aainstitutetext: George P. and Cynthia Woods Mitchell Institute for
Fundamental Physics and Astronomy
Texas A&M University, College Station, TX 77843, USAbbinstitutetext:
Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem,
PA 18018, USA

# Fully stabilized Minkowski vacua in the $2^{6}$ Landau-Ginzburg model

Muthusamy Rajaguru<EMAIL_ADDRESS>a Anindya Sengupta
<EMAIL_ADDRESS>b and Timm Wrase<EMAIL_ADDRESS>
###### Abstract

We study moduli stabilization via fluxes in the $2^{6}$ Landau-Ginzburg model.
Fluxes not only give masses to scalar fields but can also induce higher order
couplings that stabilize massless fields. We investigate this for several
different flux choices in the $2^{6}$ model and find two examples that are
inconsistent with the Refined Tadpole Conjecture. We also present, to our
knowledge, the first 4d $\mathcal{N}=1$ Minkowski solution in string theory
without any flat direction.

##  1 Introduction

Moduli stabilization is arguably one of the most important challenges in
string phenomenology. After tremendous progress since the early 2000’s, many
constructions and scenarios have been called into question in the last few
years within the swampland program. This is not only true for dS vacua with a
positive cosmological constant, but also for supersymmetric AdS vacua that
give rise to low-energy effective theories whose AdS scale is separated from
the KK-scale. Four-dimensional Minkowski vacua either contain massless scalar
fields for $\mathcal{N}\geq 2$ or are expected to receive corrections that
potentially invalidate the very existence of a solution for $\mathcal{N}\leq
1$. So, there seem to be currently no agreed-upon compactifications to lower
dimensions without a moduli space.

Probably the simplest way of generating a scalar potential for moduli is by
turning on fluxes in the internal space. Such fluxes in type II string theory
induce generically a charge that needs to be canceled by O-planes. These
O-planes together with the fluxes lead to a scalar potential that depends on
all the Kähler moduli for Calabi-Yau (CY) compactifications of massive type
IIA with smeared O6-planes DeWolfe:2005uu . Similarly, it depends on all
complex structure moduli for CY compactifications of type IIB in the presence
of an O3/O7 orientifold Giddings:2001yu . Rigid CY-manifolds have $h^{2,1}=0$
and therefore no complex structure moduli. They, therefore, allow for the
stabilization of all moduli in massive type IIA but it has not yet been
possible to find solutions with fully localized O6-planes, see
Junghans:2020acz ; Marchesano:2020qvg ; Cribiori:2021djm ; Emelin:2022cac ;
Junghans:2023yue ; Bardzell:2024anh ; Emelin:2024vug  for recent efforts in
that direction. For type IIB on the other hand it is not too difficult to find
fully localized solutions in which the backreaction of the O3-planes is
encoded in a non-trivial warp factor Giddings:2001yu . However, in that case,
the stabilization of the Kähler moduli would have to rely on non-perturbative
effects Kachru:2003aw .

An intriguing way to avoid the complication of having to deal with a
combination of fluxes and non-perturbative effects was proposed in
Becker:2006ks . The mirror dual of a rigid CY-manifold has $h^{1,1}=0$ and
therefore no Kähler moduli at all. That means one can stabilize _all_ moduli
in type IIB by using fluxes only. While there is no geometric interpretation
of the internal space in this case, one can study such setups using the
language of Landau-Ginzburg (LG) models Vafa:1988uu ; vafaOrbifoldized ;
Witten:1993yc ; Candelas:1993nd . This was done almost twenty years ago in
Becker:2006ks ; Becker:2007dn ; Becker:2007ee  and has been recently revisited
and extended in Ishiguro:2021csu ; Bardzell:2022jfh ; Becker:2022hse ;
Cremonini:2023suw ; Becker:2023rqi ; Becker:2024ijy ; Ishiguro:2024coq . This
allows one to explore flux compactifications away from the large volume, large
complex structure, and even away from the weak coupling limit and thus
provides a very interesting new place to test swampland conjectures.

While one can study these LG models in the large complex structure limit and
find AdS vacua Becker:2006ks ; Ishiguro:2021csu ; Bardzell:2022jfh , including
infinite families, the detailed properties of these solutions are currently
not under computational control unless one is also at weak coupling. However,
for Minkowski vacua, one can study many properties of the vacua at strong
coupling. This is due to the fact that string loop corrections only enter in
the Kähler potential. The superpotential $W$ in these models is exact and does
not receive perturbative or non-perturbative corrections Becker:2006ks . This
makes it, in particular, possible to calculate the number of massive fields in
Minkowski vacua Becker:2022hse  as well as the number of massless fields that
get stabilized via higher-order interactions Becker:2024ijy . Such fields are
similar to a massless scalar field $\phi$ with a $\phi^{4}$ interaction term.
In our case, higher order terms in the superpotential can constrain the
dimensionality of the solution space of the $\mathcal{N}=1$ Minkowski
equations $W=\partial_{i}W=0$ and can thereby stabilize massless fields.

These supersymmetric Minkowski vacua in the non-geometric LG models share an
important feature with their geometric analogue vacua in GKP Giddings:2001yu ,
namely the $G_{3}$ flux has to be imaginary-self-dual (ISD). This means that
it contributes to the O3-plane tadpole with the same sign as D3-branes. The
fixed number of O3-planes provides then an upper bound on the amount of flux
that we can turn on. While early studies Giryavets:2003vd ; Denef:2004dm ;
Denef:2005mm  seemed to confirm the naive expectation that generically any
chosen ISD $G_{3}$ flux would stabilize all complex structure moduli, this has
been called into question recently. For example, in the context of the sextic
CY-fourfold, there is a tension between stabilizing all complex structure
moduli and satisfying the tadpole cancellation condition Braun:2020jrx . This
and similar observations for other compactifications have led to the so-called
Tadpole Conjecture Bena:2020xrh .

The Tadpole Conjecture states that the fluxes used to stabilize complex
structure moduli contribute to the tadpole an amount $N_{\rm flux}$ that grows
unacceptably fast with the number of moduli that are massive (or get
stabilized). In particular, it is conjectured to be true that

$N_{\rm flux}>\frac{1}{3}n_{\rm stab}\,.$ (1)

For supersymmetric Minkowski vacua, this provides an upper bound on $n_{\rm
stab}$, which is the number of moduli we can stabilize. This is due to the
O3-plane tadpole cancellation condition

$N_{\rm flux}+N_{D3}=\frac{N_{O3}}{2}\,,$ (2)

where $N_{O3}$ is the finite number of O3-planes and $N_{D3}\geq 0$ the number
of D3-branes.

The Tadpole Conjecture received a lot of attention recently and has been
tested in many setups Bena:2021wyr ; Marchesano:2021gyv ; Plauschinn:2021hkp ;
Lust:2021xds ; Bena:2021qty ; Grana:2022dfw ; Tsagkaris:2022apo ; Lust:2022mhk
; Coudarchet:2023mmm ; Braun:2023pzd ; Braun:2023edp ; Becker:2024ijy . These
tests have been performed in the asymptotics as well as the interior of moduli
space. However, none of these papers found a violation of the Tadpole
Conjecture. The $2^{6}$ LG model that we study in this paper, however, allows
for violations of the conjecture as we will explain in this paper. This result
was independently obtained in Becker2024 , where the authors initiate a full
classification of all flux configurations in this model.

While the Tadpole Conjecture is mostly relevant for compactifications with a
large number of complex structure moduli, it was independently conjectured
that all 10d supergravity compactifications to 4d Minkowski solutions should
always have a massless scalar field Andriot:2022yyj . This so-called Massless
Minkowski Conjecture is satisfied in all known supergravity compactifications
but violated in this non-geometric $2^{6}$ LG model Becker2024 . The existence
of 4d $\mathcal{N}=1$ vacua without massless fields is extremely interesting
and shows how useful these non-geometric LG models are in improving our
understanding of the string theory landscape of vacua.

In this paper in section 2 we will review the $2^{6}$ LG model, mostly
following Becker:2006ks , adapted to the notations of Becker:2024ijy . Then we
will present a large number of different flux choices and calculate the number
of massive fields and some higher-order terms to check for the stabilization
of massless fields in section 3. We discuss our results in the context of
moduli stabilization and the above-mentioned swampland conjectures in section
4. We conclude by summarizing our findings in section 5.

##  2 Review of the $2^{6}$ model

The $2^{6}$ Gepner model admits a Landau-Ginzburg description and lives in the
would-be Kähler moduli space of a rigid Calabi-Yau threefold with $90$ Kähler
moduli. It is a tensor product of six minimal models, each with central charge
$c=\frac{3}{2}$, and another factor theory which is trivial in the sense that
it can be integrated out. The world-sheet superpotential of the tensor theory
is

$\mathcal{W}=\sum_{i=1}^{6}x_{i}^{4}+z^{2}~{},$ (3)

and we orbifold by the $\mathbb{Z}_{4}$ action freely generated by

$g:x_{i}\mapsto{\rm i}x_{i}\quad\quad\quad z\mapsto-z~{}.$ (4)

We will not integrate out the quadratic field $z$ since it would require the
action of the orbifold on the chiral fields $x_{i}$ to be dressed by
$(-1)^{F}$ which, in turn, would make the construction of A-branes in the
orbifoldized theory cumbersome. In order to obtain 4d $\mathcal{N}=1$
supersymmetry, this theory should now be orientifolded. There are several
choices of orientifold projections, as explained in Becker:2006ks . In this
paper we choose the canonical choice, denoted $\sigma_{0}$ in Becker:2006ks ,
which acts as follows:

$\sigma_{0}:(x_{1},\ldots,x_{6})\mapsto\omega(x_{1},\ldots
x_{6})\quad\quad\quad z\mapsto{\rm i}z$ (5)

where $\omega\equiv e^{\frac{2\pi{\rm i}}{8}}$. It is easy to see that
$\sigma_{0}^{2}=g$.

The chiral (c) and anti-chiral (a) fields in the left and right moving sectors
of this 2d, $\mathcal{N}=(2,2)$ superconformal field theory, determined
uniquely by the superpotential (3), possess a ring structure. These rings are
the LG analogs of the cohomology rings of Calabi-Yau manifolds. Spectral flow
relates the (anti-)chiral fields to Ramond ground states, and the
$U(1)$-charges of the states determine the Hodge degrees of the harmonic forms
Witten:1993yc ; vafaOrbifoldized ; Brunner:2004zd . In particular, the (c,c)
ring arises solely from the states in the untwisted sector of the Hilbert
space. Concretely, it is the ring

$\mathcal{R}=\biggl{[}\frac{\mathbb{C}\left[x_{1},\ldots,x_{6}\right]}{\partial_{x_{i}}\mathcal{W}\left(x_{1},\ldots,x_{6}\right)}\biggr{]}^{\mathbb{Z}_{4}}\,.$
(6)

It is a $182$-dimensional complex vector space, spanned by monomials

${\mathbf{x}}^{\mathbf{k}}=x_{1}^{k_{1}}\cdot x_{2}^{k_{2}}\cdots
x_{6}^{k_{6}}$ (7)

with ${\mathbf{k}}=(k_{1},\ldots,k_{6})$, $k_{i}\in\\{0,1,2\\}$ for all $i$,
and $\sum k_{i}=0\bmod 4$. The elements with $\sum k_{i}=4$ are — $15$
monomials of the form $x_{i}x_{j}x_{k}x_{l}$ with $i,j,k,l$ all distinct, $60$
monomials of the form $x_{i}^{2}x_{j}x_{k}$ with $i,j,k$ all distinct, $15$
monomials of the form $x_{i}^{2}x_{j}^{2}$ with $i,j$ distinct. These $90$
monomials span the space of marginal deformations of the theory. Concretely,
introducing deformation parameters $t^{\mathbf{k}}$, we write

$\mathcal{W}\bigl{(}\\{x_{i}\\}\bigr{)}=\sum_{i=1}^{6}x_{i}^{4}\;\;\longrightarrow\;\;\mathcal{W}\bigl{(}\\{x_{i}\\};\\{t^{\mathbf{k}}\\}\bigr{)}=\sum_{i=1}^{6}x_{i}^{4}-\sum_{\begin{subarray}{c}{\mathbf{k}}\\\\[1.42271pt]
\sum\\!k_{i}=4\end{subarray}}t^{\mathbf{k}}{\mathbf{x}}^{{\mathbf{k}}}~{}.$
(8)

The parameters $t^{\mathbf{k}}$ are analogues of complex structure moduli. In
addition, there are $3$ twisted sectors in the orbifold theory, twisted by
$e^{-2\pi{\rm i}\nu J_{0}}$, $\nu=1,2,3$. The states in the $\nu=2$ sector do
not survive the $U(1)$-projection, while the other two sectors contribute no
non-trivial elements in the (a,c) ring. Therefore, there are no Kähler moduli
in this model, making it non-geometric. The total number of moduli is $91$ —
$90$ complex structure moduli $t^{\mathbf{k}}$, and the axio-dilaton
$\tau=C_{0}+{\rm i}\,e^{-\phi}$.

###  2.1 Homology and Cohomology

A-type D-branes in a Landau-Ginzburg model with one chiral field $x$ are
contours in the $x$-space that map to a line of constant phase in the
$\mathcal{W}$-plane emanating from the critical value (assuming there is only
one critical point, which is the case for the $2^{6}$ model) of $\mathcal{W}$
HoriIqbalVafa . Using an R-rotation, we look for preimages in the $x$-plane of
the positive real line

$\mathop{\rm Im}\nolimits\mathcal{W}=0$ (9)

in the complex $\mathcal{W}$ plane.

Figure 1: The four contours $\left(V_{0},V_{1},V_{2},V_{3}\right)$ in the
complex $x$-plane.

For the minimal model of interest,

$\mathcal{W}=x^{4}~{},$ (10)

a homology basis is given by the four wedges $V_{0}$, $V_{1}$, $V_{2}$, and
$V_{3}$ (see Fig. 1), which satisfy the relation

$V_{0}+V_{1}+V_{2}+V_{3}=0~{}.$ (11)

The action (4) of $\mathbb{Z}_{4}$ on $V_{n}$, $n=0,1,2,3$, is given by

$g:V_{n}\mapsto V_{(n+1)\bmod 4}~{}.$ (12)

The matrix representation of (12) is

$\text{Mat}(g)=\begin{pmatrix}0&\hskip 8.5359pt1&\hskip 8.5359pt0&\hskip
8.5359pt0\\\ 0&\hskip 8.5359pt0&\hskip 8.5359pt1&\hskip 8.5359pt0\\\ 0&\hskip
8.5359pt0&\hskip 8.5359pt0&\hskip 8.5359pt1\\\ 1&\hskip 8.5359pt0&\hskip
8.5359pt0&\hskip 8.5359pt0\end{pmatrix}$ (13)

Following HoriIqbalVafa , the intersection $\langle V_{m}|V_{n}\rangle$ can be
geometrically computed as the geometric intersection between the two wedges,
with $V_{n}$ held fixed and $V_{m}$ rotated counter-clockwise by a small
angle:

$\bigl{(}\braket{V_{m}}{V_{n}}\bigr{)}_{m,n=0,1,2,3}=\begin{pmatrix}1&-1&0&0\\\
0&1&-1&0\\\ 0&0&1&-1\\\ -1&0&0&1\end{pmatrix}=\mathbf{I}-\textrm{Mat}(g)~{}.$
(14)

The matrix elements can be recast as

$\langle V_{m}|V_{n}\rangle=\delta_{m,n}-\delta_{m+1,n}~{},$ (15)

where the Kronecker delta is understood modulo $4$ here and everywhere else
below. The chiral ring of the minimal model (10) is
$\mathcal{R}=\mathbb{C}[x]/x^{3}$ which, as a vector space, is the span of
$1,x,x^{2}$. These basis elements correspond to the Ramond-Ramond ground
states $|l\rangle$, $l=1,2,3$,

$x^{k=0,1,2}\longleftrightarrow|l=1,2,3\rangle~{}.$ (16)

Up to an overall factor, the states $|l\rangle$ will be the most useful
ingredients in describing $3$-form fluxes in the orbifold theory. The non-
geometric analogue of the integral of a $3$-form flux on a $3$-cycle is the
pairing between the middle-dimensional homology and cohomology. To compute
this, we need the overlap between the boundary states represented by the
$V_{n}$ and the Ramond-Ramond ground states HoriIqbalVafa :

$\langle V_{n}|l\rangle=\int_{V_{n}}x^{l-1}e^{-\mathcal{W}}=\frac{1}{4}\,{\rm
i}^{nl}\,(1-{\rm i}^{l})\,\Gamma\Bigl{(}\frac{l}{4}\Bigr{)}~{}.$ (17)

Now we define the convenient “$\Omega$-basis” Becker:2006ks ; Becker:2023rqi ;
Becker:2024ijy  for the charge lattice, which enables fast computation of the
derivatives of the spacetime superpotential with respect to moduli in the
orbifoldized tensor theory. We choose the normalization

$|\Omega_{l}\rangle:=\frac{4}{\Gamma\Bigl{(}\frac{l}{4}\Bigr{)}}|l\rangle~{},$
(18)

implying

$\langle V_{n}|\Omega_{l}\rangle={\rm i}^{nl}(1-{\rm i}^{l})~{}.$ (19)

Comparing (15) and (19), we have defined

$|\Omega_{l}\rangle=\sum_{n}{\rm i}^{nl}|V_{n}\rangle~{},$ (20)

with the inverse relation

$|V_{n}\rangle=\frac{1}{4}\sum_{l}{\rm i}^{-nl}|\Omega_{l}\rangle~{}.$ (21)

For completeness, we record

$\langle\Omega_{l^{\prime}}|\Omega_{l}\rangle=4~{}\delta_{l^{\prime}+l,4}~{}(1-{\rm
i}^{l})~{}.$ (22)

Reality of $V_{n}$ and eq. (20) together imply the following conjugation rule
on $\Omega_{l}$:

$\overline{\Omega_{l}}:=\Omega_{4-l}=:\Omega_{\bar{l}}~{},$ (23)

where we have defined the conjugate index notation $\bar{l}:=4-l$. The astute
reader may have noticed that the factor of $4$ is switched in equations (20)
and (21) compared to the conventions in the recent paper Becker:2024ijy  on
the $1^{9}$ LG model. The present choice is made following the conventions
of111The concurrently published paper Becker2024  also adopts the same
normalization as ours. Becker:2006ks . We will say more on this soon.

In the full tensor theory, the Ramond ground states are tensor products of the
six factors, labeled by $|{\mathbf{l}}\rangle$,
${\mathbf{l}}=(l_{1},l_{2},\ldots,l_{6})$, $l_{i}\in\\{1,2,3\\}$. The states
that survive the orbifold projection (4) are further restricted to have
$\sum_{i}l_{i}=6,10,14,18$. Following Brunner:2004zd , these can be Hodge
decomposed into classes $H^{(p,q)}$, $p+q=3$, as shown in table 1.

$\sum_{i}l_{i}$ | 6 | 10 | 14 | 18
---|---|---|---|---
$H^{(p,q)}$ | $H^{(3,0)}$ | $H^{(2,1)}$ | $H^{(1,2)}$ | $H^{(0,3)}$
Table 1: Hodge decomposition of RR ground states in the $2^{6}$ LG model

Similar to the states $|{\mathbf{l}}\rangle$, the $3$-cycles are obtained by
tensoring the wedges222For the factor theory $\mathcal{W}=z^{2}$, there are
two wedges that are overlapping straight lines oriented in opposite
directions. We suppress these straight wedges in our notation for tensored
cycles henceforth but carefully incorporate the sign-flip due to the
$\mathbb{Z}_{4}$ action on them. $V_{n}$, and then taking a $\mathbb{Z}$-span.
Concretely, $V_{\mathbf{n}}=V_{n_{1}}\times V_{n_{2}}\times\ldots\times
V_{n_{6}}$, ${\mathbf{n}}=(n_{1},n_{2},\ldots,n_{6})$, $n_{i}\in\\{0,1,2,3\\}$
for all $i=1,2,\ldots,6$. The set $\\{V_{\mathbf{n}}\\}$ is an over-complete
basis due to the relation (11). In the orbifoldized theory, an over-complete
basis is induced by summing over $\mathbb{Z}_{4}$ images:

$|\gamma_{\mathbf{n}}\rangle:=|V_{\mathbf{n}}\rangle-|V_{\mathbf{n+1}}\rangle+|V_{\mathbf{n+2}}\rangle-|V_{\mathbf{n+3}}\rangle~{}.$
(24)

Here and in what follows, $\mathbf{1}=(1,1,1,1,1,1)$,
$\mathbf{2}=2\cdot\mathbf{1}$, and so on. The minus signs in the second and
the fourth terms above are an artifact of the $\mathbb{Z}_{4}$ action (4) on
$z$. Let us now count the number of independent cycles in the middle-
dimensional homology. With $n_{i}\in\\{0,1,2,3\\}$, there are $4^{6}=4096$
different choices of $|\gamma_{\mathbf{n}}\rangle$. This number gets reduced
by a lot because of the relation (11) on each factor. Naively, there are
${6\choose 1}\cdot 4^{5}=6144$ constraints if one imposes (11) factor-by-
factor. These are the vectors $\sum_{n_{i}}|V_{(n_{1}\ldots n_{6})}\rangle$,
$i=1,2,\ldots,6$, set to zero. Clearly, since we have $4096$ cycles
$|\gamma_{\mathbf{n}}\rangle$ to begin with, many of these $6144$ constraints
are linearly dependent. To correct for this, we reduce the number of
constraints by ${6\choose 2}\cdot 4^{4}=3840$, which is the number of
constraints satisfied by the constraints themselves333See Appendix C of
Brunner:2004zd  for the same argument applied to the quintic.. This, again, is
an over-correction which we remedy by adding ${6\choose 3}\cdot 4^{3}=1280$.
Continuing this way, it is easy to see that the number of unconstrained cycles
is444A much faster way to arrive at the number $729$ is to restrict to
$n_{i}\in\\{0,1,2\\}$ from the beginning, since $V_{3}=-(V_{0}+V_{1}+V_{2})$,
and realize that we have $3^{6}=729$ choices for ${\mathbf{n}}$. However, the
effect of orbifolding is easier to work out from (25).

$4^{6}-{6\choose 1}\cdot 4^{5}+{6\choose 2}\cdot
4^{4}-\ldots+1=(4-1)^{6}=729~{}.$ (25)

The above argument can be captured abstractly in terms of a long exact
sequence as was done for the $1^{9}$ model in Becker:2024ijy  and for the
quintic in Brunner:2004zd . Upon orbifolding, identifications take place of
cycles $|\gamma_{\mathbf{n}}\rangle$ and, consequently, the constraints
imposed on them. Each summand in the left-hand side of (25) gets divided by a
factor of $4$ except the last one which counts a $\mathbb{Z}_{4}$-invariant
constraint. Therefore, one gets $\frac{(4-1)^{6}-1}{4}+1=183$ independent
cycles in the sixth tensor power of the minimal model $\mathcal{W}=x^{4}$. Now
including the $\mathcal{W}=z^{2}$ factor, one can repeat the above counting
exercise. We start with $183\times 2$ cycles – the factor of $2$ coming from
the two wedges of the $z$-theory – and we impose the constraint that the two
straight wedges in the $z$-theory add up to zero. This yields $183\times 2-1$
cycles. Orbifolding by the $\mathbb{Z}_{4}$ action on $z$, this number reduces
to $\frac{183\times 2}{2}-1=182$. We will continue to use the same notation
$|\gamma_{\mathbf{n}}\rangle$, ${\mathbf{n}}=(n_{1},n_{2},\ldots,n_{6})$, for
the cycles even after the inclusion of the $z$-factor. In summary, the rank of
the lattice spanned by the $|\gamma_{\mathbf{n}}\rangle$ in the full
orbifoldized theory is $182$.

The overlap (19) generalizes to

$\langle\gamma_{\mathbf{n}}|\Omega_{\mathbf{l}}\rangle={\rm
i}^{{\mathbf{n}}.{\mathbf{l}}}\prod_{i=1}^{6}(1-{\rm i}^{l_{i}})~{},$ (26)

with ${\mathbf{n}}.{\mathbf{l}}:=\sum_{i}n_{i}l_{i}$, where we have divided by
a factor of $4$ due to orbifolding. The $\Omega$-basis in the tensor product
is given by

$|\Omega_{\mathbf{l}}\rangle=\sum_{{\mathbf{n}}}{\rm
i}^{{\mathbf{n}}.{\mathbf{l}}}|V_{\mathbf{n}}\rangle=\sum_{[{\mathbf{n}}]}{\rm
i}^{{\mathbf{n}}.{\mathbf{l}}}|\gamma_{\mathbf{n}}\rangle~{}.$ (27)

The intersection form in this basis is a generalization of (22):

$\langle\Omega_{{\mathbf{l}}^{\prime}}|\Omega_{\mathbf{l}}\rangle=4^{5}~{}\delta_{{\mathbf{l}}^{\prime}+{\mathbf{l}},\mathbf{4}}\prod_{i=1}^{6}(1-{\rm
i}^{i_{i}})~{},$ (28)

where we have again divided by a factor of $4$ due to orbifolding. This is the
normalization used in Sec. $5$ of Becker:2006ks . Using the
$|\gamma_{\mathbf{n}}\rangle$, we have

$\langle\gamma_{\mathbf{n}}^{\prime}|\gamma_{\mathbf{n}}\rangle=\langle
V_{{\mathbf{n}}^{\prime}}|V_{\mathbf{n}}\rangle-\langle
V_{{\mathbf{n}}^{\prime}+\mathbf{1}}|V_{\mathbf{n}}\rangle+\langle
V_{{\mathbf{n}}^{\prime}+\mathbf{2}}|V_{\mathbf{n}}\rangle-\langle
V_{{\mathbf{n}}^{\prime}+\mathbf{3}}|V_{\mathbf{n}}\rangle~{}.$ (29)

The intersection matrix in the $|\gamma_{\mathbf{n}}\rangle$-basis is most
conveniently computed by first going to a truncated set and then carving out a
full-rank sub-matrix, as described in Becker:2006ks . We skip the details here
but point out that this matrix has rank $182$, consistent with our discussion
on the rank of the lattice spanned by the $|\gamma_{\mathbf{n}}\rangle$.

###  2.2 $3$-form fluxes

Having set up the basic ingredients and normalization conventions, we can now
quickly summarize the description of $3$-form fluxes in the LG language. Just
as in the $1^{9}$ model Becker:2006ks ; Becker:2024ijy , the complexified
$3$-form flux $G_{3}=F_{3}-\tau H_{3}$ can be expressed in two equivalent
ways:

$G_{3}=\sum_{\mathbf{n}}(N^{\mathbf{n}}-\tau
M^{\mathbf{n}})\,\gamma_{\mathbf{n}}=\sum_{\mathbf{l}}A^{\mathbf{l}}\,\Omega_{\mathbf{l}}~{},$
(30)

where $N^{\mathbf{n}},M^{\mathbf{n}}\in\mathbb{Z}$, and
$A^{\mathbf{l}}\in\mathbb{C}$. Supersymmetry requires $G_{3}\in
H^{(2,1)}\oplus H^{(0,3)}$, with

$(G_{3})_{\rm Mink}\in H^{(2,1)}~{}.$ (31)

The expansion in the integral cohomology basis $\gamma_{\mathbf{n}}$ is non-
unique since $\\{\gamma_{\mathbf{n}}\\}$ is over-complete to describe
$H^{(2,1)}\oplus H^{(0,3)}$. The set of flux numbers
$N^{\mathbf{n}},M^{\mathbf{n}}\in\mathbb{Z}$ is linearly dependent – there are
$182\times 2=364$ of them, but their $\mathbb{Z}$-span is a lattice of
rank555The rank of the supersymmetric flux lattice can be derived by looking
at the set of quantized “1-$\Omega$” solutions as was done in Becker:2024ijy
for the $1^{9}$ model. $182$. The expansion in the $\Omega$-basis, with
complex coefficients $A^{\mathbf{l}}$, is the most convenient for efficiently
computing higher order terms of the spacetime superpotential. Moreover, by
allowing $A^{\mathbf{l}}$ to be non-zero only when $\sum_{i}l_{i}=10$, we will
restrict to $G_{3}\in H^{(2,1)}$ leading to Minkowski solutions. Such fluxes
span a sub-lattice of rank $180$. Dirac quantization of the fluxes reads

$\int_{\gamma_{\mathbf{n}}}G_{3}=\langle\gamma_{\mathbf{n}}|G_{3}\rangle=N_{\mathbf{n}}-\tau
M_{\mathbf{n}}~{},~{}~{}N_{\mathbf{n}},M_{\mathbf{n}}\in\mathbb{Z}~{}.$ (32)

Using (26) and the $\Omega$-basis expansion in (30),

$\langle\gamma_{\mathbf{n}}|G_{3}\rangle=\sum_{\mathbf{l}}A^{\mathbf{l}}\,{\rm
i}^{{\mathbf{n}}.{\mathbf{l}}}\prod_{i=1}^{6}(1-{\rm
i}^{l_{i}})=N_{\mathbf{n}}-\tau M_{\mathbf{n}}~{}.$ (33)

The O-plane associated with the orientifold projection (5) has charge $40$
that must be cancelled by the flux-tadpole $N_{\rm flux}$, and $N_{D3}$, the
number of mobile D3-branes in the background. Using the overlap (28) in the
$\Omega$-basis, we compute

$N_{\rm flux}=\frac{1}{\tau-\bar{\tau}}\langle
G_{3}|\bar{G}_{3}\rangle=\frac{4^{5}}{\tau-\bar{\tau}}\sum_{\mathbf{l}}|A^{\mathbf{l}}|^{2}\,\prod_{i=1}^{6}(1-{\rm
i}^{-l_{i}})~{},$ (34)

while the tadpole cancellation condition is

$N_{\rm flux}\overset{!}{=}40-N_{D3}~{}.$ (35)

The O-planes are able to contribute a much larger charge in the $2^{6}$ model
to the tadpole than in the $1^{9}$ model. This is because the $1^{9}$ model
corresponds to a toroidal orbifold with non-vanishing $B$ field, whereas, the
$2^{6}$ model corresponds to a toroidal orbifold with vanishing $B$ field
Becker:2006ks . We choose $\tau={\rm i}$ and, for $\Omega_{\mathbf{l}}\in
H^{(2,1)}\oplus H^{(0,3)}$, the product $\prod_{i=1}^{6}(1-{\rm i}^{-l_{i}})$
is purely imaginary with a positive imaginary part. This confirms $N_{\rm
flux}\geq 0$.

###  2.3 The spacetime superpotential to all orders

The Gukov-Vafa-Witten (GVW) superpotential $W$ Gukov:1999ya  engendered by a
flux $G_{3}$ in geometric compactifications is given by

$W=\int G_{3}\wedge\Omega=\int(F_{3}-\tau H_{3})\wedge\Omega~{},$ (36)

where $\Omega$ is the holomorphic $3$-form in the internal space. This has an
analogue in the LG language. Identifying $\Omega\in H^{(3,0)}$ with the RR
ground state $|{\mathbf{1}}\rangle$ according to table 1, we have for the
$2^{6}$ LG model

$W=\langle
G_{3}|{\mathbf{1}}\rangle=\sum_{\mathbf{l}}A^{\mathbf{l}}\,\langle\Omega_{\mathbf{l}}|{\mathbf{1}}\rangle~{}.$
(37)

The overlaps in the factor theory, and consequently in the orbifoldized tensor
theory, are deformed under the “complex structure” deformations parameterized
by666As per eqn. (7), the vectors ${\mathbf{k}}$ are in one-to-one
correspondence with ${\mathbf{l}}$ such that $\Omega_{\mathbf{l}}\in
H^{(2,1)}$ – ${\mathbf{k}}={\mathbf{l}}-{\mathbf{1}}$. $t^{\mathbf{k}}$. This
is through the deformation of the worldsheet superpotential via (8). In a
factor minimal model, $\mathcal{W}\mapsto x^{4}-tx$ leads to the contour
integral 777The contour integral has convergence issues unless $\mathop{\rm
Re}\nolimits t>0$. However, the derivatives are all well defined at $t=0$.
This is consistent with the fact that we can only compute the superpotential
and its derivatives at the Fermat point.

$\langle V_{n}|l\rangle_{\rm
deformed}=\int_{V_{n}}x^{l-1}e^{-x^{4}+t\,x}\,dx~{}.$ (38)

The overlap (38) can be expanded in a Taylor series around $t=0$, with
coefficients

$\Bigl{(}\frac{\partial}{\partial t}\Bigr{)}^{r}\Big{|}_{t=0}\langle
V_{n}|l\rangle_{\rm
deformed}=\int_{V_{n}}x^{r+l-1}\,e^{-x^{4}}\,dx=\frac{1}{4}\,{\rm
i}^{n(r+l)}\,(1-{\rm i}^{r+l})\,\Gamma\Bigl{(}\frac{r+l}{4}\Bigr{)}~{}.$ (39)

In the full theory, the dependence of $W$ on the deformation parameters
$t^{\mathbf{k}}$ introduced in (8) is captured in a multi-variable Taylor
expansion around $t^{\mathbf{k}}=0$, with coefficients that are derivatives of
the overlaps $\langle\Omega_{\mathbf{l}}|{\mathbf{1}}\rangle$ in (37) to be
computed using (39) as a starting point. Without belaboring this calculation,
we quote the result:

$\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial
t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial
t^{\mathbf{k}_{r}}}\bigg{|}_{\begin{subarray}{c}t^{\mathbf{k}}=0\\\\[1.42271pt]
\tau=\tau_{0}\end{subarray}}W=\sum_{\begin{subarray}{c}{\mathbf{l}}:\\\\[1.42271pt]
\bar{\mathbf{l}}={\mathbf{L}}\bmod
4\end{subarray}}\frac{1}{2}~{}A_{\mathbf{l}}~{}\prod_{i=1}^{6}(1-{\rm
i}^{L_{i}})\Gamma\left(\frac{L_{i}}{4}\right)~{},$ (40)

where we have defined

${\mathbf{L}}:=\sum_{\alpha=1}^{r}{\mathbf{k}}_{\alpha}+\mathbf{1}~{}.$ (41)

$W$ also depends linearly on the axio-dilaton $\tau$ through $G_{3}$.
Therefore, we can also compute

$\displaystyle\frac{\partial}{\partial\tau}\frac{\partial}{\partial
t^{\mathbf{k}_{1}}}\frac{\partial}{\partial
t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial
t^{\mathbf{k}_{r}}}\bigg{|}_{\begin{subarray}{c}t^{\mathbf{k}}=0\\\\[1.42271pt]
\tau=\tau_{0}\end{subarray}}W$
$\displaystyle=\sum_{\begin{subarray}{c}{\mathbf{l}}:\\\\[1.42271pt]
\bar{\mathbf{l}}={\mathbf{L}}\bmod 4\end{subarray}}\frac{-{\rm
i}}{4~{}\rm{Im}(\tau_{0})}~{}A_{\mathbf{l}}~{}\prod_{i=1}^{6}(1-{\rm
i}^{L_{i}})\Gamma\left(\frac{L_{i}}{4}\right)~{}$
$\displaystyle~{}+\sum_{\begin{subarray}{c}{\mathbf{l}}:\\\\[1.42271pt]
{\mathbf{l}}={\mathbf{L}}\bmod 4\end{subarray}}\frac{{\rm
i}}{4~{}\rm{Im}(\tau_{0})}~{}\overline{A_{\mathbf{l}}}~{}\prod_{i=1}^{6}(1-{\rm
i}^{L_{i}})\Gamma\left(\frac{L_{i}}{4}\right)~{},$ (42)

while obviously

$\displaystyle\Bigl{(}\frac{\partial}{\partial\tau}\Bigr{)}^{p\geq
2}\frac{\partial}{\partial t^{\mathbf{k}_{1}}}\frac{\partial}{\partial
t^{\mathbf{k}_{2}}}\ldots\frac{\partial}{\partial
t^{\mathbf{k}_{r}}}\bigg{|}_{\begin{subarray}{c}t^{\mathbf{k}}=0\\\\[1.42271pt]
\tau=\tau_{0}\end{subarray}}W$ $\displaystyle=0~{}.$ (43)

Using (40) and (2.3), we have

$\displaystyle W=\sum_{\bf{l}}\sum_{r=1}^{\infty}$
$\displaystyle\frac{1}{r!}\left(\sum_{\\{t^{\mathbf{k}_{\alpha}}\\}\text{ with
}\bar{\mathbf{l}}={\mathbf{L}}}\prod_{i=1}^{6}t^{\mathbf{k}_{1}}t^{\mathbf{k}_{2}}\ldots
t^{\mathbf{k}_{r}}\,~{}A_{\mathbf{l}}~{}\prod_{i=1}^{6}(1-{\rm
i}^{L_{i}})\Gamma\left(\frac{L_{i}}{4}\right)\left(\frac{1}{2}-{\rm
i}\frac{\tau-\tau_{0}}{4\mathop{\rm Im}\nolimits(\tau_{0})}\right)\right.$
(44) $\displaystyle\left.+{\rm i}\sum_{\\{t^{\mathbf{k}_{\alpha}}\\}\text{
with
}{\mathbf{l}}={\mathbf{L}}}\prod_{i=1}^{6}t^{\mathbf{k}_{1}}t^{\mathbf{k}_{2}}\ldots
t^{\mathbf{k}_{r}}\,\overline{A_{\mathbf{l}}}~{}\prod_{i=1}^{6}(1-{\rm
i}^{L_{i}})\Gamma\left(\frac{L_{i}}{4}\right)\,\frac{\tau-\tau_{0}}{{4\mathop{\rm
Im}\nolimits(\tau_{0})}}\right)\,,$ (45)

where we will set $\tau_{0}={\rm i}$ and we refer to $\tau-\tau_{0}$ as
$t^{0}$ .

##  3 Moduli stabilization at higher order

In this section, we examine the stabilization of massless fields via the
higher-order terms in the superpotential Becker:2022hse . Such an analysis was
performed recently in Becker:2024ijy  for many different flux configurations
in the $1^{9}$ LG model. The authors of that paper studied higher-order terms
in the superpotential up to 6th order in the moduli. They found that, for some
flux configurations, no massless fields get stabilized via higher-order terms
although all moduli do appear in the superpotential. For other flux choices,
more and more fields get stabilized at higher order but the number of newly
stabilized massless fields seems to quickly tend to zero (see table 2 on page
35 of Becker:2024ijy ). Here we perform a similar analysis for the $2^{6}$
model, which produces new and surprising results.

###  3.1 Review of higher-order stabilization

To illustrate the algorithm, which is explained in detail in Becker:2024ijy ,
we study a toy model. Consider the following superpotential

$W=\frac{1}{2}\left(\phi-\psi^{2}\right)^{2}\,,$ (46)

where $\phi$ and $\psi$ are complex fields. It necessarily vanishes at its
critical points and its vanishing locus is given by $\phi=\psi^{2}$. We expand
this superpotential order-by-order around a critical point to arrive at the
same conclusion. Whilst this is redundant for a function as simple as the one
in $\eqref{eq:example}$, it will be relevant for our analysis later as we will
only be able to find supersymmetric Minkowski critical points of the GVW
superpotential and compute its derivatives at a specific point in the moduli
space. The origin, $\left(\phi,\psi\right)=\left(0,0\right)$ is clearly a
critical point of the superpotential where it vanishes. As a result, expanding
around the origin we have,

$W=\frac{1}{2!}\partial_{i}\partial_{j}W|_{\\{t^{i}\\}=0}t^{i}t^{j}+\frac{1}{3!}\partial_{i}\partial_{j}\partial_{k}W|_{\\{t^{i}\\}=0}t^{i}t^{j}t^{k}+\frac{1}{4!}\partial_{i}\partial_{j}\partial_{k}\partial_{l}W|_{\\{t^{i}\\}=0}t^{i}t^{j}t^{k}t^{l}\,,$
(47)

where $t^{i}\in\\{\phi,\psi\\}$. The series truncates in this simple example
but for the supersymmetric Minkowski critical points that we will study, the
superpotential does not truncate when expanding around the critical points. As
a result, schematically we will always have

$W=\sum_{n\geq 2}W_{n}\,,$ (48)

where $W_{n}=\frac{1}{n!}(\partial_{i_{1}\ldots i_{n}}W)t^{i_{1}}\ldots
t^{i_{n}}$. We would first like to ascertain the rank of the deformation space
of the critical point after including terms up to quadratic order in the
fields in $W$. This is done by studying the critical point equations,

$\displaystyle\partial_{\phi}W_{2}$ $\displaystyle=\phi=0$ (49)
$\displaystyle\partial_{\psi}W_{2}$ $\displaystyle=0\,.$

At this quadratic order, we find that one field $\left(\phi\right)$ is fixed
up to linear order in the unfixed field $\left(\psi\right)$. Although it is
not possible to determine which field directions are fixed without knowing the
canonically normalized kinetic terms, we can count the number of fields that
are fixed. The number of fields that are fixed at the quadratic order is equal
to the rank of the Hessian of the superpotential Becker:2022hse  and so we
will refer to these fields as being massive. The field $\phi$ is fixed up to
linear order in the massless fields due to (49) but could receive higher order
corrections. The more precise statement would hence be that the critical point
equations are evaluated as order-by-order expansions of the stabilized fields
in terms of the unfixed fields,

$\phi=\phi_{(1)}+\phi_{(2)}+\phi_{(3)}+\ldots\,,$ (50)

and we have determined that $\phi_{(1)}$ vanishes. The higher-order terms will
get fixed sequentially as we include higher-order contributions in the
superpotential. Now we wish to inspect whether the massless field gets lifted
as we include higher-than-quadratic terms in $W$. First, we include cubic
terms and analyze the critical point equations after truncating to quadratic
order in the unfixed fields,

$\displaystyle\partial_{\phi}W_{2}+\partial_{\phi}W_{3}$
$\displaystyle=\phi_{(1)}+\phi_{(2)}-\psi^{2}=0$ (51)
$\displaystyle\partial_{\psi}W_{2}+\partial_{\psi}W_{3}$
$\displaystyle=-2\phi_{(1)}\psi=0$

Since $\phi_{(1)}=0$ , the first equation fixes $\phi_{(2)}=\psi^{2}$ and the
second equation trivially vanishes. The field $\psi$ remains unfixed. After
including quartic terms in the superpotential and truncating the critical
point equations to cubic order in the unfixed fields we get

$\displaystyle\partial_{\phi}W_{2}+\partial_{\phi}W_{3}+\partial_{\phi}W_{4}$
$\displaystyle=\phi_{(1)}+\phi_{(2)}-\psi^{2}+\phi_{(3)}=0$ (52)
$\displaystyle\partial_{\psi}W_{2}+\partial_{\psi}W_{3}+\partial_{\psi}W_{4}$
$\displaystyle=-2\phi_{(1)}\psi-2\phi_{(2)}\psi+2\psi^{3}=0\,.$

Once we evaluate these equations at $\phi_{(1)}=0\,,\phi_{(2)}=\psi^{2}$ , we
find $\phi_{(3)}=0$ . Since the series truncates at this order and the field
$\psi$ is unfixed, we conclude that there is one flat direction and the origin
is not an isolated critical point.

The critical points of the superpotential in equation (44) that we will study
have some similarities to the example we discussed above but in general will
be more complicated. As was explained in Becker:2024ijy , this algorithm has
some limitations. For instance, since the series expansion around the critical
point does not truncate in general, we will only be able to make definitive
statements about the number of stabilized fields at a given order. The fields
that remain unfixed are not necessarily flat directions and may be lifted at
some higher order that we have not analyzed due to computational constraints.
Another potential limitation is the appearance of branches. When solving the
polynomial equations that arise as non-trivial critical point equations we
find multiple branches of solutions which can lead to different numbers of
stabilized fields. For example, consider the two equations

$xy=0\,,\quad\quad xz=0\,.$ (53)

They are solved by either $x=0$ or $y=z=0$. We will see explicit examples
where this occurs in section 3.3. The occurrence of these branches of
solutions provides a challenge to single out the exact number of fields
stabilized up to a given order in the superpotential. As a remedy to this
issue, it was proposed in Becker:2024ijy  that the minimum number of fields
stabilized across the different branches be taken as the number of stabilized
fields at the critical point. For a more thorough mathematical discussion in
terms of the Zariski and Krull co-dimensions of the critical points, we refer
the reader to Becker:2024ijy .

###  3.2 One to three $\Omega$ flux choices

In this subsection we study the simplest flux configurations, i.e., we turn on
only 1, 2, or 3 different flux components in the $\Omega$-basis. While this
makes some things very easy, it also has severe drawbacks as we describe
below. We do not only consider physical solutions with $N_{\rm flux}\leq 40$
but also two unphysical ones.

There are three $S_{6}$ distinct single $\Omega$ solutions whose shortest,
properly quantized, flux vectors are

$\displaystyle G^{1}$ $\displaystyle=\frac{1}{32}\,\Omega_{1,1,2,2,2,2}\,,$
(54) $\displaystyle G^{2}$
$\displaystyle=\frac{1}{16}\,\Omega_{1,1,1,2,2,3}\,,$ (55) $\displaystyle
G^{3}$ $\displaystyle=\frac{1}{8}\,\Omega_{1,1,1,1,3,3}\,.$ (56)

The last solution $G^{3}$ has $N_{\rm flux}=64$ and it is thus unphysical.
However, these three solutions (and their $S_{6}$ permutations) are the
building blocks for all other solutions so we analyzed all three.

For the first flux choice $G^{1}$ which has $N_{\rm flux}=16$ we find 36
massive fields and no higher order stabilization even when going to the 6th
power in the superpotential. For the second flux choice $G^{2}$ with $N_{\rm
flux}=32$ we actually find higher order stabilization already when including
the cubic terms in the superpotential. However, we get 34 non-trivial
quadratic constraints involving 34 massless fields. Unfortunately, we have not
been able to solve these equations and therefore we do not know how many of
the massless fields actually get stabilized at 3rd power in the
superpotential. This is a recurring problem for these simple flux
configurations with a small number of $\Omega$ components turned on. If they
stabilize fields at higher orders, then there are many corresponding
constraint equations that involve a large number of fields and we cannot solve
them using standard techniques in, for example, Mathematica. Solution $G^{3}$
has $N_{\rm flux}=64$ and is therefore unphysical. It has 21 massive fields
and 52 non-trivial quadratic constraints in 52 variables when including cubic
terms in the superpotential. We are again unable to solve these. It would be
interesting to understand why for certain flux configurations we find no
moduli stabilization Grimm:2024fip , but for now we simply present it here as
an observation.

We studied the following flux choices with two $\Omega$ components and found
similar results

$\displaystyle G^{4}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{64}+\frac{{\rm
i}}{64}\right)\Omega_{1,1,2,2,2,2}+\left(\frac{1}{64}+\frac{{\rm
i}}{64}\right)\Omega_{2,2,1,1,2,2}\,,$ (57) $\displaystyle G^{5}$
$\displaystyle=$ $\displaystyle\left(-\frac{1}{64}+\frac{{\rm
i}}{64}\right)\Omega_{1,1,2,2,2,2}-\left(\frac{1}{64}-\frac{{\rm
i}}{64}\right)\Omega_{1,2,1,2,2,2}\,,$ (58) $\displaystyle G^{6}$
$\displaystyle=$
$\displaystyle\frac{1}{16}\,\Omega_{1,1,1,1,3,3}-\frac{1}{16}\,\Omega_{1,1,1,3,1,3}\,,$
(59) $\displaystyle G^{7}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\,\Omega_{1,1,1,1,3,3}-\frac{1}{16}\,\Omega_{1,1,3,3,1,1}\,,$
(60) $\displaystyle G^{8}$ $\displaystyle=$
$\displaystyle\frac{1}{32}\,\Omega_{1,1,1,2,2,3}+\frac{1}{32}\,\Omega_{1,1,3,2,2,1}\,,$
(61) $\displaystyle G^{9}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{16}+\frac{{\rm
i}}{16}\right)\Omega_{1,1,1,1,3,3}-\left(\frac{1}{64}-\frac{{\rm
i}}{64}\right)\Omega_{1,1,2,2,2,2}\,,$ (62) $\displaystyle G^{10}$
$\displaystyle=$ $\displaystyle\left(-\frac{1}{16}+\frac{{\rm
i}}{16}\right)\Omega_{1,1,1,1,3,3}-\left(\frac{1}{32}-\frac{{\rm
i}}{32}\right)\Omega_{1,1,1,2,2,3}\,,$ (63) $\displaystyle G^{11}$
$\displaystyle=$ $\displaystyle\left(-\frac{1}{16}+\frac{{\rm
i}}{16}\right)\Omega_{1,1,1,1,3,3}-\left(\frac{1}{32}-\frac{{\rm
i}}{32}\right)\Omega_{1,1,2,3,1,2}\,.$ (64)

A summary of the results for all the above flux choices and the one three
$\Omega$ flux configuration

$G^{12}=\left(\frac{1}{32}-\frac{{\rm
i}}{32}\right)\Omega_{1,1,1,1,3,3}+\frac{1}{64}\Omega_{1,2,1,2,2,2}+\left(\frac{1}{32}+\frac{{\rm
i}}{32}\right)\Omega_{1,3,1,3,1,1}$ (65)

is presented in table 2.

Model | $N_{\rm flux}$ | massive | 3rd power | 4th power | 5th power | 6th power
---|---|---|---|---|---|---
$G^{1}$ | 16 | 36 | 0 | 0 | 0 | 0
$G^{2}$ | 32 | 28 | $\neq 0$ |  |  |
$G^{3}$ | 64 | 21 | $\neq 0$ |  |  |
$G^{4}$ | 16 | 52 | 0 | 0 | 0 | 0
$G^{5}$ | 16 | 36 | 0 | 0 | 0 | 0
$G^{6}$ | 32 | 28 | $\neq 0$ |  |  |
$G^{7}$ | 32 | 36 | $\neq 0$ |  |  |
$G^{8}$ | 16 | 36 | 0 | 0 | 0 | 0
$G^{9}$ | 40 | 44 | $\neq 0$ |  |  |
$G^{10}$ | 48 | 28 | $\neq 0$ |  |  |
$G^{11}$ | 48 | 41 | $\neq 0$ |  |  |
$G^{12}$ | 20 | 48 | 0 | 0 | 0 | 0
Table 2: A summary of the different most simple flux choices that we have
analyzed. The second column lists $N_{\rm flux}$, the tadpole contribution of
the particular flux configuration. The third column lists the number of
massive fields and the following columns the number of fields that are
massless but do get fixed due to terms of the $r$-th power in the
superpotential that are polynomials. Whenever fields get stabilized we were
not able to solve the corresponding constraints and could not pursue that flux
choice to higher order.

The clear result of the table is that solutions with $N_{\rm flux}=16$, which
is really low, seem to have no further stabilization at higher order. However,
solutions with $N_{\rm flux}\geq 32$ generically lead to the stabilization of
massless fields through cubic interactions. For that reason, we have included
here the 3 $\Omega$ solution $G^{12}$ with $N_{\rm flux}=20$ to check at which
point higher order stabilization takes place. We see that for this particular
three $\Omega$ solution there are still no non-trivial higher-order
constraints that stabilize massless fields. However, as we show in the next
subsection, for a particular four $\Omega$ solution with $N_{\rm flux}=28$
there are higher order constraints that we can actually analyze.

Before ending this subsection we state the super important result that the
flux choice $G^{4}$ has $N_{\rm flux}=16$ and 52 massive fields. To our
knowledge, this is the first instance in the literature that violates the
tadpole bound: $N_{\rm flux}>\tfrac{1}{3}n_{\rm stab}$. Thus the $2^{6}$ LG
model seems to violate the Refined Tadpole Conjecture in its original form
Bena:2020xrh .888The tadpole condition in this example is not satisfied and
requires the presence of 24 D3-branes. Those might have moduli associated with
themselves but that is irrelevant for the fact that 52 complex structure
moduli are massive for this simple flux choice. We comment on this further in
section 4 below.

###  3.3 A four $\Omega$ flux choice

Given the slightly disappointing outcome for the stabilization at higher order
for the simplest solutions in the $2^{6}$ model, we now turn to an
intermediate solution that is much more interesting. This flux choice was
discovered by the authors of Becker2024  and they kindly allowed us to study
the higher-order stabilization for it, already before they published their
paper. The flux choice is

$G^{13}=\left(\frac{1}{32}+\frac{{\rm
i}}{32}\right)\Omega_{1,1,1,1,3,3}+\left(\frac{1}{32}+\frac{{\rm
i}}{32}\right)\Omega_{1,3,1,3,1,1}+\frac{1}{32}\Omega_{3,2,2,1,1,1}+\frac{{\rm
i}}{64}\Omega_{1,1,2,2,2,2}$ (66)

and it leads to $N_{\rm flux}=28$ and $75$ massive fields. After including
cubic terms in the superpotential we find the following non-trivial quadratic
constraints,

$\displaystyle t^{34}t^{87}$ $\displaystyle=0$ (67) $\displaystyle
t^{33}t^{86}$ $\displaystyle=0$ (68) $\displaystyle t^{18}t^{52}$
$\displaystyle=0$ (69) $\displaystyle t^{17}t^{51}$ $\displaystyle=0$ (70)
$\displaystyle t^{33}t^{51}$ $\displaystyle=0$ (71) $\displaystyle
t^{34}t^{52}$ $\displaystyle=0$ (72) $\displaystyle t^{18}t^{51}+t^{17}t^{52}$
$\displaystyle=0$ (73) $\displaystyle t^{50}t^{51}-(1-{\rm i})t^{52}t^{58}$
$\displaystyle=0$ (74) $\displaystyle t^{48}t^{52}-(1-{\rm i})t^{51}t^{58}$
$\displaystyle=0$ (75) $\displaystyle t^{33}t^{38}-(1-{\rm i})t^{34}t^{44}$
$\displaystyle=0$ (76) $\displaystyle t^{34}t^{36}-(1-{\rm i})t^{33}t^{44}$
$\displaystyle=0$ (77) $\displaystyle t^{51}t^{52}+2{\rm i}t^{34}t^{86}+2{\rm
i}t^{33}t^{87}$ $\displaystyle=0$ (78) $\displaystyle
t^{51}t^{62}+t^{38}t^{86}-(1-{\rm i})t^{44}t^{87}$ $\displaystyle=0$ (79)
$\displaystyle t^{52}t^{64}+t^{36}t^{87}-(1-{\rm i})t^{44}t^{86}$
$\displaystyle=0$ (80) $\displaystyle t^{44}t^{52}+(1-{\rm
i})t^{17}t^{50}+(1-{\rm i})t^{33}t^{62}+2{\rm i}t^{18}t^{58}$
$\displaystyle=0$ (81) $\displaystyle t^{44}t^{51}+(1-{\rm
i})t^{18}t^{48}+(1-{\rm i})t^{34}t^{64}+2{\rm i}t^{17}t^{58}$
$\displaystyle=0$

Solving the above constraints leads to multiple branches of solutions. There
are branches which fix $8$, $9$, $10$, and $12$ fields. The branches that
stabilize $9,10$ and, $12$ fields already violate the bound of the Refined
Tadpole Conjecture. However, the branches that stabilize $8$ fields do not
violate the bound. As explained in Becker:2024ijy , when such branches with
different numbers of stabilized fields appear, we pick the branch with the
least number of stabilized fields. So at cubic order, we do not have a
violation of the bound yet. We then analyse the non-trivial cubic constraints
arising after the inclusion of quartic terms in the superpotential. The
minimum number of stabilized fields occur on branches that had $8$ or $9$
fields stabilized at cubic order. There are branches that had stabilized $9$
fields at cubic order, that stabilize no fields at quartic or quintic order,
leaving us with $75+9+0+0=84=28\cdot 3$ stabilized fields. So, this would
violate the original Refined Tadpole Conjecture with $\alpha<1/3$. All
branches that stabilized $8$ fields at cubic order stabilize at least $1$
field at quartic order, thus leading to $75+8+1=84=28\cdot 3$ and the same
conclusion as above.

Model | $N_{\rm flux}$ | massive | 3rd power | 4th power
---|---|---|---|---
$G^{13}$ | 28 | 75 | 8 | 1
$G^{13}$ | 28 | 75 | 8 | 2
$G^{13}$ | 28 | 75 | 8 | 3
$G^{13}$ | 28 | 75 | 9 | 0
$G^{13}$ | 28 | 75 | 9 | 1
$G^{13}$ | 28 | 75 | 9 | 2
$G^{13}$ | 28 | 75 | 9 | 3
$G^{13}$ | 28 | 75 | 10 | 0
$G^{13}$ | 28 | 75 | 10 | 1
$G^{13}$ | 28 | 75 | 10 | 2
$G^{13}$ | 28 | 75 | 12 | 0
Table 3: A summary of the different branches for the flux choice $G^{13}$. The
second column lists $N_{\rm flux}$ the tadpole contribution of this particular
flux configuration. The third column lists the number of massive fields and
the following columns the number of fields that are massless but do get fixed
due to terms of the $r$-th power in the superpotential that are polynomials.
We find different components that either fix the same or different numbers of
fields, as indicated in the multiple rows for this single model.

###  3.4 Close to Full Rank Solutions

In this section, we present our study of 175 different solutions with a
tadpole contribution of $N_{\rm flux}=39$ or $N_{\rm flux}=40$. They all have
close to full mass matrix rank with between 86 and 90 massive fields. Since
all of these solutions have many different $\Omega$ components with non-
vanishing coefficients, it is not possible to present them all here and we
have thus enclosed them in a Mathematica notebook that we attached as an
ancillary file to this paper.

For the curious reader, let us briefly mention how we obtained these high rank
solutions. There is a straightforward algorithm to obtain an integral basis of
the supersymmetric flux lattice in the present model. It is the same algorithm
that was used in Becker:2023rqi  to derive an integral basis for Minkowski
fluxes in the $1^{9}$ model. See also Becker:2024ijy  for a detailed
description. We employed this algorithm in the $2^{6}$ model to obtain $180$
basis elements for the Minkowski lattice. Using “small linear combinations” –
with four or five basis vectors linearly combined with small integer
coefficients (between $-2$ and $2$) – it was possible to generate solutions
with a large variety of $(N_{\rm flux},n_{\rm massive})$ values. All of our
high rank solutions were obtained this way, with a remarkably small
computational effort. The authors of Becker2024 , in their attempt to
extensively test the tadpole conjecture, use this as one of the methods to
generate a large number of solutions.

For all of these high rank flux configurations, we have checked the potential
stabilization of massless fields via cubic and quartic terms in the
superpotential. For several, there seems to be no further stabilization, but
we also find several models where 1, 2, 3, or 4 fields get stabilized either
through cubic, quartic, or quintic terms in the superpotential. Surprisingly,
we have never found an example where fields get stabilized at cubic and
quartic levels, contrary to what was found for the $1^{9}$ model in
Becker:2024ijy .

For concreteness, we present here one example of a flux configuration with
$N_{\rm flux}=40$ that has 87 massive fields

$\displaystyle G^{14}$
$\displaystyle=\frac{1}{32}\left[-\frac{1}{2}\Omega_{1,1,1,1,3,3}+\frac{1}{2}\Omega_{1,1,3,3,1,1}-\frac{{\rm
i}}{2}\Omega_{1,3,1,3,1,1}\right.$ (82)
$\displaystyle\qquad+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,3,2,1,1,2}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,3,2,1,2,1}-\frac{1}{2}\Omega_{1,3,2,2,1,1}$ (83)
$\displaystyle\qquad-\frac{{\rm
i}}{2}\Omega_{1,3,3,1,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,1,1,2,3}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,1,1,3,2}$ (84)
$\displaystyle\qquad-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,2,3,1,1}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,3,2,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,2,1,3,1,1}$ (85) $\displaystyle\qquad-\frac{{\rm
i}}{4}\Omega_{2,2,2,1,1,2}-\frac{{\rm
i}}{4}\Omega_{2,2,2,1,2,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,2,2,2,1,1}$ (86)
$\displaystyle\qquad-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,2,3,1,1,1}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,1,1,2}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,1,2,1}$ (87)
$\displaystyle\qquad+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,2,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,2,1,1,1}-\frac{{\rm i}}{2}\Omega_{3,1,1,1,1,3}$ (88)
$\displaystyle\qquad-\frac{{\rm i}}{2}\Omega_{3,1,1,1,2,2}-\frac{{\rm
i}}{2}\Omega_{3,1,1,1,3,1}+\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,1,1,3,1,1}$ (89)
$\displaystyle\qquad-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,1,2,1,1,2}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,1,2,1,2,1}+\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,1,3,1,1,1}$ (90) $\displaystyle\qquad\left.+\frac{{\rm
i}}{2}\Omega_{3,2,1,1,1,2}+\frac{{\rm i}}{2}\Omega_{3,2,1,1,2,1}-\frac{{\rm
i}}{2}\Omega_{3,2,1,2,1,1}-\frac{{\rm i}}{2}\Omega_{3,2,2,1,1,1}\right]\,.$

The above flux choice has 32 $\Omega$ components turned on and satisfies the
tadpole cancellation condition without the requirement of adding D3-branes. It
has close to full mass matrix rank, which means that the 87 massive fields
solve 87 of the 91 equations $\partial_{i}W=0$. After solving these 87
equations for the massive fields, we can plug the answer into the remaining
four equations that are then functions of the four unstabilized fields only,
which makes them easy to solve. When calculating $\partial_{i}(W_{2}+W_{3})=0$
we find that these four equations are trivially satisfied. For
$\partial_{i}(W_{2}+W_{3}+W_{4})=0$ we find that 3 massless fields get
stabilized and there are no further constraints when including $W_{5}$. Thus
this model has, up to including $W_{5}$, a total of 87+0+3+0=90 stabilized
fields. One might be encouraged by this and look for other examples that get
to 91 or try to push this model to higher order in the fields. We did the
former and studied 140 different flux configurations with $N_{\rm flux}=40$
and found a variety of different stabilization patterns at higher order as is
summarized in table 4. In this table 4 we did not pursue all different
branches for all of these flux choices but only the one that stabilizes the
least number of massive fields.

Multiplicity | $N_{\rm flux}$ | massive | 3rd power | 4th power | 5th power
---|---|---|---|---|---
78 | 40 | 90 | 0 | 0 | 0
1 | 40 | 89 | 1 | 0 | 0
2 | 40 | 89 | 0 | 1 | 0
13 | 40 | 89 | 0 | 0 | 0
1 | 40 | 88 | 2 | 0 | 0
5 | 40 | 88 | 0 | 1 | 0
2 | 40 | 88 | 0 | 0 | 1
5 | 40 | 88 | 0 | 0 | 0
1 | 40 | 87 | 3 | 0 | 0
2 | 40 | 87 | 0 | 3 | 0
8 | 40 | 87 | 0 | 2 | 0
1 | 40 | 87 | 1 | 0 | 1
5 | 40 | 87 | 0 | 1 | 0
2 | 40 | 86 | 0 | 4 | 0
4 | 40 | 86 | 0 | 2 | 0
3 | 40 | 86 | 0 | 1 | 0
7 | 40 | 86 | 0 | 0 | 0
Table 4: A summary of 140 different flux choices with $N_{\rm flux}=40$ that
we have analyzed. The first column lists the number of $S_{6}$ distinct
solutions that exhibit the shown pattern of massive fields and massless
stabilized fields at the $r$-th power in the superpotential. It is of course
possible that these different flux choices behave differently and split
further if one were to go to even higher powers in the superpotential.

The flux choices in table 4 all have $N_{\rm flux}=40$ and a mass matrix rank
between 86-90 but are otherwise distinct and in a certain sense generated
randomly. We might thus be tempted to do some statistical interpretation of
our results. The table indicates that the stabilization of massless fields at
higher order is less likely than no stabilization (at least to the order we
work). Stabilizing fewer fields seems also more likely than stabilizing more
fields. However, probably the most surprising result is that higher-order
stabilization for all mass matrix ranks from 86-90 seems to not lead to all
fields being stabilized. The maximum number of massless, stabilized field is
given by $90$ minus the entry in the “massive” column.

We have also generated a set of 33 solutions with $N_{\rm flux}=39$ and mass
matrix ranks 86, 87, 88, and 89. We again studied higher-order stabilization
and our results are summarized in table 5. In this table 5 we did not pursue
all different branches for all of these flux choices but only the one that
stabilizes the least number of massless fields.

Multiplicity | $N_{\rm flux}$ | massive | 3rd power | 4th power | 5th power
---|---|---|---|---|---
2 | 39 | 89 | 0 | 1 | 0
3 | 39 | 89 | 0 | 0 | 0
1 | 39 | 88 | 1 | 0 | 1
1 | 39 | 88 | 1 | 0 | 0
2 | 39 | 88 | 0 | 1 | 0
2 | 39 | 88 | 0 | 0 | 1
1 | 39 | 88 | 0 | 0 | 0
3 | 39 | 87 | 0 | 2 | 0
2 | 39 | 87 | 0 | 1 | 0
1 | 39 | 87 | 0 | 0 | 0
2 | 39 | 86 | 0 | 2 | 0
2 | 39 | 86 | 1 | 0 | 0
4 | 39 | 86 | 0 | 1 | 0
7 | 39 | 86 | 0 | 0 | 0
Table 5: A summary of the different flux choices with $N_{\rm flux}=39$ that
we have analyzed. The first column lists the number of $S_{6}$ distinct
solutions that exhibit the shown pattern of massive fields and massless
stabilized fields at the $r$-th power in the superpotential. It is of course
possible that these different flux choices behave differently and split
further if one were to go to even higher powers in the superpotential.

###  3.5 Fully stabilized Minkowski vacua

Lastly, we present two solutions that were part of our randomly generated set
of solutions with $N_{\rm flux}=40$ and that actually do stabilize _all_
fields through higher order terms. Given that there are only 2 out of 142
examples, this shows that these are rare but they do exist. These two
solutions provide to our knowledge the first Minkowski vacua without any flat
directions (besides 11d supergravity). They both have 86 massive fields, 0
stabilized fields when including cubic terms via $W_{3}$, and finally 5
stabilized fields when including $W_{4}$. At that point, we do not need to
worry about higher-order terms since we cannot stabilize any more fields and
$\partial_{i}W_{n}=0$ is automatically satisfied for all $n$ since all moduli
are stabilized at the origin of moduli space. Note that these solutions have
$N_{\rm flux}=40$ and 91 stabilized fields. They are thus not in tension with
the Refined Tadpole Conjecture that would allow the stabilization of up to 119
fields for $N_{\rm flux}=40$.

The two solutions are

$\begin{split}G^{15}&=\frac{1}{32}\left[\left(-\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,1,2,3,1,2}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,2,2,1,3,1}+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{1,2,3,2,1,1}\right.\\\
&\qquad-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,3,1,2,1,2}+\frac{1}{2}\Omega_{1,3,2,1,1,2}-\frac{1}{2}\Omega_{1,3,2,1,2,1}\\\
&\qquad+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,1,1,3,1,2}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,2,1,3,1}+\frac{1}{4}\Omega_{2,1,2,2,1,2}\\\
&\qquad+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,1,2,3,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,1,3,2,1,1}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,2,1,1,3,1}\\\ &\qquad+\frac{{\rm
i}}{4}\Omega_{2,2,1,2,1,2}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,2,2,1,1,2}+\frac{1}{4}\Omega_{2,2,2,1,2,1}\\\
&\qquad-\frac{1}{4}\Omega_{2,2,2,2,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,2,3,1,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,1,1,2}\\\
&\qquad+\frac{1}{2}\Omega_{2,3,1,1,2,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,2,1,1}+\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,1,1,1,3,1}\\\ &\qquad-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{3,1,1,2,1,2}-\left(\frac{1}{2}-\frac{{\rm
i}}{2}\right)\Omega_{3,1,1,3,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,1,2,1,1,2}\\\ &\qquad+\left(\frac{1}{2}-\frac{{\rm
i}}{2}\right)\Omega_{3,1,3,1,1,1}+\frac{1}{2}\Omega_{3,2,1,1,1,2}-\frac{1}{2}\Omega_{3,2,1,1,2,1}\\\
&\qquad\left.+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,2,1,2,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{3,2,2,1,1,1}-\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,3,1,1,1,1}\right]\end{split}$ (91)

and

$\displaystyle G^{16}$
$\displaystyle=\frac{1}{32}\left[\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,1,2,3,2,1}+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{1,2,2,1,1,3}-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{1,2,3,2,1,1}\right.$ (92)
$\displaystyle\qquad-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{1,3,1,2,2,1}+\frac{{\rm
i}}{2}\Omega_{1,3,2,1,1,2}-\frac{{\rm i}}{2}\Omega_{1,3,2,1,2,1}$ (93)
$\displaystyle\qquad-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,1,3,2,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,1,2,1,1,3}-\frac{{\rm i}}{4}\Omega_{2,1,2,2,2,1}$ (94)
$\displaystyle\qquad-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,2,3,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,1,3,2,1,1}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,2,1,1,1,3}$ (95)
$\displaystyle\qquad+\frac{1}{4}\Omega_{2,2,1,2,2,1}-\frac{{\rm
i}}{4}\Omega_{2,2,2,1,1,2}-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,2,2,1,2,1}$ (96) $\displaystyle\qquad+\frac{{\rm
i}}{4}\Omega_{2,2,2,2,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,2,3,1,1,1}-\frac{{\rm i}}{2}\Omega_{2,3,1,1,1,2}$ (97)
$\displaystyle\qquad+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,1,2,1}+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{2,3,1,2,1,1}+\left(\frac{1}{2}-\frac{{\rm
i}}{2}\right)\Omega_{3,1,1,1,1,3}$ (98)
$\displaystyle\qquad-\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,1,1,2,2,1}+\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,1,1,3,1,1}+\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{3,1,2,1,2,1}$ (99)
$\displaystyle\qquad-\left(\frac{1}{2}+\frac{{\rm
i}}{2}\right)\Omega_{3,1,3,1,1,1}+\frac{{\rm
i}}{2}\Omega_{3,2,1,1,1,2}-\frac{{\rm i}}{2}\Omega_{3,2,1,1,2,1}$ (100)
$\displaystyle\qquad\left.-\left(\frac{1}{4}+\frac{{\rm
i}}{4}\right)\Omega_{3,2,1,2,1,1}+\left(\frac{1}{4}-\frac{{\rm
i}}{4}\right)\Omega_{3,2,2,1,1,1}-\left(\frac{1}{2}-\frac{{\rm
i}}{2}\right)\Omega_{3,3,1,1,1,1}\right]\,.$

In a paper that appeared at the same time as ours Becker2024 , the authors
found many flux configurations in this model that have $N_{\rm flux}=40$ and
full rank 91 for the mass matrix. So, these solutions in Becker2024  are the
first Minkowski vacua in which all scalar fields are massive. We comment
further on the significance of these and our solutions in the next section.

##  4 Implications for the swampland and the landscape

In this section, we discuss the implications of the different flux
compactifications of the $2^{6}$ LG model for our understanding of the
landscape and the swampland conjectures.

###  4.1 Full moduli stabilization and 4d $\mathcal{N}=1$ vacua

Full moduli stabilization, i.e. finding _isolated_ vacua in the string
landscape, is arguably one of the greatest challenges in string phenomenology
today. Roughly twenty years ago many scenarios were put forward like KKLT
Kachru:2003aw , LVS Balasubramanian:2005zx ; Conlon:2005ki  and DGKT
DeWolfe:2005uu  to name a few. However, throughout the last decade, all of
these constructions have been scrutinized and certain aspects have been called
into question within the swampland program. This means that there are
currently no constructions that are agreed upon by the entire string
phenomenology community which leaves us in a dire spot. Note that this is not
related to the presence or absence of supersymmetry or the value of the
cosmological constant.

Finding new simple models with full moduli stabilization has thus become an
important goal in string phenomenology. Non-geometric LG models provide an
intriguing way of doing so. On the one hand, they allow us to bring the full
strength of string theory to the problem. On the other hand, they give rise to
very simple 4d $\mathcal{N}=1$ supergravity theories that depend, in the
presence of fluxes, on all moduli. The initial study of the simplest of these
models, the $1^{9}$ model, has discovered the presence of many massless fields
and potential directions that do not get lifted even at higher order
Becker:2024ijy . However, the next more complicated example, the $2^{6}$ model
that is studied here and at the same time in Becker2024  has completely
changed the picture. Fully stabilized, isolated Minkowski vacua have been
found in this paper albeit with some massless fields. The authors of
Becker2024  found isolated Minkowski vacua without massless fields. Thus,
there are now new constructions of moduli stabilization for the community to
understand better and scrutinize in the future.

Let us already provide here a few comments on why we believe this moduli
stabilization scenario is trustworthy. We have solved the equations
$W=\partial_{i}W=0$, which do not depend on the Kähler potential at all but
only on the superpotential $W$. For non-geometric LG models it was shown in
Becker:2006ks  that the superpotential is exact and does not receive any
perturbative or non-perturbative corrections. For example, the absence of the
standard ED3 instanton corrections follows from $h^{1,1}=0$, i.e., the absence
of 4-cycles. For D(-1) instantons it was likewise argued that they vanish in
these models. This is consistent with the recent results in Kim:2022jvv ,
where it was shown that they are only non-zero in models where the D7-brane
charge is not canceled locally. Since the LG models have no D7-branes or
O7-planes there is no such charge at all. Thus, the superpotential does not
receive any corrections at all and any Minkowski vacuum is actually
trustworthy. This reasoning applies to the two solutions above in equations
(91) and (92) that have 86 massive and 5 massless but stabilized scalar
fields. It also applies equally to the Minkowski vacua found in the paper
Becker2024  that have 91 massive fields and therefore do not require any
stabilization at higher order.

###  4.2 The Massless Minkowski Conjecture

The Massless Minkowski Conjecture states that 10d supergravity
compactifications to 4d Minkowski space always have a massless scalar field.
Here we are not dealing with compactifications of type IIB supergravity so we
might be violating an important assumption of the conjecture. So, strictly
speaking, the LG models cannot violate this conjecture. However, let us
nevertheless entertain the possibility that all Minkowski vacua have a
massless scalar field.

To our knowledge, the only known Minkowski vacuum without a massless scalar
field was 11d supergravity or M-theory. When talking about compactifications
to lower dimensions, one necessarily faces the challenges of giving masses to
all the resulting scalar fields. As discussed in the previous subsection,
moduli stabilization is hard and the most trustworthy (simple) models could
therefore always contain massless fields. The presence of massless fields
would then be a lamppost effect and by studying more complicated setups one
would be able to eventually find fully stabilized models.999We are _not_
implying that the Massless Minkowski Conjecture is the result of a lamppost
effect. It is motivated by a universal tachyon in certain dS solutions
Danielsson:2012et . This tachyon becomes massless when sending the
cosmological constant to zero.

One can then set out to find more and more intricate constructions that
eventually lead to Minkowski vacua without flat directions. Before doing so
one would however have to contemplate the following problem, which we spell
out for 4d solutions. If one allows for $\mathcal{N}\geq 2$, then vector
multiplets contain scalar fields as well. Large amounts of supersymmetry do
protect the corresponding moduli space from perturbative and even non-
perturbative corrections and we can fully trust these solutions but there is
no clear way of having no massless scalar fields at all. One intriguing idea
that was pursued recently in Baykara:2023plc  is the study of asymmetric
orbifolds that give rise to models with no hypermultiplets. If among those one
would find models without vector multiplets, i.e. theories that are pure
supergravities then one would have succeeded. This has not yet been possible
although the existence of a potential pure supergravity in 5d is discussed in
that paper. With less or no supersymmetry, one faces the problem of
perturbative and non-perturbative corrections. For example, even in 4d
theories with $\mathcal{N}=1$, the superpotential is not protected from
receiving corrections. Thus, there is usually an infinite number of unknown
corrections that can affect any Minkowski vacuum. If these corrections would
change the value of the scalar potential at the minimum away from zero, then
the solution would be no longer a Minkowski vacuum. Stated differently, for
$\mathcal{N}\leq 1$ an infinite number of corrections would have to sum to
zero, making Minkowski vacua seemingly infinitely fine-tuned.

As we discussed in the previous subsection, there are no perturbative or non-
perturbative corrections for these non-geometric LG models. Thus, due to these
powerful non-renormalization theorems, we do not have to worry about infinite
numbers of corrections and can trust the fully stabilized Minkowski vacua we
found in this paper. This reasoning of course also applies to the Minkowski
vacua found in the paper Becker2024  that have 91 massive fields and do not
even require any stabilization at higher order.

###  4.3 The Tadpole Conjecture

The Tadpole Conjecture postulates a surprising relation between the length of
an ISD flux $G\in H^{2,1}$ and the number $n_{\rm stab}$ of complex structure
moduli that the flux lifts. In particular, it states that

$N_{\rm flux}=\frac{1}{\tau-\bar{\tau}}\int
G_{3}\wedge\bar{G}_{3}>\alpha\,n_{\rm stab}\,,$ (101)

with $\alpha>\tfrac{1}{3}$ in its original refined formulation Bena:2020xrh .
While this was not explicitly specified it probably makes sense to take
$n_{\rm stab}$ here to be the total number of stabilized fields and not just
the number of massive fields $n_{\rm massive}$. However, in principle, one
could also entertain the conjecture

$N_{\rm flux}=\frac{1}{\tau-\bar{\tau}}\int
G_{3}\wedge\bar{G}_{3}>\alpha\,n_{\rm massive}\,,$ (102)

which is weaker since $n_{\rm stab}\geq n_{\rm massive}$.

Either form of the conjecture is violated by our flux choice above in equation
(57). That flux choice had $N_{\rm flux}=16$ and 52 massive fields, which
gives

$\frac{N_{\rm flux}}{n_{\rm massive}}\approx.308<\frac{1}{3}\,.$ (103)

One might thus be tempted to conclude that the Tadpole Conjecture is not
correct or does not apply to this $2^{6}$ LG model.

However, the authors of Becker:2024ijy  generated a large set of flux
configurations in the $1^{9}$ LG model and found a spectacular confirmation of
the Tadpole Conjecture (see figure 2 on page 22 of that paper). A similar
study for the $2^{6}$ LG model was undertaken in the paper Becker2024  that
appeared at the same time as ours. The authors kindly shared their results
with us and those do clearly show a confirmation of a bound like the one in
equation (101) albeit with a smaller coefficient. The authors of Becker2024
propose $\alpha=1/4$ and it could be that this is the correct statement.

##  5 Conclusion

In this paper, we have studied flux compactifications on an orientifold of the
$2^{6}$ Landau-Ginzburg model. We focused on 4d $\mathcal{N}=1$ Minkowski
vacua that arise for $G_{3}$ fluxes that are imaginary-self-dual. We
demonstrated that, for several flux choices, there is non-trivial moduli
stabilization from higher-than-quadratic order terms in the expansion of the
superpotential around a critical point. This means that there are massless
fields that are stabilized by higher order interaction terms, similar to a
massless $\phi^{4}$ scalar field that has no flat direction. Unlike in the
$1^{9}$ Landau-Ginzburg model Becker:2024ijy , it is not tractable to study
all flux examples beyond terms in the superpotential that are cubic in the
complex structure moduli. However, we find two interesting solutions in the
$2^{6}$ model that violate the Refined Tadpole Conjecture. One example has 52
massive fields for a flux choice that only contributes $N_{\rm flux}=16$ to
the O3 tadpole condition. Another example has $N_{\rm flux}=28$, and 75
massive fields but has at least $84=3\cdot 28$ stabilized fields, which again
violates the Refined Tadpole Conjecture. Nevertheless, it seems that this
model is actually confirming the original Tadpole Conjecture, with a modified
bound, as is shown by studying a large number of flux configurations in
Becker2024 .

Even more interestingly, we provide examples where supersymmetric Minkowski
critical points without full mass matrix rank turn out to be isolated vacua
once we include higher order terms in the superpotential. Specifically, we
have two flux choices for which 86 out of the 91 fields are massive and the 5
remaining fields get stabilized at higher order. Due to the non-
renormalization theorem for the superpotential $W$ that was derived in
Becker:2006ks , these isolated supersymmetric Minkowski vacua should be fully
controlled and are arguably the simplest four-dimensional solutions of string
theory without a moduli space.

## Acknowledgments

We like to thank Katrin Becker, Mariana Graña, Miguel Montero, Miguel Morros,
Sav Sethi, Johannes Walcher, and Qi You for useful discussions. The work of AS
is supported in part by the NSF grant PHY-2112859. MR acknowledges the support
of the Dr. Hyo Sang Lee Graduate Fellowship from the College of Arts and
Sciences at Lehigh University. The work of MR and TW is supported in part by
the NSF grant PHY-2210271. MR and TW would like to thank the Erwin-
Schrödinger-Institute in Vienna for their hospitality; this work was completed
there during the program “The Landscape vs. the Swampland”.

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