be primarily interested in
four-dimensional toric orbifolds, with $\mathbb{M}_{4}$ represented by the
solutions in 3.2. We speculate that these $D=6,7$ solutions are
holographically dual to $d=1,2$ theories, obtained compactifying on
$\mathbb{M}_{4}$ the five-dimensional $\mathcal{N}=1$ SCFTs dual to [48] and
the six-dimensional $\mathcal{N}=(0,2)$ dual to a stack of M5-branes.
Supporting evidence is provided by the extremization of our off-shell free
energy, which reproduces exactly the entropy/central charge of the (putative)
black hole/black string with $\mathrm{AdS}_{D-4}\times\mathbb{M}_{4}$ near-
horizon.

###  5.1 Recap of the recipe

We now summarize the key ingredients necessary to construct the off-shell free
energy, whose extremization should reproduce the entropy and the central
charge of the dual gravity theories in $D=6$ and $D=7$, respectively. For
further details, see [1]. Recall that an opposite convention for the $D_{a}$
and $\hat{D}_{a}$ divisors, as well as for the other related quantities, is
employed here. In particular, the vectors $\vec{v}_{a}$ need not be primitive
and the information carried by the labels is now encoded in the vectors
themselves.

On the field theory side, the systems under exam result from the twisted
compactification of five- and six-dimensional SCFTs on a four-dimensional
toric orbifold $\mathbb{M}_{4}$, completely characterized by the counter-
clockwise ordered vectors $\vec{v}_{a}\in\mathbb{Z}^{2}$. The twisting is
realized coupling the SCFTs to two background gauge fields $A_{i}$, with field
strengths $F_{i}=\mathrm{d}A_{i}$. As a first step, we define the “physical
fluxes” as222222Here and in the following we shall rename the background gauge
fields as $g_{c}A_{i}\mapsto A_{i}$. The nomenclature “physical fluxes” should
not be confused with the fact that the natural fluxes in supergravity are
computed on the branch divisors, see (4.66).

$\mathfrak{q}_{i}^{a}\equiv\frac{1}{2\pi}\int_{D_{a}}F_{i}\,,$ (5.1)

where $a=1,\ldots,n$ runs over the number of facets. The one-forms $-A_{i}$
can also be viewed as the connections on two line orbibundles $E_{i}$. Since
the set of $c_{1}(L_{a})$ form a basis for $H^{2}(\mathbb{M}_{4},\mathbb{Q})$,
we can decompose the first Chern class of $E_{i}$ as

$c_{1}(E_{i})=-\frac{\mathrm{d}A_{i}}{2\pi}=-\sum_{a}\mathfrak{p}_{i}^{a}c_{1}(L_{a})\,,$
(5.2)

where $\mathfrak{p}_{i}^{a}\in\mathbb{Q}$. As a consequence

$\mathfrak{q}_{i}^{a}=\sum_{b}D_{ab}\,\mathfrak{p}_{i}^{b}\,,\qquad\sum_{a}\mathfrak{q}_{i}^{a}\vec{v}_{a}=0\,,$
(5.3)

where $D_{ab}$ is the intersection matrix defined in (2.8). The latter
equation, which follows from (2.5), is a constraint on the physical fluxes and
implies that, for fixed $i$, only $n-2$ of the $\mathfrak{q}_{i}^{a}$ are
independent. On the other hand, $H^{2}(\mathbb{M}_{4},\mathbb{Q})$ has
dimension $n-2$, thus, for fixed $i$, only $n-2$ of the $\mathfrak{p}_{i}^{a}$
are linearly independent—see the discussion around (2.4). Therefore, the
former system of equations can be inverted to obtain $\mathfrak{p}_{i}^{a}$ in
terms of $\mathfrak{q}_{i}^{a}$ once the redundant equations are eliminated. A
similar condition applies also to the $R$-symmetry fluxes
$\mathfrak{q}_{R}^{a}$, specifically
$\sum_{a}\mathfrak{q}_{R}^{a}\vec{v}_{a}=0$, and a possible choice to solve it
is

$\mathfrak{q}_{R}^{a}=\sum_{b}D_{ab}\,\sigma^{b}\,,$ (5.4)

where, we conjectured, $\sigma^{a}$ are $n$ arbitrary signs, i.e.
$\sigma^{a}=\pm 1$. This assumption is supported by several examples where the
spindles are explicitly involved [49, 1] in both $D=6,7$, and the solutions
presented in 3.2 are no exception, as we will see later.

We also conjecture that the entropy/central charge correspondent,
respectively, to a general class of $d=5,6$ SCFTs compactified on
$\mathbb{M}_{4}$ and with twists parameterized by $\sigma^{a}$, can be
computed by the constrained extremization of the following _off-shell free
energy_

$F(\varphi_{i},\epsilon_{i};\mathfrak{q}_{i}^{a})=k_{d}\sum_{a}\frac{\eta_{d}^{\,a}}{d_{a,a+1}}\frac{\mathcal{F}_{d}(\Phi_{i}^{a})}{\epsilon_{1}^{a}\epsilon_{2}^{a}}\,,$
(5.5)

where the sum is over the number of fixed points, that for planar polytopes
coincides with the number of facets. Here, $\mathcal{F}_{d}$ are the usual
gravitational blocks (cf. table 2 of [8])

$\mathcal{F}_{5}(\Delta_{i})=-\frac{4\sqrt{2}\pi\,N^{5/2}}{15\sqrt{8-N_{f}}}(\Delta_{1}\Delta_{2})^{3/2}\,,\qquad\mathcal{F}_{6}(\Delta_{i})=-\frac{9N^{3}}{256}(\Delta_{1}\Delta_{2})^{2}\,,$
(5.6)

the variables $\Phi_{i}^{a}$ are defined as

$\Phi_{i}^{a}=\varphi_{i}-\mathfrak{p}_{i}^{a}\epsilon_{1}^{a}-\mathfrak{p}_{i}^{a+1}\epsilon_{2}^{a}\,,$
(5.7)

and the auxiliary quantities $\epsilon_{1,2}^{a}$ are constructed from the
fugacities $\vec{\epsilon}=(\epsilon_{1},\epsilon_{2})$ associated with the
$U(1)^{2}$ rotational symmetry of $\mathbb{M}_{4}$ as

$\epsilon^{a}_{1}=-\frac{\det(\vec{v}_{a+1},\vec{\epsilon})}{d_{a,a+1}}\,,\qquad\epsilon^{a}_{2}=\frac{\det(\vec{v}_{a},\vec{\epsilon})}{d_{a,a+1}}\,.$
(5.8)

We recall that $d_{a,b}=\det(\vec{v}_{a},\vec{v}_{b})$, whereas $k_{d}$ are
numerical constants which assume, a posteriori, the values

$k_{5}=-1\,,\qquad k_{6}=\frac{64}{9}\,.$ (5.9)

Also the values of $\eta^{\,a}_{d}$ is one of the ingredients of the
prescription. Specifically, for $d=5$ we assume they are related to the type
of twist as $\eta^{\,a}_{5}=\sigma^{a}\sigma^{a+1}$, while for $d=6$ we take
all of them with the same value, which is, ultimately, related to the
chirality of the SCFT in question. In particular, $\eta^{\,a}_{6}=\kappa$,
with $\kappa=\pm 1$ so that the extremization of the off-shell free energy
reproduces the central charge of the $d=2$ SCFTs with $\mathcal{N}=(0,2)$ or
$\mathcal{N}=(2,0)$, respectively, extracted from the anomaly polynomial of
the $d=6$ theory. This $\kappa$ is also related to the sign in the equation
$\hat{\nabla}_{\hat{\mu}}\vartheta=\frac{\kappa}{2}\beta_{\hat{\mu}}\vartheta$
satisfied by the Killing spinor along $\mathrm{AdS}_{3}$.

The variables of the off-shell free energy, namely $\varphi_{i}$ and
$\epsilon_{i}$, are subject to the constraint

$\varphi_{1}+\varphi_{2}-\det(\vec{W},\vec{\epsilon})=2\,,$ (5.10)

where the auxiliary two-dimensional vector $\vec{W}$ is determined imposing

$\mathfrak{p}_{1}^{a}+\mathfrak{p}_{2}^{a}=\sigma^{a}+\det(\vec{W},\vec{v}_{a})\,.$
(5.11)

The vector $\vec{W}$ parameterizes a “gauge symmetry” featured by our
construction and can be set to any desired value by means of suitable
transformation [1]. The present construction enjoys an additional symmetry.
Once the fluxes $\mathfrak{q}_{i}^{a}$ are fixed, if an $SL(2,\mathbb{Z})$
transformation is applied to $\vec{v}_{a}$, $\vec{\epsilon}$ and $\vec{W}$,
the off-shell free energy and the constraint can be shown to preserve their
form, thus leaving the extremization problem unaltered [1].

The last step of the procedure is the constrained extremization of
$F(\varphi_{i},\epsilon_{i})$, which can be performed defining the function

$\mathcal{S}(\varphi_{i},\epsilon_{i},\Lambda;\mathfrak{q}_{i}^{a})=F(\varphi_{i},\epsilon_{i};\mathfrak{q}_{i}^{a})+\Lambda\bigl{(}\varphi_{1}+\varphi_{2}-\det(\vec{W},\vec{\epsilon})-2\bigr{)}$
(5.12)

and extremizing it with respect to $\varphi_{i}$, $\epsilon_{i}$ and the
Lagrangian multiplier $\Lambda$. As a consequence of Euler’s theorem,
$F(\varphi_{i}^{*},\epsilon_{i}^{*})=-\frac{2}{h}\Lambda^{*}$, with $h=1$ for
$d=5$, $h=2$ for $d=6$ [1].

###  5.2 Application to the new AdS${}_{D-4}\times\mathbb{M}_{4}$ solutions

We now apply the gravitational blocks prescription to the
$\mathrm{AdS}_{D-4}\times\mathbb{M}_{4}$ backgrounds with equal charges
presented in subsection 3.2. The first ingredient needed is the toric data
describing the toric orbifold $\mathbb{M}_{4}$. As we mentioned, the recipe to
construct the off-shell free energy is formally invariant under
$SL(2,\mathbb{R})$ transformations of the vectors $\vec{v}_{a}$ comprising the
toric data, although this group is broken to $SL(2,\mathbb{Z})$ when integer-
valued vectors are considered. Thanks to this property, we use vectors (4.41)
as toric data, assuming that a set of proper (integer-valued) toric data
exists and is related to (4.41) by an $SL(2,\mathbb{R})$ transformation—see
the discussion below (4.41). The physical fluxes defined in (5.1) coincide
with the magnetic fluxes (4.69), whose explicit forms are given in (4.70) and
(4.73). The vector of twists can be deduced comparing (5.4) with the quantized
expressions for the fluxes (4.72) and (4.75). When $\delta$ is positive we
obtain

$\sigma^{1}=-\tau^{(x_{-})}\,,\qquad\sigma^{2}=-\tau^{(y_{-})}\,,\qquad\sigma^{3}=-\tau^{(x_{+})}\,,\qquad\sigma^{4}=-\tau^{(y_{+})}\,,$
(5.13)

which becomes
$\sigma^{a}=[\operatorname{sign}(x^{2}-y^{2}),-\operatorname{sign}(x^{2}-y^{2}),+,+]$
if we restrict to $x,y>0$. The results of subsection 3.3 imply that three out
of the four components of $\sigma^{a}$ have always the same sign, while the
fourth one is the opposite. This property agrees with the vector of twists
found for the $\mathrm{AdS}\times\mathbbl{\Sigma}\ltimes\mathbbl{\Sigma}$
solutions studied in [1]. Taking advantage of the gauge symmetry enjoyed by
our prescription, one can always pick a gauge in which the auxiliary vector
$\vec{W}$ vanishes. When the charges are equal, as in the case of interest,
this choice is particularly useful as (5.11) implies the simple relation

$\mathfrak{p}_{i}^{a}=\frac{\sigma^{a}}{2}\,.$ (5.14)

An important feature characterizing all the systems with equal charges is that
the extremizing values $\varphi_{i}^{*}$ are equal, namely
$\varphi_{1}^{*}=\varphi_{2}^{*}$. Indeed, the off-shell free energy (5.5) is
symmetric under the exchange $\varphi_{1}\leftrightarrow\varphi_{2}$ and, in
particular, $\Phi_{1}^{a}-\Phi_{2}^{a}=\varphi_{1}-\varphi_{2}$, which does
not depend on the index $a$. Then, the necessary condition
$(\partial_{\varphi_{1}}-\partial_{\varphi_{2}})\mathcal{S}=0$ forces the
values at the extremum to be equal. Additionally, in the gauge we are
considering, the constraint (5.10) with $\vec{W}=0$ fixes them to
$\varphi_{1}^{*}=\varphi_{2}^{*}=1$. Extremizing the off-shell free energy
with respect to the remaining variables $\epsilon_{1,2}$ we obtain the
following critical values for the fugacities232323These results hold when
$\varkappa_{D}>0$. In the opposite case, we need to exchange the two
components of the vectors in (4.41) and, therefore, to exchange
$\epsilon_{1}^{*}$ and $\epsilon_{2}^{*}$ accordingly.

$\displaystyle d=5:\qquad\quad$ $\displaystyle\epsilon_{1}^{*}$
$\displaystyle=\frac{2}{g}\,\frac{2\pi}{\Delta\psi}\,,\qquad$
$\displaystyle\epsilon_{2}^{*}$
$\displaystyle=-\kappa\,2a\,\frac{2\pi}{\Delta\phi}\,,$ (5.15) $\displaystyle
d=6:\qquad\quad$ $\displaystyle\epsilon_{1}^{*}$
$\displaystyle=\frac{1}{g}\,\frac{2\pi}{\Delta\psi}\,,\qquad$
$\displaystyle\epsilon_{2}^{*}$
$\displaystyle=-\kappa\,a\,\frac{2\pi}{\Delta\phi}\,.$ (5.16)

These expressions have the same form of the results of [1] once we perform the
proper $SL(2,\mathbb{R})$ transformations connecting our basis of the torus
action $\\{\partial_{\nu_{1}},\partial_{\nu_{2}}\\}$ with the ones adopted in
[1]. Plugging these critical values in (5.5), we get the off-shell free energy
at its extremum242424In order to take into account a possible exchange in the
components of $\vec{v}_{a}$, which would affect $d_{a,a+1}$ in the denominator
of (5.5), we multiplied the off-shell free energy by $\varkappa_{D}$.

$\displaystyle F^{*}_{d=5}$ $\displaystyle=\frac{9\sqrt{2}\pi
N^{5/2}}{5\sqrt{8-N_{f}}}\,\frac{g^{3}(1-\kappa\,ag)}{|\Xi|a^{2}}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\bigl{|}(x_{+}^{2}+\tau^{(x_{-})}x_{-}^{2})\Delta
y-\Delta x(y_{+}^{2}-\tau^{(x_{-})}y_{-}^{2})\bigr{|}\,,$ (5.17)
$\displaystyle F^{*}_{d=6}$
$\displaystyle=\kappa\,4N^{3}\,\frac{g^{4}(1-\kappa\,ag)}{|\Xi|a^{3}}\frac{\Delta\psi}{2\pi}\frac{\Delta\phi}{2\pi}\bigl{|}(x_{+}^{3}+\tau^{(x_{-})}x_{-}^{3})(y_{+}^{2}-y_{-}^{2})-(x_{+}^{2}-x_{-}^{2})(y_{+}^{3}-\tau^{(x_{-})}y_{-}^{3})\bigr{|}\,.$
(5.18)

Indeed, to compare these formulas we must recall that the roots $x_{\pm}$ and
$y_{\pm}$ are related through the constraint obtained equating $a$ in (4.78)
and in (4.79). For what concerns $F^{*}_{d=6}$, its structure can be shown to
agree with the gravitational central charge (B.20), but its sign differs. We
believe that the origin of the opposite sign can be traced back to the
chirality of the dual two-dimensional SCFTs, but since we have not solved the
Killing spinor equations of $D=7$ supergravity, we have no control over this
information.

##  6 Discussion

In this paper we constructed new families of supersymmetric solutions in
$D=10$ and $D=11$ supergravities, corresponding to D4 and M5-branes wrapping
four-dimensional orbifolds $\mathbb{M}_{4}$, respectively. These generalize to
arbitrary toric orbifolds with four fixed points the examples presented in
[10, 12, 1], where the corresponding orbifolds were the total space of a
spindle fibred over another spindle. After extracting the necessary data, we
fed these into the off-shell free energies conjectured in our previous paper
[1] (and then proved in [50, 20]), obtaining a perfect match between the
gravitational observables calculated directly from the solutions and those
derived from the corresponding extremization problems. This is further
evidence that the extremal problems proposed in [1] encapsulate the necessary
and sufficient conditions for the existence of the supergravity solutions.
Furthermore, by construction, the gravitational central charge associated with
the AdS3 solutions agrees with the central charge of the dual two-dimensional
SCFTs, obtained by integrating the M5-brane anomaly polynomial on the
orbifolds [19]. It would be nice to derive in field theory the entropy
associated with the AdS2 solutions, by computing the supersymmetric partition
function of the dual five-dimensional theories compactified on
$\mathbb{M}_{4}\times S^{1}$ and then taking its large $N$ limit.

Obvious extensions of the present paper include looking for solutions
involving four-dimensional orbifolds $\mathbb{M}_{4}$ with one or more of the
following features. Firstly, one can strive to tame the solution with unequal
charges. It is also natural to wonder whether there exist solutions where the
Diophantine constraints need not be imposed, as these appear to be just an
artefact of the ansatz employed. Although explicit solutions corresponding to
toric orbifolds with more than four fixed points are likely to be
inaccessible, we expect that it should be relatively simple to construct
solutions describing branes wrapped on orbifolds with three fixed points,
namely weighted projective spaces $\mathbb{WCP}^{2}_{[n_{1},n_{2},n_{3}]}$. It
is possible that such solutions may be obtained in the special case of equal
NUT parameters ($N_{x}=N_{y}$), that we have not analysed in this
paper.252525In this scenario a singularity occurs when $x=y$ and, in order to
investigate this case, it may be useful to employ a change of coordinates
analogous to that used in [51] and [52] in different contexts. Finally, for
completeness, it would be desirable to check explicitly the supersymmetry
conditions for the $D=7$ solutions. We leave these unglamorous tasks as
possible projects for eager students.

The results of this paper add evidence that the discovery of supergravity
solutions displaying orbifold singularities has led to a vaster arena for
exploring quantum gravity via holography. We are convinced that it is worth
pursuing these explorations in different directions. On the one hand, it
should be rewarding to construct further explicit examples of orbifold
solutions, with AdS factors or otherwise. For example, using the results in
[20] one can anticipate the existence of numerous solutions in type II or
M-theory and it would be nice to construct some of these explicitly. On the
other hand, it is of uttermost importance to address the issue of proving
existence of the solutions, for the choices of topological/equivariant data
determined by the associated extremal problems. Among the several extremal
problems discussed in [19, 20], the GK geometry setup [53, 54] is perhaps the
simplest one. We are confident that the ideas of equivariant localization put
forward in [18, 19, 21, 20] will play a key role in tackling these problems,
and more generally in unveiling novel aspects of supergravity.

Finally, holography implies that the solutions discussed in this paper, as
well as all the other previous ones in the recent surge of orbifold
supergravity solutions, must have a field theory counterpart in terms of
localized partition functions on related rigidly supersymmetric backgrounds.
Based on the findings of [55, 47], we expect that this will be a fruitful
direction of work in the near future.

## Acknowledgments

We thank C. Casagrande, L. Cassia, E. Colombo, E. Martinengo, A. Pittelli and
A. Zaffaroni for useful discussions and comments. AF and DM are partially
supported by the INFN.

##  Appendix A The
AdS${}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$ limits

The $D$-dimensional system presented in section 3.1 includes, as particular
limits, the
$\mathrm{AdS}_{D-4}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$
solutions studied in [1, 12], for $D=6$, and in [10], when $D=7$. As we shall
see, these backgrounds can be retrieved only if the (putative) supersymmetry
condition $ags_{1}s_{2}=\pm 1$ is imposed, with the two charges $\delta_{i}$
having the same sign.

###  A.1 The AdS${}_{2}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$
limit

We shall now focus on $D=6$ and see how the
$\mathrm{AdS}_{2}\times\mathbbl{\Sigma}_{1}\ltimes\mathbbl{\Sigma}_{2}$
backgrounds are recovered. The first step is to perform the following
substitutions

$y\mapsto\epsilon^{2}y\,,\quad\phi\mapsto\epsilon^{2}\phi\,,\quad
a\mapsto\epsilon^{2}a\,,\quad N_{y}\mapsto\epsilon^{6}N_{y}\,,\quad
N_{x}\mapsto\epsilon^{2}N_{x}\,,\quad s_{i}\mapsto\frac{s_{i}}{\epsilon}\,,$
(A.1)

followed by the $\epsilon\to 0^{+}$ limit. The underlying idea is to take
$y\to 0$, but keeping the ratio $y/a$ finite, preserving the supersymmetry
condition and preventing many of the functions to become trivial.

A thorough analysis shows that the desired configurations can be obtained only
when the two charges $\delta_{i}$ have the same sign. For the sake of
simplicity, we shall restrict to $\delta_{1},\delta_{2}>0$ and, in this case,
introduce the new coordinates

$\tilde{y}=x\,,\qquad\tilde{x}=\frac{2y}{a}\,,\qquad\tilde{z}=\psi+\frac{\kappa\,\phi}{2ag}\,,\qquad\tilde{\psi}=\frac{\kappa\,\phi}{a}\,,$
(A.2)

with $\kappa=\pm 1$. Imposing the supersymmetry condition
$ags_{1}s_{2}=-\kappa$, the metric and the scalar fields reduce to the ones in
(2.40) of [1], when $m=2g_{c}/3$ and with $q_{i}$ and $\mathtt{a}$ defined as

$q_{i}=-2N_{x}s_{i}^{2}\,,\qquad\mathtt{a}=\frac{8N_{y}}{a^{3}}\,.$ (A.3)

In order to reconstruct the gauge potentials of [1], we first need to perform
the gauge transformation $A_{i}\mapsto A_{i}-\mathrm{d}\psi$ and, then, the
correct expression follows. Computing the limit of the two-form $B$ we obtain

$B=\kappa\,\frac{\mathtt{a}\tilde{y}}{2g}\,\mathrm{vol}(\mathrm{AdS}_{2})\,.$
(A.4)

When $\kappa=-1$, the origin of the opposite sign with respect to [1] can be
traced in a different orientation of the spacetime induced by the change of
coordinates (A.2).