August 31, 2024

Equivariant volume extremization and holography

Edoardo Colomboa,b, Federico Faedoc,d, Dario Martellic,d and Alberto
Zaffaronia,b

aDipartimento di Fisica, Università di Milano-Bicocca
Piazza della Scienza 3, 20126 Milano, Italy

bINFN, sezione di Milano-Bicocca, Piazza della Scienza 3, 20126 Milano, Italy

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

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

In a previous paper two of us (D.M. and A.Z.) proposed that a vast class of
gravitational extremization problems in holography can be formulated in terms
of the equivariant volume of the internal geometry, or of the cone over it. We
substantiate this claim by analysing supergravity solutions corresponding to
branes partially or totally wrapped on a four-dimensional orbifold, both in
M-theory as well as in type II supergravities. We show that our approach
recovers the relevant gravitational central charges/free energies of several
known supergravity solutions and can be used to compute these also for
solutions that are not known explicitly. Moreover, we demonstrate the validity
of previously conjectured gravitational block formulas for M5 and D4 branes.
In the case of M5 branes we make contact with a recent approach based on
localization of equivariant forms, constructed with Killing spinor bilinears.

###### Contents

  1. 1 Introduction
  2. 2 Equivariant volume with higher times
  3. 3 $\mathrm{AdS}_{3}$ and $\mathrm{AdS}_{5}$ solutions in M theory
    1. 3.1 $\mathrm{AdS}_{3}\times M_{8}$ solutions
    2. 3.2 AdS${}_{5}\times M_{6}$ solutions
    3. 3.3 Comparison with other approaches
  4. 4 $\mathrm{AdS}_{2}$ and $\mathrm{AdS}_{3}$ solutions in type II supergravities
    1. 4.1 $\mathrm{AdS}_{2}\times M_{8}$ solutions in massive type IIA
    2. 4.2 AdS${}_{3}\times M_{7}$ solutions in type IIB
  5. 5 Conclusions
  6. A Fixing the Kähler moduli of $\mathrm{AdS}_{5}\times M_{6}$ solutions with $\mathbb{Z}_{2}$ symmetry
  7. B $\mathrm{AdS}_{3}\times M_{7}$ solutions with the addition of higher times

##  1 Introduction

In this paper we propose a general prescription to write extremal functions
for supergravity solutions with a holographic dual. The extremal functions
depend on equivariant parameters for the expected abelian isometries of the
background and a set of parameters describing the geometry. The extremization
with respect to the parameters gives the free energy of the supergravity
solution, that is holographically equal to the central charge or free-energy
of the dual conformal field theory. On the quantum field theory side, this
construction is the gravitational dual of extremizing the central charge (in
even dimension) or the sphere partition function (in odd dimensions) in order
to find the exact R-symmetry. There is a huge literature about extremal
functions for black hole, and more generally for holographic solutions.
Extremal functions of known black holes and black strings can be expressed in
terms of gravitational blocks Hosseini:2019iad  and strongly suggest that some
equivariant localization is at work. Following Martelli:2023oqk  we will
indeed express the extremal functions in terms of a universal geometrical
quantity, the equivariant volume of the internal supergravity geometry.

Given a symplectic orbifold $\mathbb{M}_{2m}$ of real dimension $2m$ with a
toric111The toric assumption is not essential, but is made for two reasons.
Firstly, if a geometry has a symmetry group that contains
$\mathbb{T}^{m}=U(1)^{m}$, we need to extremize over all the corresponding
equivariant parameters, otherwise the critical point found would not be a
_bona fide_ extremum of the gravitational action. Secondly, in this case the
fixed point theorem simplifies to a sum of contribution at isolated fixed
points. More generally, it would be straightforward to proceed assuming a
$\mathbb{T}^{k}=U(1)^{k}$ Hamiltonian action, with $1<k<m$. action of
$\mathbb{T}^{m}=U(1)^{m}$ generated by the Hamiltonian $H$, we can define the
equivariant volume

$\mathds{V}=(-1)^{m}\int_{\mathbb{M}_{2m}}{\mathrm{e}}^{-\frac{\omega{T}}{2\pi}}\,,$
(1)

where $\omega{T}=\omega+2\pi H$ is the equivariant Kälher form. It is a
function of the $m$ equivariant parameters $\epsilon_{I}$ for the torus
$\mathbb{T}^{m}$ action and of the Kälher parameter $\lambda_{A}$ of the
geometry. The latter enter in the expansion of the Kälher class in a sum of
Chern classes of toric line bundles

$-\frac{[\omega]}{2\pi}=\sum_{A}\lambda_{A}c_{1}(L_{A})\,.$ (2)

The equivariant volume of toric orbifolds is a basic topological object. It
can be computed using a fixed point formula and it is only sensitive to the
degenerations of the torus $\mathbb{T}^{m}$ near the fixed points. In the
applications to holography one encounters metrics that are not Kähler and not
even symplectic, but with underlying spaces that are in fact symplectic toric
orbifolds and one can nevertheless define $\mathds{V}$ and use it to compute
topological quantities that ultimately will not depend on the metric. In many
examples when the underlying geometry is not strictly symplectic or toric we
can also define a natural generalization of $\mathds{V}$ by a sort of
analytical continuation.222This happens for geometries where the fan is not
strictly convex or geometries involving $S^{4}$. Given these properties, the
equivariant volume is the gravitational analogue of quantum field theory
quantities like t’Hooft anomalies and supersymmetric indices that are
invariant under small deformations of the theory once symmetries and matter
content are fixed. In Martelli:2023oqk  it was argued therefore that all
extremization problems in gravity can be reformulated in terms of the
equivariant volume. It was shown that this is true for volume minimization
Martelli:2005tp ; Martelli:2006yb  (dual to $a$ Intriligator:2003jj  and
$F$-maximization Jafferis:2010un ) and the formalism of GK geometry
Couzens:2018wnk ; Gauntlett:2018dpc  (dual to $c$ Benini:2012cz  and ${\cal
I}$-extremization Benini:2015eyy ). It was proposed that this should be true
more generally.

As a partial check of this proposal, it has been shown in Martelli:2023oqk
that all known extremization problems for branes wrapped over a sphere or a
spindle in type II and M theory can be reformulated in terms of an extremal
function

$F=\mathds{V}^{(\alpha)}(\lambda_{A},\epsilon_{I})$

subject to a set of flux constraints

$\nu M_{A}=-\frac{\partial\mathds{V}^{(\beta)}}{\partial\lambda_{A}}\,,$ (3)

where $M_{A}$ are the integer fluxes of the relevant RR or M theory
antisymmetric form, $\nu$ is a normalization constant that depends on the type
of brane, and $\mathds{V}^{(\gamma)}$ is the homogeneous piece of $\mathds{V}$
of degree $\gamma$ in $\lambda_{A}$. The integers $\alpha$ and $\beta$ depend
on the type of brane. By a simple scaling argument, it was found that

D3 branes in type IIB: $\displaystyle\alpha$ $\displaystyle=2\,,$
$\displaystyle\beta$ $\displaystyle=2$ (4) M2 branes in M theory:
$\displaystyle\alpha$ $\displaystyle=3\,,$ $\displaystyle\beta$
$\displaystyle=3$ M5 branes in M theory: $\displaystyle\alpha$
$\displaystyle=3\,,$ $\displaystyle\beta$ $\displaystyle=2$ D4 branes in
massive type IIA: $\displaystyle\alpha$ $\displaystyle=5\,,$
$\displaystyle\beta$ $\displaystyle=3$ D2 branes in massive type IIA:
$\displaystyle\alpha$ $\displaystyle=5\,,$ $\displaystyle\beta$
$\displaystyle=4\,.$

In this paper we show that this construction also holds for known
extremization problems for branes (partially or totally) wrapped over four-
dimensional toric orbifolds. We need to generalize our construction by
introducing higher times in the equivariant volume

$\mathds{V}=(-1)^{m}\int_{\mathbb{M}_{2m}}{\mathrm{e}}^{-\frac{\omega{T}}{2\pi}+\sum_{k=2}\lambda_{A_{1}\ldots
A_{k}}c_{1}{T}(L_{A_{1}})\ldots c_{1}{T}(L_{A_{k}})}\,,$ (5)

where $\lambda_{A_{1}\ldots A_{k}}$ are symmetric tensors and a sum over
repeated indices $A_{i}$ is understood. Higher times have appeared only
recently in the literature Nekrasov:2021ked  and are still poorly studied. The
previous expression has a large gauge invariance and many parameters are
redundant. As we will see, the equivariant volume with higher times contains
all the information needed to fully capture the topological properties and the
quantization of fluxes for a very large class of supergravity solutions.

The above construction relies on even-dimensional toric orbifolds. For
supergravity backgrounds AdS${}_{d}\times M_{k}$ with odd-dimensional internal
space $M_{k}$ the geometry to consider is the cone over $M_{k}$, as familiar
from holography. This cone is often a non-compact toric Calabi-Yau, or, in the
case of supersymmetry preserved with an anti-twist, a non-convex
generalization.333See for example Boido:2022mbe . When $M_{k}$ is even-
dimensional, we consider the equivariant volume of the compact $M_{k}$ itself.
Some M5 branes solutions have a $\mathbb{Z}_{2}$ symmetry that allows to cut
into half the number of fixed point and consider an equivalent problem for a
non-compact Calabi-Yau (half of the manifold). This was done in
Martelli:2023oqk  for M5 branes wrapped on a spindle.

Our approach naturally incorporates the GMS construction based on GK geometry
Couzens:2018wnk ; Gauntlett:2018dpc  as well as the recent localization
technique based on Killing spinor bilinears in M theory
BenettiGenolini:2023kxp . We will show indeed, that for M5 solutions, with
even-dimensional $M_{6}$ or $M_{8}$, our approach is effectively equivalent to
the one in BenettiGenolini:2023kxp . In particular, all the geometrical
constraints that must be imposed on a case-by-case analysis in order to find
the free-energy in BenettiGenolini:2023kxp  appear naturally in our
construction as an extremization with respect to all parameters that are not
fixed by the flux quantization conditions. On one side, this is a nice
confirmation of our prescription. On the other hand, our approach for the
toric case is more general, it covers in a simple and universal way the even
and odd-dimensional case, it naturally extends to massive type IIA solutions,
which are not yet covered by the previous techniques, and expresses everything
in terms of the extremization of a universal quantity, the equivariant volume
of the associated geometry, without referring to supergravity quantities. We
are confident that when the explicit case by case supergravity analysis will
be performed for the missing backgrounds it will confirm our general
prescription.

The paper is organized as follows. In section 2 we define the equivariant
volume of a general toric orbifold and we review some of its basic properties
following Martelli:2023oqk . We also introduce the concept of higher times
which are necessary to parameterize all the fluxes supported by a given
geometry. In section 3 we analyse M theory solutions with M5 brane flux. In
section 3.1 we consider solutions associated with M5 branes wrapped over a
four-dimensional orbifold $\mathbb{M}_{4}$. We show that the free energy can
be obtained by extremizing the appropriate term in the equivariant volume and
that the result agrees with the field theory computation in Martelli:2023oqk ,
obtained by integrating the anomaly of the M5 brane theory over
$\mathbb{M}_{4}$. In section 3.2 we consider solutions that are potentially
related to M5 branes wrapped on a two-cycle in $\mathbb{M}_{4}$. By
extremizing the appropriate term in the equivariant volume, we reproduce known
results in the literature and extend them to predictions for solutions still
to be found. In section 3.3 we compare our prescription with the recent
approach based on Killing spinor bilinears in M theory BenettiGenolini:2023kxp
. In section 4 we consider solutions in type II string theory with geometries
that are fibrations over a four-dimensional orbifold $\mathbb{M}_{4}$. In
section 4.1 we consider massive type IIA solutions associated with D4 branes
wrapped around a four-dimensional toric orbifold $\mathbb{M}_{4}$ and derive
the free energy proposed in Faedo:2022rqx . In section 4.2 we consider type
IIB solutions with D3 flux associated with $S^{3}/\mathbb{Z}_{p}$ fibrations
over $\mathbb{M}_{4}$, which could potentially arise as the near horizon limit
of a system of D3 branes wrapped on a two-cycle of the four-dimensional
orbifold $\mathbb{M}_{4}$. This example can be covered by the formalism of GK
geometry, that we here extend to the case of fibrations over orbifolds, using
the equivariance with respect to the full four-torus $\mathbb{T}^{4}$. In this
and other previous examples with M5 branes, we observe that, in order to
obtain the correct critical point, one should allow all the equivariant
parameters not fixed by symmetries to vary, thus rectifying some previous
results in the literature. We conclude with a discussion of open problems and
future perspectives. Two appendices contain technical aspects of some
computations.

##  2 Equivariant volume with higher times

In this section we review and generalize some basic facts about the
equivariant volume of general toric orbifolds that will be used in the
following. We adopt the conventions of Martelli:2023oqk  to which we refer for
more details and a review of equivariant localization.

We consider a toric orbifold $\mathbb{M}_{2m}$ with an action $\mathbb{T}^{m}$
generated by the $m$ vector fields $\partial_{\phi_{I}}$. We introduce $m$
equivariant parameters $\epsilon_{I}$, with $I=1,\ldots m$ and the vector
field $\xi=\epsilon_{I}\partial_{\phi_{I}}$ and consider equivariantly closed
forms $\alpha^{\mathbb{T}}$ satisfying

$(\mathrm{d}+2\pi i_{\xi})\alpha^{\mathbb{T}}=0\,.$ (6)

We will be dealing with varieties and orbifolds of different dimension and,
when needed, we will also write $\alpha^{\mathbb{T}^{m}}$ to specify the
dimension. Each toric orbifold comes equipped with a fan, a collection of
integer vectors $V^{A}$ that exhibits $\mathbb{M}_{2m}$ as a $\mathbb{T}^{m}$
fibration over a convex polytope

$\mathcal{P}=\\{y_{I}\in\mathbb{R}^{m}\>|\>y_{I}V_{I}^{A}\geq\lambda_{A}\\}\>.$
(7)

On each of the facets $y^{I}V^{A}_{I}=\lambda_{A}$ a particular circle in
$\mathbb{T}^{m}$ degenerates, as familiar from toric geometry.444Notice that
for toric varieties the vectors $V^{A}$ are primitive, while in the case of
generic symplectic toric orbifolds they are not. We can define for each
$V^{A}$ a primitive vector $\hat{V}^{A}$ and a positive integer $n_{A}$ such
that $v^{A}=n_{A}\hat{V}^{A}$. The symplectic divisor $D_{A}$ has a local
$\mathbb{Z}_{n_{A}}$ singularity. For more details we refer to
Martelli:2023oqk . Each facet defines a toric divisor $D_{A}$, with associated
line bundle $L_{A}$, and an equivariant Chern class

$c_{1}^{\mathbb{T}}(L_{A})=c_{1}(L_{A})+2\pi\epsilon_{I}\mu_{I}^{A}=\mathrm{d}(\mu_{I}^{A}\mathrm{d}\phi_{I})+2\pi\epsilon_{I}\mu_{I}^{A}\,\,.$
(8)

The functions $\mu^{A}$ play the role of moment maps. We will not need their
explicit expression, which is discussed in Martelli:2023oqk  but we will
frequently use the relation

$\sum_{A}V^{A}_{I}c_{1}^{\mathbb{T}}(L_{A})=-\epsilon_{I}\,,$ (9)

which is the equivariant version of the co-homological statement
$\sum_{A}V^{A}_{I}c_{1}(L_{A})=0$ following from the toric equivalence
relations among divisors $\sum_{A}V^{A}_{I}D_{A}=0$.

The equivariant volume with higher times (5) can be computed with a fixed
point formula that can be also taken as an operative definition

$\mathds{V}(\lambda_{A_{1}\ldots
A_{K}},\epsilon_{I})=(-1)^{m}\sum_{\alpha=1}^{n}\frac{{\mathrm{e}}^{\tau{T}|_{y_{\alpha}}}}{d_{\alpha}\,e^{\mathbb{T}}|_{y_{\alpha}}}\,,$
(10)

where $y_{\alpha}$ are the fixed points of the $\mathbb{T}^{m}$ action,
$d_{\alpha}$ the order of the orbifold singularity at $y_{\alpha}$,
$e^{\mathbb{T}}$ is the equivariant Euler class of the tangent bundle at
$y_{\alpha}$ and

$\tau{T}=\sum_{k=1}\lambda_{A_{1}\ldots A_{k}}c_{1}{T}(L_{A_{1}})\ldots
c_{1}{T}(L_{A_{k}})\,.$ (11)

To use the localization formula we assume that $\mathbb{M}_{2m}$ has only
isolated orbifold singularities. This is the case if the fan is the union of
$m$-dimensional cones $\\{v^{A_{1}},\ldots,v^{A_{m}}\\}$.555We can always
resolve the fan by adding vectors if this condition is not met. Each cone
$\alpha=(A_{1},\ldots,A_{m})$ corresponds to an isolated fixed point
$y_{\alpha}$ with a local orbifold singularity of order

$d_{\alpha}=|\det(v^{A_{1}},\ldots,v^{A_{m}})|\,.$ (12)

The restriction of the Euler class is given by

$e^{\mathbb{T}}|_{y_{\alpha}}=\prod_{A_{i}\in\alpha}c^{\mathbb{T}}_{1}(L_{A_{i}})|_{y_{\alpha}}\,.$
(13)

The restriction of the Chern classes at the fixed points $y_{\alpha}$ are
computed as follows Martelli:2023oqk

$c^{\mathbb{T}}_{1}(L_{A})\Big{|}_{y_{\alpha}}=\begin{cases}-\frac{\epsilon_{I}u_{I}^{A}}{d_{A}}\,\qquad&{\rm
if}\,A\in\alpha\\\ 0\,\qquad&{\rm if}\,A\notin\alpha\end{cases}\,,$ (14)

where $u_{I}^{A}$ are the inwards normal to the facets of the cone $\alpha$
defined and normalized by the relation

$u^{A_{i}}_{\alpha}\cdot v^{A_{j}}=d_{\alpha}\delta_{ij}\,.$ (15)

Now the fixed point formula gives

$\mathds{V}(\lambda_{A_{1}\ldots
A_{k}},\epsilon_{I})=\sum_{\alpha=(A_{1},\ldots,A_{m})}\frac{{\mathrm{e}}^{\tau_{\alpha}}}{d_{A}\prod_{i=1}^{m}\frac{\epsilon\cdot
u^{a_{i}}_{A}}{d_{A}}}\,,$ (16)

where $\tau_{\alpha}$ is the restriction of the equivariant form (11) to the
fixed point $y_{\alpha}$ and explicitly reads

$\tau_{\alpha}=-\sum_{i=1}^{m}\lambda_{A_{i}}(\frac{\epsilon\cdot
u^{A_{i}}_{\alpha}}{d_{\alpha}})+\sum_{i,j=1}^{m}\lambda_{A_{i}A_{j}}(\frac{\epsilon\cdot
u^{A_{i}}_{\alpha}}{d_{\alpha}})(\frac{\epsilon\cdot
u^{A_{j}}_{\alpha}}{d_{\alpha}})+\ldots\,$ (17)

The equivariant volume can be expanded in power series of the higher times,

$\mathds{V}(\lambda_{A_{1}\ldots
A_{k}},\epsilon_{I})=\sum_{n=0}^{\infty}\mathds{V}^{(n)}(\lambda_{A_{1}\ldots
A_{k}},\epsilon_{I})\,,$ (18)

where we denote with $\mathds{V}^{(n)}$ the homogeneous component of degree
$n$ in the set of higher times $\lambda_{A_{1}\ldots A_{k}}$ for all $k$.
$\mathds{V}^{(n)}$ is a polynomial in $\epsilon_{I}$ in the compact case,
while it can be a rational function of $\epsilon_{I}$ when $\mathbb{M}_{2m}$
is non-compact.

In the examples in Martelli:2023oqk  were used only single times
($\lambda_{A}$). In this paper we will use single and double times
($\lambda_{A}$ and $\lambda_{AB}$). As a general rule, to fully capture the
parameters of the supergravity solution, we need a number of independent
parameters at least equal to the number of fixed points. Functionally, indeed
$\mathds{V}$ is a function of $\epsilon_{I}$ and $\tau_{\alpha}$ only. Notice
that there is a large redundancy in the description with higher times. Due to
the relation (9), $\tau{T}$ is invariant under the gauge transformations

$\displaystyle\lambda_{A_{1}\ldots A_{k+1}}\rightarrow\lambda_{A_{1}\ldots
A_{k+1}}+\beta_{I}^{(A_{1}\ldots
A_{k}}V^{A_{k+1})}_{I}\,,\qquad\lambda_{A_{1}\ldots
A_{k}}\rightarrow\lambda_{A_{1}\ldots
A_{k}}+\epsilon_{I}\beta_{I}^{A_{1}\ldots A_{k}}\,,$ (19)

where $\beta_{I}^{A_{1}\ldots A_{k}}$ is symmetric in the indices $A_{1}\ldots
A_{k}$. Notice that the subgroup with $\epsilon_{I}\beta_{I}^{A_{1}\ldots
A_{k}}=0$ acts only on $\lambda_{A_{1}\ldots A_{k+1}}$ without mixing times of
different degree and it is the only transformation allowed for single times.
In the Calabi-Yau case, where the vectors in the fan lie on a plane identified
by the direction $I=CY$, say $V^{A}_{CY}=1$,666In this paper $\epsilon_{CY}$
will be identified with $\epsilon_{3}$ in all examples. this subgroup can also
be written as

$\lambda_{A_{1}\ldots A_{k}}\rightarrow\lambda_{A_{1}\ldots
A_{k}}+(\epsilon_{CY}V^{(A_{1}}_{I}-\epsilon_{I})\gamma^{A_{2}\ldots
A_{k})}_{I}\,,$ (20)

generalizing the results in Martelli:2023oqk . Many times can be therefore
gauged fixed to zero.

As an example we give some more explicit expressions about the four-
dimensional case. These will be heavily used in the following since in this
paper we will mostly consider geometries that are fibrations over four-
dimensional compact orbifolds $\mathbb{M}_{4}$. For clarity of notation, we
will use capital letters $(V^{A}_{I},A,I)$ for the higher-dimensional
geometry, and lower case letters $(v^{a}_{i},a,i)$ for $\mathbb{M}_{4}$. The
fan of a four-dimensional compact orbifolds is just a collections of two-
dimensional integers vectors $v^{a}$, $a=1,\dots,d$ that define a convex
polygon in the plane. The fixed points are associated with the cones
$(v^{a},v^{a+1})$, where we take a counter-clock-wise order for the vector and
identify cyclically $v^{d+a}=v^{a}$. Notice that in the compact four-
dimensional case the number of fixed points is equal to the number of vectors
in the fan and we can use the index $a=1,\ldots,d$ to label both. With the
notations of Martelli:2023oqk  we define the quantities

$\epsilon_{i}^{a}=\frac{\epsilon\cdot u^{a}_{i}}{d_{a,a+1}}\,,\qquad i=1,2\,,$
(21)

where $u^{a}_{i}$ are the inward normals to the cones $(v^{a},v^{a+1})$.
explicitly

$\epsilon^{a}_{1}=-\frac{\det(v^{a+1},\epsilon)}{\det(v^{a},v^{a+1})}\,,\qquad\epsilon^{a}_{2}=\frac{\det(v^{a},\epsilon)}{\det(v^{a},v^{a+1})}\,,$
(22)

where $\epsilon\equiv(\epsilon_{1},\epsilon_{2})$. In particular, the
equivariant Euler class of the tangent bundle at a fixed point $y_{a}$ reads

$e^{\mathbb{T}}|_{y_{a}}=\epsilon^{a}_{1}\,\epsilon^{a}_{2}\,,$ (23)

and the order of the local orbifold singularity is

$d_{a,a+1}=\det(v^{a},v^{a+1})\,.$ (24)

The restriction to the fixed points of the equivariant Chern classes
$c_{1}^{\mathbb{T}}(L_{a})$ can be written as

$c_{1}^{\mathbb{T}}(L_{a})|_{y_{b}}=-(\delta_{a,b}\epsilon_{1}^{b}+\delta_{a,b+1}\epsilon_{2}^{b})\,.$
(25)

The fixed point formula (16) for the equivariant volume specializes to the
expression777Notice that there is no summation on $a$ in the exponent.

$\displaystyle\mathds{V}(\lambda_{a_{1}\ldots a_{k}},\epsilon_{i})$
$\displaystyle=\sum_{a=1}^{d}\frac{{\mathrm{e}}^{-\lambda_{a}\epsilon^{a}_{1}-\lambda_{a+1}\epsilon^{a}_{2}+\lambda_{a,a}(\epsilon^{a}_{1})^{2}+2\lambda_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}+\lambda_{a+1,a+1}(\epsilon^{a}_{2})^{2}+\ldots}}{d_{a,a+1}\epsilon_{1}^{a}\epsilon_{2}^{a}}\,.$
(26)

We will also need in the following the intersections matrix of divisors, which
is independent of the equivariant parameters $\epsilon_{1},\epsilon_{2}$
Martelli:2023oqk :

$D_{a}\cdot
D_{b}=D_{ab}=\int_{\mathbb{M}_{4}}c_{1}^{\mathbb{T}}(L_{a})c_{1}^{\mathbb{T}}(L_{b})=\begin{cases}\frac{1}{d_{a-1,a}}&\text{if
$b=a-1$}\,,\\\ \frac{1}{d_{a,a+1}}&\text{if $b=a+1$}\,,\\\
-\frac{d_{a-1,a+1}}{d_{a-1,a}d_{a,a+1}}&\text{if $b=a$}\,,\\\
0&\text{otherwise}\,.\end{cases}$ (27)

##  3 $\mathrm{AdS}_{3}$ and $\mathrm{AdS}_{5}$ solutions in M theory

We start by analyzing M theory solutions with M5 brane flux and show that the
free energy can be obtained by extremizing the appropriate term in the
equivariant volume. The case of M5 branes wrapped on a spindle have been
already studied in Martelli:2023oqk . Here we focus on geometries that are
fibrations over a four-dimensional toric orbifold $\mathbb{M}_{4}$.

###  3.1 $\mathrm{AdS}_{3}\times M_{8}$ solutions

In this section we consider $\mathrm{AdS}_{3}\times M_{8}$ solutions in M
theory, where888In general, $M_{8}$ is itself an orbifold. $M_{8}$ is a
$S^{4}$ fibration over the four-dimensional orbifold $\mathbb{M}_{4}$.
Examples of this form have been found in Cheung:2022ilc  and further discussed
in Faedo:2022rqx ; Couzens:2022lvg ; Bomans:2023ouw . They are obtained by
uplifting $\mathrm{AdS}_{3}\times\mathbb{M}_{4}$ solutions of $D=7$ maximal
gauged supergravity to eleven dimensions. These $\mathrm{AdS}_{3}\times M_{8}$
solutions can be interpreted as the near-horizon geometry of a system of M5
branes wrapped around $\mathbb{M}_{4}$.

We need first to identify the topological structure of the underlying
geometry. We will focus on the case of toric $\mathbb{M}_{4}$. The eight-
dimensional geometry $M_{8}$ is not strictly toric but it admits an action of
$\mathbb{T}^{4}=U(1)^{4}$. If $d$ is the dimension of the fan of
$\mathbb{M}_{4}$, there are $2d$ fixed points of the torus action obtained by
selecting a fixed point on $\mathbb{M}_{4}$ and combining it with the North
and South pole of $S^{4}$. We will assume that there is a $\mathbb{Z}_{2}$
symmetry of the fibration that identifies the North and South pole
contributions to the fixed point formula. In this situation we can consider
half of the geometry, a $\mathbb{C}^{2}$ fibration over $\mathbb{M}_{4}$ with
the geometry of a non-compact toric CY4. One can understand the appearance of
the fibre $\mathbb{C}^{2}$ from the transverse geometry of the brane system,
which is $\mathbb{C}^{2}\times\mathbb{R}$, with $S^{4}$ embedded inside. We
then consider a CY4 with fan generated by the vectors

$V^{a}=(v^{a},1,\mathfrak{t}_{a})\,,\qquad V^{d+1}=(0,0,1,0)\,,\qquad
V^{d+2}=(0,0,1,1)\,,$ (28)

where $v^{a}$, $a=1,\dots,d$ are the vectors of the fan of $\mathbb{M}_{4}$
and $\mathfrak{t}_{a}$ are integers specifying the twisting of
$\mathbb{C}^{2}$ over $\mathbb{M}_{4}$. When supersymmetry is preserved with
an anti-twist Ferrero:2021etw , the toric diagram is not convex and it does
not strictly define a toric geometry. We will nevertheless proceed also in
this case, considering it as an extrapolation from the twist case. The non-
convex case is obtained from the formulas in this paper by sending
$v^{a}\rightarrow\sigma^{a}v^{a}$, where $\sigma^{a}=\pm 1$.

In addition to the metric, the supergravity solution is specified by the
integer fluxes of the M theory four-form along all the non-trivial four-
cycles. The toric four-cycles of the geometry are $\mathbb{M}_{4}$ itself, the
sphere $S^{4}$ and $\mathbb{P}^{1}$ fibrations over the toric two-cycles
$\Sigma_{a}\subset\mathbb{M}_{4}$. In our half-geometry, the sphere $S^{4}$
and $\mathbb{P}^{1}\subset S^{4}$ are replaced with copies of $\mathbb{C}^{2}$
and $\mathbb{C}$. All together, the toric four-cycles correspond to all the
possible intersections of the toric divisors $D_{A}\cap D_{B}$ and we can
therefore introduce a matrix of fluxes $M_{AB}$. As usual, not all toric
divisor are inequivalent in co-homology. The relations
$\sum_{A}V^{A}_{I}D_{A}=0$ imply that the matrix of fluxes satisfy

$\sum_{A}V^{A}_{I}M_{AB}=0\,.$ (29)

We are now ready to formulate our prescription for the extremal function. For
M5 branes in M theory, as discussed in Martelli:2023oqk  and in the
introduction, we define the free-energy to extremize as

$F=\mathds{V}^{(3)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\,,$ (30)

and impose the flux constraints999We put a bar on top of $\nu_{M5}$ to stress
that we are using a half-geometry. To have the correct normalization of the
free energy when using half of the geometry, the parameter $\nu_{M5}$ must be
rescaled as in formula (74) as we will discuss more extensively in section
3.2.

$\overline{\nu}_{M5}\,(2-\delta_{AB})\,M_{AB}=-\frac{\partial}{\partial\lambda_{AB}}\mathds{V}^{(2)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\,.$
(31)

Here the index $A=1,\ldots,d+2$ runs over all the vectors of the fan of the
CY4, whereas we reserve the lowercase index $a=1,\ldots,d$ for the vectors of
the fan of the base $\mathbb{M}_{4}$. The index $I=1,2,3,4$ on the other end
runs over the equivariant parameters of the CY4 and we will use $i=1,2$ for
the directions inside $\mathbb{M}_{4}$. We have added a $(2-\delta_{AB})$
factor in the equation for the fluxes for convenience. It is easy to see using
(16) that this equation can be equivalently rewritten as

$\overline{\nu}_{M5}\,M_{AB}=-\frac{\partial^{2}}{\partial\lambda_{A}\partial\lambda_{B}}\mathds{V}^{(3)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\>,$
(32)

and one may wonder if we really need higher times. The answer is yes. As we
discuss later, with only single times the previous equation can not be
solved.101010One would need to restrict the $\mathfrak{t}_{a}$ in order to
find solutions.

In the rest of this section we will show that $F$ reproduces the expected
extremal function and its factorization in gravitational blocks discussed in
Faedo:2022rqx ; Martelli:2023oqk .

####  3.1.1 The equivariant volume with double times

The $\mathbb{T}^{4}$ torus action on the CY4 has $d$ fixed points, each one
corresponding to a cone in the fan with generators
$(V^{a},V^{a+1},V^{d+1},V^{d+2})$, $a=1,\ldots,d$. In particular, there is a
one-to-one correspondence between these fixed points and the ones of the base
orbifold $\mathbb{M}_{4}$; for the latter the fixed points correspond to two-
dimensional cones of the form $(v^{a},v^{a+1})$ and they can be labelled by
the index $a$. The orders of the orbifold singularity associated to the fixed
points of CY4 and $\mathbb{M}_{4}$ also match:

$d_{a,a+1,d+1,d+2}=\big{|}\det(V^{a},V^{a+1},V^{d+1},V^{d+2})\,\big{|}=\big{|}\det(v^{a},v^{a+1})\,\big{|}=d_{a,a+1}\>.$
(33)

Thus the fixed point formula for the equivariant volume with higher times of
CY4 takes the following form:

$\mathds{V}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\sum_{a}\frac{e^{\tau_{a}}}{d_{a,a+1}\>e^{\mathbb{T}_{4}}|_{a}}\>.$
(34)

Here the $\tau_{a}$ are the restriction to the fixed point $a$ of the form
(11)

$\tau_{a}=\left(\sum_{A}\lambda_{A}\,c_{1}^{\mathbb{T}_{4}}(L_{A})+\sum_{AB}\lambda_{AB}\,c_{1}^{\mathbb{T}_{4}}(L_{A})\,c_{1}^{\mathbb{T}_{4}}(L_{B})\right)\Bigg{|}_{a}\>,$
(35)

while at the denominator we have the restriction of the Euler class
$e^{\mathbb{T}_{4}}$

$e^{\mathbb{T}_{4}}|_{a}=\Big{(}c_{1}^{\mathbb{T}_{4}}(L_{a})\,c_{1}^{\mathbb{T}_{4}}(L_{a+1})\,c_{1}^{\mathbb{T}_{4}}(L_{d+1})\,c_{1}^{\mathbb{T}_{4}}(L_{d+2})\Big{)}\Big{|}_{a}\>.$
(36)

The restrictions of the Chern classes can be computed using (14). The inward
normals to the faces of the cone generated by
$(V_{a},V_{a+1},V_{d+1},V_{d+2})$ are

$\displaystyle U^{a}=(u^{a}_{1},0,0)\,,$ (37) $\displaystyle
U^{a+1}=(u^{a}_{2},0,0)\,,$ $\displaystyle
U^{d+1}=\big{(}(\mathfrak{t}_{a}-1)u^{a}_{1}+(\mathfrak{t}_{a+1}-1)u^{a}_{2}\,,d_{a,a+1},-d_{a,a+1}\big{)}\,,$
$\displaystyle
U^{d+2}=(-\mathfrak{t}_{a}u^{a}_{1}-\mathfrak{t}_{a+1}u^{a}_{2}\,,0,d_{a,a+1})\,,$

where $u^{a}_{1}$ and $u^{a}_{2}$ are the two-dimensional normals to the cone
$(v^{a},v^{a+1})$. Using the notations introduced in (21) we find

$\displaystyle
c_{1}^{\mathbb{T}_{4}}(L_{a})|_{a}=-\frac{\epsilon_{i}\cdot(u^{a}_{1})^{i}}{d_{a,a+1}}=-\epsilon_{1}^{a}\,,$
$\displaystyle
c_{1}^{\mathbb{T}_{4}}(L_{a+1})|_{a}=-\frac{\epsilon_{i}\cdot(u^{a}_{2})^{i}}{d_{a,a+1}}=-\epsilon_{2}^{a}\,,$
$\displaystyle c_{1}^{\mathbb{T}_{4}}(L_{b})|_{a}=0\,,\qquad b\neq a,a+1\,,$
(38) $\displaystyle
c_{1}^{\mathbb{T}_{4}}(L_{d+1})|_{a}=-\epsilon_{3}+\epsilon_{4}-(\mathfrak{t}_{a}-1)\epsilon_{1}^{a}-(\mathfrak{t}_{a+1}-1)\epsilon_{2}^{a}\,,$
$\displaystyle
c_{1}^{\mathbb{T}_{4}}(L_{d+2})|_{a}=-\epsilon_{4}+\mathfrak{t}_{a}\epsilon_{1}^{a}+\mathfrak{t}_{a+1}\epsilon_{2}^{a}\,,$

where for simplicity we have used Einstein notation for the sums over the
index $i=1,2$.

We can write the equivariant volume of the CY4 as an integral over the base
orbifold $\mathbb{M}_{4}$ of four-dimensional equivariant forms with
$\epsilon_{3}$ and $\epsilon_{4}$ as parameters. Let us denote with
$\mathbb{T}$ the two-dimensional torus associated with $\epsilon_{1}$ and
$\epsilon_{2}$, and let $c_{1}^{\mathbb{T}}(L_{a})$ be the equivariant Chern
classes associated to the restrictions of the line bundles $L_{a}$ to the base
$\mathbb{M}_{4}$. We can then take advantage of the one-to-one correspondence
between fixed point of the CY4 and fixed points of $\mathbb{M}_{4}$ and, using
(25), we can rewrite (34) as

$\mathds{V}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\int_{\mathbb{M}_{4}}\frac{e^{\tau{T}}}{\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}\>,$
(39)

where

$\displaystyle\tau{T}=\sum_{A}\lambda_{A}\>\mathcal{C}_{A}+\sum_{A,B}\lambda_{AB}\>\mathcal{C}_{A}\>\mathcal{C}_{B}\,,$
(40) $\displaystyle\mathcal{C}_{a}\,=\,c_{1}{T}(L_{a})\>,\qquad a=1,\ldots,d$
$\displaystyle\mathcal{C}_{d+1}\,=\,-\epsilon_{3}+\epsilon_{4}+\sum_{a}(\mathfrak{t}_{a}-1)c_{1}{T}(L_{a})\,,$
$\displaystyle\mathcal{C}_{d+2}\,=\,-\epsilon_{4}-\sum_{a}\mathfrak{t}_{a}c_{1}{T}(L_{a})\>.$

Notice the relations $\sum_{a}v^{a}_{i}\,c_{1}{T}(L_{a})=-\epsilon_{i}$ and
$\sum_{A}V_{I}^{A}\,\mathcal{C}_{A}=-\epsilon_{I}$, following from
(9).111111The second relation, which can be checked by direct computation, is
obviously the restriction of
$\sum_{A}V^{A}_{I}\,c_{1}^{\mathbb{T}_{4}}(L_{A})=-\epsilon_{I}$ to
$\mathbb{M}_{4}$.

The homogeneous component of degree $\alpha$ of the equivariant volume with
higher times can be expressed as

$\mathds{V}^{(\alpha)}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\int_{\mathbb{M}_{4}}\frac{(\tau{T})^{\alpha}}{\alpha!\>\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}=\sum_{a}\frac{B^{(\alpha)}_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(41)

where we have defined $B^{(\alpha)}_{a}$ to be the restriction over the $a$-th
fixed point of $\mathbb{M}_{4}$ of the following equivariant form:

$B^{(\alpha)}=\frac{(\tau{T})^{\alpha}}{\alpha!\>\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}\>.$
(42)

For later reference we derive the relation between $B^{(\alpha)}_{a}$ and
$B^{(\beta)}_{a}$

$\displaystyle B^{(\beta)}_{a}=$
$\displaystyle\,\frac{(\tau_{a})^{\beta}}{\beta!\,\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}}=\frac{(\alpha!)^{\frac{\beta}{\alpha}}}{\beta!}\left[\frac{(\tau_{a})^{\alpha}}{\alpha!\,\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}}\right]^{\frac{\beta}{\alpha}}\Big{[}\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}\Big{]}^{\frac{\beta}{\alpha}-1}$
(43) $\displaystyle=$
$\displaystyle\,\frac{(\alpha!)^{\frac{\beta}{\alpha}}}{\beta!}\big{(}B_{a}^{(\alpha)}\big{)}^{\frac{\beta}{\alpha}}\big{(}(1-\mathfrak{t}_{a})\epsilon_{1}^{a}+(1-\mathfrak{t}_{a+1})\epsilon_{2}^{a}-\epsilon_{3}+\epsilon_{4}\big{)}^{\frac{\beta}{\alpha}-1}\big{(}\mathfrak{t}_{a}\epsilon_{1}^{a}+\mathfrak{t}_{a+1}\epsilon_{2}^{a}-\epsilon_{4}\big{)}^{\frac{\beta}{\alpha}-1}.$

When $\alpha$ is even this formula holds in terms of absolute values, and the
signs must be fixed separately. This will not be the case for the computation
of this section, so we postpone the discussion about the signs to section 4.1.

####  3.1.2 Solving the flux constraints

The flux constraints (31) reads

$\overline{\nu}_{M5}\,(2-\delta_{AB})\,M_{AB}=-\frac{\partial\mathds{V}^{(2)}}{\partial\lambda_{AB}}=-(2-\delta_{AB})\int_{\mathbb{M}_{4}}\frac{\mathcal{C}_{A}\,\mathcal{C}_{B}\,\tau{T}}{\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}\>,$
(44)

or, equivalently

$\overline{\nu}_{M5}\,M_{AB}=-\int_{\mathbb{M}_{4}}\frac{\mathcal{C}_{A}\,\mathcal{C}_{B}\,\tau{T}}{\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}=-\sum_{a}\frac{B^{(1)}_{a}\cdot\big{(}\mathcal{C}_{A}\,\mathcal{C}_{B}\big{)}|_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>.$
(45)

Let us focus on the $A,B\in\\{1,\ldots,d\\}$ sector. Using (25) we find

$\displaystyle\overline{\nu}_{M5}\,M_{a,a+1}=-\frac{B^{(1)}_{a}}{d_{a,a+1}}\,,$
(46)
$\displaystyle\overline{\nu}_{M5}\,M_{a,a}=-\frac{B^{(1)}_{a}\>\epsilon_{1}^{a}}{d_{a,a+1}\>\epsilon_{2}^{a}}-\frac{B^{(1)}_{a-1}\>\epsilon_{2}^{a-1}}{d_{a-1,a}\>\epsilon_{1}^{a-1}}\,,$
$\displaystyle\overline{\nu}_{M5}\,M_{ab}=0\qquad\text{when }b\neq
a,\,a+1,\,a-1\>.$

These equations give constraints on the fluxes but they have a very simple
solution

$\displaystyle B^{(1)}_{a}=-\overline{\nu}_{M5}\,N\,,$ (47) $\displaystyle
M_{ab}=N\,D_{ab}\>,$

where $D_{ab}$ is the intersection matrix of divisors (27) and $N$ is any
integer that is a multiple of all the products $d_{a-1,a}\,d_{a,a+1}$.

This can be seen as follows. By combining the first two equations we obtain

$M_{a,a}=M_{a,a+1}\>\frac{\epsilon_{1}^{a}}{\epsilon_{2}^{a}}+M_{a,a-1}\>\frac{\epsilon_{2}^{a-1}}{\epsilon_{1}^{a-1}}\>,$
(48)

and using the relation Martelli:2023oqk

$\frac{\epsilon_{1}^{a}}{d_{a,a+1}\>\epsilon_{2}^{a}}+\frac{\epsilon_{2}^{a-1}}{d_{a-1,a}\>\epsilon_{1}^{a-1}}\,=\,-\>\frac{d_{a-1,a+1}}{d_{a-1,a}\>\>d_{a,a+1}}\,,$
(49)

we can rewrite this as

$M_{a,a}\,d_{a,a+1}+M_{a,a-1}\,d_{a-1,a+1}=\frac{\epsilon_{1}^{a}}{\epsilon_{2}^{a}}\,\big{(}M_{a,a+1}\,d_{a,a+1}-M_{a,a-1}\,d_{a-1,a}\big{)}\>.$
(50)

Given that the fluxes $M_{AB}$ and the orders of the orbifold singularity
$d_{a,a+1}$ are just integers, the only way that this equation can be true for
general values of $\epsilon$ is for both sides to vanish. This implies that
$M_{ab}$ is proportional to the intersections $D_{ab}$ given in (27). We can
then conclude that the only solution to equations
(LABEL:M_ab_and_B_a_equations) is (LABEL:AdS3xM8_flux_equations_solution).
Notice that there is just one independent flux associated with the $M_{ab}$
components of the flux matrix. This was to be expected since this corresponds
to the M theory four-form flux on $S^{4}$.

The values of the remaining entries of the matrix of fluxes $M_{AB}$ are
related to the fibration parameters. By substituting
$B^{(1)}_{a}=-\overline{\nu}_{M5}\,N$ in (45) we find

$M_{AB}=N\sum_{a}\frac{\big{(}\mathcal{C}_{A}\,\mathcal{C}_{B}\big{)}|_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}=N\int_{\mathbb{M}_{4}}\mathcal{C}_{A}\,\mathcal{C}_{B}=N\sum_{c,d}\mathfrak{t}_{A}^{c}\,\mathfrak{t}_{B}^{d}\,D_{cd}\>.$
(51)

In the last step we have used (27) and for convenience we have defined
$\mathfrak{t}_{A}^{c}$ as

$\mathfrak{t}_{A}^{c}=\begin{cases}\delta^{c}_{A}\qquad&A\in\\{1,\ldots,d\\}\\\
\mathfrak{t}_{c}-1\qquad&A=d+1\\\ -\mathfrak{t}_{c}\qquad&A=d+2\end{cases}\>.$
(52)

Given that the $\mathfrak{t}_{a}$ are integers, the fluxes $M_{AB}$ in (51)
are all integers.

We note that the expression (51) for $M_{AB}$ satisfies the relation required
to be considered a matrix of fluxes,

$\sum_{A}V_{I}^{A}M_{AB}=0\>.$ (53)

This can easily be verified by noting that

$\sum_{A}V_{I}^{A}\,\mathfrak{t}^{c}_{A}=\begin{cases}v_{i}^{c}\quad&I\equiv
i=1,2\\\ 0\quad&I=3,4\end{cases}\>,\qquad\sum_{a}v^{c}_{i}D_{cd}=0\>.$ (54)

The simplest solution to the equations

$B^{(1)}_{a}\equiv\frac{\tau_{a}(\lambda_{A}(\epsilon_{I}),\lambda_{AB}(\epsilon_{I}),\epsilon_{I})}{\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}}=-\overline{\nu}_{M5}\,N$
(55)

is to set $\lambda_{d+1,d+2}=-\overline{\nu}_{M5}\,N$ while setting all the
other $\lambda_{A}$ and $\lambda_{AB}$ to zero. We note that in general there
exist no solutions to these equations with $\lambda_{AB}=0$ for all $A,B$,
meaning that the inclusion of the higher times to the equivariant volume is
necessary. This stems from the fact that when $\lambda_{AB}=0$ only $d-1$ of
the $\tau_{a}$ are independent: using the gauge invariance (20)

$\lambda_{A}\rightarrow\lambda_{A}+\sum_{I=1}^{4}\gamma_{I}(\epsilon_{3}V^{A}_{I}-\epsilon_{I})\,,$
(56)

three out of the $d+2$ Kähler moduli $\lambda_{A}$ can be set to zero.

####  3.1.3 The extremal function and $c$-extremization

We are now ready to compute the extremal function

$F(\epsilon_{I})=\mathds{V}^{(3)}(\lambda_{A}(\epsilon_{I}),\lambda_{AB}(\epsilon_{I}),\epsilon_{I})\,.$
(57)

The dual field theory is supposed to be the two-dimensional SCFT obtained
compactifying on $\mathbb{M}_{4}$ the $(2,0)$ theory living on a stack of $N$
M5 branes. The gravitational extremization problem should correspond to
$c$-extremization in the dual two-dimensional SCFT.

Using relations (41) and (43) we find

$\displaystyle
F=\sum_{a}\frac{B^{(3)}_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(58) $\displaystyle
B^{(3)}_{a}=\frac{1}{6}(-\overline{\nu}_{M5}\,N)^{3}\,\Big{(}\epsilon_{4}-\epsilon_{3}+(1-\mathfrak{t}_{a})\epsilon_{1}^{a}+(1-\mathfrak{t}_{a+1})\epsilon_{2}^{a}\Big{)}^{2}\Big{(}-\epsilon_{4}+\mathfrak{t}_{a}\epsilon_{1}^{a}+\mathfrak{t}_{a+1}\epsilon_{2}^{a}\Big{)}^{2}\,,$

thus recovering the conjectured formula of Faedo:2022rqx  in terms of
gravitational blocks Hosseini:2019iad .121212We need to set
$\epsilon_{3}^{2}\overline{\nu}_{M5}^{3}=-6$. The convention for the sign of
the free energy in Faedo:2022rqx  is the opposite of ours.

To make contact with the dual field theory, we can also write our result in
terms of an integral of equivariant forms over the base $\mathbb{M}_{4}$ as
follows:

$\displaystyle F$
$\displaystyle=-\frac{1}{6}\,\overline{\nu}_{M5}^{3}\,N^{3}\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}^{2}\,\mathcal{C}_{d+2}^{2}$
(59)
$\displaystyle=-\frac{1}{6}\,\overline{\nu}_{M5}^{3}\,N^{3}\int_{\mathbb{M}_{4}}\Big{(}\epsilon_{3}-\epsilon_{4}+\sum_{a}(1-\mathfrak{t}_{a})c_{1}{T}(L_{a})\Big{)}^{2}\Big{(}\epsilon_{4}+\sum_{a}\mathfrak{t}_{a}c_{1}{T}(L_{a})\Big{)}^{2}\>.$

This expression correctly reproduces the M5-brane anomaly polynomial
integrated over the four-dimensional orbifold $\mathbb{M}_{4}$ as computed in
Martelli:2023oqk .

A general comment that applies to all the examples in this paper is the
following. The free energy must be extremized with respect to all but one of
the parameters $\epsilon_{I}$ in order to find the critical point. The value
of the remaining parameter must be instead fixed by requiring the correct
scaling of the supercharge under the R-symmetry vector field $\xi$. In this
case, we extremize with respect $\epsilon_{4}$, $\epsilon_{1}$ and
$\epsilon_{2}$ with $\epsilon_{3}$ fixed to a canonical value.131313Notice
that (LABEL:AdS3xM8_prescription_result0) is homogeneous of degree two in the
parameters $\epsilon_{I}$, so it makes no sense to extremize with respect to
all parameters.

Let us briefly review the comparison with field theory, referring to
Martelli:2023oqk  for details. The anomaly polynomial of the 2d SCFT is
obtained by integrating the eight-form anomaly polynomial of the six-
dimensional theory over $\mathbb{M}_{4}$, which, at large $N$, gives

${\cal A}_{2{\rm d}}=\int_{\mathbb{M}_{4}}{\cal A}_{6{\rm
d}}=\frac{N^{3}}{24}\int_{\mathbb{M}_{4}}c_{1}(F_{1})^{2}c_{1}(F_{2})^{2}\,,$
(60)

where $F_{I}$ are the generators of the $U(1)\times U(1)\subset SO(5)_{R}$
Cartan subgroup of the $(2,0)$ theory R-symmetry. The $c_{1}(F_{I})$ can be
decomposed as

$c_{1}(F_{I})=\Delta_{I}c_{1}(F_{R}^{2{\rm
d}})-\mathfrak{p}_{I}^{a}\Big{(}c_{1}({L}_{a})+2\pi\mu_{a}^{i}c_{1}({\cal
J}_{i})\Big{)}\,,$ (61)

where $F_{R}^{2{\rm d}}$, ${\cal J}_{1},{\cal J}_{2}$ are line bundles
associated with the $2{\rm d}$ R-symmetry and the two global symmetries coming
from the isometries of $\mathbb{M}_{4}$. They correspond to background fields
for the two-dimensional theory with no legs along $\mathbb{M}_{4}$.
Substituting (61) in (60) and setting $c_{1}({\cal
J}_{i})=\epsilon_{i}c_{1}(F_{R}^{2{\rm d}})$, leads to the equivariant
integral

${\cal A}_{2{\rm d}}=\frac{c_{r}}{6}c_{1}(F_{R}^{2{\rm
d}})^{2}=\frac{N^{3}}{24}c_{1}(F_{R}^{2{\rm
d}})^{2}\int_{\mathbb{M}_{4}}(\Delta_{1}-\mathfrak{p}_{1}^{a}c_{1}^{\mathbb{T}}(L_{a}))^{2}(\Delta_{2}-\mathfrak{p}_{2}^{a}c_{1}^{\mathbb{T}}(L_{a}))^{2}\,.$
(62)

Preserving supersymmetry with a twist requires
$c_{1}(F_{1})+c_{1}(F_{2})=2c_{1}(F_{R}^{2{\rm d}})-\sum_{a}c_{1}(L_{a})$
which gives Martelli:2023oqk

$\Delta_{1}+\Delta_{2}=2+\det(W,\epsilon)\,,\qquad\mathfrak{p}_{1}^{a}+\mathfrak{p}_{2}^{a}=1+\det(W,v^{a})\,,$
(63)

where $\epsilon=(\epsilon_{1},\epsilon_{2})$ and $W\in\mathbb{R}^{2}$ is a
two-dimensional constant vector.141414$W$ can be gauged away, see
Martelli:2023oqk . The two-dimensional central charge $c_{r}$ is extracted
from (62) and should be extremized with respect to $\epsilon_{i}$ and
$\Delta_{I}$ subject to the previous constraint. We then see that the
extremization of the gravitational free-energy is equivalent to
$c$-extremization under the identifications

$\Delta_{1}=\epsilon_{4}\,,\qquad\Delta_{2}=\epsilon_{3}-\epsilon_{4}\,,\qquad\mathfrak{p}_{1}^{a}=\mathfrak{t}_{a}\,,\qquad\mathfrak{p}_{2}^{a}=1-\mathfrak{t}_{a}\,,\qquad
W=0\,.$ (64)

The condition $\epsilon_{3}=2$ is enforced by the correct scaling of the
supersymmetry spinor, in analogy to Martelli:2005tp ; Gauntlett:2018dpc . The
case of anti-twist is similar and can be discussed by taking a non convex fan
for $\mathbb{M}_{4}$. The most general supersymmetry condition is now
$c_{1}(F_{1})+c_{1}(F_{2})=2c_{1}(F_{R}^{2{\rm
d}})-\sum_{a}\sigma_{a}c_{1}(L_{a})$ where $\sigma_{a}=\pm 1$ as discussed in
Faedo:2022rqx  and requires

$\Delta_{1}+\Delta_{2}=2+\det(W,\epsilon)\,,\qquad\mathfrak{p}_{1}^{a}+\mathfrak{p}_{2}^{a}=\sigma_{a}+\det(W,v^{a})\,.$
(65)

This case can be just obtained by formally sending
$v^{a}\rightarrow\sigma^{a}v^{a}$ everywhere, implying
$\epsilon_{1}^{a}\rightarrow\sigma^{a}\epsilon_{1}^{a}$ and
$\epsilon_{2}^{a}\rightarrow\sigma^{a+1}\epsilon_{2}^{a}$.

###  3.2 AdS${}_{5}\times M_{6}$ solutions

In this section we consider a generalization of the family of M theory
solutions found in Gauntlett:2004zh  and further studied in Gauntlett:2006ai .
Their geometry is AdS${}_{5}\times M_{6}$ where $M_{6}$ is a manifold obtained
as a $\mathbb{P}^{1}$ bundle over a four-dimensional compact manifold $B_{4}$,
that can be either a Kähler-Einstein manifold ($B_{4}=\mathrm{KE}_{4}$) or the
product of two KE2 ($B_{4}=\Sigma_{1}\times\Sigma_{2})$. The bundle is the
projectivization of the canonical bundle over $B_{4}$, $\mathbb{P}(K\oplus\cal
O)$. Here we consider the case where $B_{4}$ is replaced by a generic four-
dimensional toric orbifold $\mathbb{M}_{4}$. Notice that generically $M_{6}$
can be an orbifold,151515Using our formalism, we could easily study the case
that $M_{6}$ is a generic toric six-dimensional orbifold. It would be
interesting to understand what kinds of orbifold admit an holographic
interpretation. like in the solutions discussed of Ferrero:2021wvk . In
addition to recovering the gravitational central charges of the existing
solutions, we give a prediction for these more general backgrounds that are
still to be found. These solutions are potentially interpreted as M5 branes
wrapped over a two-cycle in $\mathbb{M}_{4}$ (see for example Bah:2019rgq ;
Bah:2019vmq ).

The topological structure of the underlying geometry can be encoded in the fan

$\displaystyle V^{a}=(v^{a},1)\,,\qquad V^{d+1}=(0,0,1)\,,\qquad
V^{d+2}=(0,0,-1)\ \,,\qquad a=1,\ldots d\,\,,$ (66)

where $v^{a}$ are the two-dimensional vectors in the fan of $\mathbb{M}_{4}$.
We will use a capital index $A$ to run over $a=1,\ldots,d$, $d+1$ and $d+2$.
That this is the right geometry can be seen looking at the symplectic
reduction presentation $\mathbb{C}^{d+2}//G$ of $M_{6}$. Here $G$ is the
subgroup of the torus $\mathbb{T}^{d+2}=U(1)^{d+2}$ generated by the GLSM
charges

$Q_{A}^{k}V^{A}_{I}=0\,,\qquad\qquad k=1,\ldots d-1\,.$ (67)

We can choose the following basis of GLSM charges

$(q_{a}^{p},-\sum_{a}q_{a}^{p},0)\,,\qquad(0,\ldots,0,1,1)\,,$ (68)

where $q_{a}^{p}$ are the $d-2$ charges for $\mathbb{M}_{4}$,
$\sum_{a}q_{a}^{p}v^{a}_{i}=0$. The first $d-2$ vectors define the canonical
bundle $K$ of $\mathbb{M}_{4}$ with an extra copy of $\mathbb{C}$. The final
charge vector projectivizes it and gives indeed the geometry we are interested
in:

$\mathbb{P}(K\oplus\cal O)\,.$ (69)

We need also to specify the integer fluxes of the M theory four-form along all
the non-trivial four-cycles. There are $d+2$ toric four-cycles in the
geometry, associated with the divisors $D_{A}$. The divisors $D_{a}$ are
$\mathbb{P}^{1}$ fibrations over the toric two-cycles
$\Sigma_{a}\subset\mathbb{M}_{4}$, while $D_{d+1}$ and $D_{d+2}$ are copies of
$\mathbb{M}_{4}$ sitting at the North and South pole of $\mathbb{P}^{1}$,
respectively. All together, they define a vector of fluxes $M_{A}$. The
relations $\sum_{A}V^{A}_{I}D_{A}=0$ imply that not all toric divisors are
inequivalent and that the vector of fluxes satisfies

$\sum_{A}V^{A}_{I}M_{A}=0\,.$ (70)

Since we are dealing with M5 branes in M theory, we define the free-energy as
in section 3.1

$F=\mathds{V}^{(3)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\,,$ (71)

and, since now we have a vector of fluxes, we impose the flux constraints

$\nu_{M5}\,M_{A}=-\frac{\partial}{\partial\lambda_{A}}\mathds{V}^{(2)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\,.$
(72)

Differently from the case discussed in the previous section, for these
geometries there is no general field theory result for the central charge of
the dual four-dimensional SCFTs. Our results here can therefore be seen as a
prediction for the general form of the off-shell central charge, which
presumably can be obtained by integrating the M5 brane anomaly polynomial on a
suitable two-cycle inside $M_{6}$, or using the method of Bah:2019rgq . In
order to compare with the existing literature, we will therefore consider in
some detail a number of explicit examples of $\mathbb{M}_{4}$, including KE4
and $\Sigma_{1}\times\Sigma_{2}$, but also other examples for which there is
known supergravity solution, nor field-theoretic understanding. The equations
to be solved in the extremization problem typically lead to finding the zeroes
of symultanous polynomials of high degree and are therefore not manageable.
For this reason, we will proceed by making different technical assumptions to
simplify the algebra. One such general assumption is the existence of a
$\mathbb{Z}_{2}$ symmetry acting on the $\mathbb{P}^{1}$ fiber, as we discuss
below. Furthermore, all the examples that we discuss will comprise four fixed
points, and occasionally we will restrict to non-generic fluxes in order to
simplify the otherwise unwieldy expressions.

If we restrict to a class of geometry with a $\mathbb{Z}_{2}$ symmetry that
exchanges the North and South pole of $\mathbb{P}^{1}$, we can consider a
simplified geometry obtained by cutting $\mathbb{P}^{1}$ into half. We thus
obtain a non-compact Calabi-Yau geometry given by the canonical bundle over
$\mathbb{M}_{4}$. The corresponding fan is obtained by dropping $V^{d+2}$:

$\displaystyle V^{a}=(v^{a},1)\,,\qquad V^{d+1}=(0,0,1)\ \,,\qquad a=1,\ldots
d\,.$ (73)

Notice that this is a (partial) resolution of a CY3 cone where $V^{d+1}$ is
associated with a compact divisor. Supergravity solutions with
$\mathbb{Z}_{2}$ symmetry have been considered in Gauntlett:2004zh ;
Gauntlett:2006ai  where they correspond to set the parameter called $c$ to
zero. Effectively, the $\mathbb{Z}_{2}$ symmetry reduces by one the number of
independent fluxes we can turn on, thus simplifying the calculations. Notice
that the on-shell equivariant volume $\mathds{V}$ for the half-geometry is
half of the one for the total geometry. The relation between the parameters to
use in the two cases, in order to have the same normalization for the free
energy, is the following

$\overline{\nu}_{M5}=2^{-2/3}\nu_{M5}\,,$ (74)

where $\overline{\nu}_{M5}$ is the correct one for half-geometries.

Notice that we introduced single and double times in (72). We can immediately
understand the need for higher times. In a compact geometry
$\mathds{V}^{(2)}(\lambda_{A})$ with only single times would vanish
identically.161616For a compact geometry
$\mathds{V}^{(2)}(\lambda_{A})=-\frac{1}{2}\sum_{AB}\lambda_{A}\lambda_{B}\int_{M_{6}}c_{1}{T}(L_{A})c_{1}{T}(L_{B})=0$
since it is the integral of a four-form at most on a six-dimensional manifold.
In the non-compact case, this condition is evaded and
$\mathds{V}^{(2)}(\lambda_{A})$ is a rational function of $\epsilon_{I}$. See
Martelli:2023oqk  for details. As we will discuss later, the double times are
generically necessary also when imposing the $\mathbb{Z}_{2}$ symmetry in
order to have enough parameters to solve the equations.171717In the case of
compactification on a spindle they are not necessary Martelli:2023oqk .

####  3.2.1 Geometries with $\mathbb{Z}_{2}$ symmetry

We consider first geometries with $\mathbb{Z}_{2}$ symmetry. Cutting $M_{6}$
into half we consider the non-compact CY3 specified by the fan (73). The $I=3$
condition in (70) gives

$M_{d+1}=-\sum_{a}M_{a}\,,$ (75)

thus fixing the flux along $\mathbb{M}_{4}$ in terms of the other fluxes. The
$I=1,2$ conditions in (70) give two linear relations among the $M_{a}$,
leaving a total number $d-2$ of independent fluxes. Notice that geometries
without $\mathbb{Z}_{2}$ symmetry have one additional independent flux, as we
discuss later.

The fan is the union of $d$ cones $(V^{a},V^{a+1},V^{d+1})$ and we see that
the number of fixed points is the same of that of the base $\mathbb{M}_{4}$.
It is then easy to write the equivariant volume with higher times as a sum
over the fixed points of $\mathbb{M}_{4}$

$\mathds{V}=\sum_{a}\frac{{\mathrm{e}}^{\tau_{a}}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}\,,$
(76)

where

$\tau_{a}=\left(\sum_{B}\lambda_{B}c_{1}^{\mathbb{T}_{3}}(L_{B})+\sum_{B,C}\lambda_{BC}c_{1}^{\mathbb{T}_{3}}(L_{B})c_{1}^{\mathbb{T}_{3}}(L_{C})\right)\Big{|}_{a}\,$
(77)

or, more explicitly,

$\displaystyle\tau_{a}=$
$\displaystyle-\lambda_{a}\epsilon_{1}^{a}-\lambda_{a+1}\epsilon_{2}^{a}-\lambda_{d+1}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})+\lambda_{aa}(\epsilon_{1}^{a})^{2}+2\lambda_{a,a+1}(\epsilon_{1}^{a})\epsilon_{2}^{a}+\lambda_{a+1,a+1}(\epsilon_{2}^{a})^{2}+$
(78) $\displaystyle
2(\lambda_{a,d+1}\epsilon_{1}^{a}+\lambda_{a+1,d+1}\epsilon_{2}^{a})(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})+\lambda_{d+1,d+1}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})^{2}\,.$

Notice that the equations (72) are not solvable with only single times.
$M_{d+1}=-\sum_{a}M_{a}\neq 0$ while, for $\lambda_{AB}=0$,

$\displaystyle-\frac{\partial\mathds{V}^{(2)}}{\partial\lambda_{d+1}}$
$\displaystyle=\sum_{a}\frac{(-\lambda_{a}\epsilon_{1}^{a}-\lambda_{a+1}\epsilon_{2}^{a}-\lambda_{d+1}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2}))}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}}$
(79)
$\displaystyle=\int_{\mathbb{M}_{4}}\left(\sum_{a}\lambda_{a}c_{1}^{\mathbb{T}}(L_{a})-\lambda_{d+1}(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))\right)=0\,,$

being the integral of a two-form at most.

The equations (72) explicitly read

$\displaystyle(I)$
$\displaystyle\overline{\nu}_{M5}M_{a}=\frac{\epsilon_{1}^{a}\tau_{a}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}+\frac{\epsilon_{2}^{a-1}\tau_{a-1}}{d_{a-1,a}\epsilon^{a-1}_{1}\epsilon^{a-1}_{2}(\epsilon_{3}-\epsilon^{a-1}_{1}-\epsilon^{a-2}_{2})}$
(80) $\displaystyle(II)$
$\displaystyle-\overline{\nu}_{M5}\sum_{a}M_{a}=\sum_{a}\frac{\tau_{a}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}}$

These equations are not independent. In particular, $(II)$ follows from
$(I)$.181818Using $\sum_{a}v^{a}M_{a}=0$ and the vector identity
$v^{a}_{i}\epsilon^{a}_{1}+v^{a+1}_{i}\epsilon^{a}_{2}=\epsilon_{i}$, one
derives
$\sum_{a}\frac{\tau_{a}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}=0$
from $(I)$. Then, summing over $a$ in $(I)$, and using the previous identity:
$\overline{\nu}_{M5}\sum_{a}M_{a}=\sum_{a}\frac{(\epsilon^{a}_{1}+\epsilon^{a}_{2})\tau_{a}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}=-\sum_{a}\frac{\tau_{a}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}}\,.$
(81) The equations $(I)$ can be written as

$B_{a-1}^{(1)}-B_{a}^{(1)}=d_{a,a+1}\epsilon_{2}^{a}\overline{\nu}_{M5}M_{a}\,,\qquad\qquad
B_{a}^{(1)}=-\frac{\tau_{a}}{\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2}}\,.$
(82)

It is then clear that these equations can be solved for $\tau_{a}$, but one
“time”, say $\tau_{1}$, remains undetermined. Our prescription is to extremize
the free energy with respect to all parameters that are left undetermined
after imposing the flux constraints. In this case then we extremize

$\mathds{V}^{(3)}(\epsilon_{i},\tau_{1})$ (83)

with respect to $\epsilon_{1}$, $\epsilon_{2}$ and $\tau_{1}$, with
$\epsilon_{3}$ set to some canonical value, fixed by the scaling of the
supercharge under the R-symmetry vector field. In the next subsection we will
parameterize the free energy in a more convenient way.

####  3.2.2 The extremal function for geometries with $\mathbb{Z}_{2}$
symmetry

We can write the general form of the extremal function for geometries with
$\mathbb{Z}_{2}$ symmetry. Let us define

$\tau^{\mathbb{T}_{3}}_{CY_{3}}=\sum_{B}\lambda_{B}c_{1}^{\mathbb{T}_{3}}(L_{B})+\sum_{B,C}\lambda_{BC}c_{1}^{\mathbb{T}_{3}}(L_{B})c_{1}^{\mathbb{T}_{3}}(L_{C})\,,$
(84)

the equivariant form with restriction $\tau_{a}$ at the fixed points. By
restricting the form to $\mathbb{M}_{4}$ and considering $\epsilon_{3}$ as a
parameter, we obtain

$\displaystyle\tau{T}=\sum_{A}\lambda_{A}\>\mathcal{C}_{A}+\sum_{A,B}\lambda_{AB}\>\mathcal{C}_{A}\>\mathcal{C}_{B}\>,$
(85) $\displaystyle\mathcal{C}_{a}\,=c_{1}{T}(L_{a})\>,\qquad a=1,\ldots,d\>,$
$\displaystyle\mathcal{C}_{d+1}\,=\,-(\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a}))\>,$

where $c_{1}^{\mathbb{T}}(L_{a})$ are the restrictions of the line bundles
$L_{a}$ to the base $\mathbb{M}_{4}$ and $\mathbb{T}$ is the two-dimensional
torus spanned by $\epsilon_{1}$ and $\epsilon_{2}$. From now on, unless
explicitly said, all classes will refer to the base $\mathbb{M}_{4}$. In terms
of $\tau{T}$ the quadratic piece of the equivariant volume can be written as

$\mathds{V}^{(2)}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\frac{1}{2}\int_{\mathbb{M}_{4}}\frac{(\tau^{\mathbb{T}})^{2}}{(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))}\,.$
(86)

The flux constraints (72) give

$\displaystyle(I)$
$\displaystyle-\overline{\nu}_{M5}M_{a}=\int_{\mathbb{M}_{4}}\frac{c_{1}^{\mathbb{T}}(L_{a})\,\tau^{\mathbb{T}}}{(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))}\,,$
(87) $\displaystyle(II)$
$\displaystyle-\overline{\nu}_{M5}\sum_{a}M_{a}=\int_{\mathbb{M}_{4}}\tau^{\mathbb{T}}\,.$

For a generic fan, using the gauge transformations (19) and (20) we can set
all $\lambda_{a}=\lambda_{a,a}=\lambda_{a,a+1}=0$.191919For special symmetric
fans, like $\mathbb{P}^{2}$ and $\mathbb{P}^{1}\times\mathbb{P}^{1}$, and
other simple example with low $d$, one of the single times $\lambda_{a}$
remains unfixed. However, from the combined conditions $(I)$+$(II)$ we obtain
$\sum_{ab}\frac{D_{ab}\lambda_{a}}{\epsilon_{3}-\epsilon_{1}^{a}-\epsilon_{2}^{a}}=0$
which implies that the remaining single time must vanish. We will show more
formally in appendix A that $\mathds{V}^{(3)}$ has a critical point at
$\lambda_{a}=\lambda_{a,a}=\lambda_{a,a+1}=0$. Then condition $(I)$ becomes

$\displaystyle-\overline{\nu}_{M5}M_{a}$
$\displaystyle=\int_{\mathbb{M}_{4}}c_{1}^{\mathbb{T}}(L_{a})(-\lambda_{d+1}-2\sum_{b}\lambda_{b,d+1}c_{1}^{\mathbb{T}}(L_{b})+\lambda_{d+1,d+1}(\epsilon_{3}+\sum_{b}c_{1}^{\mathbb{T}}(L_{b})))$
(88) $\displaystyle=\sum_{b}D_{ab}(-2\lambda_{b,d+1}+\lambda_{d+1,d+1})\,.$

We can similarly compute $(II)$ as an integral

$-\overline{\nu}_{M5}\sum_{a}M_{a}=\int_{\mathbb{M}_{4}}\tau^{\mathbb{T}}=\sum_{a,b}D_{ab}(-2\lambda_{b,d+1}+\lambda_{d+1,d+1})\,,$
(89)

and see that it is automatically satisfied if $(I)$ is. Since
$v^{a}_{i}M_{a}=0$, the flux constraints fix the $\lambda_{b,d+1}$ only up to
the ambiguities

$\displaystyle\lambda_{a,d+1}\rightarrow\lambda_{a,d+1}+\sum_{i=1}^{2}\delta_{i}v^{a}_{i}+\gamma\,,$
(90) $\displaystyle\lambda_{d+1,d+1}\rightarrow\lambda_{d+1,d+1}+2\gamma\,,$

where $\delta_{i}$ and $\gamma$ are free parameters. However, these free
parameters can be all reabsorbed in a redefinition of

$\lambda_{d+1}\rightarrow\lambda_{d+1}+2\gamma\epsilon_{3}+2\sum_{i=1}^{2}\delta_{i}\epsilon_{i}\,,$
(91)

since $v^{a}_{i}c_{1}^{\mathbb{T}}(L_{a})=-\epsilon_{i}$ and they are not
really independent.

The free energy is then given by

$\mathds{V}^{(3)}=\frac{1}{6}\int_{\mathbb{M}_{4}}\frac{(\tau^{\mathbb{T}})^{3}}{(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))}\,,$
(92)

which explicitly gives

$\mathds{V}^{(3)}=\frac{1}{6}\int_{\mathbb{M}_{4}}(\epsilon_{3}+\sum_{a}c_{1}^{\mathbb{T}}(L_{a}))^{2}(\bar{\lambda}_{d+1}+\sum_{b}\bar{\lambda}_{b,d+1}c_{1}^{\mathbb{T}}(L_{b}))^{3}\,,$
(93)

where we defined

$\bar{\lambda}_{d+1}=-\lambda_{d+1}+\lambda_{d+1,d+1}\epsilon_{3}\,\,\qquad\bar{\lambda}_{a,d+1}=-2\lambda_{a,d+1}+\lambda_{d+1,d+1}\,,$
(94)

which are subject to the constraints

$-\overline{\nu}_{M5}M_{a}=\sum_{b}D_{ab}\bar{\lambda}_{b,d+1}\,.$ (95)

Substituting the solution of the flux constraints, $\mathds{V}^{(3)}$ becomes
a function of $\epsilon_{i}$ and the extra parameter $\bar{\lambda}_{d+1}$.
Indeed, as we have seen, the ambiguites (LABEL:amb) can be reabsorbed in a
redefinition of $\bar{\lambda}_{d+1}$. A direct evaluation gives

$\displaystyle
6\mathds{V}^{(3)}=\bar{\lambda}_{d+1}^{3}\sum_{ab}D_{ab}+\bar{\lambda}_{d+1}^{2}\Bigl{(}6\epsilon_{3}\sum_{ab}D_{ab}\bar{\lambda}_{a,d+1}+3\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}\Bigr{)}$
$\displaystyle\quad+3\bar{\lambda}_{d+1}\Bigl{(}\epsilon_{3}^{2}\sum_{ab}D_{ab}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}+2\epsilon_{3}\\!\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}+\sum_{abcd}D_{abcd}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}\Bigr{)}$
$\displaystyle\quad+\Bigl{(}\epsilon_{3}^{2}\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}\bar{\lambda}_{c,d+1}+2\epsilon_{3}\sum_{abcd}D_{abcd}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}\bar{\lambda}_{c,d+1}$
$\displaystyle\quad+\sum_{abcde}D_{abcde}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}\bar{\lambda}_{c,d+1}\Bigr{)}\,,$
(96)

where the generalized intersection numbers are defined by

$D_{a_{1}\ldots
a_{p}}=\int_{\mathbb{M}_{4}}c_{1}^{\mathbb{T}}(L_{a_{1}})\ldots
c_{1}^{\mathbb{T}}(L_{a_{p}})\,.$ (97)

Notice that $D_{ab}$ is $\epsilon$-independent, while $D_{a_{1}\cdots a_{p}}$
is a homogeneous function of degree $p-2$ in $\epsilon_{1}$ and
$\epsilon_{2}$. $\mathds{V}^{(3)}$ need to be extremized with respect to
$\epsilon_{1}$, $\epsilon_{2}$ and $\bar{\lambda}_{d+1}$, with $\epsilon_{3}$
set to the canonical value.

The critical point is generically at a non-zero value of $\epsilon_{1}$ and
$\epsilon_{2}$. We can expect a critical point202020In the opposite direction,
of course one would have as critical point $\epsilon_{1}=\epsilon_{2}=0$ if
the base $B_{4}$ has _no_ continous symmetries. This is the case for examples
for del Pezzo surfaces dPk with $k>3$, which we do not treat here. This would
lead one to suspect that all KE4 have $\epsilon_{1}=\epsilon_{2}=0$ as
critical point, but this is actually incorrect, as the example of the toric
dP3 will show. at $\epsilon_{1}=\epsilon_{2}=0$ only if the background and the
fluxes have some extra symmetry, as for examples in the cases where all $U(1)$
isometries are enhanced to a non-abelian group. In these particular cases, we
can further simplify the expression

$\displaystyle 6\mathds{V}^{(3)}$
$\displaystyle=\bar{\lambda}_{d+1}^{3}\sum_{ab}D_{ab}+6\epsilon_{3}\bar{\lambda}_{d+1}^{2}\sum_{ab}D_{ab}\bar{\lambda}_{a,d+1}+3\epsilon_{3}^{2}\bar{\lambda}_{d+1}\sum_{ab}D_{ab}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}+O(\epsilon_{i}^{2})$
(98)
$\displaystyle=\bar{\lambda}_{d+1}^{3}\sum_{ab}D_{ab}-6\overline{\nu}_{M5}\sum_{a}M_{a}\epsilon_{3}\bar{\lambda}_{d+1}^{2}+3\epsilon_{3}^{2}\bar{\lambda}_{d+1}\sum_{ab}D_{ab}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}+O(\epsilon_{i}^{2})\,,$

and extremize it with respect to $\bar{\lambda}_{d+1}$.

As a check of our expression, we can reproduce the central charge of the
existing solutions with Kälher-Einstein metrics and fluxes all equal
Gauntlett:2006ai . The only toric four-manifolds that are also KE are
$\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3, with fans

$\displaystyle\mathbb{P}^{2}:\,\,\,\,v^{1}=(1,1)\,,\,v^{2}=(-1,0)\,,\,v^{3}=(0,-1)\,,$
(99)
$\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{1}:\,\,v^{1}=(1,0)\,,\,v^{2}=(0,1)\,,\,v^{3}=(-1,0)\,,\,v^{4}=(0,-1)\,,$
$\displaystyle\text{dP}_{3}:\,v^{1}=(1,0)\,,\,v^{2}=(1,1)\,,\,v^{3}=(0,1)\,,\,v^{4}=(-1,0)\,,\,v^{5}=(-1,-1)\,,\,v^{6}=(0,-1)\,,$

and intersection matrices

$\displaystyle\mathbb{P}^{2}:\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,D_{ab}=1\,,$ (100)
$\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{1}:\,D_{a,b}=1\,\text{if}\,|a-b|=1(\text{mod}\,2)\,\text{and\,
zero\, otherwise}\,,$
$\displaystyle\text{dP}_{3}:\,\,\,\,\,\,\,\,\,\,\,D_{aa}=-1\,,D_{a,a\pm
1}=1\,\text{and\, zero\, otherwise}\,,$

where the indices are cyclically identified. To compare with the KE4
solutions, we set all $M_{a}\equiv N$. We can then choose all
$\bar{\lambda}_{a,d+1}$ equal and we find

$\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}=\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}=\sum_{abc}D_{abc}\bar{\lambda}_{a,d+1}\bar{\lambda}_{b,d+1}\bar{\lambda}_{c,d+1}=0\,,$
(101)

thus ensuring that the linear terms in $\epsilon_{1}$ and $\epsilon_{2}$ in
$\mathds{V}^{(3)}$ vanish, and that there is indeed a critical point at
$\epsilon_{1}=\epsilon_{2}=0$. Extremizing (98) we get

$\mathds{V}^{(3)}=\epsilon_{3}^{3}\overline{\nu}_{M5}^{3}(-5+3\sqrt{3})N^{3}\\{\frac{1}{9},\frac{1}{3},2\\}\,,$
(102)

for $\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3,
respectively, which agrees with (2.16) in Gauntlett:2006ai  for
$\epsilon_{3}\overline{\nu}_{M5}=3$.212121$N_{C_{N}}$ in Gauntlett:2006ai  can
be identified with $M_{d+1}=-\sum M_{a}$, so that
$N_{there}=-hN_{C_{N}}/M=h\sum M_{a}/M$ where $(h,M)$ are defined in
Gauntlett:2006ai  and they have value $(3,9),(4,8)$ and $(2,6)$ for
$\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3, respectively.

For $\mathbb{P}^{1}\times\mathbb{P}^{1}$ we can introduce a second flux. The
general solution to $\sum_{A}V^{A}_{I}M_{A}=0$ is indeed

$M_{A}=(N_{1},N_{2},N_{1},N_{2},-2N_{1}-2N_{2})\,.$ (103)

The background has an expected $SU(2)\times SU(2)$ symmetry that it is
realized in the supergravity solution Gauntlett:2006ai , that now is not in
the KE class. Using the gauge transformation (19) we can set
$\bar{\lambda}_{a+2,d+1}=\bar{\lambda}_{a,d+1}$. A simple computation then
shows that the free energy extremized has a critical point in
$\epsilon_{1}=\epsilon_{2}=0$, consistently with the non-abelian isometry of
the solution, with critical value

$\mathds{V}^{(3)}=\frac{\epsilon_{3}^{3}\overline{\nu}_{M5}^{3}}{6}\left(2(N_{1}^{2}+N_{1}N_{2}+N_{2}^{2})^{3/2}-(2N_{1}^{3}+3N_{1}^{2}N_{2}+3N_{1}N_{2}^{2}+2N_{2}^{3})\right)\,,$
(104)

which should be compared with (2.29) in Gauntlett:2006ai  with $N_{1}=pN$ and
$N_{2}=qN$. This looks superficially different, but it can be rewritten in the
form above (_cf_. for example (F.14) in Bah:2019rgq ).

####  3.2.3 Examples of geometries with non-zero critical $\epsilon$

So far, in all the explicit examples we have discussed we found that
$\epsilon_{1}=\epsilon_{2}=0$ is a critical point. However, we have already
pointed out that for generic toric $\mathbb{M}_{4}$ and/or with generic fluxes
this will not be the case. In this subsection we will investigate situations
in which at least one of $\epsilon_{1},\epsilon_{2}$ is different from zero at
the critical point, by considering geometries with $SU(2)\times U(1)$
symmetry, as well as the case of dP3 with generic fluxes. Interestingly, it
turns out that for dP3 there exist two special configurations of fluxes
(different from the case where they are all equal) where the critical point is
again $\epsilon_{1}=\epsilon_{2}=0$, but the corresponding supergravity
solutions are not known. For four independent generic fluxes, instead,
$\epsilon_{1}=\epsilon_{2}=0$ is not a critical point.

#### dP3 with unequal fluxes

The symmetry of dP3 is just $U(1)\times U(1)$ and the existence of the
critical point $\epsilon_{1}=\epsilon_{2}=0$ of the extremization problem is
not obviously implied by the fact that there exist a KE metric on dP3. In the
basis of the fan as in (LABEL:fanKE), the general assignment of fluxes
compatible with $\sum_{A}V_{I}^{A}M_{A}=0$ can be parameterized as

$\displaystyle
M_{A}=(N_{1},N_{2},N_{3},N_{4},N_{5},N_{6},-2N_{1}-3N_{2}-2N_{3}+N_{5})\,,$
(105)

where we choose $N_{1},N_{2},N_{3},N_{5}$ as independent, with
$N_{4}=N_{1}+N_{2}-N_{5}$ and $N_{6}=N_{2}+N_{3}-N_{5}$. Upon setting
$\lambda_{a}=\lambda_{a,a}=\lambda_{a,a+1}=0$ using the gauge freedom, as
discussed before, the constraint (95) on $\bar{\lambda}_{a,d+1}$ can be solved
by taking, for example

$\displaystyle\bar{\lambda}_{1,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{2}+N_{3}}{2}\,,\qquad$
$\displaystyle\bar{\lambda}_{2,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{3}+N_{1}}{2}\,,$ (106)
$\displaystyle\bar{\lambda}_{3,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{1}+N_{2}}{2}\,,\qquad$
$\displaystyle\bar{\lambda}_{4,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{2}+N_{3}}{2}\,,$
$\displaystyle\bar{\lambda}_{5,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{3}+N_{1}+2(N_{2}-N_{5})}{2}\,,\qquad$
$\displaystyle\bar{\lambda}_{6,d+1}$
$\displaystyle=-\bar{\nu}_{M5}\frac{N_{1}+N_{2}}{2}\,.$

Writing out the free energy (3.2.2), up to linear order in
$\epsilon_{1},\epsilon_{2}$, we have

$\mathds{V}^{(3)}=\left.\mathds{V}^{(3)}\right|_{\epsilon_{i}=0}+\left.\partial_{\epsilon_{1}}\mathds{V}^{(3)}\right|_{\epsilon_{i}=0}\epsilon_{1}+\left.\partial_{\epsilon_{2}}\mathds{V}^{(3)}\right|_{\epsilon_{i}=0}\epsilon_{2}+O(\epsilon_{i}^{2})$
(107)

where the constant term is not particularly interesting and

$\begin{split}\left.\partial_{\epsilon_{1}}\mathds{V}^{(3)}\right|_{\epsilon_{i}=0}&=\bar{\nu}_{M5}\frac{N_{5}-N_{2}}{2}\Bigl{[}6\bar{\lambda}_{d+1}^{2}-12(N_{2}+N_{3})\bar{\nu}_{M5}\epsilon_{3}\bar{\lambda}_{d+1}\\\
&+\bigl{(}N_{2}^{2}-2N_{5}^{2}+3N_{3}(N_{2}+N_{5})+N_{5}N_{2}\bigr{)}\bar{\nu}_{M5}^{2}\epsilon_{3}^{2}\Bigr{]}\,,\\\
\left.\partial_{\epsilon_{2}}\mathds{V}^{(3)}\right|_{\epsilon_{i}=0}&=\bar{\nu}_{M5}\frac{N_{5}-N_{2}}{2}\Bigl{[}6\bar{\lambda}_{d+1}^{2}-12(N_{2}+N_{1})\bar{\nu}_{M5}\epsilon_{3}\bar{\lambda}_{d+1}\\\
&+\bigl{(}N_{2}^{2}-2N_{5}^{2}+3N_{1}(N_{2}+N_{5})+N_{5}N_{2}\bigr{)}\bar{\nu}_{M5}^{2}\epsilon_{3}^{2}\Bigr{]}\,.\end{split}$
(108)

We see that for generic values of the fluxes the expressions above cannot be
zero simultaneously, implying that $\epsilon_{i}=0$ is not a critical point of
the extremization. The complete extremization equations are unwieldy, so in
the following we will instead concentrate on two special configurations of
fluxes, with enhanced symmetry, for which $\epsilon_{1}=\epsilon_{2}=0$ turns
out to be a critical point.

The first special value of fluxes is clearly obtained for $N_{5}=N_{2}$, that
leaves three fluxes $N_{1},N_{2},N_{3}$ free. In this case the parameters
(106) acquire the cyclic symmetry
$\bar{\lambda}_{a,d+1}=\bar{\lambda}_{a+3,d+1}$, analogously to the
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ discussed in the previous section and
indeed the linear terms in $\mathds{V}^{(3)}$ manifestly vanish, so that
$\epsilon_{i}=0$ is a critical point. The fluxes display an enhanced symmetry:

$\displaystyle
M_{A}=(N_{1},N_{2},N_{3},N_{1},N_{2},N_{3},-2N_{1}-2N_{2}-2N_{3})\,.$ (109)

Extremizing $\mathbb{V}^{(3)}$ with respect to $\bar{\lambda}_{d+1}$ yields

$\begin{split}\bar{\lambda}_{d+1}^{*}&=\frac{2\bar{\nu}_{M5}\epsilon_{3}}{3}(N_{1}+N_{2}+N_{3})\\\
&-\frac{\bar{\nu}_{M5}\epsilon_{3}}{3}\sqrt{4(N_{1}^{2}+N_{2}^{2}+N_{3}^{2})+5(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{1})}\,,\end{split}$
(110)

and the corresponding value of the on-shell central charge is

$\begin{split}\mathds{V}^{(3)}&=\frac{2\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}}{27}\Bigl{[}\bigl{(}4(N_{1}^{2}+N_{2}^{2}+N_{3}^{2})+5(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{1})\bigr{)}^{3/2}\\\
&-(N_{1}+N_{2}+N_{3})\bigl{(}8(N_{1}^{2}+N_{2}^{2}+N_{3}^{2})+7(N_{1}N_{2}+N_{2}N_{3}+N_{3}N_{1})\bigr{)}\Bigr{]}\,.\end{split}$
(111)

It can be checked that this expression agrees precisely with the central
charge given in eq. (3.79) of BenettiGenolini:2023ndb  and it correctly
reduces to (102) upon setting $N_{1}=N_{2}=N_{3}=N$.

Notice that while the expression of $\bar{\lambda}_{d+1}^{*}$ depends on the
specific gauge chosen for the parameters $\bar{\lambda}_{a,d+1}$, the critical
values $\epsilon_{i}^{*}=0$ and the central charge (111) do not rely on this.

The second special value of fluxes that we found is $N_{1}=N_{3}=N_{5}$, which
implies $N_{a}=N_{a+2}$, so that the fluxes have again an enhanced symmetry:

$\displaystyle M_{A}=(N_{1},N_{2},N_{1},N_{2},N_{1},N_{2},-3N_{1}-3N_{2})\,.$
(112)

In this case, notice that the two expressions in (108) coincide, so that it is
possible that both linear terms vanish, for a particular value of
$\bar{\lambda}_{d+1}^{*}$, despite $N_{2}\neq N_{5}$. However, the parameters
in (106) do not enjoy this new symmetry, so it is better to look for a
different gauge, where the parameters respect the additional symmetry, namely
$\bar{\lambda}_{a,d+1}=\bar{\lambda}_{a+2,d+1}$. This can be achieved choosing

$\bar{\lambda}_{1,d+1}=-\bar{\nu}_{M5}\frac{N_{1}+2N_{2}}{3}\,,\qquad\bar{\lambda}_{2,d+1}=-\bar{\nu}_{M5}\frac{2N_{1}+N_{2}}{3}\,,$
(113)

and cyclic permutations. In this gauge, we can now check that
$\mathds{V}^{(3)}$ has no linear terms in $\epsilon_{1}$ and $\epsilon_{2}$.
Therefore, extremizing $\mathds{V}^{(3)}$ with respect to
$\bar{\lambda}_{d+1}$, $\epsilon_{1}$ and $\epsilon_{2}$, we obtain the
critical values $\epsilon_{1,2}^{*}=0$ and

$\bar{\lambda}_{d+1}^{*}=\frac{6(N_{1}+N_{2})-\sqrt{6(5N_{1}^{2}+8N_{1}N_{2}+5N_{2}^{2})}}{6}\,\bar{\nu}_{M5}\epsilon_{3}\,,$
(114)

and the corresponding value of the on-shell central charge is

$\mathbb{V}^{(3)}=\frac{\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}}{4}\Bigl{[}\frac{\bigl{(}6(5N_{1}^{2}+8N_{1}N_{2}+5N_{2}^{2})\bigr{)}^{3/2}}{27}-2(N_{1}+N_{2})\bigl{(}3N_{1}^{2}+4N_{1}N_{2}+3N_{2}^{2}\bigr{)}\Bigr{]}\,.$
(115)

It can be checked that this expression agrees precisely with the central
charge given in eq. (3.79) of BenettiGenolini:2023ndb  and it correctly
reduces to (102) upon setting $N_{1}=N_{2}=N$.

It would be interesting to construct explicit supergravity solutions
corresponding to the two special configurations of fluxes we found. If they
exist, they should lie outside the KE class considered in Gauntlett:2004zh .

#### $\mathbb{M}_{4}=S^{2}\ltimes\mathbbl{\Sigma}$

We now consider the toric orbifold
$\mathbb{M}_{4}=S^{2}\ltimes\mathbbl{\Sigma}$, namely a spindle
$\mathbbl{\Sigma}=\mathbb{W}\mathbb{P}^{1}_{[n_{+},n_{-}]}$ fibered over a
two-sphere, which is a case with only a $SU(1)\times U(1)$ symmetry. We take
the following fan

$v^{1}=(n_{-},0)\,,\qquad v^{2}=(-k,1)\,,\qquad v^{3}=(-n_{+},0)\,,\qquad
v^{4}=(0,-1)\,,$ (116)

and refer to Faedo:2022rqx  for more details about this orbifold. The total
fan is as in (73) and the constraint (70) is solved by

$M_{a}=\left(\frac{N_{1}}{n_{-}},N_{2},\frac{N_{1}-k\,N_{2}}{n_{+}},N_{2}\right)\,,\qquad
M_{d+1}=-\sum_{a}M_{a}\,,$ (117)

where $N_{1},N_{2}$ parameterize the two independent fluxes, and notice that
$N_{2}=N_{4}$ is implied by the $SU(2)$ symmetry acting on the base $S^{2}$.
The constraint (95) on $\bar{\lambda}_{a,d+1}$ is solved by taking

$\bar{\lambda}_{3,d+1}=-n_{+}\biggl{(}\bar{\nu}_{M5}N_{2}+\frac{\bar{\lambda}_{1,d+1}}{n_{-}}\biggr{)}\,,\quad\bar{\lambda}_{4,d+1}=-\biggl{(}\bar{\nu}_{M5}N_{1}+\frac{k\,\bar{\lambda}_{1,d+1}}{n_{-}}+\bar{\lambda}_{2,d+1}\biggr{)}\,,$
(118)

and we can choose a gauge in which

$\bar{\lambda}_{2,d+1}=-\frac{1}{2}\biggl{(}\bar{\nu}_{M5}N_{1}+\frac{k\,\bar{\lambda}_{1,d+1}}{n_{-}}\biggr{)}\,.$
(119)

After using the remaining gauge freedom to fix $\bar{\lambda}_{1,d+1}$, we are
then left to extremize $\mathds{V}^{(3)}$ with respect to
$\epsilon_{1},\epsilon_{2},\bar{\lambda}_{d+1}$. One can show that the
combination

$k\frac{\partial\mathds{V}^{(3)}}{\partial\epsilon_{1}}-2\frac{\partial\mathds{V}^{(3)}}{\partial\epsilon_{2}}=0$
(120)

implies $\epsilon_{2}^{*}=0$, as expected from the $SU(2)$ symmetry, while
generically $\epsilon_{1}^{*}\neq 0$. In particular, $\epsilon_{1}^{*}$ is
determined solving a quartic equation, which takes about half a page to be
written, so we will refrain from reporting this. The on-shell central charge
can then written in terms of the parameters $N_{1},N_{2},k,n_{+},n_{-}$ and
$\epsilon_{1}^{*}$. For simplicity we shall present the results in three
special cases, where the equations are qualitatively unchanged, but simpler to
write.

Firstly, let us set $k=0$. This leads to the direct product
$\mathbb{M}_{4}=S^{2}\times\mathbbl{\Sigma}$ and in this case, defining

$\chi\equiv\frac{n_{+}+n_{-}}{n_{+}n_{-}}\,,\qquad\mu\equiv\frac{n_{+}-n_{-}}{n_{+}+n_{-}}\,,$
(121)

it is convenient to use the remaining gauge freedom to set

$\bar{\lambda}_{1,d+1}=-\bar{\nu}_{M5}\frac{2(1-\mu)N_{2}-\mu\chi
N_{1}}{2(1+\mu)\chi}\,.$ (122)

Upon extremizing we find that indeed $\epsilon_{2}^{*}=0$ and

$\bar{\lambda}_{d+1}^{*}=\frac{\bar{\nu}_{M5}\epsilon_{3}}{4\chi}\bigl{[}2(\chi
N_{1}+2N_{2})\pm\mathtt{s}_{1}^{1/2}\bigr{]}\,,$ (123)

where we defined the quantity

$\mathtt{s}_{1}=N_{1}^{2}\chi^{2}(\mu\chi\hat{\epsilon}_{1}^{*}-2)^{2}-2N_{2}(\chi
N_{1}+2N_{2})\bigl{(}(1-\mu)\chi\hat{\epsilon}_{1}^{*}+2\bigr{)}\bigl{(}(1+\mu)\chi\hat{\epsilon}_{1}^{*}-2\bigr{)}\,.$
(124)

Here, $\hat{\epsilon}_{1}^{*}$ is solution to the quartic equation

$\begin{split}&3\bigl{[}N_{1}^{2}\mu\chi^{2}(2-\mu\chi\hat{\epsilon}_{1})+2N_{2}(\chi
N_{1}+2N_{2})(2\mu+(1-\mu^{2})\chi\hat{\epsilon}_{1})\bigr{]}\mathtt{s}_{1}^{1/2}-3N_{1}^{3}\mu\chi^{3}(2-\mu\chi\hat{\epsilon}_{1})^{2}\\\
&\qquad-N_{2}(3\chi^{2}N_{1}^{2}+6\chi
N_{1}N_{2}+8N_{2}^{2})\bigl{(}12\mu+4(1-3\mu^{2})\chi\hat{\epsilon}_{1}-3\mu(1-\mu^{2})\chi^{2}\hat{\epsilon}_{1}^{2}\bigr{)}=0\,,\end{split}$
(125)

the critical value of $\epsilon_{1}$ is given by
$\epsilon_{1}^{*}=\hat{\epsilon}_{1}^{*}\epsilon_{3}$ and the on-shell central
charge reads

$\begin{split}\mathbb{V}^{(3)}&=\frac{\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}}{48\chi^{2}}\Bigl{\\{}\mathtt{s}_{1}^{3/2}-(2-\mu\chi\hat{\epsilon}_{1}^{*})\bigl{[}N_{1}^{3}\chi^{3}(2-\mu\chi\hat{\epsilon}_{1}^{*})^{2}\\\
&+N_{2}(3\chi^{2}N_{1}^{2}+6\chi
N_{1}N_{2}+8N_{2}^{2})\bigl{(}2+(1-\mu)\chi\hat{\epsilon}_{1}^{*}\bigr{)}\bigl{(}2-(1+\mu)\chi\hat{\epsilon}_{1}^{*}\bigr{)}\bigr{]}\Bigr{\\}}\,.\end{split}$
(126)

Notice that setting $n_{+}=n_{-}=1$ in the above expressions we get
$\epsilon_{1}^{*}=0$ and reproduce the expression (104) for the central charge
of the $\mathbb{P}^{1}\times\mathbb{P}^{1}$ case.

Following the reasoning in Bah:2019rgq , the total space $M_{6}$ may be also
viewed as an $\mathbb{F}_{2}$ fibered over the spindle $\mathbbl{\Sigma}$ and
we therefore interpret the corresponding putative AdS${}_{5}\times M_{6}$
solution as arising from a stack of M5 branes at
$\mathbb{C}^{2}/\mathbb{Z}_{2}$ singularity, further wrapped on the spindle
$\mathbbl{\Sigma}$. It would be very interesting to reproduce the above
central charge from an anomaly computation, or to construct the explicit
AdS${}_{5}\times M_{6}$ supergravity solution.

A second sub-case is obtained setting $n_{+}=n_{-}=1$, with $k>0$, and
corresponds to the Hirzebruch surfaces $\mathbb{F}_{k}$. Using the remaining
gauge freedom now we can set

$\bar{\lambda}_{1,d+1}=-\bar{\nu}_{M5}\frac{(2-k)N_{2}}{4}$ (127)

and we find that the remaining two extremization equations are solved by

$\bar{\lambda}_{d+1}^{*}=\frac{\bar{\nu}_{M5}\epsilon_{3}}{4}\bigl{[}(2N_{1}+(2-k)N_{2})\pm\mathtt{s}_{2}^{1/2}\bigr{]}\,,$
(128)

where

$\mathtt{s}_{2}=4N_{1}^{2}-4N_{1}N_{2}(\hat{\epsilon}_{1}^{*}+1)(\hat{\epsilon}_{1}^{*}-1+k)-N_{2}^{2}(\hat{\epsilon}_{1}^{*}+1)\bigl{(}(4-2k-k^{2})\hat{\epsilon}_{1}^{*}-4+2k-k^{2}\bigr{)}$
(129)

and $\hat{\epsilon}_{1}^{*}$ is the solution to the quartic equation

$\displaystyle
12N_{1}^{2}(2\hat{\epsilon}_{1}+k)-6N_{1}N_{2}\bigl{(}3k\hat{\epsilon}_{1}^{2}-2(2-2k-k^{2})\hat{\epsilon}_{1}-k(1-2k)\bigr{)}$
$\displaystyle\qquad-
N_{2}^{2}\bigl{(}3k(2-3k-k^{2})\hat{\epsilon}_{1}^{2}-2(8-6k+3k^{2}+3k^{3})\hat{\epsilon}_{1}-k(2-3k+3k^{2})\bigr{)}$
$\displaystyle\qquad-3\bigl{[}2N_{1}(2\hat{\epsilon}_{1}+k)+N_{2}\bigl{(}(4-2k-k^{2})\hat{\epsilon}_{1}-k^{2}\bigr{)}\bigr{]}\mathtt{s}_{2}^{1/2}=0\,.$
(130)

The critical value of $\epsilon_{1}$ is again given by
$\epsilon_{1}^{*}=\hat{\epsilon}_{1}^{*}\epsilon_{3}$ and the on-shell central
charge reads

$\displaystyle\mathbb{V}^{(3)}$
$\displaystyle=\frac{\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}}{24}\Bigl{\\{}\mathtt{s}_{2}^{3/2}-8N_{1}^{3}+12N_{1}^{2}N_{2}(\hat{\epsilon}_{1}^{*}+1)(\hat{\epsilon}_{1}^{*}-1+k)$
(131)
$\displaystyle-6N_{1}N_{2}^{2}(\hat{\epsilon}_{1}^{*}+1)\bigl{(}k\hat{\epsilon}_{1}^{*}{}^{2}-(2-k-k^{2})\hat{\epsilon}_{1}^{*}+2-2k-k^{2}\bigr{)}$
$\displaystyle-
N_{2}^{3}(\hat{\epsilon}_{1}^{*}+1)\bigl{(}k(2-3k-k^{2})\hat{\epsilon}_{1}^{*}{}^{2}-2(4-2k+k^{3})\hat{\epsilon}_{1}^{*}+(8-6k+3k^{2}-k^{3})\bigr{)}\Bigr{\\}}\,.$

Again, setting $k=0$ in the above expressions we get $\epsilon_{1}^{*}=0$ and
reproduce the expression (104) for the central charge of the
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ case.

This result is manifestly not agreement with the central charge given in eq.
(3.79) of BenettiGenolini:2023ndb , where by construction
$\epsilon_{1}^{*}=\epsilon_{2}^{*}=0$. In fact, we can reproduce this result
if we impose by hand that $\epsilon_{1}^{*}=\epsilon_{2}^{*}=0$ so that

$\mathds{V}^{(3)}(\bar{\lambda}_{d+1})=\frac{\bar{\lambda}_{d+1}}{6}\bigl{[}8\bar{\lambda}_{d+1}^{2}-6(2N_{1}+(2-k)N_{2})\bar{\nu}_{M5}\epsilon_{3}\bar{\lambda}_{d+1}+3N_{2}(2N_{1}-k\,N_{2})\bar{\nu}_{M5}^{2}\epsilon_{3}^{2}\bigr{]}$
(132)

and then extremizing this with respect to the remaining parameter
$\bar{\lambda}_{d+1}$ yields

$\bar{\lambda}_{d+1}^{*}=\frac{2N_{1}+(2-k)N_{2}-\sqrt{4N_{1}^{2}+4(1-k)N_{1}N_{2}+(4-2k+k^{2})N_{2}^{2}}}{4}\,\bar{\nu}_{M5}\epsilon_{3}\,,$
(133)

giving the on-shell central charge

$\displaystyle\mathds{V}^{(3)}(\bar{\lambda}_{d+1}^{*})$
$\displaystyle=\frac{\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}}{24}\Bigl{[}\bigl{(}4N_{1}^{2}+4(1-k)N_{1}N_{2}+(4-2k+k^{2})N_{2}^{2}\bigr{)}^{3/2}$
(134)
$\displaystyle-(2N_{1}+(2-k)N_{2})\bigl{(}4N_{1}^{2}+2(1-2k)N_{1}N_{2}+(4-k+k^{2})N_{2}^{2}\bigr{)}\Bigr{]}\,,$

coinciding with the expression given in eq. (3.79) of BenettiGenolini:2023ndb
. This, however, does not correspond to a true extremum of
$\mathds{V}^{(3)}$and therefore it is unlikely that there exist corresponding
supergravity solutions, nor dual SCFTs.

Finally, let us also present the results for the particular configuration of
fluxes $N_{1}=\frac{k}{2}N$, $N_{2}=N$ implying that222222We take $k$ to be
even in this case.

$M_{a}=\left(\frac{k\,N}{2n_{-}},N,-\frac{k\,N}{2n_{+}},N\right)\,,$ (135)

without any assumption on $k$ and $n_{+},n_{-}$. In order to simplify the
expressions, we make use of the remaining gauge freedom to set

$\bar{\lambda}_{1,d+1}=-\bar{\nu}_{M5}\frac{((1-\mu)(4+k\mu\chi)-k\chi)N}{(1+\mu)\chi(4+k\mu\chi)}\,.$
(136)

The extremization problem is then solved by

$\bar{\lambda}_{d+1}^{*}=\frac{\bar{\nu}_{M5}\epsilon_{3}N}{2\chi\xi}\bigl{(}2\xi\pm\mathtt{s}_{3}^{1/2}\bigr{)}\,,$
(137)

where we defined $\xi=4+k\mu\chi$ and

$\mathtt{s}_{3}=4\xi^{2}-4\mu\chi\xi^{2}\hat{\epsilon}_{1}^{*}-\chi^{2}(8(2+k\mu\chi)(1-\mu^{2})-(1-\mu^{2}+\mu^{4})k^{2}\chi^{2})\hat{\epsilon}_{1}^{*}{}^{2}\,.$
(138)

Here, $\hat{\epsilon}_{1}^{*}$ is solution to the quartic equation

$\begin{split}&2\xi^{2}(k\chi+6\mu\xi)+4\chi\xi^{2}(4-3\mu^{2}\xi)\hat{\epsilon}_{1}-3\chi^{2}\Bigl{(}\frac{\xi^{2}(1-\mu^{2})(k\chi+2\mu\xi)}{2}-k^{3}\chi^{3}\Bigr{)}\hat{\epsilon}_{1}^{2}\\\
&\qquad-3\bigl{(}2\mu\xi^{2}+\chi(8(2+k\mu\chi)(1-\mu^{2})-(1-\mu^{2}+\mu^{4})k^{2}\chi^{2})\hat{\epsilon}_{1}\bigr{)}\mathtt{s}_{3}^{1/2}=0\,,\end{split}$
(139)

and the critical value of $\epsilon_{1}$ is
$\epsilon_{1}^{*}=\hat{\epsilon}_{1}^{*}\epsilon_{3}$. The central charge in
terms of $\hat{\epsilon}_{1}^{*}$ is given by

$\begin{split}\mathbb{V}^{(3)}&=\frac{\bar{\nu}_{M5}^{3}\epsilon_{3}^{3}N^{3}}{24\chi^{2}\xi^{2}}\Bigl{\\{}\mathtt{s}_{3}^{3/2}-\chi^{3}\Bigl{(}\frac{\xi^{2}(1-\mu^{2})(k\chi+2\mu\xi)}{2}-k^{3}\chi^{3}\Bigr{)}\hat{\epsilon}_{1}^{*}{}^{3}\\\
&+2\chi^{2}\xi^{2}(4-3\mu^{2}\xi)\hat{\epsilon}_{1}^{*}{}^{2}+2\chi\xi^{2}(k\chi+6\mu\xi)\hat{\epsilon}_{1}^{*}-8\xi^{3}\Bigr{\\}}\,.\end{split}$
(140)

####  3.2.4 General geometries

We now discuss the general AdS${}_{5}\times M_{6}$ solution with no
$\mathbb{Z}_{2}$ symmetry. The fan (66) corresponds now to a compact geometry.
The fan is the union of $2d$ cones $(V^{a},V^{a+1},V^{d+1})$ and
$(V^{a},V^{a+1},V^{d+2})$ corresponding to the fixed points of the torus
action, that are specified by selecting a fixed point on $\mathbb{M}_{4}$ and
simultaneously the North or South pole of the fibre $\mathbb{P}^{1}$.

The equivariant volume is now given by

$\mathds{V}=\sum_{a}\frac{{\mathrm{e}}^{-\lambda_{a}\epsilon_{1}^{a}-\lambda_{a+1}\epsilon_{2}^{a}-\lambda_{d+1}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})+\ldots}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}-\sum_{a}\frac{{\mathrm{e}}^{-\lambda_{a}\epsilon_{1}^{a}-\lambda_{a+1}\epsilon_{2}^{a}+\lambda_{d+2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})+\ldots}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}\,,$
(141)

where the dots at the exponents contain the higher times.

This expression can be also written as an integral over $\mathbb{M}_{4}$

$\mathds{V}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\int_{\mathbb{M}_{4}}\frac{e^{\tau_{N}{T}}-e^{\tau_{S}{T}}}{\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})}\>,$
(142)

where we have defined the North pole equivariant form $\tau{T}_{N}$ and South
pole equivariant form $\tau{T}_{S}$ as

$\displaystyle\tau_{N}{T}=\sum_{A}\lambda_{A}\>\mathcal{C}_{A}^{N}+\sum_{A,B}\lambda_{AB}\>\mathcal{C}_{A}^{N}\>\mathcal{C}_{B}^{N}\>,$
(143)
$\displaystyle\tau_{S}{T}=\sum_{A}\lambda_{A}\>\mathcal{C}_{A}^{S}+\sum_{A,B}\lambda_{AB}\>\mathcal{C}_{A}^{S}\>\mathcal{C}_{B}^{S}\>,$
$\displaystyle\mathcal{C}_{a}^{N}\,=\,\mathcal{C}_{a}^{S}\,=\,c_{1}{T}(L_{a})\>,\qquad
a=1,\ldots,d\>,$
$\displaystyle\mathcal{C}_{d+2}^{S}\,=\,-\,\mathcal{C}_{d+1}^{N}\,=\,\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})\>,$
$\displaystyle\mathcal{C}_{d+2}^{N}\,=\,\mathcal{C}_{d+1}^{S}\,=0\>.$

The flux equations are the following:

$-\nu_{M5}M_{A}=\partial_{\lambda_{A}}\mathds{V}^{(2)}=\int_{\mathbb{M}_{4}}\frac{\mathcal{C}_{A}^{N}\,\tau_{N}{T}-\mathcal{C}_{A}^{S}\,\tau_{S}{T}}{\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})}\>.$
(144)

For a generic fan, using the gauge transformations (19) and (20) we can set
all $\lambda_{a}=\lambda_{a,a}=\lambda_{a,a+1}=0$. However, as already
mentioned, for special fans, including $\mathbb{P}^{2}$ and
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, one of the single times $\lambda_{a}$
remains unfixed. As a difference with the the $\mathbb{Z}_{2}$ symmetric case,
an arbitrary $\lambda_{a}$ solves trivially the flux equations. Therefore we
set $\lambda_{a,a}=\lambda_{a,a+1}=0$ and keep $\lambda_{a}$ with the
understanding that the latter can be partially or totally gauged fixed to
zero. The forms $\tau{T}_{N}$ and $\tau{T}_{S}$ with all variables can then be
written as

$\displaystyle\tau_{N}{T}=$
$\displaystyle\sum_{a}\lambda_{a}\,c_{1}{T}(L_{a})+\,\Big{(}\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})\Big{)}\Big{(}\overline{\lambda}_{d+1}+\sum_{a}\overline{\lambda}_{a,d+1}\,c_{1}{T}(L_{a})\Big{)}\>,$
(145) $\displaystyle\tau_{S}{T}=$
$\displaystyle\sum_{a}\lambda_{a}\,c_{1}{T}(L_{a})\,-\Big{(}\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})\Big{)}\Big{(}\overline{\lambda}_{d+2}+\sum_{a}\overline{\lambda}_{a,d+2}\,c_{1}{T}(L_{a})\Big{)}\>,$

where we have defined the $\overline{\lambda}$ variables as

$\displaystyle\overline{\lambda}_{d+1}=\epsilon_{3}\lambda_{d+1,d+1}-\lambda_{d+1}\>,\qquad$
$\displaystyle\overline{\lambda}_{d+2}=-\epsilon_{3}\lambda_{d+2,d+2}-\lambda_{d+2}\>,$
(146)
$\displaystyle\overline{\lambda}_{b,d+1}=\lambda_{d+1,d+1}-2\lambda_{b,d+1}\>,\qquad$
$\displaystyle\overline{\lambda}_{b,d+2}=-\lambda_{d+2,d+2}-2\lambda_{b,d+2}\>.$

Then equations (144) become

$\displaystyle-\nu_{M5}M_{a}=\sum_{b}D_{ab}(\bar{\lambda}_{b,d+1}+\bar{\lambda}_{b,d+2})\>,$
(147)
$\displaystyle-\nu_{M5}M_{d+1}=-\sum_{ab}D_{ab}\,\bar{\lambda}_{b,d+1}\>,$
$\displaystyle-\nu_{M5}M_{d+2}=\sum_{ab}D_{ab}\,\bar{\lambda}_{b,d+2}\>.$

The expression for $\mathds{V}^{(3)}$ is

$\displaystyle\mathds{V}^{(3)}$
$\displaystyle=\frac{1}{6}\int_{\mathbb{M}_{4}}\Big{(}\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})\Big{)}^{2}\Big{[}\bigl{(}\Lambda^{N}\bigr{)}^{3}+\bigl{(}\Lambda^{S}\bigr{)}^{3}\Big{]}$
(148)
$\displaystyle+\frac{1}{2}\int_{\mathbb{M}_{4}}\sum_{a}\lambda_{a}\,c_{1}{T}(L_{a})\Big{(}\epsilon_{3}+\sum_{a}c_{1}{T}(L_{a})\Big{)}\Big{[}\bigl{(}\Lambda^{N}\bigr{)}^{2}-\bigl{(}\Lambda^{S}\bigr{)}^{2}\Big{]}$
$\displaystyle+\frac{1}{2}\int_{\mathbb{M}_{4}}\Big{(}\sum_{a}\lambda_{a}\,c_{1}{T}(L_{a})\Big{)}^{2}\Big{[}\Lambda^{N}+\Lambda^{S}\Big{]}\>,$

where we defined

$\Lambda^{N}=\overline{\lambda}_{d+1}+\sum_{a}\overline{\lambda}_{a,d+1}\,c_{1}{T}(L_{a})\>,\qquad\Lambda^{S}=\overline{\lambda}_{d+2}+\sum_{a}\overline{\lambda}_{a,d+2}\,c_{1}{T}(L_{a})\>.$
(149)

The flux constraints are not enough to fix all the $\overline{\lambda}$, so
the idea is again to extremize $\mathds{V}^{(3)}$ with respect to the
remaining variables. It is convenient to define $\overline{\lambda}_{b,+}$ and
$\overline{\lambda}_{b,-}$ as

$\overline{\lambda}_{b,\pm}=\overline{\lambda}_{b,d+1}\pm\overline{\lambda}_{b,d+2}\>,$
(150)

so that all the $\overline{\lambda}_{b,+}$ are fixed (up to gauge
transformations) by

$-\nu_{M5}M_{a}=\sum_{b}D_{ab}\,\overline{\lambda}_{b,+}\>,$ (151)

whereas the $\overline{\lambda}_{b,-}$ are only subject to the following
constraint:

$\nu_{M5}(M_{d+1}+M_{d+2})=\,\sum_{ab}D_{ab}\,\overline{\lambda}_{b,-}\>.$
(152)

The extremization conditions then are

$0=\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{a}}=\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{d+1}}=\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{d+2}}=\sum_{a}\rho^{a}\,\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{a,-}}\>,\quad\forall\>\rho^{a}\text{
such that }\sum_{ab}D_{ab}\,\rho^{b}=0\>.$ (153)

In general these equations do not look easy, but in the special case
$M_{d+1}+M_{d+2}=0$ there is a simple solution: we can set $\lambda_{a}=0$ and
$\overline{\lambda}_{d+1}=\overline{\lambda}_{d+2}$ and
$\overline{\lambda}_{b,d+1}=\overline{\lambda}_{b,d+2}$ so that the equations
with $\rho^{a}$ are trivially solved. The rest of the computation reduces to
that of section 3.2.2 for $\mathbb{Z}_{2}$ symmetric geometries. When
$M_{d+1}+M_{d+2}\neq 0$ this simple solution is not possible because the
constraint (152) would not be satisfied.

####  3.2.5 Examples of general geometries

In this section we consider again the examples based on $\mathbb{P}^{2}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3 and compare with the supergravity
solutions in Gauntlett:2006ai , where the additional parameter “$c$” is turned
on. Since the solutions in Gauntlett:2006ai  all correspond to a critical
point at $\epsilon_{1}=\epsilon_{2}=0$, for simplicity in this section we
restrict again to configurations with this feature, which as we discussed
requires a special choice of fluxes for the case of dP3, while it is automatic
for generic fluxes for $\mathbb{P}^{2}$ and
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. The explicit value of the central charge
has been written in Bah:2019rgq ; Bah:2019vmq ; BenettiGenolini:2023ndb .

The free energy (148) can be expanded in a sum of integrals of equivariant
Chern classes. Since the multiple intersections (97) are homogeneous function
of degree $p-2$ in $\epsilon_{1}$ and $\epsilon_{2}$, all the terms involving
$D_{a_{1},\ldots a_{p}}$ with $p>2$ in (148) vanish for
$\epsilon_{1}=\epsilon_{2}=0$, and the free energy simplifies to

$\displaystyle 6\mathds{V}^{(3)}$
$\displaystyle=\bar{\lambda}_{d+1}^{3}\sum_{ab}D_{ab}+3\bar{\lambda}_{d+1}^{2}\sum_{ab}D_{ab}(2\epsilon_{3}\bar{\lambda}_{a,d+1}+\lambda_{a})$
(154)
$\displaystyle+3\bar{\lambda}_{d+1}\sum_{ab}D_{ab}(\epsilon_{3}\bar{\lambda}_{a,d+1}+\lambda_{a})(\epsilon_{3}\bar{\lambda}_{b,d+1}+\lambda_{b})$
$\displaystyle+\bar{\lambda}_{d+2}^{3}\sum_{ab}D_{ab}+3\bar{\lambda}_{d+2}^{2}\sum_{ab}D_{ab}(2\epsilon_{3}\bar{\lambda}_{a,d+2}-\lambda_{a})$
$\displaystyle+3\bar{\lambda}_{d+2}\sum_{ab}D_{ab}(\epsilon_{3}\bar{\lambda}_{a,d+2}-\lambda_{a})(\epsilon_{3}\bar{\lambda}_{b,d+2}-\lambda_{b})\,.$

Using the flux constraints we can also write

$\displaystyle\mathds{V}^{(3)}$
$\displaystyle=\bar{\lambda}_{d+1}^{3}\sum_{ab}D_{ab}+3\bar{\lambda}_{d+1}^{2}\epsilon_{3}\nu_{M5}M_{d+1}+3\bar{\lambda}_{d+1}^{2}\sum_{ab}D_{ab}\lambda_{a}^{+}+3\bar{\lambda}_{d+1}\sum_{ab}D_{ab}\lambda_{a}^{+}\lambda_{b}^{+}$
(155)
$\displaystyle+\bar{\lambda}_{d+2}^{3}\sum_{ab}D_{ab}-3\bar{\lambda}_{d+2}^{2}\epsilon_{3}\nu_{M5}M_{d+2}+3\bar{\lambda}_{d+2}^{2}\sum_{ab}D_{ab}\lambda_{a}^{-}+3\bar{\lambda}_{d+2}\sum_{ab}D_{ab}\lambda_{a}^{-}\lambda_{b}^{-}\,,$

where

$\lambda_{a}^{+}=\epsilon_{3}\bar{\lambda}_{a,d+1}+\lambda_{a}\,,\qquad\lambda_{a}^{-}=\epsilon_{3}\bar{\lambda}_{a,d+2}-\lambda_{a}\,,$
(156)

are constrained variables.

We consider first the general case ($\mathbb{P}^{2}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3) with all fluxes associated to
the fan of the KE4 set equal to $N$. The fan and intersection matrix are given
in (LABEL:fanKE) and (LABEL:intersKE). There are, in principle, $d-2$
independent fluxes on $\mathbb{M}_{4}$ that we can turn on but in the
supergravity solution with KE metric they are equal and we first restrict to
this case. The relations $\sum_{A}V_{I}^{A}M_{A}=0$ require
$M_{A}=(N,\ldots,N,N_{N},N_{S})$ with $dN+N_{N}-N_{S}=0$ and we can
parameterize $N_{N}=M-\frac{d}{2}N$ and $N_{S}=M+\frac{d}{2}N$, possibly
allowing an half-integer $M$. Given the symmetry of the problem, we take all
$\lambda_{a}$ to be equal, and similarly for the $\bar{\lambda}_{a,d+1}$ and
$\bar{\lambda}_{a,d+2}$. The condition $\sum_{abc}D_{abc}=0$ holds for these
models, and therefore all linear terms in $\epsilon_{1},\epsilon_{2}$ in
$\mathds{V}^{(3)}$ vanish, guaranteeing a critical point at
$\epsilon_{1}=\epsilon_{2}=0$. The flux conditions are solved by

$\bar{\lambda}_{a,d+1}=\nu_{M5}\frac{\left(M-\frac{dN}{2}\right)}{dm_{k}}\,,\qquad\bar{\lambda}_{a,d+2}=-\nu_{M5}\frac{\left(M+\frac{dN}{2}\right)}{dm_{k}}\,,$
(157)

where $\sum_{ab}D_{ab}=dm_{k}$ so that $m_{k}=3,2,1$ for $\mathbb{P}^{2}$,
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3, respectively. Extremizing with
respect to $\lambda_{a}$ and defining $\bar{\lambda}_{d+1}=\nu_{M5}(H+K)$ and
$\bar{\lambda}_{d+2}=\nu_{M5}(H-K)$ we find

$6\nu_{M5}^{-3}\mathds{V}^{(3)}=2dm_{k}H^{3}-\frac{3}{2}d\epsilon_{3}^{2}N^{2}\frac{K^{2}}{m_{k}H}-6d\epsilon_{3}NH^{2}+3\epsilon_{3}H(4KM+\frac{d}{2m_{k}}\epsilon_{3}N^{2})\,,$
(158)

which after extremization gives232323Recall that to compare with section 3.2.2
we need to use the rescaling (74).

$\mathds{V}^{(3)}=\frac{d^{2}\nu_{M5}^{3}\epsilon_{3}^{3}N^{4}}{12m_{k}^{2}(d^{2}N^{2}+12M^{2})^{2}}\left((3d^{2}N^{2}-12M^{2})^{3/2}-dN(5d^{2}N^{2}-36M^{2})\right)\,.$
(159)

An analogous formula for non-necessarily toric KE has recently appeared in
BenettiGenolini:2023ndb .

In the case $\mathbb{P}^{1}\times\mathbb{P}^{1}$ we can turn on two
independent fluxes and have round metrics on the $\mathbb{P}^{1}$s. We take
the general assignment of fluxes compatible with $\sum_{A}V_{I}^{A}M_{A}=0$:

$\displaystyle M_{A}=(N_{1},N_{2},N_{1},N_{2},N_{N},N_{S})\,,$ (160)

where

$2N_{1}+2N_{2}+N_{N}-N_{S}=0$

and we can parameterize $N_{N}=M-N_{1}-N_{2}$ and $N_{S}=M+N_{1}+N_{2}$. Using
the gauge transformations (19) we can also reduce to the case

$\displaystyle\lambda_{a+2}=\lambda_{a}\,,\,\bar{\lambda}_{a+2,d+1}=\bar{\lambda}_{a,d+1}\,,\,\bar{\lambda}_{a+2,d+2}=\bar{\lambda}_{a,d+2}\,.$
(161)

Notice that, in this gauge, all the linear terms in
$\epsilon_{1},\epsilon_{2}$ in $\mathds{V}^{(3)}$ vanishes since, as one can
check,

$\sum_{abc}D_{abc}l_{a}^{(1)}l_{b}^{(2)}l_{c}^{(3)}=0$ (162)

provided the vectors $l_{a}^{(k)}$ satisfy $l_{a}^{(k)}=l_{a+2}^{(k)}$. We can
solve the flux constraints

$\displaystyle
2\bar{\lambda}_{1,d+1}+2\bar{\lambda}_{1,d+2}+\nu_{M5}N_{1}=0\,,\,\,$
$\displaystyle
4\bar{\lambda}_{1,d+1}+4\bar{\lambda}_{2,d+1}+\nu_{M5}(-M+N_{1}+N_{2})=0\,,$
(163) $\displaystyle
2\bar{\lambda}_{2,d+1}+2\bar{\lambda}_{2,d+2}+\nu_{M5}N_{2}=0\,,\,\,$
$\displaystyle
4\bar{\lambda}_{1,d+2}+4\bar{\lambda}_{2,d+2}+\nu_{M5}(M+N_{1}+N_{2})=0\,,$

by

$\displaystyle\bar{\lambda}_{2,d+1}=-\bar{\lambda}_{1,d+1}+\frac{1}{4}\nu_{M5}(M-N_{1}-N_{2})\,,$
(164)
$\displaystyle\bar{\lambda}_{1,d+2}=-\bar{\lambda}_{1,d+1}-\nu_{M5}\frac{N_{2}}{2}\,,$
$\displaystyle\bar{\lambda}_{2,d+2}=\bar{\lambda}_{1,d+1}+\frac{1}{4}\nu_{M5}(-M-N_{1}+N_{2})\,.$

Extremizing with respect to $\lambda_{1,2}$ and $\bar{\lambda}_{1,d+1}$ and
defining $\bar{\lambda}_{d+1}=\nu_{M5}(H+K)$ and
$\bar{\lambda}_{d+2}=\nu_{M5}(H-K)$ we find

$6\nu_{M5}^{-3}\mathds{V}^{(3)}=16H^{3}-3\epsilon_{3}^{2}N_{1}N_{2}\frac{K^{2}}{H}-12\epsilon_{3}(N_{1}+N_{2})H^{2}+3\epsilon_{3}H(4KM+\epsilon_{3}N_{1}N_{2})\,,$
(165)

which after extremization gives

$\displaystyle\mathds{V}^{(3)}$
$\displaystyle=\frac{\nu_{M5}^{3}\epsilon_{3}^{3}N_{1}^{2}N_{2}^{2}(4N_{1}^{2}+4N_{1}N_{2}+4N_{2}^{2}-3M^{2})^{3/2}}{6(4N_{1}N_{2}+3M^{2})^{2}}$
(166)
$\displaystyle-\frac{\nu_{M5}^{3}\epsilon_{3}^{3}N_{1}^{2}N_{2}^{2}(N_{1}+N_{2})(8N_{1}^{2}+4N_{1}N_{2}+8N_{2}^{2}-9M^{2})}{6(4N_{1}N_{2}+3M^{2})^{2}}\,,$

reproducing (5.7) of Bah:2019vmq .

Finally, let us mention that in the case of dP3 we can turn on four
independent fluxes along the base plus one additional flux $M$, and the
general extremization problem is intractable. It is possible to to solve it
for the two special configurations of fluxes with enhanced symmetry discussed
previously. We leave this as an instructive exercise for the reader.

The case $\mathbb{P}^{1}\times\mathbb{P}^{1}$ has been interpreted in
Bah:2019rgq ; Bah:2019vmq  as a solution for M5 branes sitting at the orbifold
$\mathbb{C}^{2}/\mathbb{Z}_{2}$ wrapped over one of the $\mathbb{P}^{1}$. The
interpretation follows by deriving the central charge from an anomaly
polynomial computation. It would be very interesting to understand if our
general formula (148) can be written as the integral of the anomaly polynomial
for some M5 brane theory wrapped over a two-cycle in $\mathbb{M}_{4}$ and give
a field theory interpretation of the solution.

###  3.3 Comparison with other approaches

It is interesting to compare with the recent approach based on Killing spinor
bilinears in M theory BenettiGenolini:2023kxp . The technique consists in
considering a set of equivariantly closed differential forms which can be
constructed from Killing spinor bilinears. Three such forms have been
explicitly constructed for AdS${}_{5}\times M_{6}$ in BenettiGenolini:2023kxp
and for AdS${}_{3}\times M_{8}$ in BenettiGenolini:2023ndb ;
BenettiGenolini:2023yfe . Our results in sections 3.1 and 3.2 partially
overlap with those in BenettiGenolini:2023ndb ; BenettiGenolini:2023yfe  and
it is interesting to compare the two methods. We will show that they are
actually equivalent, when they can be compared, although in a non-trivial way.

For both cases, AdS${}_{11-k}\times M_{k}$ with $k=6,8$, the authors of
BenettiGenolini:2023ndb ; BenettiGenolini:2023yfe  define an equivariant
$k$-form $\Phi$ whose higher degree component is the warped volume of $M_{k}$
and the lowest component the third power of a special locally defined function
$y$. Up to coefficients, the integral of $\Phi$ is the free energy, so we have

$F=\int_{M_{K}}\Phi=\sum_{a}\frac{y^{3}|_{\alpha}}{d_{\alpha}\>e^{\mathbb{T}_{4}}|_{\alpha}}\>,$
(167)

where $\alpha$ are the fixed points of the geometry, and we recognize our
expression for $\mathds{V}^{(3)}$ for M theory solutions242424We are omitting
a $(-1)^{k/2}$ sign in the expression for $\mathds{V}^{(3)}$. In this
discussion we are ignoring all such overall numerical factors.

$F=\mathds{V}^{(3)}=\sum_{a}\frac{(\tau_{\alpha})^{3}}{d_{\alpha}\>e^{\mathbb{T}_{4}}|_{\alpha}}\>,$
(168)

upon identifying

$y|_{\alpha}=\tau_{\alpha}\,.$ (169)

There exists also an equivariant four-form $\Phi^{F}$ whose higher degree
component is the M theory four-form and the lowest component the first power
of the function $y$. The flux quantization conditions give then

$\displaystyle M_{AB}$
$\displaystyle=\int_{M_{8}}\Phi^{F}c_{1}(L_{A})c_{1}(L_{B})=\sum_{\alpha}\frac{(c_{1}(L_{A}){T}c_{1}{T}(L_{B})y)|_{a}}{d_{\alpha}\>e^{\mathbb{T}_{4}}|_{\alpha}}\>,$
(170) $\displaystyle M_{A}$
$\displaystyle=\int_{M_{6}}\Phi^{F}c_{1}(L_{A})=\sum_{\alpha}\frac{(c_{1}(L_{A}){T}y)|_{a}}{d_{\alpha}\>e^{\mathbb{T}_{4}}|_{\alpha}}\>,$

for AdS${}_{3}\times M_{8}$ and AdS${}_{5}\times M_{6}$, respectively and it
easy to see that these conditions are equivalent, up to coefficients, to our
(31) and (72) with the same identification $y|_{\alpha}=\tau_{\alpha}$.

Finally there exists another auxiliary form, a four-form $\Phi^{*F}$ in
AdS${}_{3}\times M_{8}$ and a two-form $\Phi^{Y}$ in AdS${}_{5}\times M_{6}$,
whose lowest component is the second power of the function $y$.

Consider first the AdS${}_{3}\times M_{8}$ solutions with wrapped M5 branes of
section 3.1. The vanishing of the $\Phi^{*F}$ flux along $S^{4}$ is used in
BenettiGenolini:2023ndb ; BenettiGenolini:2023yfe  to enforce a
$\mathbb{Z}_{2}$ symmetry of the solution by identifying
$y^{N}|_{a}=-y^{S}|_{a}$, thus effectively cutting by half the number of fixed
points. With the identification $y|_{a}=\tau_{a}$, our construction in section
3.1 is then equivalent to the one in BenettiGenolini:2023ndb .

Consider next the AdS${}_{5}\times M_{6}$ solutions of section 3.2. The
approaches are complementary. While we consider toric orbifolds and the action
of the full torus $\mathbb{T}^{3}=U(1)^{3}$, the authors of
BenettiGenolini:2023yfe  consider $\mathbb{P}^{1}$ bundles over a smooth four-
manifold $B_{4}$ and assume that the R-symmetry vector has no legs along
$B_{4}$. Let us observe that this assumption can fail in general. For a
generic $B_{4}$ with abelian isometries there is no reason to expect that the
R-symmetry does not mix with the isometries of $B_{4}$ and a full-fledged
computation considering the torus action on $B_{4}$ is necessary. Also for the
toric $B_{4}=$dP3 with a generic choice of fluxes we expect a mixing with the
isometries of $B_{4}$, as discussed in section 3.2.2. Under this condition,
the central charge given in BenettiGenolini:2023yfe  is not necessarily the
extremum of the free-energy. Obviously, whenever the two approaches can be
compared and the assumption in BenettiGenolini:2023yfe  is satisfied, we find
agreement.

From a technical point of view, this might be surprising. Recall indeed that
the flux constraints do not completely fix the values of the times
$\tau_{a}=y|_{a}$. In our construction, we just extremize the free-energy $F$
with respect to the remaining parameters. In BenettiGenolini:2023ndb ;
BenettiGenolini:2023yfe  instead, in a case-by-case analysis, the auxiliary
form $\Phi^{Y}$ is used to find additional conditions to fix the
$y|_{\alpha}$. The two methods look superficially different, but we now show
that they are effectively equivalent.

The extremization conditions with respect to the Kähler parameters that are
not fixed by the flux constraints are written in (153). The first three
conditions

$\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{A}}=0\,,\qquad
A=a,d+1,d+2\,,$ (171)

can be also rewritten as

$\sum_{B}V^{B}_{i}\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{BA}}(1+\delta_{AB})=0\,,\qquad
i=1,2\,,\qquad A=a,d+1,d+2\,.$ (172)

Indeed

$\displaystyle\sum_{B}V^{B}_{i}\frac{\partial\mathds{V}}{\partial\lambda_{BA}}(1+\delta_{AB})=-2\int_{M_{6}}{\mathrm{e}}^{\tau^{\mathbb{T}_{3}}}c_{1}^{\mathbb{T}_{3}}(L_{A})\sum_{B}V^{B}_{i}c_{1}^{\mathbb{T}_{3}}(L_{B})$
(173)
$\displaystyle\qquad=2\epsilon_{i}\int_{M_{6}}{\mathrm{e}}^{\tau^{\mathbb{T}_{3}}}c_{1}^{\mathbb{T}_{3}}(L_{A})=-2\epsilon_{i}\frac{\partial\mathds{V}}{\partial\lambda_{A}}\,,$

and taking the degree two component of this equation we see that all the
conditions (172) collapse to the extremization of the free energy with respect
to the parameter $\lambda_{A}$.

The conditions (153) can be then written as

$0=\sum_{b}v^{b}_{i}\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{b,a}}(1+\delta_{ba})=\sum_{b}v^{b}_{i}\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{b,d+1}}=\sum_{b}v^{b}_{i}\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{b,d+2}}=\sum_{a}\rho^{a}\,\frac{\partial\mathds{V}^{(3)}}{\partial\overline{\lambda}_{a,-}}\>,$
(174)

where $\rho^{a}$ is such that $\sum_{ab}D_{ab}\,\rho^{b}=0$. Now, the
equations

$\displaystyle(1+\delta_{bc})\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{b,c}}$
$\displaystyle=\sum_{a}(c_{1}(L_{b})c_{1}(L_{c}))|_{a}\frac{(\tau_{a}^{N})^{2}-(\tau_{a}^{S})^{2}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}(\epsilon_{3}-\epsilon^{a}_{1}-\epsilon^{a}_{2})}\,,$
(175) $\displaystyle\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{b,d+1}}$
$\displaystyle=\sum_{a}c_{1}(L_{b})|_{a}\frac{(\tau_{a}^{N})^{2}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}}\,,$
$\displaystyle\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{b,d+2}}$
$\displaystyle=\sum_{a}c_{1}(L_{b})|_{a}\frac{(\tau_{a}^{S})^{2}}{d_{a,a+1}\epsilon^{a}_{1}\epsilon^{a}_{2}}\,,$

given the identification $y|_{a}=\tau_{a}$ and the fact that $\Phi^{Y}$ has
lowest component $y^{2}$, translate into the localization formulas for

$\displaystyle\int_{\mathbb{P}^{1}_{bc}}\Phi^{Y}$
$\displaystyle=\int_{M_{6}}c_{1}(L_{c})c_{1}(L_{b})\Phi^{Y}\,,$ (176)
$\displaystyle\int_{D_{b}^{N}}\Phi^{Y}$
$\displaystyle=\int_{M_{6}}c_{1}(L_{d+1})c_{1}(L_{b})\Phi^{Y}\,,$
$\displaystyle\int_{D_{b}^{S}}\Phi^{Y}$
$\displaystyle=\int_{M_{6}}c_{1}(L_{d+2})c_{1}(L_{b})\Phi^{Y}\,,$

respectively, where $\mathbb{P}^{1}_{bc}$ is the fibre taken at the fixed
points $D_{c}\cap D_{b}$ on the base ($b=c\pm 1$ necessarily) and the
$D_{b}^{N,S}$ are the divisor on the base taken at the North and South pole of
the fibre, respectively. The extremization constraints are then equivalent to
the following co-homological relations

$\displaystyle 0$
$\displaystyle=\sum_{b}v^{b}_{i}\int_{\mathbb{P}^{1}_{bc}}\Phi^{Y}=\sum_{b}v^{b}_{i}\int_{D_{b}^{N}}\Phi^{Y}=\sum_{b}v^{b}_{i}\int_{D_{b}^{S}}\Phi^{Y}$
(177)
$\displaystyle=\sum_{a}\rho^{a}\,\Big{(}\int_{D_{a}^{N}}\Phi^{Y}-\int_{D_{a}^{S}}\Phi^{Y}\Big{)}\>,\quad\forall\>\rho^{a}\text{
such that }\sum_{ab}D_{ab}\,\rho^{b}=0\>.$

The first three conditions are obvious: the cycles
$\sum_{b}v^{b}_{i}\,\mathbb{P}^{1}_{bc}$, $\sum_{b}v^{b}_{i}\,D_{b}^{N,S}$ are
trivial in homology. The last equation equates cycles sitting at the North and
South pole. The corresponding fluxes of $\Phi^{Y}$ do not need to be equal but
they must be related. We know that
$c_{1}(L_{d+2})=c_{1}(L_{d+1})+\sum_{a}c_{1}(L_{a})$.252525The $I=3$ condition
of $\sum_{A}V^{A}_{I}c_{1}(L_{A})=0$. Then

$\displaystyle\sum_{a}\rho^{a}\,\Big{(}\int_{D_{a}^{N}}\Phi^{Y}-\int_{D_{a}^{S}}\Phi^{Y}\Big{)}=\sum_{a}\rho^{a}\int_{M_{6}}\Big{(}c_{1}(L_{d+1})-c_{1}(L_{d+2})\Big{)}c_{1}(L_{a})\Phi^{Y}$
(178)
$\displaystyle=-\sum_{ab}\rho^{a}\int_{M_{6}}c_{1}(L_{b})c_{1}(L_{a})\Phi^{Y}\propto\sum_{ab}\rho^{a}D_{ab}=0\,.$

The last step follows by expanding $\Phi^{Y}$ in a sum of Chern classes, and
by writing $\int_{M_{6}}c_{1}(L_{b})c_{1}(L_{a})\Phi^{Y}$ as a sum of triple
intersections $D_{ABC}^{M_{6}}$ on $M_{6}$. But $D_{abc}^{M_{6}}=0$ and
$D_{d+1,a,b}^{M_{6}}=D_{d+2,a,b}^{M_{6}}=D_{ab}$.262626The triple
intersections on $M_{6}$ are easily computed as
$D_{ABC}^{M_{6}}=\frac{\partial\mathds{V}^{(3)}}{\partial\lambda_{A}\partial\lambda_{B}\partial\lambda_{C}}$
from (141).

We see that our construction based on the equivariant volume naturally
incorporates the localization approach of BenettiGenolini:2023kxp ;
BenettiGenolini:2023ndb ; BenettiGenolini:2023yfe , with the advantage that
all the geometrical constraints that must be imposed case-by-case in order to
find the free-energy in BenettiGenolini:2023kxp ; BenettiGenolini:2023ndb ;
BenettiGenolini:2023yfe  appear naturally in our construction: they correspond
to the extremization with respect to all parameters that remain after imposing
the flux constraints. This avoids an analysis based on the specific topology
of the background.

##  4 $\mathrm{AdS}_{2}$ and $\mathrm{AdS}_{3}$ solutions in type II
supergravities

In this section we consider solutions in type II string theory with geometries
that are fibrations over a four-dimensional orbifold $\mathbb{M}_{4}$. We
consider the case of massive type IIA solutions with D4 brane flux,
corresponding to D4 branes wrapped $\mathbb{M}_{4}$ and the case of type IIB
solutions with D3 brane flux. In all cases, we show that the free energy can
be obtained by extremizing the appropriate term in the equivariant volume.

###  4.1 $\mathrm{AdS}_{2}\times M_{8}$ solutions in massive type IIA

In this section we turn our attention to D4 branes wrapped around a generic
four-dimensional toric orbifold $\mathbb{M}_{4}$ Faedo:2022rqx ;
Couzens:2022lvg ; Bomans:2023ouw . Specifically the brane system we study
corresponds to $\mathrm{AdS}_{2}\times M_{8}$ solutions in massive type IIA,
where $M_{8}$ is a $S_{4}$ fibration over $\mathbb{M}_{4}$. The geometry is
similar to the case of M5 branes wrapped around $\mathbb{M}_{4}$ considered in
section 3.1 and we can borrow most of the computations. Here, due to the
orientifold projection,272727The brane system is actually D4 in the presence
of D8, which generate the cosmological constant, and an orientifold plane O8
that cuts $S^{4}$ into half. the $\mathbb{Z}_{2}$ projection used in section
3.1 is automatically implemented and there is only one set of fixed points, at
one of the poles of $S^{4}$. The geometry to consider is then a CY4, a
$\mathbb{C}^{2}$ fibration over $\mathbb{M}_{4}$ with toric fan generated by
the vectors (28). As discussed in Martelli:2023oqk  and in the introduction,
the prescription for D4 in massive type IIA is also similar to (31), with
different degrees of homogeneity:

$\nu_{D4}\,(2-\delta_{AB})\,M_{AB}=-\frac{\partial}{\partial\lambda_{AB}}\mathds{V}^{(3)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\,,\qquad
F=\mathds{V}^{(5)}(\lambda_{A},\lambda_{AB},\epsilon_{I})\>.$ (179)

The rest of the discussion is very similar to the section 3.1. We can write
the flux equations as

$\nu_{D4}\,M_{AB}=-\int_{\mathbb{M}_{4}}\frac{\mathcal{C}_{A}\,\mathcal{C}_{B}\big{(}\tau{T}\big{)}^{2}}{2\,\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}=-\sum_{a}\frac{B^{(2)}_{a}\cdot\big{(}\mathcal{C}_{A}\,\mathcal{C}_{B}\big{)}|_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(180)

where the equivariant forms $\mathcal{C}_{A}$, $\tau{T}$ and the
$B^{(\alpha)}_{a}$ are defined respectively by
(LABEL:AdS3xM8_equivariant_forms) and (42). These equations are identical to
the ones of section 3.1, with the only difference being that $B^{(2)}_{a}$
takes the place of $B^{(1)}_{a}$. The solution can be read from
(LABEL:AdS3xM8_flux_equations_solution) and (51):

$\displaystyle B^{(2)}_{a}=-\nu_{D4}\,N\>,$ (181) $\displaystyle
M_{AB}=N\sum_{c,d}\mathfrak{t}_{A}^{c}\,\mathfrak{t}_{B}^{d}\,D_{cd}\>,$

with $\mathfrak{t}_{A}^{c}$ given by (52).

Following prescription (179), the solution
(LABEL:AdS2xM8_flux_equations_solution) must be substituted in the expression
for $\mathds{V}^{(5)}$, which depends on the $B^{(5)}_{a}$ as

$\mathds{V}^{(5)}(\lambda_{A},\lambda_{AB},\epsilon_{I})=\sum_{a}\frac{B^{(5)}_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>.$
(182)

The relation between $B^{(5)}_{a}$ and $B^{(2)}_{a}$ is given in equation
(43), with the added complication that the exponents are half integers and
thus we need to be careful about the signs:

$\displaystyle
B^{(5)}_{a}=\eta_{a}\,\frac{2^{\frac{5}{2}}}{5!}\,\big{|}\nu_{D4}\,N\big{|}^{\frac{5}{2}}\,\Big{|}\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}\Big{|}^{\frac{3}{2}}\>,$
(183)
$\displaystyle\eta_{a}=\operatorname{sign}\Big{(}\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}\Big{)}\cdot\operatorname{sign}\big{(}\tau_{a}\big{)}\>.$

We note that the sign of
$\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}$ is the same as the
sign of $B^{(2)}_{a}$, and thus is fixed:

$\operatorname{sign}\Big{(}\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}\Big{)}=\operatorname{sign}\left(\frac{(\tau_{a})^{2}}{2\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}}\right)=\operatorname{sign}\big{(}-\nu_{D4}\,N\big{)}\equiv\,\sigma\>.$
(184)

The sign of the $\tau_{a}$ however is not fixed by
(LABEL:AdS2xM8_flux_equations_solution). We can rewrite the equations
$B^{(2)}_{a}=-\nu_{D4}\,N$ as

$\tau_{a}=\sigma\,\eta_{a}\,\sqrt{-2\,\nu_{D4}\,N\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}}\>.$
(185)

It is always possible to find $\lambda_{A}$ and $\lambda_{AB}$ that solve
these equations, whatever the value of $\eta_{a}$ might be.

For the free energy we can write

$F=\frac{2^{\frac{5}{2}}}{5!}\big{(}-\nu_{D4}\,N\big{)}^{\frac{5}{2}}\sum_{a}\frac{\eta_{a}\Big{(}(\epsilon_{3}-\epsilon_{4}+(\mathfrak{t}_{a}-1)\epsilon_{1}^{a}+(\mathfrak{t}_{a+1}-1)\epsilon_{2}^{a})(\epsilon_{4}-\mathfrak{t}_{a}\epsilon_{1}^{a}-\mathfrak{t}_{a+1}\epsilon_{2}^{a})\Big{)}^{\frac{3}{2}}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(186)

thus reproducing the extremal function in Faedo:2022rqx , upon setting
$\epsilon_{3}=2$.282828Compare formula (5.7) in Faedo:2022rqx  and set
$\varphi_{1}=\epsilon_{4}\,,\varphi_{2}=\epsilon_{3}-\epsilon_{4}\,,m_{a}\mathfrak{p}_{1}^{a}=\mathfrak{t}_{a}\,,m_{a}\mathfrak{p}_{2}^{a}=1-\mathfrak{t}_{a}\,,W=0$.
Our result for the free energy then matches theirs (up to an overall sign due
to different conventions) upon choosing
$(-\nu_{D4})^{\frac{5}{3}}=\frac{8\pi}{\sqrt{8-N_{f}}}$. Notice that in
Faedo:2022rqx  the vectors $v^{a}$ are taken to be primitive, contrary to the
convections we are using in this paper. Our $v^{a}$ are their $\hat{v}^{a}$.

The sign ambiguities remain to be fixed by a more careful analysis. For a
convex fan, supersymmetry is preserved with a topological twist and we expect
that all the $\eta_{a}$ have the same sign Faedo:2022rqx . This could follow
from a generalization of the following argument valid for the equivariant
volume with single times only. The $\lambda_{A}$ determine the polytope

$\mathcal{P}=\\{y_{I}\in\mathbb{R}^{4}\>|\>y_{I}V_{I}^{A}\geq\lambda_{A}\\}\>.$
(187)

Naturally $\mathcal{P}$ must be non-empty, so let us take
$y_{I}\in\mathcal{P}$. If we contract the inequalities
$y_{I}V_{I}^{A}\geq\lambda_{A}$ with $c_{1}^{\mathbb{T}_{4}}(L_{A})|_{a}$ we
get

$-y_{I}\epsilon_{I}\geq\tau_{a}\quad\forall a\in\\{1,\ldots,d\\}\>.$ (188)

Given that $\mathcal{P}$ is a resolved cone and that the equivariant volume is
given by an integral over $\mathcal{P}$

$\mathds{V}(\lambda_{A},\epsilon_{I})=\int_{\mathcal{P}}\mathrm{d}^{4}y\>{\mathrm{e}}^{-y_{I}\epsilon_{I}}\>,$
(189)

then the exponent $-y_{I}\epsilon_{I}\geq\tau_{a}$ must be negative for
convergence. This implies $\tau_{a}\leq 0$, and thus $\eta_{a}=-\sigma$. By
choosing $\sigma=1$ we would find the result of Faedo:2022rqx .

The case of an anti-twist requires taking a non convex fan for
$\mathbb{M}_{4}$. This can be obtained by formally sending
$v^{a}\rightarrow\sigma^{a}v^{a}$ everywhere, implying
$\epsilon_{1}^{a}\rightarrow\sigma^{a}\epsilon_{1}^{a}$ and
$\epsilon_{2}^{a}\rightarrow\sigma^{a+1}\epsilon_{2}^{a}$. It was proposed in
Faedo:2022rqx  that the correct assignment of signs is
$\eta_{a}=-\sigma^{a}\sigma^{a+1}$, and it would be interesting to understand
this by a geometrical argument.

###  4.2 AdS${}_{3}\times M_{7}$ solutions in type IIB

In this section we consider AdS${}_{3}\times M_{7}$ solutions in type IIB,
where $M_{7}$ is a $S^{3}/\mathbb{Z}_{p}$ fibration over $B_{4}$, which could
potentially arise as the near horizon limit of a system of D3 branes wrapped
on a two-cycle in $B_{4}$. Explicit solutions of this type have been found in
Gauntlett:2006af ; Gauntlett:2006ns  for Kähler-Einstein $B_{4}$ or products
of Kähler-Einstein spaces. The case of _smooth_ Kähler $B_{4}$ has been
studied in Gauntlett:2019pqg  using the formalism of GK geometry and the GMS
construction Gauntlett:2018dpc . The orbifold case has not been considered in
the literature as of yet, so in this section we will take $B_{4}$ to be a
generic toric orbifold $B_{4}\equiv\mathbb{M}_{4}$ and we also allow a general
dependence on all the equivariant parameters, including those on the base
$\mathbb{M}_{4}$. As we already discussed, this is important to obtain the
correct critical point for generic $\mathbb{M}_{4}$ without particular
symmetries, even in the smooth case. We also hope that our general formulas in
terms of four-dimensional integrals will be useful to find a field theory
interpretation of these solutions.

With odd-dimensional $M_{7}$ we need to add one real dimension, the radial
one, as familiar in holography. The relevant CY4 geometry is given by the
fibration with $\mathbb{M}_{4}$ as the base and the Kähler cone over the Lens
space as the fiber, that is
$\mathbb{C}^{2}/\mathbb{Z}_{p}\hookrightarrow\text{CY}_{4}\to\mathbb{M}_{4}$.
This CY4 is toric and its fan is generated by the vectors

$V^{a}=(v^{a},1,\mathfrak{t}_{a})\,,\qquad V^{d+1}=(0,0,1,0)\,,\qquad
V^{d+2}=(0,0,1,p)\,,$ (190)

where as usual the vectors $v^{a}$ generate the fan of $\mathbb{M}_{4}$,
$a=1,\dots,d$.

The prescription reduces to the GMS construction Gauntlett:2019pqg

$\nu_{D3}\,M_{A}=-\frac{\partial}{\partial\lambda_{A}}\mathds{V}^{(2)}(\lambda_{A},\epsilon_{I})\,,\qquad
F=\mathds{V}^{(2)}(\lambda_{A},\epsilon_{I})\,,\qquad\sum_{A}M_{A}V_{I}^{A}=0\,.$
(191)

Here $\mathds{V}^{(2)}$ matches the “supersymmetric action” introduced in
Gauntlett:2018dpc . We know from Gauntlett:2018dpc  that there is no need to
use higher times for these solutions.

When $p=1$ the CY4 matches exactly the one of sections 3.1 and 4.1. The
equivariant volume is computed in the same manner, with only minor
corrections. The one-to-one correspondence between the fixed points of CY4 and
$\mathbb{M}_{4}$ given by
$(V^{a},V^{a+1},V^{d+1},V^{d+2})\leftrightarrow(v^{a},v^{a+1})$ still holds,
but the orders of the orbifold singularities now differ by a factor of $p$:

$d_{a,a+1,d+1,d+2}=p\,d_{a,a+1}\>.$ (192)

The inward normals to the faces of the cone generated by
$(V^{a},V^{a+1},V^{d+1},V^{d+2})$ are now given by

$\displaystyle U^{a}=(p\,u^{a}_{1},0,0)\,,$ (193) $\displaystyle
U^{a+1}=(p\,u^{a}_{2},0,0)\,,$ $\displaystyle
U^{d+1}=\big{(}(\mathfrak{t}_{a}-p)u^{a}_{1}+(\mathfrak{t}_{a+1}-p)u^{a}_{2}\,,p\,d_{a,a+1},-d_{a,a+1}\big{)}\,,$
$\displaystyle
U^{d+2}=(-\mathfrak{t}_{a}u^{a}_{1}-\mathfrak{t}_{a+1}u_{a}^{2}\,,0,d_{a,a+1})\,.$

From these we can derive the restriction of the equivariant Chern forms of CY4
to the fixed points

$\begin{split}&c_{1}^{\mathbb{T}_{4}}(L_{b})|_{a}=-(\epsilon_{1}^{a}\,\delta_{a,b}+\epsilon_{2}^{a}\,\delta_{a+1,b})\,,\\\\[5.69054pt]
&c_{1}^{\mathbb{T}_{4}}(L_{d+1})|_{a}=\frac{-p\,\epsilon_{3}+\epsilon_{4}-(\mathfrak{t}_{a}-p)\epsilon_{1}^{a}-(\mathfrak{t}_{a+1}-p)\epsilon_{2}^{a}}{p}\,,\\\
&c_{1}^{\mathbb{T}_{4}}(L_{d+2})|_{a}=\frac{-\epsilon_{4}+\mathfrak{t}_{a}\epsilon_{1}^{a}+\mathfrak{t}_{a+1}\epsilon_{2}^{a}}{p}\,,\end{split}$
(194)

and the respective restrictions to the base $\mathbb{M}_{4}$

$\displaystyle\mathcal{C}_{a}\,=\,c_{1}{T}(L_{a})\>,\qquad a=1,\ldots,d\>,$
(195)
$\displaystyle\mathcal{C}_{d+1}\,=\,\frac{-p\,\epsilon_{3}+\epsilon_{4}+\sum_{a}(\mathfrak{t}_{a}-p)c_{1}{T}(L_{a})}{p}\>,$
$\displaystyle\mathcal{C}_{d+2}\,=\,\frac{-\epsilon_{4}-\sum_{a}\mathfrak{t}_{a}c_{1}{T}(L_{a})}{p}\>.$

It is easy to verify that these forms satisfy
$\sum_{a}v_{i}^{a}\,c_{1}{T}(L_{a})=-\epsilon_{i}$ and
$\sum_{A}V_{I}^{A}\,\mathcal{C}_{A}=-\epsilon_{I}$.

The second degree homogeneous component of the equivariant volume can be
written as an integral on the base $\mathbb{M}_{4}$ as follows:

$\mathbb{V}^{\,(2)}(\lambda_{A}\,,\epsilon_{I})=\int_{\mathbb{M}_{4}}\,\frac{(\tau{T})^{2}}{2\,p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}\>,\qquad\tau{T}=\sum_{A}\lambda_{A}\,\mathcal{C}_{A}\>.$
(196)

The flux constraints then read

$-\nu_{D3}\,M_{A}=\partial_{\lambda_{A}}\mathbb{V}^{\,(2)}(\lambda_{A}\,,\epsilon_{I})=\int_{\mathbb{M}_{4}}\,\frac{\mathcal{C}_{A}\>\tau{T}}{p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}=\sum_{a}\frac{B^{(1)}_{a}\cdot\big{(}\mathcal{C}_{A}\big{)}|_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(197)

where $B_{a}^{(1)}$ is the restriction to the $a$-th fixed point of the form

$B^{(1)}\equiv\frac{\tau{T}}{p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}\>.$
(198)

The solution to equations (197) takes the following form:

$\nu_{D3}^{-1}\,B^{(1)}_{a}=b(\epsilon_{I})-\sum_{b}m_{b}\,c_{1}{T}(L_{b})|_{a}\>,$
(199)

where the $m_{b}$ are such that $M_{a}=\sum_{b}D_{ab}\,m_{b}$. Indeed, if we
substitute this expression into the right hand side of (197) for $A\equiv
b\in\\{1,\ldots,d\\}$ we obtain

$\sum_{a}\frac{B^{(1)}_{a}\cdot\big{(}\mathcal{C}_{b}\big{)}|_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}=\nu_{D3}\int_{\mathbb{M}_{4}}\big{(}b(\epsilon_{I})-\sum_{a}m_{a}\,c_{1}{T}(L_{a})\big{)}\,\mathcal{C}_{b}=-\nu_{D3}\sum_{a}D_{ab}\,m_{a}\>,$
(200)

thus recovering the left hand side of (197). When $A=d+1,d+2$ from a similar
computation we find that

$M_{d+1}=\sum_{a}\frac{(\mathfrak{t}_{a}-p)M_{a}}{p}\,,\qquad
M_{d+2}=-\sum_{a}\frac{\mathfrak{t}_{a}M_{a}}{p}\>,$ (201)

which are precisely the values of $M_{d+1}$ and $M_{d+2}$ necessary to satisfy
the relation expected from the fluxes, $\sum_{A}V_{I}^{A}M_{A}=0$.

So far we have not specified the value of $b(\epsilon_{I})$ in (199). Using
the gauge invariance (20) we can fix three parameters $\lambda_{A}$. Therefore
only $d-1$ of the restrictions of $\tau{T}$ to the fixed points are
independent, which translates into a relation among the $B_{a}^{(1)}$ that we
use to fix the value of $b(\epsilon_{I})$. This can be seen by observing that
$\tau{T}$ is an equivariant two-form and thus its integral over
$\mathbb{M}_{4}$ vanishes, giving us

$0=\int_{\mathbb{M}_{4}}\tau{T}=\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}B^{(1)}\,.$
(202)

The value of $b(\epsilon_{I})$ that satisfies this relation can then be
written as

$b(\epsilon_{I})=\frac{\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\sum_{a}m_{a}\,c_{1}{T}(L_{a})}{\int_{\mathbb{M}_{4}}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}}\>.$
(203)

We observe that the reason why we had to turn on the higher times in the cases
studied in the previous sections was related to the fact that $d-1$
independent parameters were not enough to solve the flux constraints. In the
case considered in this section however the $d-1$ independent restrictions of
$\tau{T}=\sum_{A}\lambda_{A}\,\mathcal{C}_{A}$ are sufficient and there is no
necessity to include higher times. Nonetheless, it is interesting to repeat
the same computation of this section with the addition of higher times, which
we report in appendix B.

The free energy is given by the second degree homogeneous component of the
equivariant volume, which we write as

$F=\mathbb{V}^{\,(2)}(\lambda_{A}\,,\epsilon_{I})=\int_{\mathbb{M}_{4}}\,\frac{(\tau{T})^{2}}{2\,p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}=\sum_{a}\frac{B^{(2)}_{a}}{d_{a,a+1}\>\epsilon_{1}^{a}\>\epsilon_{2}^{a}}\>,$
(204)

where the $B^{(2)}_{a}$ are the restrictions of the integrand to each fixed
point. The value of the Kähler moduli, and consequently of the $B^{(2)}_{a}$,
is fixed by the flux constraints. We can easily do this by employing the same
strategy as formula (43) to relate the $B_{a}^{(2)}$ to the $B_{a}^{(1)}$:

$\displaystyle
B^{(2)}_{a}\equiv\bigg{(}\frac{\big{(}\tau{T}\big{)}^{2}}{2\,p\>\mathcal{C}_{d+1}\>\mathcal{C}_{d+2}}\bigg{)}\Big{|}_{a}=\frac{p}{2}\,\big{(}B^{(1)}_{a}\big{)}^{2}\,\big{(}\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\big{)}|_{a}$
(205)
$\displaystyle=\frac{\nu_{D3}^{2}}{2p}\big{(}b(\epsilon_{I})+m_{a}\epsilon_{1}^{a}+m_{a+1}\epsilon_{2}^{a}\big{)}^{2}\big{(}p\epsilon_{3}-\epsilon_{4}+(\mathfrak{t}_{a}-p)\epsilon_{1}^{a}+(\mathfrak{t}_{a+1}-p)\epsilon_{2}^{a}\big{)}\big{(}\epsilon_{4}-\mathfrak{t}_{a}\epsilon_{1}^{a}-\mathfrak{t}_{a+1}\epsilon_{2}^{a}\big{)}.$

We can also write the free energy as an integral over $\mathbb{M}_{4}$ as
follows:

$F=\frac{p}{2}\,\nu_{D3}^{2}\int_{\mathbb{M}_{4}}\big{(}b(\epsilon_{I})-\sum_{a}m_{a}\,c_{1}{T}(L_{a})\big{)}^{2}\>\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\>.$
(206)

Notice that the equation that fixes $b(\epsilon_{I})$ can be written as

$\int_{\mathbb{M}_{4}}\big{(}b(\epsilon_{I})-\sum_{a}m_{a}\,c_{1}{T}(L_{a})\big{)}\>\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}=0$
(207)

and can be used to further rewrite the free energy as

$F=-\frac{p}{2}\,\nu_{D3}^{2}\int_{\mathbb{M}_{4}}\sum_{a}m_{a}\,c_{1}{T}(L_{a})\big{(}b(\epsilon_{I})-\sum_{a}m_{a}\,c_{1}{T}(L_{a})\big{)}\>\mathcal{C}_{d+1}\,\mathcal{C}_{d+2}\>.$
(208)

It would be very interesting to understand if our formulas can be written as
the integral of the anomaly polynomial for some D3 brane theory wrapped over a
two-cycle in $\mathbb{M}_{4}$ and thus providing a field theory interpretation
of the solution.

####  4.2.1 Examples: Kähler-Einstein and Hirzebruch surfaces

We can check that our general formalism reproduces the know expressions for
the toric cases $\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3
with equal fluxes. The fan and intersection matrix are given in (LABEL:fanKE)
and (LABEL:intersKE). We take all the $M_{a}\equiv M$,
$\mathfrak{t}_{a}\equiv\mathfrak{t}$ and $m_{a}\equiv m$ equal. We find
$\sum_{ab}D_{ab}=dm_{k}$ and $M=mm_{k}$ with $m_{k}=3,2,1$ for
$\mathbb{P}^{2}$, $\mathbb{P}^{1}\times\mathbb{P}^{1}$ and dP3, respectively.
Since $\sum_{abc}D_{abc}=0$, there is no linear term in $\epsilon_{1,2}$ in
$\mathds{V}^{(2)}$ which is extremized at $\epsilon_{1,2}=0$. By expanding
(207) in integrals of Chern classes we find

$b=\frac{m[(p\epsilon_{3}-\epsilon_{4})\mathfrak{t}+\epsilon_{4}(p-\mathfrak{t})]}{\mathfrak{t}(p-\mathfrak{t})}\,$
(209)

and

$\mathds{V}^{(2)}=-\nu_{D3}^{2}\frac{dM^{2}[\epsilon_{3}\epsilon_{4}p(p-3\mathfrak{t})\mathfrak{t}+\epsilon_{3}^{2}p^{2}\mathfrak{t}^{2}+\epsilon_{4}^{2}(p^{2}-3p\mathfrak{t}+3\mathfrak{t}^{2})]}{2pm_{k}(p-\mathfrak{t})\mathfrak{t}}\,,$
(210)

which reproduces formula (5.5) in Gauntlett:2019pqg  with $\epsilon_{3}=2$ and
$\epsilon_{4}=b_{2}$.292929In Gauntlett:2019pqg  $N$ is the flux of the five-
cycle fibered over $c_{1}/m_{k}$ in $\mathbb{M}_{4}$. To compare the formulas
we need to identify $M=\frac{m_{k}N}{d}$ and
$\mathfrak{t}=\frac{n_{2}}{m_{k}}$, which follows from (5.5). The formulas
match for $\nu_{D3}=\sqrt{6}$. This still needs to be extremized with respect
to $\epsilon_{4}$.

As we already discussed in section 3.2, the critical point is generically at a
non-zero value of $\epsilon_{1}$ and $\epsilon_{2}$, unless there is some
extra symmetry in the background and the fluxes. As an example where the
critical point is not at $\epsilon_{1}=\epsilon_{2}=0$ we consider the case of
the Hirzebruch surface $\mathbb{M}_{4}=\mathbb{F}_{k}$ with fan

$v^{1}=(1,0)\,,\qquad v^{2}=(-k,1)\,,\qquad v^{3}=(-1,0)\,,\qquad
v^{4}=(0,-1)\,.$ (211)

The constraint $\sum_{A}V^{A}_{I}M_{A}=0$ leaves two independent fluxes on the
base $\mathbb{M}_{4}$ and two fluxes associated with the fibre

$\displaystyle M_{3}=M_{1}-k\,M_{2}\,,\qquad M_{4}=M_{2}\,,$ (212)
$\displaystyle
M_{5}=\frac{M_{1}(\mathfrak{t}_{1}+\mathfrak{t}_{3}-12)+M_{2}(6k-12+\mathfrak{t}_{2}+\mathfrak{t}_{4}-k\mathfrak{t}_{3})}{6}\,,$
$\displaystyle
M_{6}=-\frac{M_{1}(\mathfrak{t}_{1}+\mathfrak{t}_{3})+M_{2}(\mathfrak{t}_{2}+\mathfrak{t}_{4}-k\mathfrak{t}_{3})}{6}\,.$