monthdayyeardate[] ,

Tadpole conjecture
in non-geometric backgrounds

Katrin Becker1, Nathan Brady1, Mariana Graña2, Miguel Morros2,

Anindya Sengupta1 and Qi You1

1 Mitchell Institute for Fundamental Physics and Astronomy, Texas A$\&$M
University College Station, TX 77843, USA

2 Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS Orme des
Merisiers, 91191 Gif-sur-Yvette CEDEX, France

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Abstract

Calabi-Yau compactifications have typically a large number of complex
structure and/or Kähler moduli that have to be stabilised in
phenomenologically-relevant vacua. The former can in principle be done by
fluxes in type IIB solutions. However, the tadpole conjecture proposes that
the number of stabilised moduli can at most grow linearly with the tadpole
charge of the fluxes required for stabilisation. We scrutinise this conjecture
in the $2^{6}$ Gepner model: a non-geometric background mirror dual to a rigid
Calabi-Yau manifold, in the deep interior of moduli space. By constructing an
extensive set of supersymmetric Minkowski flux solutions, we spectacularly
confirm the linear growth, while achieving a slightly higher ratio of
stabilised moduli to flux charge than the conjectured upper bound. As a
byproduct, we obtain for the first time a set of solutions within the tadpole
bound where all complex structure moduli are massive. Since the $2^{6}$ model
has no Kähler moduli, these show that the massless Minkowski conjecture does
not hold beyond supergravity.

###### Contents

  1. 1 Introduction
  2. 2 Flux compactifications and tadpole conjecture
    1. 2.1 Flux compactifications
    2. 2.2 Moduli stabilisation and tadpole conjecture
  3. 3 The $2^{6}$ model
    1. 3.1 Orientifold and Hodge numbers
    2. 3.2 Cycles and periods
    3. 3.3 Fluxes and tadpole cancellation condition
    4. 3.4 Mass matrix
  4. 4 Solutions
    1. 4.1 $n$-$\chi_{{\mathbf{l}}}$ solutions
    2. 4.2 Integral bases for the flux lattice
    3. 4.3 Solutions with all moduli massive
  5. 5 Conclusions
  6. A Homology and cohomology of the $2^{6}$ model
    1. A.1 Cohomology
    2. A.2 Homology
  7. B Two and three $\chi$ solutions

##  1 Introduction

Calabi-Yau (CY) flux compactifications have been one of the main focuses of
string phenomenology in the past years, for multiple reasons. On one hand, the
Standard Model can easily be embedded in the gauge group of compactifications
of F-theory or the heterotic theory to four-dimensions (see e.g. [1, 2]), and
on the other, they can be used as the building block of de Sitter vacua (see
[3] for a review). Furthermore, there are powerful algebraic techniques to
build Calabi-Yau manifolds, which have been used to construct thousands of
them (see [4] for a list of complete intersection Calabi-Yau three-folds).
However, Calabi-Yau manifolds have moduli spaces with typical dimensions of
the order of hundreds. Moduli are scalar fields in the four-dimensional
effective theory, which, without a potential, would lead to long range “fifth
forces”. Since there are very tight experimental bounds on such forces, any
phenomenologically-relevant compactification should have no moduli.

Fluxes are the only known mechanism to generate a potential for the moduli at
the perturbative level. However, fluxes have an associated energy-momentum
tensor which acts as a source for the Ricci curvature, forcing one to explore
other non-Ricci flat backgrounds. There is nonetheless a way to combine NSNS
and RR three-form fluxes in type IIB, or more generally four-form fluxes in
F-theory, such that Calabi-Yau manifolds (with a metric that is related to the
Ricci flat one by an overall function) are still a solution [5, 6]. Fluxes
then generate a potential for the complex structure moduli and dilaton [7, 8],
encoded in the Gukov-Vafa-Witten (GVW) superpotential [9]. Contrarily, Kähler
moduli can only be stabilised non-perturbatively in this scenario.

Fluxes generate D-brane charges that have to be cancelled globally as required
by the tadpole cancellation condition. It is easy to see that in these
compactifications the flux generates a positive D3 charge, requiring O-planes
for a consistent global solution111Generically, a supergravity analysis shows
that any flux configuration gives a positive contribution to the four-
dimensional energy, making orientifolds a necessary ingredient of four-
dimensional Minkowski vacua [10].. The total amount of negative D3 charge
(coming from O3-planes and/or D7/O7 planes wrapped on 4-cycles) gives an upper
bound to the flux charge $N_{\rm flux}$, leading to a finite number of vacua
in a given Calabi-Yau manifold [11, 12].

While it was widely believed that turning on a few fluxes with an order one
$N_{\rm flux}$ was enough to fix all moduli, the tadpole conjecture [13],
challenged this common lore by stipulating a linear relation between the
number of moduli stabilised (at a generic point in moduli space) and $N_{\rm
flux}$. More concretely, the proposal is that for a large number of moduli

$n_{\rm stab}<\beta\,N_{\rm flux}\,,$

with $\beta$ an order 1 number. The refined version of the conjecture proposes
an upper bound $\beta<3$. Besides the “experimental support”, the rationale
behind this bound comes from F-theory, where the negative O3-charge of
7-branes (typically much larger than that of O3 planes, which was found to be
at most 25 in complete intersection Calabi-Yau three-folds [14]) is given by
the Euler characteristic of the Calabi-Yau 4-fold. For a large number of
complex structure moduli, the latter is proportional to 1/4 times the number
of moduli. Then, as long as $\beta<4$ one cannot stabilise all complex
structure moduli in F-theory compactifications with a large number of them.

The tadpole conjecture (including its original bound on $\beta$) was shown to
hold in different setups. The original one was F-theory on K3 $\times$ K3
four-folds, as well as in toy models with lower-dimensional cohomology
lattices [15]. In K3 $\times$ K3, full moduli stabilisation at points in
moduli space where the gauge symmetry is purely Abelian, was achieved beyond
the tadpole bound, with ${n_{\rm stab}}/{N_{\rm flux}}=57/25=2.28$. Other
examples of full complex structure moduli stabilisation in IIB at symmetric
points in moduli space (points with discrete symmetries) exist, with $\beta$
ranging from $272/124\simeq 2.19$ [16] to $2.86$ [17]. Note that these are
within the tadpole bound but include a contribution from D7-branes, which
either have unstabilised moduli or lead to non-Abelian gauge groups222On the
contrary, full stabilisation at symmetric points (in the deep interior of
moduli space) was shown to be in principle possible in supersymmetry-breaking
backgrounds with $N_{\rm flux}$ a priori independent of the number of
invariant and non-invariant moduli [18], and an explicit example was given
with very low $N_{\rm flux}$. The stabilisation points are not generic, and as
such do not violate the tadpole conjecture, but they lead to discrete, rather
than non-Abelian, gauge symmetries. We come back to this point in the
conclusions.. Stabilisation of F-theory on the sextic Calabi-Yau at the Fermat
point was studied in detail in [19, 20], where all complex structure moduli
are stabilised (beyond the tadpole bound) with the largest $\beta=2.94$. Note
that in F-theory, stabilisation of complex structure moduli implies
stabilisation of both the complex structure moduli of the base manifold and
its fiber, as well as the stabilisation of 7-brane moduli. Re. the latter, the
linear relation between the number of 7-brane moduli and $N_{\rm flux}$ was
proven for any weak Fano base [21], where $\beta=16/7\simeq 2.29$. Similarly,
the linear relation between (any) number of complex structure moduli
stabilised at large values or close to conifold point(s) can be very neatly
proven using the tools of asymptotic Hodge structure [22]. The parameter
$\beta$ is generically less than 1, but a more precise bound requires
knowledge of the structure of the asymptotic limit. In the examples analysed,
a very crude estimate gives $\beta$ well below 1.4.

By finding a very large set of supersymmetric Minkowski solutions, in this
paper we spectacularly confirm the linear growth of $N_{\rm flux}$ with the
number of stabilised moduli, while achieving a slightly higher bound for
$\beta$, in a completely different setup: non-geometric backgrounds in a
substringy and strongly coupled region of moduli space, accessed via their
world-sheet description in terms of Landau-Ginzburg (LG) orbifolds [23, 24].
More precisely, we analyse the so-called $2^{6}$ Gepner model, corresponding
to a tensor product of 6 minimal models at level 2. Besides being at the deep
interior of moduli space, this model is particularly interesting as it has no
chiral-antichiral deformations, and consequently the associated cohomolgy,
$h^{1,1}$, is trivial. There are thus no Kähler moduli, and as such there is
no notion of volume, forbidding a geometric description in this setup. Mirror
symmetry relates this model to a (rigid)
$T^{6}/({\mathbb{Z}}_{4}\times{\mathbb{Z}}_{4}$) orbifold. Despite being non-
geometric, supersymmetric cycles, D-branes and O-planes are well defined [25].
Furthermore, using perturbative non-renormalisation theorems for the Gukov-
Vafa-Witten flux superpotential, as well as the absence of Kähler moduli to
argue for its non-renormalisation at the non-perturbative level as well, one
can claim that this superpotential correctly describes the four-dimensional
effective theory also for these non-geometric vacua [26]. Using then a
combination of world-sheet and space-time techniques, [26] studied complex
structure/dilaton moduli stabilisation in the $2^{6}$ and $1^{9}$ models, none
of which has Kähler moduli.

The absence of Kähler moduli led to the belief that all moduli were stabilised
in the supersymmetric Minkowski vacua discovered. However, in light of the
tadpole conjecture, the $1^{9}$ model was revisited in [27, 28], showing that
full moduli stabilisation actually requires going beyond the tadpole bound.
Furthermore, the linear growth proposed by the tadpole conjecture was
remarkably verified [29], with the largest $\beta=57/21\simeq 2.71$ ($N_{\rm
flux}=21$ and $57$ massive moduli). A larger ratio has been achieved in the
$1^{9}$ model after the publication of [29], namely $\beta=63/23\simeq 2.74$,
very close to 3.

The $2^{6}$ model (under the specific orientifold we are considering) has the
advantage of a smaller ratio $h^{2,1}/|Q_{O3}|=90/40=2.25$ (versus
$63/12=5.25$ for the $1^{9}$), allowing in principle for full moduli
stabilisation within the tadpole bound. We will show that this is indeed the
case: we find some solutions with all moduli stabilised in a supersymemtric
Minkowski vacua exactly at (as well as slightly below) the tadpole bound. This
shows that the massless Minkowski conjecture [30] proposing that in
supersymmetric Minkowski supergravity solutions there are always left-over
moduli, does not hold beyond the supergravity regime.

The spacetime superpotential $W$ à la Gukov-Vafa-Witten, although
perturbatively and non-perturbatively exact, is an infinite series in the
moduli deformation parameters of the model and one has to resort to an order-
by-order computation. The mass matrix, related to the Hessian
$\partial_{I}\partial_{J}W$, contains information up to order $2$ in this
Taylor series expansion. The rank of the Hessian is the number of massive
moduli which is a lower bound on the total number of fields stabilised by the
superpotential including interaction terms to all orders. The mechanism of
stabilising massless fields at higher order in $W$ has been studied in detail
for the $1^{9}$ model most recently in [29]. In the paper [31] jointly
submitted with ours, the authors study higher-order stabilisation in the
$2^{6}$ model. Remarkably, they find examples where less-than-full-rank
solutions get lifted to full stabilisation upon including order $4$ terms in
$W$. These are solutions with massless fields but no flat directions in the
moduli space, and nicely complement the solutions we present in section 4.3
where all moduli are stable by acquiring a mass. Additionally, [31] finds a
solution where the bound $\beta<3$ is satisfied at quadratic level but
slightly violated at higher order, which also complements the handful of
solutions we find here where this upper bound on $\beta$ is also slightly
violated. All solutions studied in [31], up to orders $4,5$ in $W$, as well as
ours, are consistent with the bound $\beta<4$, as suggested by the F-theory
tadpole cancellation condition.

The paper is organised as follows: in section 2 we review the basic notions of
flux compactifications and present the tadpole conjecture. In section 3 we
present the $2^{6}$ Landau-Ginzburg orbifold, relegating the detailed
computation of its homology and cohomology to Appendix A. The extensive
results obtained using the simplest flux configurations, as well as those
using a strategy for the search of solutions with low $N_{\rm flux}$ are
exposed in section 4. We also present there one solution with all moduli
stabilised. Further solutions obtained by an intensive search are given in
Appendix B. All solutions found are collected in Figure 1.

##  2 Flux compactifications and tadpole conjecture

Even though the $2^{6}$ Landau-Ginzburg orbifold model does not have a Calabi-
Yau analogue, much of the tools of geometric flux compactifications apply, as
argued in [26]. We briefly review those arguments.

In LG models, orientifold planes and D-branes wrapped on cycles are well
understood [25, 32, 33]. The topologically non-trivial 3-cycles, associated to
the (co)homology $h^{(2,1)}$ of chiral-chiral marginal deformations of the
superpotential, are well defined (see Appendix A for a brief review). Despite
the fact that the world-sheet formulation of RR fluxes is not completely
understood, one can resort to a space-time geometric description where each
distinct homology cycle can support a quantised flux. Combinations of NS and
RR fluxes induce a D-brane charge which must be cancelled globally by D-branes
or orientifold planes. In other words, these fluxes must satisfy the same
tadpole condition as in geometric compactifications.

In a geometric setup, odd (even) fluxes generate a potential for the complex
structure (Kähler) moduli of the Calabi-Yau manifold. In the four-dimensional
${\cal N}=1$ low energy effective theory of Calabi-Yau orientifold
compactifications, the potential can be derived from a holomorphic
superpotential (one also needs the Kähler potential, but in the case we are
interested in the Kähler potential plays no role). The superpotential can also
be computed as the tension of a domain wall separating two vacua with
different units of flux, which extends to LG models where D-branes are also
well-understood. The superpotential does not receive perturbative corrections,
while the non-perturbative instanton corrections depend on the Kähler moduli
in type IIB compactifications. Since these are absent in the present setup,
the classical flux potential is not corrected, and should therefore be
applicable to LG models.

Below we review the basic features of type IIB flux compactifications,
pointing out an actual important difference with the geometric setup, which
will play no role in the present paper, but is worth emphasizing: the absence
of Kähler moduli allows for supersymmetric AdS vacua.

###  2.1 Flux compactifications

We take type IIB backgrounds with NSNS and RR fluxes

$\int_{\Gamma_{n}}F_{3}=N_{n}\in{\mathbb{Z}}\,,\quad\int_{\Gamma_{n}}H_{3}=M_{n}\in{\mathbb{Z}}$
(2.1)

where $\Gamma_{n}$, $n=1,...,2h^{2,1}+2$, is a basis of integral 3-cycles
($\Gamma_{n}\in H_{3}(M,{\mathbb{Z}})$).Three-form fluxes generate a
superpotential, that as argued above is the same as for geometric CY
compactifications with O3 planes, given by the Gukov-Vafa-Witten form

$W=\int G_{3}\wedge\Omega\ ,$ (2.2)

where

$G_{3}=F_{3}-\tau H_{3}=(N^{n}-\tau M^{n})\gamma_{n}\ ,$ (2.3)

$\Omega$ is the holomorphic $(3,0)$-form and $\gamma_{n}$ is a dual basis to
the cycles $\Gamma_{n}$. The combination of RR and NSNS three-form fluxes has
an induced D3-charge $N_{\rm flux}$, which has to be canceled by orientifold
planes and eventually D-branes:

$N_{\rm flux}=\int_{M}F_{3}\wedge
H_{3}=\frac{1}{\tau-\bar{\tau}}\int_{M}G_{3}\wedge\bar{G}_{3}=|Q_{O3}|-N_{D3}\
.$ (2.4)

$N_{\rm flux}$ is then bounded by the absolute value of the (negative) charge
of orientifold planes

$N_{\rm flux}\leq|Q_{O3}|\ .$ (2.5)

In this paper we focus on supersymmetric, Minskowski vacua, satisfying
$DW=0,W=0$, where the derivative is done with respect to all moduli. For
complex structure moduli $t^{I}$, $I=1,...,h^{2,1}$, this reads

$0=D_{t^{I}}W=\int_{M}G_{3}\wedge\chi_{I}\,\quad$ (2.6)

where $\chi_{I}$ are the (2,1) forms in cohomology.

In compactifications on ordinary Calabi-Yau manifolds, the F-term equations
for the Kähler moduli $D_{\rho}W=0$, require $\langle W\rangle\equiv W_{0}=0$,
and therefore supersymmetric vacua are necessarily Minkowski. There are no
Kähler moduli in our situation, and the flux superpotential allows for
supersymmetric AdS solutions with $W_{0}\neq 0$. However, we will be
restricting to Minkowski vacua, and the condition $W_{0}=0$ together with
$D_{\tau}W=0$ sets both $(0,3)$ and $(3,0)$ pieces of the 3-form fluxes to
zero333The condition $D_{\tau}W$ in the present setup is actually different
than in ordinary ISD flux compactifications, since the Kähler potential is
$K=-4\log[-i(\tau-\bar{\tau})]$ instead of the usual one [27]. The extra
factor of 4 can intuitively be understood as the dilaton piece “eating” the
$-3\log[-i(\rho-\bar{\rho})]$ of the volume modulus. As a consequence,
$D_{\tau}W$ involves a particular linear combination of (3,0) and (0,3) pieces
(to be more precise $(3G+\bar{G})\wedge\Omega$), and thus supersymmetric AdS
vacua have a (3,0) piece, and as such are not purely ISD.. In summary,

$G_{\rm Mink}\in H^{(2,1)}\ .$ (2.7)

This flux component is imaginary self-dual and, as such, it is straightforward
to show that $N_{\rm flux}>0$.

###  2.2 Moduli stabilisation and tadpole conjecture

The requirement that $G_{3}$ should be (2,1) is a complex structure moduli-
dependent condition (and also a condition on the axion-dilaton, since it
appears in the definition of $G_{3}$). If, for a given complex structure
moduli and dilaton, the flux is (2,1) and there are no infinitesimal
deformations that maintain this property, then all moduli are fixed. If on the
contrary there are $n$ independent ways of deforming the moduli without
altering this property, then the left-over moduli space has dimension $n$. At
linear order in the deformations, the number of fixed moduli ($h^{2,1}+1-n$,
where the $+1$ comes from the axion-dilaton) is determined by (half of) the
rank of the mass matrix

$M=\begin{pmatrix}D_{I}D_{J}W&\bar{D}_{\bar{I}}D_{J}W\\\
D_{I}\bar{D}_{\bar{J}}\bar{W}&\bar{D}_{\bar{I}}\bar{D}_{\bar{J}}\bar{W}\end{pmatrix}\bigg{|}_{D_{I}W=0,W=0}=\begin{pmatrix}\partial_{I}\partial_{J}W&0\\\
0&\bar{\partial}_{\bar{I}}\bar{\partial}_{\bar{J}}\bar{W}\end{pmatrix}\ .$
(2.8)

The number of massive moduli is then given by

$n_{\rm massive}={\rm rank}\,(\partial_{I}\partial_{J}W)\ .$ (2.9)

While it was believed that turning on fluxes for a few cycles should be enough
to stabilise all moduli, the tadpole conjecture [13] put forward a direct
linear relation between the number of moduli stabilised $n_{\rm stabilised}$
and the flux contribution to the tadpole $N_{\rm flux}$ for a given flux
vacua. More precisely, it says that at a given $N_{\rm flux}$, the number of
moduli that can be stabilised at a generic point in moduli space is

$n_{\rm stabilised}<\beta N_{\rm flux}\,\ .$ (2.10)

The refined version says $\beta<3$. As we will show, our results spectacularly
confirm this conjecture in the $2^{6}$ LG model, but with a value of $\beta$
slightly above this bound, though still well bellow $\beta=4$. The latter has
a clear meaning when there are Kahler moduli and D7-branes wrapping their
associated 4-cycles. D7-branes provide extra negative contributions to the
3-brane charge, and at the same time have associated moduli. D7-brane moduli
and complex structure moduli combine in the F-theory description into the
$h^{3,1}$ complex structure moduli of a Calabi-Yau four-fold. Similarly,
3-form fluxes and D7-brane fluxes combine into four-form fluxes, subject to
the tadpole cancellation condition

$N_{\rm
flux}\leq\frac{\chi}{24}=\frac{1}{4}(h^{3,1}+h^{1,1}-h^{2,1}+8)\sim\frac{1}{4}h^{3,1}\,,$
(2.11)

for $h^{3,1}\gg 1$, and $h^{3,1}\gg h^{1,1},h^{2,1}$. Hence, if $n_{\rm
stabilised}<4N_{\rm flux}$, necessarily $n_{\rm stabilised}<h^{3,1}$. Hence,
the bound $\beta<4$ forbids full moduli stabilisation in F-theory.

There are two caveats that are worth emphasizing. The first is that moduli can
be stabilised at higher orders (i.e., have no quadratic term in the potential,
but e.g. a quartic one), see [27] for a discussion in the context of Landau-
Ginzburg models, and [29, 31] for detailed analyses in the $1^{9}$ and $2^{6}$
LG models respectively. Considering only massive moduli as we do here, allows
to test a weaker form of the conjecture, namely

$n_{\rm massive}\,(\leq n_{\rm stabilised})<\beta N_{\rm flux}\ .$ (2.12)

The second caveat is that we are studying stabilisation at a particular,
rather than generic, point in moduli space, namely the Fermat point.
Therefore, in principle a violation of (2.10) in our setup would in principle
not provide a counter-example to the tadpole conjecture. We find however that
it is satisfied, which suggests that points where there is a left-over
discrete symmetry, rather than a non-Abelian as suggested in [13, 15], should
still be considered generic.

##  3 The $2^{6}$ model

Orbifolds of Landau-Ginzburg (LG) models are examples of ${\cal N}=(2,2)$ two-
dimensional conformal field theories that for central charges $c=\bar{c}=9$
can realise four-dimensional ${\cal N}=2$ string vacua [24]. These models also
admit orientifold projections, yielding phenomenologically interesting ${\cal
N}=1$ vacua. As in Calabi-Yau compactifications, the chiral-chiral and chiral-
antichiral ring of marginal deformations of the SCFT correspond to Kähler and
complex structure moduli of the four-dimensional theory. Of particular
interest are the models that have no (c,c) marginal deformations, as from the
Calabi-Yau point of view they would have no Kähler moduli. These cannot have a
conventional geometric description, as they have $h^{1,1}=0$, and therefore no
notion of volume. From the point of view of mirror symmetry, they are mirror
to “rigid” ($h^{2,1}=0$) Calabi-Yau manifolds.

In more detail, LG orbifold models are the IR fixed points of 2-dimensional
${\cal N}=(2,2)$ theories of a number $r$ of chiral fields $\Phi_{i}$ with a
relevant deformation coming from a world-sheet superpotential ${\cal
W}(\Phi_{i})$ that is a homogeneous function (or more generally weighted
homogeneous function)444This means the superpotential satisfies ${\cal
W}(\lambda^{\omega_{i}}\Phi_{i})=\lambda^{d}{\cal W}(\Phi_{i})$. Note that
${\cal W}=0$ looks very much like the defining equation for a Calabi-Yau
manifold. Indeed, for 5 chiral fields, the corresponding LG model can be
thought of as the “substringy regime” of a Calabi-Yau sigma model, where the
CY 3-manifold is defined by the equation ${\cal W}=0$ in ${\mathbb{C}P}^{4}$.
There are however orbifolds of LG models that do not have a Calabi-Yau
analogue, as the one we analyse here.. For any given superpotential with this
property, there exists a choice of Kähler potential ${\cal K}$ such that the
theory defined by the action

$S=\int d^{2}zd^{4}\theta{\cal K}(\Phi_{i},\bar{\Phi}_{i})+\int
d^{2}zd^{2}\theta\,{\cal W}(\Phi_{i})+{\rm c.c.}$ (3.1)

has an IR conformal fixed point [23]. For superpotentials of the form

${\cal W}=\sum_{i=1}^{r}\Phi_{i}^{k+2}\,,$ (3.2)

the SCFT at the IR fixed point is the Gepner model that is a tensor product of
$r$ minimal models at level $k$. These models are referred to as $k^{r}$ and
have a central charge

$c=\sum_{i=1}^{r}\frac{3k}{k+2}\ .$ (3.3)

When $c=9$ the 2d SCFTs can, in principle, be used as four-dimensional string
vacua. However, one still needs to demand that the charges of the NS states
under the $U(1)$ action that rotates the two supersymmetries is integer. This
requires quotienting out the theory by a discrete ${\mathbb{Z}}_{k+2}$ action.
We present below the $2^{6}$ model that is the focus of this paper.

###  3.1 Orientifold and Hodge numbers

The $2^{6}$ Gepner model has six scalar fields and a superpotential given by

$\mathcal{W}=\Phi_{1}^{4}+\ldots+\Phi_{6}^{4}~{}.$ (3.4)

divided by the $\mathbb{Z}_{4}$ orbifold action

$g:\Phi_{i}\rightarrow\omega\,\Phi_{i}~{},\quad{\rm with}\,\,\omega={\rm
e}^{\tfrac{2\pi{\rm i}}{4}}={\rm i}~{}.$ (3.5)

The IR fixed point corresponds to the tensor product of 6 minimal models at
level 2, with total central charge $c=6\times\tfrac{3}{2}=9$.

From the worldsheet point of view, the orientifold projection is realised by
dividing out by worldsheet parity and an involution $\sigma$ such that

${\cal W}(\sigma\Phi_{i})=-{\cal W}(\Phi_{i})~{}.$ (3.6)

The simplest such involution, that we choose in the following, is

$\sigma\,:\Phi_{i}\to e^{\frac{\pi{\rm i}}{4}}\,\Phi_{i}~{}\,,$ (3.7)

leading to O3 planes. The total charge of the O3-planes for this involution
can be computed using their overlap with the Ramond ground states [26] and is

$|Q_{O3}|=40~{}.$ (3.8)

In the framework of LG/CY correspondence, deformations of the compact space
are logically given by appropriate deformations of the worldsheet
superpotential. The cohomology ring of the compact space corresponds to the
ring of chiral-chiral primary fields

${\cal
R}=\quantity[\mathbb{C}[\Phi_{1},...,\Phi_{6}]/\langle\partial_{\Phi_{i}}{\cal
W}(\Phi_{1},...,\Phi_{6})\rangle]^{\mathbb{Z}_{4}}~{}.$ (3.9)

The marginal deformations are of the form
$\Phi_{1}^{l^{1}-1}...\Phi_{6}^{l^{6}-1}$ with $l^{i}\in\\{1,2,3\\}$ and
$\sum_{i}l^{i}=2\mod 4$. The correspondence between the value of this sum and
the degree of the corresponding harmonic forms at the Fermat point
(superpotential of the form 3.4) is given in Table 1 (see Appendix A.1 for
details).

$\sum l_{i}$ | 6 | 10 | 14 | 18
---|---|---|---|---
$H^{(p,q)}$ | $H^{(3,0)}$ | $H^{(2,1)}$ | $H^{(1,2)}$ | $H^{(0,3)}$
Number of forms | $1$ | 90 | 90 | 1
| $\Omega$ | $\chi_{\mathbf{l}}$ | $\bar{\chi}_{\mathbf{l}}$ | $\bar{\Omega}$
Table 1: Complex 3-forms at the Fermat point $t^{I}=0$, where
${\mathbf{l}}=(l_{1}...l_{6})$, $l^{i}\in\\{1,2,3\\}$

For each (2,1) form $\chi_{\mathbf{l}}$ there is an associated complex
structure deformation $t^{\mathbf{l}}$, and the correspondingly deformed
superpotential is

${\cal
W}(t^{\mathbf{l}})=\sum_{i=1}^{6}\Phi_{i}^{4}-\sum_{\mathbf{l}}t^{\mathbf{l}}\,\mathbf{\Phi}^{{\mathbf{l}}-1}$
(3.10)

where $\mathbf{\Phi}^{{\mathbf{l}}-1}\equiv\prod_{i=1}^{6}\Phi_{i}^{l^{i}-1}$.

It is easy to see that all 3-forms are odd under the orientifold projection
(3.7). The Hodge numbers for this orientifold model are then

$h^{1,1}=0~{},\ \ ~{}h^{2,1}_{-}=h^{2,1}=90~{}.$ (3.11)

The former implies this Landau-Ginzburg model cannot be associated to an
ordinary Calabi-Yau manifold, but rather to the mirror of a rigid one. The
mirror manifold is the orbifold
$T^{6}/({\mathbb{Z}}_{4}\times{\mathbb{Z}}_{4})$.

###  3.2 Cycles and periods

As explained in detail in Appendix A.2, a basis of 182 integral three-cycles
is given by

$\Gamma_{\mathbf{n}}\in H_{3}(M,{\mathbb{Z}}),\ \ {\mathbf{n}}=(n_{1}\ldots
n_{6})\,,\ \ n_{i}=0,1,2\,,$ (3.12)

where the ${\mathbf{n}}$ correspond to the first 182 integers written in base
3, completing with 0’s on the left to build a length six vector. Thus they go
from (000000) to (020201). To each $\Gamma_{\mathbf{n}}$ we associate its
Poincaré dual, the real 3-form $\gamma_{\mathbf{n}}$. The intersection form on
the charge lattice can be expressed in matrix form [26] as

$\mathbf{I}_{{\mathbf{n}}{\mathbf{m}}}=\int_{\Gamma_{\mathbf{n}}}\gamma_{\mathbf{m}}=\Gamma_{\mathbf{n}}\cap\Gamma_{\mathbf{m}}~{}.$
(3.13)

See eqn. (A.10) in Appendix A.2 for a more explicit form. The integral of a
complex 3-form $\gamma_{\mathbf{l}}$ over any real 3-cycle
$\Gamma_{\mathbf{n}}$ can be expressed from the worldsheet perspective via the
correspondence between Landau-Ginzburg A-branes/chiral primaries
$\Phi^{{\mathbf{l}}-{\mathbf{1}}}$ and spacetime 3-cycles/3-forms
$\gamma_{\mathbf{l}}$ (see Appendix A.2)

$\int_{\Gamma_{\mathbf{n}}}\gamma_{\mathbf{l}}=\int_{\Gamma_{\mathbf{n}}}d^{6}\Phi\,\,\mathbf{\Phi}^{{\mathbf{l}}-\mathbf{1}}e^{-{\cal
W}}\overset{t=0}{=}{\rm i}^{{\mathbf{n}}\cdot{\mathbf{l}}}~{}\prod_{i}(-1+{\rm
i}^{l^{i}})~{}.$ (3.14)

Let us also mention the existence of the so-called homogeneous basis of
(complex) cycles of the compact space $\\{C_{\mathbf{l}}\\}$ that are the
support of the (2,1) forms. They have the property

$\int_{C_{{\mathbf{l}}^{\prime}}}\chi_{{\mathbf{l}}}~{}=~{}2^{11}~{}\delta_{{\mathbf{l}},{\mathbf{l}}^{\prime}}\prod_{i}(1-{\rm
i}^{l^{i}})~{}$ (3.15)

which makes them very convenient for computations. Its explicit construction
can be found in Appendix A.2. Besides, these cycles have non-zero intersection
only with their complex conjugate, with value

$C_{{\mathbf{l}}}\cap C_{{\mathbf{l}}}^{*}=2^{12}\prod_{i}(1-{\rm
i}^{l^{i}})~{}.$ (3.16)

###  3.3 Fluxes and tadpole cancellation condition

As argued in section 2, RR and NSNS fluxes in LG orientifold models can be
treated using usual space-time descriptions. The basis
$\\{\Gamma_{\mathbf{n}}\\}$ defined in (3.12) supports quantised fluxes as in
(2.1), leading to a complex 3-form flux

$G_{3}=(N^{\mathbf{n}}-\tau M^{\mathbf{n}})\gamma_{\mathbf{n}}\ .$ (3.17)

However for convenience we mainly express the fluxes in the complex basis.
Since supersymmetric Minkowski solutions require $G_{3}$ to be (2,1), we
should have

$G_{3}=\sum_{\mathbf{l}}A_{\mathbf{l}}~{}\chi_{{\mathbf{l}}}~{}.$ (3.18)

The coefficients $A_{\mathbf{l}}$ should be such that fluxes are properly
quantised:

$\int_{\Gamma_{\mathbf{n}}}G=\sum_{\mathbf{l}}\big{(}A_{\mathbf{l}}~{}{\rm
i}^{{\mathbf{n}}.{\mathbf{l}}}\prod_{i}(1-{\rm
i}^{l_{i}})\big{)}=N_{{\mathbf{n}}}-\tau M_{{\mathbf{n}}}\ ,\ \ \
\forall\,{\mathbf{n}}~{}.\ \ $ (3.19)

Tadpole cancellation condition takes the same form as for geometric
compactifications, given in (2.4). For this LG orientifold model we have the
upper bound

$N_{\rm flux}\leq 40$ (3.20)

Note that

$\frac{h^{2,1}+1}{|Q_{O3}|}=\frac{91}{40}=2.275\,,$ (3.21)

and since this is less than 3, a solution within the tadpole bound where all
moduli are stabilised would not violate the tadpole conjecture.

Using the Riemann bilinear identity we can write $N_{\rm flux}$ in terms of
the complex coefficients $A_{\mathbf{l}}$ in (3.18)

$\displaystyle N_{\rm flux}$ $\displaystyle=\frac{1}{\tau-\bar{\tau}}\int
G_{3}\wedge\bar{G}_{3}$
$\displaystyle=\frac{1}{\tau-\bar{\tau}}\sum_{{\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3}}\frac{A_{{\mathbf{l}}_{1}}\bar{A}_{{\mathbf{l}}_{2}}}{C_{{\mathbf{l}}_{3}}\cap
C_{{\mathbf{l}}_{3}}^{*}}\int_{C_{{\mathbf{l}}_{3}}}\chi_{{\mathbf{l}}_{1}}\int_{C_{{\mathbf{l}}_{3}}^{*}}\bar{\chi}_{{\mathbf{l}}_{2}}~{},$
(3.22)
$\displaystyle=2^{12}\quantity(\sum_{{\mathbf{l}}=s(111133)}|A_{\mathbf{l}}|^{2}+\sum_{{\mathbf{l}}=s(111223)}2~{}|A_{\mathbf{l}}|^{2}+\sum_{{\mathbf{l}}=s(112222)}4~{}|A_{\mathbf{l}}|^{2})$

where $s$ stands for any permutation of the 6 elements of the vector
${\mathbf{l}}$, as these possibilities are all the ones that give $\sum
l^{i}=10$, and in the last line we have substituted $\tau={\rm i}$ (we comment
on this choice in section 4).

###  3.4 Mass matrix

As shown in (2.9), for supersymmetric Minkowski solutions the number of
massive moduli is given by the rank of the Hessian of the superpotential.

Using the flux expression (3.18), the GVW spacetime superpotential (2.2) reads

$W=\int
G_{3}\wedge\Omega=\sum_{{\mathbf{l}}_{1},{\mathbf{l}}_{2}}\frac{A_{{\mathbf{l}}_{1}}}{C_{{\mathbf{l}}_{2}}\cap
C_{{\mathbf{l}}_{2}}^{*}}\int_{C_{{\mathbf{l}}_{2}}}\chi_{{\mathbf{l}}_{1}}\int_{C_{{\mathbf{l}}_{2}}^{*}}\Omega~{}.$
(3.23)

This depends on the dilaton via the definition of $G_{3}$, such that

$\partial_{\tau}W=-\int
H_{3}\wedge\Omega=\frac{1}{\tau-\bar{\tau}}\int(G-\bar{G})\wedge\Omega$ (3.24)

The dependence on $t^{\mathbf{l}}$ can be found using

$\int_{C_{{\mathbf{l}}^{\prime}}}\chi_{{\mathbf{l}}}=\int_{C_{{\mathbf{l}}^{\prime}}}d^{6}\Phi~{}\mathbf{\Phi}^{{\mathbf{l}}-{\bf
1}}~{}e^{-{\cal W}(t^{\mathbf{l}})}~{}.$ (3.25)

where ${\cal W}(t^{\mathbf{l}})$ is the deformed superpotential (3.10), and
${\bf 1}=(111111)$. This expression is also valid for the holomorphic 3-form
$\Omega$, with ${\mathbf{l}}={\bf 1}$

We can now compute the second derivative of the superpotential with respect to
all moduli. At $\tau={\rm i}$ and at the Fermat point $t^{\mathbf{l}}=0$ we
get

$\displaystyle\left.\frac{\partial^{2}}{\partial t^{{\mathbf{l}}_{1}}\partial
t^{{\mathbf{l}}_{2}}}W~{}\right|_{t=0}$
$\displaystyle=\frac{1}{2}\sum_{{\mathbf{l}}_{3}}A_{{\mathbf{l}}_{3}}\delta({\mathbf{l}}_{1}+{\mathbf{l}}_{2}+{\mathbf{l}}_{3}\\!\\!\\!\\!\mod\mathbf{4}~{},\mathbf{1})\prod_{i=1}^{6}({\rm
i}^{l_{1}^{i}+l_{2}^{i}-1}-1)\Gamma\quantity(\frac{l_{1}^{i}+l_{2}^{i}-1}{4})$
$\displaystyle\left.\frac{\partial^{2}}{\partial
t^{{\mathbf{l}}_{1}}\partial\tau}W~{}\right|_{t=0}$ $\displaystyle=\frac{{\rm
i}}{4}\sum_{{\mathbf{l}}_{2}}\quantity[A_{{\mathbf{l}}_{2}}^{*}\delta({\mathbf{l}}_{1},{\mathbf{l}}_{2})-A_{{\mathbf{l}}_{2}}\delta({\mathbf{l}}_{1}+{\mathbf{l}}_{2}\\!\\!\\!\\!\mod\mathbf{4}~{},\mathbf{0})]\prod_{i=1}^{6}({\rm
i}^{l_{1}^{i}}-1)\Gamma\quantity(\frac{l_{1}^{i}}{4})$
$\displaystyle\left.\frac{\partial^{2}}{\partial\tau^{2}}W~{}\right|_{t=0}$
$\displaystyle=0\ .$ (3.26)

##  4 Solutions

In this section, we explicitly present some supersymmetric Minkowski solutions
with quantised flux and $N_{\rm flux}$ below the tadpole bound.

A generic flux $G_{3}$ can be expressed using the $3$-forms
$\gamma_{\mathbf{n}}$ that are Poincaré dual to the cycles
$\Gamma_{\mathbf{n}}$. This is the expression (3.17). We note that this
expansion is non-unique since the $\gamma_{\mathbf{n}}$ are overcomplete to
describe $G_{3}\in H^{(2,1)}$. However, it is straightforward to choose an
independent set of linear combinations and thus parametrise a (2,1) form
$G_{3}$ by the associated _independent_ flux numbers. Alternatively, $G_{3}$
can be expanded in the complex $\chi_{{\mathbf{l}}}$-basis as in (3.18). The
expansion in the $\chi_{{\mathbf{l}}}$-basis is rather useful since it
manifestly constrains the Hodge-type of the form. These two parametrisations
of $G_{3}$ are obviously equivalent and it is easy to compute the linear
transformations that relate the complex coefficients $A_{\mathbf{l}}$ to the
independent flux numbers, and vice versa.

While generating solutions, we make use of both (3.17) and (3.18). In one
approach, we “turn on” the coefficients $A_{\mathbf{l}}$ one by one. That is,
we allow a certain number of $A_{\mathbf{l}}$’s to be non-zero, while the rest
of the $A_{\mathbf{l}}$’s are set to zero. The actual value of non-zero
$A_{\mathbf{l}}$’s is then determined by imposing flux quantisation and the
tadpole cancellation bound $N_{\rm flux}\leq 40$. We refer to a flux with $n$
non-zero $A_{\mathbf{l}}$ coefficients in (3.18) as an
$n$-$\chi_{{\mathbf{l}}}$ solution. Proceeding this way, we find a large
number of solutions that we present in this section and in Appendix B. Already
at $n\leq 4$, one finds solutions that are on or above the line $n_{\rm
massive}=3N_{\rm flux}$ in Figure 1. Computational limit notwithstanding, one
can chart the entire (finite) set of solutions within the tadpole cancellation
bound using this strategy. We did this exhaustively up to $n=3$, and also
explored some solutions with $n=4$.

The other approach is to use (3.17) by first restricting to a
$\mathbb{Z}$-linearly independent spanning set of $\gamma_{\mathbf{n}}$’s (or
linear combinations thereof). We describe in section 4.2 how to obtain such an
integral basis of the supersymmetric flux lattice, following the algorithm
designed in [29]. The rank of the supersymmetric flux lattice turns out to
be555We will justify this statement shortly. $182$. Once an integral basis,
say $\\{\mathbf{e}_{i}:i=1,2,\ldots,182\\}$, is obtained, $G_{3}$ is
_uniquely_ expressed as

$G_{3}=\sum_{i}k^{i}\,\mathbf{e}_{i}~{},$ (4.1)

where the coefficients $k^{i}\in\mathbb{Z}$ are the independent flux numbers
we alluded to above. We then generate solutions by “turning on” a fixed number
of the $k^{i}$, and setting the rest to zero. We refer to a flux with $n$ non-
zero $k^{i}$ coefficients in (4.1) as a “level-$n$” solution. In practice, we
have generated many solutions by going up to level-$5$, with “small” values
for the non-zero $k^{i}$’s, typically $|k^{i}|\leq 2$.

In both of the approaches described above, we restrict ourselves to Minkowski
solutions. This is a rank $180$ sublattice in the full lattice (rank $182$) of
supersymmetric fluxes. The tadpole cancellation bound $N_{\rm flux}\leq 40$
picks out a finite region in this infinite sublattice, so one can reach all
physical solutions in a finite time. Also, the basis $\\{\mathbf{e}_{i}\\}$ we
used for generating Minkowski solutions level-by-level can be transformed by
judicious $SL(180,\mathbb{Z})$ transformations to obtain equivalent integral
bases that may be more or less useful for select purposes. We present a couple
of bases that we have made use of in this work in the GitHub repository [34].

Before proceeding to the list of solutions, let us pause to make an important
observation. Using e.g. (3.14) it is not hard to see that quantisation
condition can only be satisfied if

$\tau=\frac{a\,\omega+b}{c\,\omega+d}\,,$ (4.2)

where $\omega$ is the phase that defines the orbifold action (3.5) (in the
$2^{6}$ model $\omega={\rm i}$), and $a,b,c,d\in{\mathbb{Z}}$. We make the
choice

$\tau=\omega={\rm i}\ .$ (4.3)

###  4.1 $n$-$\chi_{{\mathbf{l}}}$ solutions

As described in the introduction to this section, we now set out to generate
the $n$-$\chi_{{\mathbf{l}}}$ solutions. To exhaustively get all Minkowski
solutions with $n$ $\chi_{{\mathbf{l}}}$’s turned on and $N_{\rm flux}\leq
40$, one first notes that, at first blush, there are $\binom{180}{n}$ distinct
subspaces of complex dimension $n$ to deal with. However, there is an $S_{6}$
symmetry of permuting the factors in the tensored model, which induces an
action of $S_{6}$ on this set of subspaces. We use this symmetry to identify
the subspaces in the same $S_{6}$ orbit, and are left with a smaller number of
$n$-dimensional subspaces to search through. Note that there can still be some
redundancy in the solutions derived from each subspace – in general there are
non-trivial $S_{6}$ elements that fix a subspace but permute the
$\chi_{{\mathbf{l}}}$’s that span it. The solution set obtained from a
subspace can be modded by this residual symmetry to arrive at a list of
$S_{6}$-distinct solutions. We have performed this exercise for $n=1,2,3$
exhaustively, and carried out the $n=4$ case partially. The
$1$-$\chi_{{\mathbf{l}}}$ solutions are presented below, and the complete list
of $2,3$-$\chi_{{\mathbf{l}}}$ solutions can be found in appendix B. In what
follows, the rank of the mass matrix of a given solution is denoted by $r$.

$1$-$\chi_{{\mathbf{l}}}$ solutions: There are $3\times 4=12$ $S_{6}$-distinct
$1$-$\chi_{{\mathbf{l}}}$ solutions within the bound $N_{\rm flux}\leq 40$.
These are:

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{n}}{16}~{}\chi_{111223}~{},~{}r=28~{},~{}N_{\rm flux}=32~{},$ (4.4a)
$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{n}}{32}~{}\chi_{112222}~{},~{}r=36~{},~{}N_{\rm flux}=16~{},$ (4.4b)
$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{n}}{32}(1+{\rm
i})~{}\chi_{112222}~{},~{}r=36~{},~{}N_{\rm flux}=32~{},$ (4.4c)

where $n=0,1,2,3$. The third $1$-$\chi_{{\mathbf{l}}}$ family of $S_{6}$
orbits yields solutions whose lowest $N_{\rm flux}$ value is beyond the
tadpole cancellation bound. These are $G=\frac{{\rm
i}^{n}}{8}\,\chi_{111133}\,$, $r=21,~{}N_{\rm flux}=64$. An overall factor of
${\rm i}^{n}$ does not change the physical characteristics of a solution – the
flux remains quantised, and has the same values of $n_{\rm massive}$ (also
$n_{\rm stab}$ at all orders in $W$), and $N_{\rm flux}$. So one should
actually not distinguish such fluxes even if they are not related by $S_{6}$.
On the contrary, an overall integer factor would increase $N_{\rm flux}$,
while leaving $n_{\rm massive}$, $n_{\rm stab}$ unaltered, and are not of
interest in our chase of solutions with highest $n_{\rm massive}/N_{\rm flux}$
ratio.

Already with the help of $1$-$\chi_{{\mathbf{l}}}$ solutions, we can establish
that the rank of the supersymmetric flux lattice, restricted to Minkowski
solutions, is $180$ – the set

$\Bigl{\\{}\frac{{\rm i}^{n}}{16}~{}\chi_{s(111223)},\frac{{\rm
i}^{n}}{32}~{}\chi_{s(112222)},\frac{{\rm
i}^{n}}{8}\,\chi_{s(111133)}\Big{|}s\in S_{6},\,n=0,1\Bigr{\\}}$ (4.5)

contains precisely $180$ $\mathbb{Z}$-linearly independent, integer quantised
fluxes. It does not form an integral basis, since its $\mathbb{Z}$-span is
only a proper sublattice. We will present in the next subsection the
derivation of an integral basis.

$2,3$-$\chi_{{\mathbf{l}}}$ solutions: Continuing in this fashion the use of
$S_{6}$ symmetry to reduce the search space, we derive the complete list of
$S_{6}$-distinct $2,3$-$\chi_{{\mathbf{l}}}$ solutions with very little
effort. These are presented in Appendix B, with the code used to generate them
made available in [34].

###  4.2 Integral bases for the flux lattice

Ref. [28] presented an integral basis of Minkowski fluxes in the $1^{9}$ LG
model, and the algorithm used there was spelled out explicitly in [29]. For
the readers’ convenience, we paraphrase here the argument from [29] which we
use to obtain an integral basis for Minkowski fluxes in the present model.

We have already described two equivalent parametrisations666Here, (4.1) is
restricted to Minkowski solutions, so there are $180$ independent flux numbers
$y^{i}$ instead of $182$. of the $3$-form flux:

$G_{3}=\sum_{{\mathbf{l}}:\sum
l_{i}=10}A_{\mathbf{l}}\,\chi_{{\mathbf{l}}}=\sum_{i=1}^{180}k^{i}\,\mathbf{e}_{i}~{}.$
(4.6)

Let us split the $90$ complex coefficients $A_{\mathbf{l}}$ into real and
imaginary parts to represent $G_{3}$ as a vector of $\mathbb{R}^{180}$. The
flux quantisation conditions (3.19) are constraints on the complex
coefficients $A_{\mathbf{l}}$. Taking $\tau={\rm i}$, and separating (3.19)
into real and imaginary parts, we get $182\times 2=364$ real constraints on
the $180$ real and imaginary parts of $A_{\mathbf{l}}$’s. Not all these
constraints are linearly independent. In fact, the rank of the system of
linear constraints is $180$. A (non-unique) set of independent constraints can
be used to express the independent flux numbers $k^{i}$ in terms of the real
and imaginary parts of the complex coefficients $A_{\mathbf{l}}$’s. Using the
same notation as [29],

$\sum_{{\mathbf{l}}}\mathbf{B}^{i}{}_{\mathbf{l}}A_{\mathbf{l}}=k^{i}~{}.$
(4.7)

In the equation above, $A_{\mathbf{l}}$ stands (temporarily) for both real and
imaginary parts of the complex coefficients $A_{\mathbf{l}}$ – we choose not
to introduce new notation for this. The matrix $\mathbf{B}^{i}{}_{\mathbf{l}}$
is real, $180\times 180$, and has full rank (we already know that the
Minkowski flux lattice has rank $180$). Inverting it, we obtain the
$A_{\mathbf{l}}$ as linear functions of $k^{i}$:
$A_{\mathbf{l}}=A_{\mathbf{l}}(k^{1},\ldots,k^{180})$. Going back to the flux
quantisation equation in the form (3.19), one can explicitly check that the
integers $\\{N^{\mathbf{n}},M^{\mathbf{n}}\\}$ are $\mathbb{Z}$-linear
combinations of the integers $k^{i}$.

The columns of $[\mathbf{B}_{i{\mathbf{l}}}]^{-1}$ yield an integral basis of
the Minkowski flux lattice. In fact, these correspond to the fluxes
$\mathbf{e}_{i}$ obtained at “level-$1$” with $k^{j}=\delta^{ji}$. The
explicit form of the basis we constructed in this way can be found in the
GitHub repository [34]. Any two integral bases have to be related by a
similarity transformation in ${\it SL}(180,\mathbb{Z})$.

The “level-$n$” solutions described at the beginning of this section are
simply elements of the $\mathbb{Z}$-span of precisely $n$ of the
$\mathbf{e}_{i}$’s. This way of classifying solutions is highly basis-
dependent, but serves us tremendously well in generating data for Figure 1.
Many of the elements in the integral basis we find this way have large $N_{\rm
flux}$ values, which makes it prone to generating solutions with large $N_{\rm
flux}$. Let us call this a “long” basis. To find solutions with small $N_{\rm
flux}$ more efficiently, one would try to find a basis of shorter vectors
(where the norm is $N_{\rm flux}$). One question is what is the shortest
possible vector. The answer (according to the database of solutions we
generated by an intense search) turns out to be 9. These solutions are
$9$-$\chi_{{\mathbf{l}}}$ solutions with $N_{\rm flux}=9$ and are elements of
the long basis. This is deeply evocative of the shortest vectors in the
$1^{9}$ model [28] which are $8$-$\chi_{{\mathbf{l}}}$ solutions with $N_{\rm
flux}=8$. Taking the $N_{\rm flux}=9$ vectors in the long basis together with
their $S_{6}$ orbits, we find a large set of vectors with $N_{\rm flux}=9$.
The number of linearly independent vectors in this set turns out to be $156$.
We then searched for the elements in the original long basis that can be
expressed as $\mathbb{Z}$-linear combinations of these $156$ (possibly)
shortest vectors, and replaced them by the latter. The remaining $24$ vectors
of the long basis were kept. Let us call this set a “short” basis. As a sanity
check, we computed the transformation matrix between the long and the short
bases described above, and it is indeed in $SL(180,\mathbb{Z})$. The short
basis is computationally much more efficient in finding solutions with small
$N_{\rm flux}$. We again refer interested readers to the GitHub repository
[34] for explicit expressions of these basis vectors.

We have described two strategies of generating solutions – one organised by
the number of $\chi_{{\mathbf{l}}}$’s turned on, the other organised by the
number of independent flux numbers $k^{i}$’s turned on. The latter strategy
has been referred to as “finding solutions level-by-level”. It is facilitated
by first building an integral basis of the flux lattice. We have constructed
two integral bases – one long and one short – which have been used to find
numerous data-points at the high777We filtered out solutions with $N_{\rm
flux}>70$ from the plot. and low $N_{\rm flux}$ regions of Figure 1
respectively. Figure 1 also includes the data from the
$1,2,3,4$-$\chi_{{\mathbf{l}}}$ solutions. One of the solutions above the
green line $(n_{\rm massive}=52,N_{\rm flux}=16$) is a $2-\chi_{{\mathbf{l}}}$
solution, and is given explicitly in Appendix B. The four other above the line
(as well as the one exactly on the line) were found in the (incomplete) list
of $4$-$\chi_{{\mathbf{l}}}$ solutions. The highest ratio obtained is
$\beta=68/20=3.4$, in the $4$-$\chi_{{\mathbf{l}}}$ solutions

$\displaystyle G$ $\displaystyle=\left\\{\begin{array}[]{ll}\frac{{\rm
i}^{m}}{64}\left[(1+{\rm i})\chi_{111223}+(1+{\rm i})\chi_{113221}+{\rm
i}^{n}\left(2\chi_{221131}\pm{\rm
i}\,\chi_{222112}\right)\right]\\\\[5.69046pt] \frac{{\rm
i}^{m}}{64}\left[(1+{\rm i})\chi_{111223}-(1+{\rm i})\chi_{113221}+{\rm
i}^{n}\left(2\chi_{221131}\pm\chi_{222112}\right)\right]~{}.\end{array}\right.$
(4.10)

We also find solutions exactly on the line $n_{\rm massive}=3\,N_{\rm flux}$
with $(n_{\rm massive}=72,N_{\rm flux}=24)$:

$\displaystyle G$ $\displaystyle=\left\\{\begin{array}[]{ll}\frac{{\rm
i}^{m}}{64}\left[2(1+{\rm i})\left(\chi_{111133}+\chi_{331111}\right)+{\rm
i}^{n}\left(\chi_{112222}\pm{\rm i}\chi_{222211}\right)\right]\\\\[5.69046pt]
\frac{{\rm i}^{m}}{64}\left[2(1+{\rm
i})\left(\chi_{111133}-\chi_{331111}\right)+{\rm
i}^{n}\left(\chi_{112222}\pm\chi_{222211}\right)\right]\end{array}\right.~{}.$
(4.13)

Figure 1: Tadpole and number of massive moduli for solutions found in the
$2^{6}$ model using the strategies described in the text.

###  4.3 Solutions with all moduli massive

A very important question is whether there exists a string vacuum in which all
moduli are stabilised by fluxes. This is highly non-trivial, and in fact has
never been realised before in a supersymmetric Minkoswki solution, to the best
of our knowledge. There is even a _massless Minkowski conjecture_ stating that
there are always massless moduli in supersymmetric, Minkowski supergravity
solutions [30].

In the present model, we actually find within the tadpole bound many flux
configurations that lead to Minkowski vacua with all moduli massive. Below is
one such example with $N_{\rm flux}=40$, and many more can be found in the
GitHub repository [34].

$\displaystyle G=$ $\displaystyle\frac{1}{128}\left[-2{\rm
i}\chi_{111133}+(1+{\rm i})\chi_{111232}-(1+{\rm i})\chi_{112231}+2{\rm
i}\chi_{113131}-(1+{\rm i})\chi_{121321}\right.$ (4.14)
$\displaystyle-\chi_{122212}+{\rm i}\chi_{122221}+(1-{\rm
i})\chi_{123121}-2{\rm i}\chi_{123211}+2{\rm i}\chi_{131311}-2{\rm
i}\chi_{132112}$ $\displaystyle+(1-{\rm
i})\chi_{132211}+2\chi_{133111}+(1+{\rm i})\chi_{211123}-\chi_{211222}+(1-{\rm
i})\chi_{211321}+\chi_{212212}$ $\displaystyle+2{\rm
i}\chi_{213211}+\chi_{221212}+{\rm i}\chi_{222112}+{\rm
i}\chi_{222211}-2\chi_{223111}+2{\rm i}\chi_{231112}-2\chi_{232111}$
$\displaystyle-2\chi_{311113}-(1+{\rm i})\chi_{311212}+2{\rm
i}\chi_{311311}-2{\rm i}\chi_{312211}-2{\rm
i}\chi_{321112}+2\chi_{322111}\left.\right]$

Most of the other full-rank solutions we find also have $N_{\rm flux}=40$.
These do not require mobile D3-branes to satisfy tadpole cancellation. In our
dataset, the smallest value of $N_{\rm flux}$ where full rank is achieved is
$N_{\rm flux}=39$. These backgrounds need one D3-brane. Note that these full-
rank solutions are far from being the ones with the highest $n_{\rm
massive}/N_{\rm flux}$ ratio, as can be seen in Figure 1.

##  5 Conclusions

We have generated a plethora of supersymmetric Minkowski solutions in an
orientifold of the $2^{6}$ LG model using two separate solution generating
strategies described in detail in section 4. The most significant achievements
of this work were two-fold. On one hand, the discovery of quantised flux
backgrounds leading to 4d Minkowski spacetime with all moduli massive. On the
other, the spectacular confirmation of the linear behavior of the number of
massive moduli relative to the flux charge.

The possible existence of moduli-free solutions in the non-geometric LG models
was anticipated in [26], where the authors gave a systematic description of
fluxes in these models for the first time. However, the check of the actual
number of stabilised moduli was only done 20 years later [27], motivated by
the tadpole conjecture. Surprisingly (or not, depending on the perspective),
none of the original solutions turned out to be moduli-free.

Here we found $13$ distinct $S_{6}$ orbits of moduli-free Minkwski solutions
with flux charge excatly at the tadpole bonund, as well as $3$
($S_{6}$-distinct) solutions that require one extra D3-brane for tadpole
cancellation.

Our findings demonstrate that the massless Minkowski conjecture of [30] does
not hold beyond the supergravity regime. To be fair, the original conjecture
is stipulated to apply to supergravity solutions, so our solutions are not
counter-examples to it. However, they do limit the scope of the conjecture to
its original formulation.

Concerning solutions with partial moduli stabilisation, our results
spectacularly confirm the linear behavior of the tadpole conjecture. As for
its refined version, we found a handful of solutions with a slightly higher
ratio $n_{\rm massive}/N_{\rm flux}=\beta$ than the original upper bound
proposed in [13] ($\beta<3$). The highest value found, $\beta=3.4$, is still
comfortably within the bound $\beta<4$, whose violation is required for full
moduli stabilisation in F-theory. This behavior persists even if higher-order
corrections are included [31]888As emphasised in the introduction,
stabilisation of moduli can take place due to higher order terms in the
expansion of the spacetime superpotential $W$ in powers of the deformation
parameters. This possibility has been examined thoroughly in the papers [29,
31] for the $1^{9}$ and the $2^{6}$ models respectively. Despite the fact that
some moduli are indeed lifted at higher orders, no violation of the 3 bound
was found in the $1^{9}$ model. As for the $2^{6}$, a solution that satisfies
the 3 bound at quadratic order was shown to violate it at higher orders
(though never going beyond the 4 bound).. The actual bound should therefore be
in between those two.

There are two subtle points of the tadpole conjecture. One is to pinpoint what
counts as a non-generic point in moduli space (such that the conjecture, which
excludes them, does not apply). The other is whether it applies to vacua with
$W_{0}\neq 0$. Regarding the former, points in moduli space that lead to non-
Abelian gauge symmetries were excluded from the start, since stabilisation at
such points can be achieved easily with low $N_{\rm flux}$[13, 15]999However,
non-Abelian gauge symmetries typically come hand-in hand with their own
moduli, which also have to be stabilised. When the symmetries arise from
7-branes, the tadpole conjecture leads to a dead end: on one hand in type IIB
it was shown that there is a fixed ratio between the number total of D7-moduli
and the flux charge needed for full stabilisation, namely 16/7$\simeq 2.27$
[21], well below the upper bound allowed. On the other, for more general (p,q)
7-branes, the most suitable description is that of F-theory, combining brane
moduli and bulk complex structure moduli, and full moduli stabilisation
requires $\beta\geq 4$. (and the highest the rank of the non-Abelian gauge
group, the smallest the $N_{\rm flux}$ needed to stabilise there [20]). On the
other hand, the question whether points in moduli space with left-over
discrete symmetries bypass the tadpole conjecture bound is more subtle. Our
results, as well as those in [27, 28, 29, 35], show that the conjecture holds
at the Fermat point, despite its high level of symmetry. Conversely, Ref. [18]
finds examples of stabilisation at points with discrete symmetry at the cost
of very little $N_{\rm flux}$, irrespective of the number of moduli. However,
these examples have $W_{0}\neq 0$, and we suspect this to be the reason for
violation of the tadpole conjecture, rather than the presence of discrete
symmetries. Indeed, as [18] argues, when $W_{0}\neq 0$ the off-diagonal
components of the mass matrix (2.8) are non-zero, and furthermore have full
rank. Hence, there can be massless moduli only if there are cancellations
between the diagonal and off-diagonal blocks. On the contrary, in the 609
Calabi-Yau manifolds explored in [18], the maximum rank of the mass matrix in
$W_{0}=0$ solutions is less than $0.8$ times the number of non-invariant
moduli. These results provide strong evidence that the tadpole conjecture
holds despite discrete symmetries, provided $W_{0}=0$. Note that, in usual
compactifications with Kähler moduli, perturbative flux solutions with
$W_{0}\neq 0$ are supersymmetry-breaking (while, once accounting for non-
perturbative corrections, these become supersymmetric AdS vacua). In our
context, solutions with $W_{0}\neq 0$ are supersymmetric AdS from the start,
there are no non-perturbative corrections, and indeed one can easily find
examples with all moduli stabilised at low $N_{\rm flux}$ [27].

A number of obvious extensions to our work can be anticipated. The
$\mathbb{Z}_{2}\times S_{7}$ and $S_{6}$ symmetries of the $1^{9}$ and the
$2^{6}$ orientifolds respectively can be exploited in the manner described in
this paper, along with clever parallel computation tricks, to classify all
physical solutions in these models. This will allow for an exhaustive check of
the tadpole conjecture in these models. Short of this, one can also use modern
training algorithms to maximise the $n_{\rm massive}/N_{\rm flux}$ ratio. The
relatively small orientifold charge in the $1^{9}$ model may allow a full
classification of all physical solutions with modest computer resources (see
[29] where the shortest vectors have been fully classified). Some of the
present authors are pursuing this problem. Furthermore, Gepner models obtained
by tensoring $N$ minimal models have to satisfy $c=9$ in order to serve as a
compactification “manifold” leading to a $4$d spacetime. Visit this link for a
complete list of possibilities satisfying this constraint. The combination of
world-sheet and spacetime techniques used in this paper can also be applied to
any of these examples. We leave this exploration for future work.

## Acknowledgements

We would like to thank Iosif Bena, Antoine Bourget, Andreas Braun, Severin
Lust, Ilarion Melnikov, Jakob Moritz, Hector Parra De Freitas, Muthusamy
Rajaguru, Johannes Walcher, Alexander Whestphal and Timm Wrase for interesting
discussions and useful comments. The work of KB, NB, AS and QY is supported in
part by the NSF grant PHY-2112859. MG is partly supported by the ERC
Consolidator Grant 772408-Stringlandscape.

##  Appendix A Homology and cohomology of the $2^{6}$ model

###  A.1 Cohomology

Harmonic forms are in one-one correspondence with the ground states in the
R-sector, with their degree associated to the $U(1)$ R-charges $q,\bar{q}$.
For a CY 3-fold admitting a LG description, states with charge $(q,\bar{q})$
are identified with elements of the Dolbeaut cohomology
$H^{\frac{3}{2}-q,\tfrac{3}{2}+\bar{q}}$ [33]. Below we will count RR ground
states and compute the $U(1)$ charges to derive the Hodge diamond.

The $2^{6}$ model is a product of 6 minimal models at level $k=2$. A minimal
model at level $k$ has $(k+1)$ supersymmetric ground states in the untwisted
sector, labelled by a subscript $0$: $|l\rangle_{0}$, $l=1,2,\ldots,k+1$,
which correspond to chiral primaries $\Phi^{l-1}$. The RR charges of a state
$|l\rangle_{0}$ are given by $q=\bar{q}=\tfrac{l}{k+2}-\tfrac{1}{2}$. We are
interested in minimal models at $k=2$, which have the following ground states

$\displaystyle|1\rangle_{0}$ $\displaystyle:~{}~{}q=\bar{q}=-\tfrac{1}{4}$
$\displaystyle|2\rangle_{0}$ $\displaystyle:~{}~{}q=\bar{q}=0$ (A.1)
$\displaystyle|3\rangle_{0}$ $\displaystyle:~{}~{}q=\bar{q}=\tfrac{1}{4}$

A ${\mathbb{Z}}_{k+2}$ orbifold action given by ${\rm e}^{-2\pi iJ_{0}}$
results in $k+1$ twisted sectors (labelled by an index $\nu=1,...,k+1$), with
a unique ground state in each twisted sector $|0\rangle_{\nu}$ with charge
$q=\bar{q}=\tfrac{\nu}{k+2}-\tfrac{1}{2}$. For $k=2$ this gives the states

$\displaystyle|0\rangle_{1}$ $\displaystyle:~{}~{}q=-\bar{q}=\tfrac{1}{4}$
$\displaystyle|0\rangle_{2}$ $\displaystyle:~{}~{}q=-\bar{q}=0$ (A.2)
$\displaystyle|0\rangle_{3}$ $\displaystyle:~{}~{}q=-\bar{q}=-\tfrac{1}{4}$

We now tensor 6 minimal models at level $k=2$. From the untwisted sector we
get the states:

$\otimes_{i=1}^{6}|l^{i}\rangle_{0}\equiv|{\mathbf{l}}\,\rangle_{0}\
,\quad{\rm with}q=\bar{q}=\tfrac{1}{4}\sum_{i=1}^{6}l^{i}-3~{}$ (A.3)

Out of these states, we need to project the ones that are invariant under the
${\mathbb{Z}}_{4}$ orbifold action, which enforces
$q,\bar{q}\in\tfrac{1}{2}+\mathbb{Z}$. This implies that we only have the
possibilities

$\sum_{i=1}^{6}l^{i}=6,10,14,18,$ (A.4)

with corresponding charges
$q=\bar{q}=-\tfrac{3}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2}$, Hodge
degrees $(3,0),(2,1),(1,2),(0,3)$ respectively. The numbers of states in the
classes are $1,90,90,1$ respectively. These exhaust all the harmonic
$3$-forms.

From the twisted sectors we have the ground states , $(|0\rangle_{1})^{\otimes
6}$, $(|0\rangle_{2})^{\otimes 6}$ and $(|0\rangle_{3})^{\otimes 6}$ with
corresponding charges $q=-\bar{q}=\tfrac{3}{2},0,-\tfrac{3}{2}$. The middle
one has integer charge, and does not survive the ${\mathbb{Z}}_{4}$
projection. The first one is a $(0,0)$ form, while the last one a $(3,3)$
form.

In conclusion, the Hodge diamond for the $2^{6}$ model is given by

$\begin{matrix}&&&1&&&\\\ &&0&&0&&\\\ &0&&0&&0&\\\ 1&&90&&90&&1\\\
&0&&0&&0&\\\ &&0&&0&&\\\ &&&1&&&\\\ \end{matrix}$

It is not hard to see that all these forms are odd under the orientifold
projection (3.7), and as such are not projected out in the orientifold model.

###  A.2 Homology

The supersymmetric cycles are the ones wrapped by A and B-branes,
corresponding to special Lagrangian (A) and holomorphic (B) cycles. We are
interested in the former, which are the cycles that can support 3-form fluxes.
A-cycles are given by the preimages of the positive real axis of the
worldsheet superpotential. As usual, we start with one minimal model ${\cal
W}=\Phi^{4}$ and then tensor 6 of them. The condition

$\textrm{Im}(\mathcal{W})=0~{}$ (A.5)

gives four directed contours $\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}$
whose union makes up this preimage. These are depicted in Figure 2. A-type
D-branes in the minimal model are spanned by the $k+1=3$ non-compact cycles

$V_{0}:=\gamma_{1}-\gamma_{0},~{}~{}V_{1}:=\gamma_{2}-\gamma_{1},~{}~{}V_{2}:=\gamma_{3}-\gamma_{2},~{}~{}V_{3}:=\gamma_{0}-\gamma_{3}~{}.$
(A.6)

These cycles clearly satisfy

$V_{0}+V_{1}+V_{2}+V_{3}=0$ (A.7)

and have intersection matrix

$V_{i}\cap V_{j}=\begin{pmatrix}1&-1&0&0\\\ 0&1&-1&0\\\ 0&0&1&-1\\\
-1&0&0&1\end{pmatrix}~{}.$ (A.8)

For the building block minimal model, the truncated $\mathbb{Z}_{4}$ generator
and the intersection matrix in the normalisation used in [26] are:

$A=\begin{pmatrix}0&0&-1\\\ 1&0&-1\\\
0&1&-1\end{pmatrix}~{},~{}~{}~{}I=\begin{pmatrix}1&-1&0\\\ 0&1&-1\\\
0&0&1\end{pmatrix}~{}$ (A.9)

The intersection matrix in the full tensored theory is:

$\mathbf{I}=I^{\otimes 6}(1-A^{\otimes 6}+(A^{\otimes 6})^{2}-(A^{\otimes
6})^{3})~{}.$ (A.10)

This matrix is $3^{6}\times 3^{6}$, but has rank $182$. Truncating it to the
top-left $182\times 182$ block yields a full-ranked sub-matrix.

$\gamma_{0}$$\gamma_{1}$$\gamma_{2}$$\gamma_{3}$

$V_{0}$$V_{1}$$V_{2}$$V_{3}$

Figure 2: The “gamma”-contours (left) and V-cycles (right) in the
$\Phi$-plane.

A set of integral three-cycles for the $2^{6}/\mathbb{Z}_{4}$ model is built
by tensoring six $V_{n}$’s, and then $\mathbb{Z}_{4}$-completing them.
Explicitly, these branes are

$\displaystyle\Gamma_{{\mathbf{n}}}$
$\displaystyle=\frac{1}{\sqrt{4}}\left(V_{{\mathbf{n}}}-V_{{\mathbf{n}}+{\bf
1}}+V_{{\mathbf{n}}+{\bf 2}}-V_{{\mathbf{n}}+{\bf
3}}\right)=\frac{1}{\sqrt{4}}\left(\otimes_{i}V_{n_{i}}-\otimes_{i}V_{n_{i}+1}+\otimes_{i}V_{n_{i}+2}-\otimes_{i}V_{n_{i}+3}\right)~{},$
(A.11) $\displaystyle\qquad\qquad{\mathbf{n}}=(n_{1},\ldots,n_{6}),\qquad
n_{i}=0,1,2,3.$

$\mathbb{Z}_{4}$ acts on $\otimes_{i}V_{n_{i}}$ as a tensor product on each of
the factors. On a factor $V_{n}$, it acts as $V_{n}\to
V_{(n+1)~{}\textrm{mod}~{}4}$. The set of cycles $\\{\Gamma_{{\mathbf{n}}}\\}$
defined by is linearly dependent. It turns out that one can constrain $n_{i}$
to $n_{i}=0,1,2,$ and further restrict ${\mathbf{n}}$’s to be the ternary
representations101010written with six ternary digits, padding with zeroes on
the left when necessary. of the first $182$ non-negative integers to obtain an
integral basis of three-cycles in the $2^{6}/\mathbb{Z}_{4}$ orbifold.

There is a nice way to derive the fact that the rank of the lattice $\Lambda$
of three cycles is $182$. The worldsheet superpotential of the $2^{6}$ model
is

$\mathcal{W}=\sum_{i=1}^{6}\Phi_{i}^{4}+z^{2}~{},$ (A.12)

where the factor $z$ can be integrated out. However, it is best to not do so
because the orbifold action becomes awkward to implement at the level of the
$A$-branes. The $\mathbb{Z}_{4}$ action on $\Phi_{i},z$ is freely generated
by:

$\Phi_{i}\mapsto{\rm i}\Phi_{i}~{},~{}~{}z\mapsto-z~{}.$ (A.13)

In the factor $z$-theory, the $A$-branes are two straight wedges, say
$V_{z_{1}},V_{z_{2}}$, which coincide but are oppositely oriented:

$V_{z_{1}}+V_{z_{2}}=0~{}.$ (A.14)

On a single $\Phi$-factor, the charge lattice $\Lambda_{\Phi}$ of $A$-branes
fits into the exact sequence

$0\to\mathbb{Z}\to\mathbb{Z}^{4}\to\Lambda_{\Phi}\to 0~{},$ (A.15)

where the $\mathbb{Z}^{4}$ is coordinatised by the $V_{n}$, while the
$\mathbb{Z}$ can be thought of as the constraint (A.7), namely
$V_{0}+V_{1}+V_{2}+V_{3}=0$. On the sixth tensor power of this model, this
generalizes to the long exact sequence:

$0\to\mathbb{Z}\to 6\mathbb{Z}^{4}\to\ldots\to
6(\mathbb{Z}^{4})^{5}\to(\mathbb{Z}^{4})^{6}\to\Lambda^{\prime}\to 0~{},$
(A.16)

where $\Lambda^{\prime}$ denotes the lattice of $A$-branes in the tensored
model. The rank of $\Lambda^{\prime}$ before orbifolding is,

${\rm rank}(\Lambda^{\prime})|_{\rm{pre-
orb}}=4^{6}-6(4^{5})+\ldots-6(4)+1=(4-1)^{6}=729~{}.$ (A.17)

$\mathbb{Z}_{4}$ acts on each element of the sequence freely, except for the
first $\mathbb{Z}$ on which the action is trivial. Hence, in the
$\mathbb{Z}_{4}$-orbifold of the tensor product, we ought to divide each
summand in the above sum by $4$ except the last one:

${\rm
rank}(\Lambda^{\prime})=\tfrac{4^{6}-6(4^{5})+\ldots-6(4)}{4}+1=\tfrac{(4-1)^{6}-1}{4}+1=183~{}.$
(A.18)

Similarly, on the $z$-factor, the charge lattice $\Lambda_{z}$ of $A$-branes
fits into the exact sequence

$0\to\mathbb{Z}\to\mathbb{Z}^{2}\to\Lambda_{z}\to 0~{}.$ (A.19)

Tensoring this sequence with the lattice $\Lambda^{\prime}$ of the
$\Phi$-factors, one gets

$0\to\mathbb{Z}\to\Lambda^{\prime}\otimes\mathbb{Z}^{2}\to\Lambda\to 0~{},$
(A.20)

where $\Lambda$ is the lattice of $A$-branes in the full, orbifolded theory,
and the first $\mathbb{Z}$ is the relation (A.14). And, before orbifolding the
$z$-factor

${\rm rank}(\Lambda)|_{\rm{pre-z-orb}}=183(2)-1=365~{}.$ (A.21)

The action of $\mathbb{Z}_{4}$ on $\Lambda^{\prime}\otimes\mathbb{Z}^{2}$ is
neither free nor trivial but reduces its dimension by a factor of $2$, and the
$\mathbb{Z}_{4}$ action on $\mathbb{Z}$ is trivial, hence

${\rm rank}(\Lambda)=\tfrac{183(2)}{2}-1=182~{}.$ (A.22)

The RR charges of the $V_{n}$’s can be computed as the overlaps $\langle
V_{n}|l\rangle$. Since the Ramond ground states are in one-to-one
correspondence to the chiral primary deformations of the superpotential of the
minimal model, we have

$|l\rangle\leftrightarrow\Phi^{l-1}~{},$ (A.23)

which yields

$\langle V_{n}|l\rangle=\int_{V_{n}}d\Phi~{}\Phi^{l-1}{\rm
e}^{-\Phi^{4}}=\tfrac{{\rm i}^{l}-1}{4}~{}\Gamma(\tfrac{l}{4})~{}{\rm
i}^{nl}~{}.$ (A.24)

For the tensored states and cycles, there is a correspondence between Ramond
ground states and chiral primaries associated to complex 3-forms
$\chi_{{\mathbf{l}}}$. Consistency with the expression of the intersection
matrix $\mathbf{I}$ in [26] requires the normalization:

$\chi_{{\mathbf{l}}}~{}~{}\leftrightarrow~{}~{}|{\mathbf{l}}~{}\rangle~{}~{}\leftrightarrow~{}~{}2^{11}\prod_{i=1}^{6}\tfrac{1}{\Gamma\left(\tfrac{l^{i}}{4}\right)}~{}\Phi_{i}^{l^{i}-1}~{},$
(A.25)

with

$\displaystyle\langle
V_{{\mathbf{n}}}|{\mathbf{l}}\rangle=\int_{V_{{\mathbf{n}}}}\chi_{{\mathbf{l}}}$
$\displaystyle=2^{11}\prod_{i=1}^{6}\int_{V_{n_{i}}}d\Phi_{i}~{}\frac{\Phi_{i}^{l^{i}-1}}{\Gamma\left(\tfrac{l^{i}}{4}\right)}~{}e^{-\Phi_{i}^{4}}=\tfrac{1}{2}~{}{\rm
i}^{{\mathbf{n}}\cdot{\mathbf{l}}}~{}\prod_{i=1}^{6}(-1+{\rm i}^{l^{i}})~{},$
(A.26) $\displaystyle{\rm
and}~{}\int_{\Gamma_{{\mathbf{n}}}}\chi_{{\mathbf{l}}}$ $\displaystyle={\rm
i}^{{\mathbf{n}}\cdot{\mathbf{l}}}~{}\prod_{i=1}^{6}(-1+{\rm i}^{l^{i}})$
(A.27)

We now define another homology basis called the homogeneous basis, which is
very adapted to computations involving the basis of complex 3-forms
$\chi_{{\mathbf{l}}}$. We start with a basis of cycles in the minimal model

$\displaystyle W_{1}$ $\displaystyle:=V_{0}+{\rm i}V_{1}-V_{2}-{\rm i}V_{3}$
(A.28a) $\displaystyle W_{2}$ $\displaystyle:=V_{0}-V_{1}+V_{2}-V_{3}$ (A.28b)
$\displaystyle W_{3}$ $\displaystyle:=V_{0}-{\rm i}V_{1}-V_{2}+{\rm
i}V_{3}~{},$ (A.28c)

which can be succinctly written as:

$W_{l}:=\sum_{n=0}^{3}{\rm i}^{nl}V_{n}~{},~{}~{}l=1,2,3~{}\iff
V_{n}=\frac{1}{4}\sum_{l=1}^{3}{\rm i}^{-nl}W_{l}$ (A.29)

Equation (A.9) implies the intersection matrix

$W_{l^{\prime}}\cap W_{l}=4\begin{pmatrix}0&0&1+{\rm i}\\\ 0&1-{\rm
i}^{2}&0\\\ 1-{\rm
i}&0&0\end{pmatrix}~{}=4~{}\delta_{l^{\prime}+l,4}~{}(1-{\rm i}^{l})~{},$
(A.30)

and therefore

$C_{l^{\prime}}\cap C_{l}=4~{}\delta_{l^{\prime}+l,4}~{}(1-{\rm
i}^{-l})~{}\text{ which simplifies to }~{}C_{l}\cap C_{\bar{l}}=4~{}(1-{\rm
i}^{l})~{}.$ (A.31)

Reality of the $V_{n}$ imply $\overline{W_{l}}=W_{4-l}=:W_{\bar{l}}$, where we
have defined the conjugate index $\bar{l}=4-l$. In [28] some of the present
authors had adopted the notation

$C_{l}:=W_{4-l}=W_{\bar{l}}~{}.$ (A.32)

These cycles satisfy, for $l,l^{\prime}=1,2,3$,

$\int_{W_{l^{\prime}}}\Phi^{l-1}{\rm
e}^{-\Phi^{4}}=\delta_{l^{\prime}+l,4}~{}(-1+{\rm
i}^{l})\Gamma(\tfrac{l}{4})~{}\Leftrightarrow~{}\int_{C_{l^{\prime}}}\Phi^{l-1}{\rm
e}^{-\Phi^{4}}=\delta_{l,l^{\prime}}~{}(-1+{\rm
i}^{l})\Gamma(\tfrac{l}{4})~{}.$ (A.33)

Just like before, to obtain the homogeneous cycles of the full theory we
tensor six cycles from the minimal models such that

$C_{{\mathbf{l}}}~{}=~{}\bigotimes_{i=1}^{6}C_{l^{i}}~{}\text{ and
}~{}C_{{\mathbf{l}}}^{*}~{}=~{}\bigotimes_{i=1}^{6}C_{\bar{l}^{i}}~{},$ (A.34)

with symplectic intersection matrix

$C_{{\mathbf{l}}}\cap C_{{\mathbf{l}}}^{*}=\prod_{i=1}^{6}C_{l}\cap
C_{\bar{l}}=2^{12}\prod_{i=1}^{6}(1-{\rm i}^{l^{i}})~{}.$ (A.35)

The integration of the complex 3-forms $\chi_{{\mathbf{l}}}$ over the
homogeneous cycles now has the convenient form

$\int_{C_{{\mathbf{l}}^{\prime}}}\chi_{{\mathbf{l}}}~{}=~{}2^{11}~{}\delta_{{\mathbf{l}},{\mathbf{l}}^{\prime}}\prod_{i=1}^{6}(1-{\rm
i}^{l^{i}})~{}.$ (A.36)

##  Appendix B Two and three $\chi$ solutions

We present in this appendix a complete list of $S_{6}$-distinct
$2,3$-$\chi_{{\mathbf{l}}}$ solutions. For $2$-$\chi_{{\mathbf{l}}}$
solutions, we have modded out the solution set in
$\textrm{span}_{\mathbb{Z}}\\{\chi_{{\mathbf{l}}_{{\mathbf{l}}_{1}}},\chi_{{\mathbf{l}}_{{\mathbf{l}}_{2}}}\\}$
by the stabilizer subgroup (under the action of $S_{6}$) of the set
$\\{\chi_{{\mathbf{l}}_{{\mathbf{l}}_{1}}},\chi_{{\mathbf{l}}_{{\mathbf{l}}_{2}}}\\}$,
and only enumerate representatives from distinct $S_{6}$ orbits. This
complicates the presentation of the solutions since some patterns are lost
after modding by the residual $S_{6}$. To avoid the proliferation of
equations, we do not perform this final modding out for the
$3$-$\chi_{{\mathbf{l}}}$ solutions, and present an $S_{6}$-redundant set that
can be expressed more compactly.

$2$-$\chi_{{\mathbf{l}}}$ solutions: The list of $2$-$\chi_{{\mathbf{l}}}$
solutions below has been broken into $24$ families. In general, $N_{\rm flux}$
is a homogeneous quadratic in the variables $k^{i}$, $i=1,2,\ldots,180$, but
for $2$-$\chi_{{\mathbf{l}}}$ solutions it reduces to a quadratic in $4$
integer variables. The $24$ families below are categorised by the form of this
quadratic in $4$ variables.

$\displaystyle G$ $\displaystyle=\left\\{\begin{array}[]{ll}\frac{{\rm
i}^{m}}{16}\left(\chi_{111133}+\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{16}\left(-\chi_{111133}+\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{{\rm
i}}{16}\left(-\chi_{111133}+\chi_{\mathbf{l}}\right)\end{array}\right.~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}111313,~{}~{}r=28\\\
113311,~{}~{}r=36\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1f)
$\displaystyle G$ $\displaystyle={\rm i}^{m}\left(\tfrac{(1+{\rm
i})}{16}\chi_{111133}+{\rm i}^{n}\tfrac{(1+{\rm
i})}{64}\chi_{{\mathbf{l}}}\right)~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}112222,~{}~{}r=44\\\
122212,~{}~{}r=50\end{array}\right.,~{}~{}~{}N_{\rm flux}=40$ (B.1i)
$\displaystyle G$ $\displaystyle={\rm i}^{m}\left(\tfrac{(1+{\rm
i})}{16}\chi_{111133}+{\rm i}^{n}\tfrac{(1+{\rm
i})}{64}\chi_{222211}\right)~{},~{}~{}r=55,~{}~{}~{}N_{\rm flux}=40$ (B.1j)
$\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{32}\left((-1-{\rm
i})\chi_{111223}+{\rm i}^{m}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}\left((-1+{\rm i})\chi_{111223}+{\rm i}^{q}(1-{\rm
i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt] \frac{1}{32}\left((1-{\rm
i})\chi_{111223}+{\rm i}^{s}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}(1+{\rm
i})\left(\chi_{111223}+\chi_{\mathbf{l}}\right)\end{array}\right.,~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}111232,~{}~{}r=28\\\
112132,~{}~{}r=38\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1q)
$\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{32}\left((-1-{\rm
i})\chi_{111223}+{\rm i}^{m}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}\left((-1+{\rm i})\chi_{111223}+{\rm i}^{q}(1-{\rm
i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt] \frac{1}{32}\left((1-{\rm
i})\chi_{111223}+{\rm i}^{s}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}(1+{\rm
i})\left(\chi_{111223}+\chi_{\mathbf{l}}\right)\end{array}\right.,~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}112123,~{}~{}r=28\\\
122113,~{}~{}r=38\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1x)
$\displaystyle G$ $\displaystyle={\rm i}^{m}\left(\tfrac{(1+{\rm
i})}{32}\chi_{111223}+{\rm i}^{n}\tfrac{(1+{\rm
i})}{64}\chi_{\mathbf{l}}\right)~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}112222,~{}~{}r=38\\\
122122,~{}~{}r=46\\\ 222112,~{}~{}r=56\end{array}\right.,~{}~{}~{}N_{\rm
flux}=24$ (B.1ab) $\displaystyle G$ $\displaystyle={\rm
i}^{m}\left(\tfrac{(1+{\rm i})}{32}\chi_{111223}+{\rm i}^{n}\tfrac{(1+{\rm
i})}{32}\chi_{\mathbf{l}}\right)~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}112231,~{}~{}r=38\\\
122131,~{}~{}r=46\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1ae)
$\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{32}\left((-1-{\rm
i})\chi_{111223}+{\rm i}^{m}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}\left((-1+{\rm i})\chi_{111223}+{\rm i}^{q}(1-{\rm
i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt] \frac{1}{32}\left((1-{\rm
i})\chi_{111223}+{\rm i}^{s}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}\left((1+{\rm i})\chi_{111223}+(1+{\rm
i})\chi_{\mathbf{l}}\right)\end{array}\right.,~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}123121,~{}~{}r=44\\\
223111,~{}~{}r=50\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1al)
$\displaystyle G$ $\displaystyle=\left\\{\begin{array}[]{ll}\frac{{\rm
i}^{m}}{32}\left(\chi_{111223}+\chi_{113221}\right)\\\\[5.69046pt] \frac{{\rm
i}^{s}}{32}\left(-\chi_{111223}+\chi_{113221}\right)\end{array}\right.~{},~{}~{}r=36~{}~{}~{}N_{\rm
flux}=16$ (B.1ao) $\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{32}\left((-1-{\rm
i})\chi_{111223}+{\rm i}^{m}(1-{\rm i})\chi_{113221}\right)\\\\[5.69046pt]
\frac{1}{32}\left((-1+{\rm i})\chi_{111223}+{\rm i}^{q}(1-{\rm
i})\chi_{113221}\right)\\\\[5.69046pt] \frac{1}{32}\left((1-{\rm
i})\chi_{111223}+{\rm i}^{s}(1-{\rm i})\chi_{113221}\right)\\\\[5.69046pt]
\frac{1}{32}(1+{\rm
i})\left(\chi_{111223}+\chi_{113221}\right)\end{array}\right.~{},~{}~{}r=36~{}~{}~{}N_{\rm
flux}=32$ (B.1at) $\displaystyle G$ $\displaystyle={\rm
i}^{m}\left(\tfrac{(1+{\rm i})}{32}\chi_{111223}+{\rm i}^{n}\tfrac{(1+{\rm
i})}{64}\chi_{\mathbf{l}}\right)~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}122221,~{}~{}r=44\\\
222121,~{}~{}r=54\end{array}\right.,~{}~{}~{}N_{\rm flux}=24$ (B.1aw)
$\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{64}\left((-1-{\rm
i})\chi_{112222}+{\rm i}^{m}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{64}\left((-1+{\rm i})\chi_{112222}+{\rm i}^{q}(1-{\rm
i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt] \frac{1}{64}\left((1-{\rm
i})\chi_{112222}+{\rm i}^{s}(1-{\rm i})\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{64}(1+{\rm
i})\left(\chi_{112222}+\chi_{\mathbf{l}}\right)\end{array}\right.~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}121222,~{}~{}r=36\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}221122,~{}~{}r=52}\end{array}\right.,~{}~{}~{}N_{\rm
flux}=16$ (B.1bd) $\displaystyle G$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{32}\left({\rm
i}^{m}~{}.~{}{\rm i}~{}\chi_{112222}+\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
\frac{1}{32}\left({\rm i}^{q}~{}.~{}{\rm i}~{}\chi_{112222}+{\rm
i}~{}\chi_{\mathbf{l}}\right)\\\\[5.69046pt] \frac{1}{32}\left({\rm
i}^{s}~{}.~{}(-1)~{}\chi_{112222}-{\rm
i}~{}\chi_{\mathbf{l}}\right)\\\\[5.69046pt]
-\frac{1}{32}\left(\chi_{112222}+\chi_{\mathbf{l}}\right)\end{array}\right.~{},~{}~{}{\mathbf{l}}=\left\\{\begin{array}[]{ll}121222,~{}~{}r=36\\\
221122,~{}~{}r=52\end{array}\right.,~{}~{}~{}N_{\rm flux}=32$ (B.1bk)

where $m,n=0,1,2,3$, $q=0,1,2$, and $s=0,1$.

The solution in blue has $\beta=3.25$, above the lower bound of the refined
tadpole conjecture.

$3$-$\chi_{{\mathbf{l}}}$ solutions: The $3$-$\chi_{{\mathbf{l}}}$ solutions
below are organised by their $N_{\rm flux}$ values.

$N_{\rm flux}=12$:

$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{m}}{64}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(1-{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]$ (B.2)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,112222,113221),\end{array}~{}\right.$

with rank $26$.

$N_{\rm flux}=20$:

$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{m}}{32}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm
i}^{n}\left(\tfrac{1}{2}\chi_{{\mathbf{l}}_{2}}+{\rm i}^{n}(1-{\rm
i})\chi_{{\mathbf{l}}_{3}}\right)\right]$ (B.3)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111133,112222,113311),\end{array}\right.$

with rank $48$.

$N_{\rm flux}=24$:

$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{m}}{32}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(1-{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]$ (B.4)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111133,111223,111313),~{}~{}(111133,113221,111313),\end{array}~{}\right.$

with ranks $21$, $44$.

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{64}\left[2\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}(1+{\rm
i})\chi_{{\mathbf{l}}_{2}}\pm 2{\rm i}\chi_{{\mathbf{l}}_{3}}\right]$ (B.5)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,112222,131221),~{}~{}(111223,122122,113221),\\\
(111223,221122,113221),~{}~{}(111223,222112,113221),\\\
(111223,221221,113221),\end{array}~{}\right.$

with ranks $44$, $44$, $58$, $62$, $52$.

$G=\left\\{\begin{array}[]{ll}\frac{{\rm i}^{m}}{64}\left[2\chi_{111223}+{\rm
i}^{n}\left((1+{\rm i})\chi_{112222}+2{\rm
i}^{n+1}\chi_{113221}\right)\right]~{},~{}~{}~{}r=26,\\\\[5.69046pt]
\frac{{\rm i}^{m}}{64}\left[2\chi_{111223}+{\rm i}^{n}\left((1+{\rm
i})\chi_{112222}-2{\rm
i}^{n+1}\chi_{113221}\right)\right]~{},~{}~{}~{}r=38~{}.\end{array}\right.$
(B.6)

$N_{\rm flux}=28$:

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{64}\left\\{\begin{array}[]{ll}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(-3-{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]\\\ \left[(3+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(-1-{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]\\\ \left[(1+3{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(1+{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]\\\ \left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{n}(1+3{\rm i})\chi_{{\mathbf{l}}_{3}}\right)\right]\\\ \left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left((1+2{\rm
i})\chi_{{\mathbf{l}}_{2}}+{\rm i}^{n}(1-{\rm
i})\chi_{{\mathbf{l}}_{3}}\right)\right]\\\ \left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left((2+{\rm
i})\chi_{{\mathbf{l}}_{2}}+{\rm i}^{n}(-1+{\rm
i})\chi_{{\mathbf{l}}_{3}}\right)\right]\end{array}\right.$ (B.7)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,112222,113221),{}{}\end{array}~{}\right.$

with rank $38$.

$N_{\rm flux}=32$:

$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{m}}{32}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left(\chi_{{\mathbf{l}}_{2}}\pm{\rm
i}\chi_{{\mathbf{l}}_{3}}\right)\right]$ (B.8)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,111232,113212),~{}~{}(111223,112132,112312),\\\
(111223,112123,112321),~{}~{}(112123,111223,131221),\\\
(122113,111223,113221),~{}~{}(221113,111223,113221),\\\
(112132,111223,131221),~{}~{}(132112,111223,113221),\\\
(231112,111223,113221),~{}~{}(112231,131221,111223),\\\
(221131,113221,111223),~{}~{}(111223,112231,132211),\\\
\\{(113221,131221,111223),(131221,111223,113221),(111223,113221,131221)\\},\\\
(231121,113221,111223),{}{}\end{array}\right.$

with ranks $38$, $44$, $38$, $46$, $48$, $58$, $50$, $52$, $60$, $46$, $56$,
$50$, $\\{44\\}$, $54$.

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{64}\left[2\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}(1+{\rm
i})\chi_{{\mathbf{l}}_{2}}+{\rm i}^{p}(1+{\rm
i})\chi_{{\mathbf{l}}_{3}}\right]$ (B.9)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}\\{(112222,121222,122122),(121222,122122,112222),(122122,112222,121222)\\},\\\
\\{(112222,121222,212122),(121222,212122,112222),(212122,112222,121222)\\},\\\
\\{(112222,121222,222112),(121222,222112,112222),(222112,112222,121222)\\},\\\
\\{(112222,221122,222211),(221122,222211,112222),(222211,112222,221122)\\},\\\
\\{(112222,121222,211222),(121222,211222,112222),(211222,112222,121222)\\}~{},\end{array}\right.$

with ranks $\\{44\\}$, $\\{52\\}$, $\\{58\\}$, $\\{66\\}$, $\\{42\\}$.

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{32}\left[\chi_{{\mathbf{l}}_{1}}+{\rm
i}^{n}\chi_{{\mathbf{l}}_{2}}\pm\chi_{{\mathbf{l}}_{3}}\right]$ (B.10)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,112222,131221),~{}~{}(111223,122122,113221),\\\
(111223,221122,113221),~{}~{}(111223,222112,113221),\\\
(111223,221221,113221),\end{array}~{}\right.$

with ranks $44$, $44$, $58$, $62$, $52$.

$G=\left\\{\begin{array}[]{ll}\frac{{\rm i}^{m}}{32}\left[\chi_{111223}+{\rm
i}^{n}\left(\chi_{112222}+{\rm
i}^{n}\chi_{113221}\right)\right]~{},~{}~{}~{}r=36,\\\\[5.69046pt] \frac{{\rm
i}^{m}}{32}\left[\chi_{111223}+{\rm i}^{n}\left(\chi_{112222}-{\rm
i}^{n}\chi_{113221}\right)\right]~{},~{}~{}~{}r=38~{}.\end{array}\right.$
(B.11)

$N_{\rm flux}=36$:

$\displaystyle G$ $\displaystyle=\frac{{\rm i}^{m}}{32}\left[(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}\left((\tfrac{1}{2}\pm{\rm
i})\chi_{{\mathbf{l}}_{2}}+{\rm i}^{n}(1-{\rm
i})\chi_{{\mathbf{l}}_{3}}\right)\right]$ (B.12)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111133,112222,113311),{}{}\end{array}\right.$

with rank $50$.

$N_{\rm flux}=40$:

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{64}\left[4\chi_{{\mathbf{l}}_{1}}\pm 4{\rm
i}\chi_{{\mathbf{l}}_{2}}+(\pm 1\pm{\rm i})\chi_{{\mathbf{l}}_{3}}\right]$
(B.13) $\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111133,111313,112222),~{}~{}(111133,111313,122122),\\\
(111133,111313,122221),~{}~{}(111133,111313,222121),\\\
(111133,111313,222112),~{}~{}(111133,131311,112222),\\\
(111133,331111,112222),~{}~{}(111133,113311,221212),\end{array}~{}\right.$

with ranks $44$, $52$, $52$, $62$, $56$, $55$, $60$, $62$.

$\displaystyle G$ $\displaystyle=\frac{{\rm
i}^{m}}{64}\left\\{\begin{array}[]{ll}\left[2(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+2{\rm i}^{n}\chi_{{\mathbf{l}}_{2}}+{\rm
i}^{p}(1+{\rm i})\chi_{{\mathbf{l}}_{3}}\right]\\\ \left[2(1+{\rm
i})\chi_{{\mathbf{l}}_{1}}+{\rm i}^{n}(1+{\rm i})\chi_{{\mathbf{l}}_{2}}+2{\rm
i}^{p}\chi_{{\mathbf{l}}_{3}}\right]\end{array}\right.$ (B.14)
$\displaystyle({\mathbf{l}}_{1},{\mathbf{l}}_{2},{\mathbf{l}}_{3})$
$\displaystyle=\left\\{\begin{array}[]{ll}(111223,112222,121222),~{}~{}(111223,122122,212122),\\\
(111223,112222,122122),~{}~{}(111223,112222,221122),\\\
(111223,112222,222112),~{}~{}(111223,122122,122212),\\\
(111223,122122,212212),~{}~{}(111223,122122,222112),\\\
(111223,112222,122221),~{}~{}(111223,112222,221221),\\\
(111223,112222,222121),~{}~{}(111223,122122,122221),\\\
(111223,122122,212221),~{}~{}(111223,122122,222121),\\\
(111223,122122,222211),~{}~{}(111223,122221,222112),\\\
(111223,222112,222121),~{}~{}(111223,122221,212221),\\\
(111223,122221,222121),~{}~{}(111223,222121,222211),\end{array}~{}\right.$

with ranks $42$, $48$, $46$, $54$, $64$, $52$, $64$, $56$, $44$, $52$, $58$,
$46$, $56$, $54$, $64$, $60$, $56$, $46$, $54$, $62$.

$G=\left\\{\begin{array}[]{ll}\frac{{\rm
i}^{m}}{64}\left[4\chi_{111133}+(-1)^{s}4{\rm
i}\chi_{113311}\pm(1+(-1)^{s}{\rm
i})\chi_{112222}\right]~{},~{}~{}~{}r=48,\\\\[5.69046pt] \frac{{\rm
i}^{m}}{64}\left[4\chi_{111133}+(-1)^{s}4{\rm
i}\chi_{113311}\pm(1-(-1)^{s}{\rm
i})\chi_{112222}\right]~{},~{}~{}~{}r=50~{}.\end{array}\right.$ (B.15)

All these solutions are below the bound $\beta<3$.

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