IFT-UAM/CSIC-24-62

End of The World brane networks for infinite distance limits in CY moduli
space

Roberta Angius
Instituto de Física Teórica IFT-UAM/CSIC,
C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain
<EMAIL_ADDRESS>
Abstract

Dynamical Cobordism provides a powerful method to probe infinite distance
limits in moduli/field spaces parameterized by scalars constrained by generic
potentials, employing configurations of codimension-1 end of the world (ETW)
branes. These branes, characterized in terms of critical exponents, mark
codimension-1 boundaries in the spacetime in correspondence of finite
spacetime distance singularities at which the scalars diverge. Using these
tools, we explore the network of infinite distance singularities in the
complex structure moduli space of Calabi-Yau fourfolds compactifications in
M-theory with a four-form flux turned on, which is described in terms of
normal intersecting divisors classified by asymptotic Hodge theory. We provide
spacetime realizations for these loci in terms of networks of intersecting
codimension-1 ETW branes classified by specific critical exponents which
encapsulate the relevant information of the asymptotic Hodge structure
characterizing the corresponding divisors.

###### Contents

  1. 1 Introduction
  2. 2 Generalities on Calabi Yau Moduli Space and Flux potentials
    1. 2.1 Variation of Hodge structure
    2. 2.2 Asymptotic regime and N.O.T.
    3. 2.3 Strict Asymptotic regime and $SL(2)-$orbit theorem
    4. 2.4 Classification of singularities
    5. 2.5 Allowed enhancements
  3. 3 Dynamical Cobordisms
    1. 3.1 Codimension-1 ETW branes
    2. 3.2 Intersecting ETW branes
    3. 3.3 Intersecting configurations for decoupled scalar fields
    4. 3.4 A new solution beyond the conformal flatness
  4. 4 ETW networks for infinite distance limits in CY moduli space
    1. 4.1 ETW branes for simple singularities
    2. 4.2 ETW networks for the enhancements
    3. 4.3 Application to Swampland Distance Conjecture
  5. 5 Conclusions

##  1 Introduction

The study of compactifications in String Theory represents a powerful method
for understanding the rich landscape of possible derived vacua and their
corresponding phenomenology (see Polchinski:1998rr ; Ibanez:2012zz ;
Blumenhagen:2013blt ). In this realm, the most interesting classes of
compactifications are those that preserve only a small number of
supersymmetries in the non-compact dimensions. Among these classes, a
prominent one arises from compactifying M-theory on a Calabi-Yau fourfold
manifold $Y_{4}$ Becker:1996gj ; Dasgupta:1999ss ; Haack:2001hl , which leads
to three-dimensional effective supergravity theory with $\mathcal{N}=2$
supersymmetry. Some of these vacua can be lifted to four-dimensional
$\mathcal{N}=1$ F-vacua up to T-duality (see Heckman:2010hjj ; Weigand:2010wti
for applications to phenomenology). Considering F-theory on $Y_{4}$, for
specific choices of the compactification background, in particular when the
manifold $Y_{4}$ admits an elliptic fibration with base $\mathcal{B}$, the
limit in which the elliptic fiber shrinks to zero can be read as a four-
dimensional vacuum obtained compactifying type IIB on the base $\mathcal{B}$.

A central aspect in the study of Calabi-Yau compactifications controlled by
several moduli is the intricate structure of their moduli space. These moduli
parameterize geometric deformations of the compact manifold $Y_{4}$
controlling its total volume or the size of the internal cycles. It has been
proven in Grimm:2018cpv  that infinite distance limits of this moduli space
are singular loci corresponding to (de-)compactification regimes in the Kähler
sector or points in the complex structure sector where some internal cycle
shrinks to zero size. For effective field theories with exact moduli space it
is possible to explore these singular loci using spacetime independent scalar
vevs.
In the context of the swampland program Vafa:2005ui  (see also Palti:2019pca ;
vanBeest:2021lhn  for more recent reviews), many of the conjectures ruling out
effective field theories that cannot be lifted into a consistent quantum
gravity theory put constraints in the asymptotic behavior of these theories
near the boundaries of their corresponding moduli space. This is the case of
the Distance conjecture Ooguri:2006in , which predicts a tower of
exponentially light particles emerging when we approach one of these limits,
and its sharpened versions Lee:2019wij ; Etheredge:2022opl , which provide
information about the nature of these towers and set a lower bound on the
decay rate of their masses. Similar constrains are imposed in terms of a
Convex Hull condition Calderon-Infante:2020dhm  for the geodesic trajectories
approaching the asymptotic regions of the field spaces of scalars constrained
by non-trivial potentials.

In the presence of effective scalar potentials the adiabatic approach to
investigate infinite distance limits with constant vevs is in general
inconsistent Mininno:2011sdb  or forbidden Gonzalo:2019gjp  and the way to
probe these limits is to make use of spacetime dependent solutions describing
scalars that go to infinity in a finite distance in spacetime. These tools,
first proposed in Buratti:2021yia ; Buratti:2021fiv  (see also
Etheredge:2022opl ; Rudelius:2021azq ; Calderon-Infante:2022nxb  for recent
discussions on spacetime dependent solutions), have been subsequently defined
within the framework of Dynamical Cobordisms, Angius:2022aeq ; Angius:2022mgh
; Blumenhagen:2022mqw ; Angius:2023xtu ; Blumenhagen:2023abk ; Huertas:2023syg
; Angius:2023rma  (see also Dudas:2000ff ; Blumenhagen:2000dc ; Dudas:2002dg ;
Dudas:2004nd  for early related works and Basile:2018irz ; Antonelli:2019nar ;
Basile:2020xwi ; Basile:2021mkd ; Mourad:2021mas ; Mourad:2021mas2 ;
Mourad:2022loy ; Basile:2022ypo  for more recent developments and
Charmousis:2010cke ; Kiritsis:2017knp  for holographic applications),
describing configurations where the scalars run to infinity along a spacetime
direction that ends at a finite distance in spacetime at which the spacetime
metric features a Ricci singularity. These solutions can be regarded as
networks of codimension-1 boundaries dressed by extended objects, i.e. End of
The World (ETW) branes, sourcing the singularities and allowing spacetime to
end. In this sense, Dynamical Cobordism solutions are especially interesting
for the bottom-up exploration of these infinite distance limits.

Beyond these motivations, Dynamical Cobordism solutions represent the natural
way to implement the Cobordism Conjecture McNamara:2019rup  in the framework
of the effective field theories. In particular, they provide effective
realizations of the End of The World configurations predicted by that
conjecture at the topological level and hint at the presence of defects in the
complete theory, realized as ETW branes in this effective approach, which are
able to trivialize the corresponding cobordism group and make the
compactification background bordant to nothing.

In the context of three dimensional supergravity theories, obtained by
M-theory compactification on Calabi-Yau fourfolds, the effective potential
admits a purely geometric description in terms of a $G_{4}$-flux turned on in
some internal cycles of the compact manifold $Y_{4}$. The complex structure
sector $\mathcal{M}_{cs}$ of the field space of these theories still holds a
very intricate net of infinite distance singularities described in terms of
normal crossing divisors Hironaka:1964  at which some internal four-cycle,
maybe dressed with a $G_{4}$ flux, shrinks to zero size and produces a
singular geometry for the corresponding $Y_{4}$. The mathematical formalism
best suited to explore the structure of this network is encoded in the
asymptotic Hodge theory Schmid:1973 ; Cattani:1986cks . While in the bulk of
the moduli space the middle cohomology $H^{4}\left(Y_{4},\mathbb{C}\right)$
admits a pure Hodge decomposition, as we approach a point in the boundary this
structure is no longer valid and we need to define a finer structure, known as
Deligne splitting, which includes the possibility to enhance four-forms to
higher forms. The new splitting is completely determined by the nilpotent
orbit approximation of the period vector and by the properties of local
monodromy around the putative singular locus. This structure allows to provide
a classification of the types of possible singular divisors forming the
network and the allowed enhanced singularities occurring at their
intersections Cattani:1986cks ; kerr2019polarized . Remarkably, the asymptotic
structure of the middle cohomology also furnishes a suitable framework to
formulate a growth theorem that is able to capture the leading growth of the
Hodge norm of the four-forms near the boundary. The theorem allows to compute
all the terms to construct the three-dimensional effective action controlling
the dynamics of the scalars parameterizing the divisors involved in the
putative local patch of the network. Applications of this mathematical
machinery to the study of Calabi-Yau compactifications have been developed in
several worksGrimm:2019ixq ; Grimm:2018cpv ; Grimm:2018ohb ; Grimm:2019grh ;
Bastian:2020bgh , as well as applications to the computations of scattering
amplitudes are starting to generate some interest Bonisch:2022 ; Vanhove:2014
.

In this paper we will start to probe the network of infinite distance
singularities of the complex structure sector of the moduli space associated
with Calabi-Yau four-folds flux compactifications of M-theory using Dynamical
Cobordism solutions of the corresponding three-dimensional effective action.
We will present a dictionary associating to each singular divisor in
$\mathcal{M}_{cs}$, classified in terms of its asymptotic Hodge-Deligne
structure and equipped with the flux information, a specific codimension-1 ETW
brane in spacetime, classified in terms of its critical exponent. Highly non-
trivially, we will show that the consistency of the construction requires that
the spacetime solution be time-dependent, and the codimension-1 ETW brane mark
a boundary for a timelike coordinate.

In order to explore the intersections between distinct singular divisors in
the moduli space, we will construct a new class of Dynamical Cobordism
solutions involving intersecting ETW branes associated to two scalars
attaining infinite field space distance in finite spacetime distance. The
solution explores the infinite distance network of intersecting divisors,
albeit in a subtle way, different from the naive expectation to associate to
each intersecting ETW brane an intersecting divisor in the moduli space. The
resulting spacetime picture nicely matches the moduli space view in
Grimm:2018cpv ; Grimm:2019ixq  that the intersection of divisors can be
described as the enhancement of singularities in specific growth sectors of
the moduli along infinite distance paths. We will show that the structure of
the scalar flux potentials has exactly the structure required to support the
new spacetime dependent intersecting ETW brane Dynamical Cobordism solution.
Although we are dealing with singular solutions in a regime where the
effective field theory exhibits a lowered cutoff that limits its validity,
these solutions describe the EFT version of objects that are well defined in
the UV. This was also checked in a large classes of examples in
Buratti:2021yia ; Buratti:2021fiv ; Angius:2022aeq  (see also Saracco:2012sat
; Marchesano:2020rnd  for other setups in which singularities in supergravity
are resolved by sources in the complete theory).

The paper is organized as follows. In section 2 we review the main
mathematical tools of the asymptotic Hodge theory necessary to characterize
the network of infinite distance singularities in the complex structure sector
of the Calabi-Yau moduli space. In section 3 we will start reviewing the
codimension-1 ETW brane solutions, following Angius:2022aeq , and their
intersecting configurations introduced in Angius:2023xtu . In section 3.3 we
will show how these intersecting configurations can be read in terms of
decoupled scalar fields up to an appropriate redefinition of the fields. In
section 3.4 we will construct a new class of Dynamical Cobordism solutions
involving two distinct divergent scalar fields and a non-conformally flat
ansatz for the spacetime metric. These solutions can still be interpreted as
intersecting configurations of two codimension-1 ETW branes up to an
appropriate redefinition of the fields. In section 4 we will exploit these
Dynamical Cobordism solutions to probe the network of infinite distance
singularities of $\mathcal{M}_{cs}$. In section 4.1 we will present the
dictionary between singular divisors and ETW branes; in section 4.2 we will
apply the results of section 3.4 to the enhanced singularities occurring at
the loci of intersections of singular divisors in $\mathcal{M}_{cs}$. Finally,
we will provide some final considerations in section 5.

##  2 Generalities on Calabi Yau Moduli Space and Flux potentials

In this introductory section we will review the mathematical tools necessary
to provide a powerful local description of the complex structure moduli space
of Calabi-Yau manifolds around its singular loci. Although the results are
already in the literature, we review them to make our discussion self-
contained, and to emphasize the key ideas relevant for the construction of our
solutions in later sections. We keep the discussion brief, and refer the
reader to Schmid:1973 ; Cattani:1986cks ; Kashiwara:1985 ; kerr2019polarized
for more detailed explanations in the mathematical part and to Grimm:2018cpv ;
Grimm:2018ohb ; Grimm:2019grh ; Grimm:2019ixq  for the physical
interpretation. The reader not interested in the mathematical details may take
the results summarized in figure 4 and table 1 and safely jump to section 3.

A Calabi-Yau D-folds manifold is a Kähler manifold of complex dimension $D$
admitting everywhere a non-vanishing $\left(D,0\right)$ form $\Omega$. One way
to fully determine a particular Calabi-Yau manifold $Y_{D}$ in a family of
$CY_{D}$ manifolds is to specify its holomorphic form $\Omega$ and its Kähler
$(1,1)-$form $J$. Deformations of these choices that keep the manifold in the
same family can be of two different types: we can have complex structure
deformations parameterized by the choice of a $(D-1,1)$ form, or Kähler
structure deformations encoding all the possibilities to fix the Kähler form
$J$. Geometrically, deformations of the first type control the size of the
internal $D-$cycles, while deformations of the second type control the sizes
of the even internal cycles of the Calabi-Yau.

From now on we will consider the Kähler moduli to be fixed and we will focus
our attention on the complex structure sector of the moduli space
parameterized by deformations of $D-$cycles. For Calabi-Yau D-folds the
complex structure manifold $\mathcal{M}_{cs}$ is still a Kähler manifold of
complex dimension $h^{D-1,1}$, and it has the property of being neither smooth
nor compact, Tian:1987  Todorov:1989 . This implies the existence of singular
loci on it, whose corresponding Calabi-Yau manifolds are singular. Let us call
this set of points $\Delta$ the discriminant locus and assume that we can
resolve it in terms of the union of normally intersecting divisors:

$\Delta=\bigcup_{k}\Delta_{k}.$ (2.1)

Far away from these critical divisors, in the bulk of the moduli space, the
Kähler potential $\mathcal{K}^{cs}(z,\bar{z})$ is always a well-defined
function:

$\mathcal{K}^{cs}(z,\bar{z})=-\log\int_{Y_{D}}\Omega(z)\wedge\ast\overline{\Omega}(\bar{z})$
(2.2)

where the dependence on the moduli $\left\\{z^{I}\right\\}$, with
$I=1,...,h^{D-1,1}$, is made explicit. This function induces a natural metric
in $\mathcal{M}_{cs}$:

$G_{I\overline{J}}=\partial_{z^{I}}\partial_{\overline{z}^{J}}\mathcal{K}^{cs}$
(2.3)

known as Weil-Petersson metric.

In the rest of the paper we will concentrate our attention on the study of
$\mathcal{M}_{cs}$ associated with Calabi-Yau 4-folds. M-theory
compactifications on this class of manifolds leads to three-dimensional
supergravity with $\mathcal{N}=2$ supersymmetry where the complex structure
moduli become the scalar fields for the effective action with kinetic terms
induced by the metric (2.3). An effective three-dimensional potential for
these fields can be generated turning-on $G_{4}$ fluxes in the four-cycles of
the internal $CY_{4}$ Becker:1996gj ; Gukov:2000gvw ; Dasgupta:1999ss ;
Giddings:2001yu . Its definition in the three-dimensional Einstein frame is
Haack:2001hl :

$V_{M}=\frac{1}{\mathcal{V}_{4}^{3}}\left(\int_{Y_{4}}G_{4}\wedge\star\overline{G}_{4}-\int_{Y_{4}}G_{4}\wedge
G_{4}\right).$ (2.4)

The second term of this expression is topological and it is constrained by a
tadpole cancellation condition:

$\frac{1}{2}\int_{Y_{4}}G_{4}\wedge G_{4}=\frac{\chi(Y_{4})}{24},$ (2.5)

where $\chi(Y_{4})$ is the Euler characteristic of $Y_{4}$. Assuming that this
condition is satisfied, for the rest of the discussion we will limit our
attention on the first contribution.
The expression (2.4) depends on both the complex structure and the Kähler
moduli through the Hodge star operator and the volume $\mathcal{V}_{4}$ of
$Y_{4}$. To separate these contributions and keep only the complex structure
dependence, we can impose the following condition on the flux $G_{4}$:

$G_{4}\wedge J=0$ (2.6)

which limit our possibilities on the primitive cohomology
$H_{p}^{4}\left(Y_{4},\mathbb{R}\right)$.111For the rest of the paper we
restrict our attention on this primitive cohomology but we will omit the
subindex $p$.

###  2.1 Variation of Hodge structure

All the relevant quantities we introduced so far to construct the three-
dimensional physical action depend on the complex structure through the
holomorphic form $\Omega$ or the Hodge star operator in (2.4). In this section
we will review some essential mathematical tools that allow us to encode all
these dependences in the language of variation of the Hodge structure.

The nice property that the $k-th$ cohomology groups
$H^{k}\left(Y_{D},\mathbb{C}\right)$ of a smooth Calabi-Yau manifold admit a
pure Hodge structure of weight $k$ means that for each level $k$ we can define
a vector space $V_{\mathbb{C}}=H^{k}\left(Y_{D},\mathbb{C}\right)$, which
always admits an Hodge decomposition:

$V_{\mathbb{C}}=H^{k,0}\oplus H^{k-1,1}\oplus...\oplus H^{1,k-1}\oplus
H^{0,k}=\bigoplus_{k=p+q}H^{p,q}$ (2.7)

where the building subspaces satisfy the following complex-conjugation
property:

$H^{p,q}=\overline{H}^{q,p},$ (2.8)

and the weight $k$ is the constant sum of the indices $p+q=k$ of all the
blocks in (2.7).
An equivalent way to rephrase the same property is saying that each cohomology
group $H^{k}\left(Y_{D},\mathbb{C}\right)$ defines a decreasing Hodge
filtration:

$V_{\mathbb{C}}=F^{0}\subset F^{1}\subset...\subset F^{k-1}\subset
F^{k}=H^{k,0}$ (2.9)

where $H^{p,q}=F^{p}\cap\overline{F}^{q}$ and such that
$F^{p}\oplus\overline{F}^{k+1-p}\cong H^{k}$.
If we can further equip our Hodge structure of a bilinear form
$S(\cdot,\cdot)$ on $V_{\mathbb{C}}$ satisfying the following two properties:

$\begin{split}&(i)\quad\textsc{orthogonality}\quad
S\left(H^{p,q},H^{r,s}\right)=0\quad\text{for}\quad p\neq s,q\neq r;\\\
&(ii)\quad\textsc{non-degeneracy}\quad
i^{p-q}S\left(v,\overline{v}\right)>0\quad\text{for}\quad v\in H^{p,q},\quad
v\neq 0,\\\ \end{split}$ (2.10)

the structure is called polarized.

Since we are dealing with four-form in Calabi-Yau fourfolds we are interested
in a deeper exploration of the middle cohomology
$H^{4}\left(Y_{4},\mathbb{C}\right)$. In such a case the bilinear form is the
cup product:

$S(v,w)=\int_{Y_{4}}v\wedge w\quad\quad\quad v,w\in
H^{4}\left(Y_{4},\mathbb{C}\right),$ (2.11)

and it induces a norm for the vectors of the whole filtration:

$||v||^{2}=S(v,\ast\overline{v})$ (2.12)

called Hodge norm.

When we move on the moduli space the Hodge decomposition (2.7) changes.
Roughly speaking due to change of what we call holomorphic and anti-
holomorphic. One way to express this variation is in terms of the variation of
the holomorphic four form $\Omega$ with respect to a fixed basis
$\left\\{\gamma^{\mathcal{I}}\right\\}$ of $F^{0}$, with
$\mathcal{I}=1,...,dimH^{4}\left(Y_{4},\mathbb{C}\right)=2d$, such that:

$\eta_{\mathcal{I}\mathcal{J}}=-\int_{Y_{4}}\gamma^{\mathcal{I}}\wedge\gamma^{\mathcal{J}}=\left(\begin{matrix}0&\mathbb{I}_{d\times
d}\\\ -\mathbb{I}_{d\times d}&0\\\ \end{matrix}\right)$ (2.13)

In particular we can expand the form $\Omega$ along this basis writing:

$\Omega=\Pi^{\mathcal{I}}(z)\gamma_{\mathcal{I}}.$ (2.14)

The coefficients of such expansion are called periods and they are in general
complicated holomorphic trascendental functions of the moduli. Equivalently,
they are defined by:

$\Pi^{\mathcal{I}}(z)=\int_{\Gamma^{\mathcal{I}}}\Omega,$ (2.15)

where $\Gamma^{\mathcal{I}}$ are the 4-cycles Poincaré-dual of the forms
$\gamma^{\mathcal{I}}$.
At this point we can understand the variations of the spaces $F^{p}$ on
$V_{\mathbb{C}}$ over the space $\mathcal{M}_{cs}$ in terms of the variations
of the periods (2.15) with respect to the coordinates $z$.

This whole structure is completely well defined in the bulk of the moduli
space, as long as the manifold $Y_{4}$ is smooth. However, when we approach a
singular locus in $\mathcal{M}_{cs}$ the periods (2.15) diverge, the Hodge
filtration $F^{p}$ and the corresponding Hodge decomposition $H^{p,q}$ are not
well defined anymore and we lose all information about what happens at these
points. This pushed the mathematical research to define analogous quantities
at these limits, truncating the divergences of $F^{p}$ and $H^{p,q}$.

###  2.2 Asymptotic regime and N.O.T.

In this section we start exploring the boundaries of the complex structure
moduli space $\mathcal{M}_{cs}$, in particular studying the asymptotic
behavior of the Hodge structure of the middle cohomology within the nilpotent
orbit approximation.

As we pointed out above these asymptotic regions are resolved in terms of
normal crossing divisors (2.1), where each individual critical divisor
$\Delta_{k}$ identifies a codimension-1222Note that when we talk about
codimension in the moduli space we mean a complex codimension. In later
sections, when we will refer to spacetime codimension, it will mean a real
codimension.  locus in $\mathcal{M}_{cs}$ and each normal intersection
$\Delta_{k_{1}...k_{n}}=\Delta_{k_{1}}\cap...\cap\Delta_{k_{n}}$ identifies a
codimension-$n$ locus. See figure 1 for an illustrative representation in a
complex two-dimensional moduli space.

Figure 1: Intersection of two codimension-1 singular divisors $\Delta_{1}$ and
$\Delta_{2}$ in the codimension-2 locus $\Delta_{12}$ of the two-dimensional
complex moduli space $\mathcal{M}_{cs}$.

In order to understand what happens around these loci it is useful to
introduce an adapted set of local coordinates
$\left\\{z^{i},\xi^{\bar{j}}\right\\}$ describing the patch
$\mathcal{L}\subset\mathcal{M}_{cs}$ that contain the sub-discriminant locus
$\Delta_{k_{1}...k_{n}}$ identified by the conditions $z^{i}=0$, with
$i=1,...,n$ and $\bar{j}=n+1,...,h^{3,1}$. The structure of $\mathcal{L}$ is
given by the product of $(h^{3,1}-n)$ disks
$\mathcal{D}=\left\\{\xi\in\mathbb{C}\big{|}|\xi|<1\right\\}$, spanning the
transverse directions to the singularity, and $n$ punctured disks
$\mathcal{D}^{\ast}=\left\\{z\in\mathbb{C}\big{|}0<|z|<1\right\\}$, which span
the longitudinal directions:

$\mathcal{L}=\left(\mathcal{D}^{\ast}\right)^{n}\times\left(\mathcal{D}\right)^{h^{3,1}-n}.$
(2.16)

Using a conformal transformation we can map the punctured disks
$\mathcal{D}^{\ast}$ in the upper half complex plane via:

$z^{i}\quad\mapsto\quad t^{i}=\frac{1}{2\pi i}\log z^{i}=a^{i}+is^{i}.$ (2.17)

With respect to these new suggestive coordinates the boundary $z^{i}=0$ is
reached sending to infinity the imaginary part $s^{i}$ of $t^{i}$ and leaving
the real part $a^{i}$ constant. Moreover, the periods are multivalued
functions of the moduli, due to monodromies in the phase of the coordinates
$z^{i}$, which in the new coordinates are thus associated to shifts of
$a^{i}$. Let us encode the transformation properties of the periods around
each singular divisor $\Delta_{k}$ in the monodromy matrices $T_{k}$ acting
conventionally as:

$\mathbf{\Pi}\left(...,e^{2\pi
i}z^{k},...\right)=T_{k}^{-1}\Pi\left(...,z^{k},...\right).$ (2.18)

These local monodromies will be sufficient to classify all the types of
singularities in $\mathcal{M}_{cs}$ and to extract information about the
behavior of the Hodge structure of the middle cohomology around these limits.
In particular, the crucial information for this analysis is contained in the
infinite order part of these matrices, which can be extracted through the
factorization:

$T_{k}=T_{k}^{(s)}\cdot T_{k}^{(u)},$ (2.19)

where $T_{k}^{(s)}$ is the finite order part and $T_{k}^{(u)}$ is the
unipotent part. For the rest of the discussion we will only care about this
last factor defining the following related matrix:

$N_{k}=\log\left(T_{k}^{(u)}\right),$ (2.20)

which is Nilpotent, namely there exists a positive integer $n_{k}$ such that
$N_{k}^{n_{k}+1}=0$. Matrices $N_{k}$ associated with different divisors
$\Delta_{k}$ are commutative.

In the asymptotic regime, reached in the limit
$t_{1},...,t_{n}\mapsto+\infty$, the Nilpotent Orbit Theorem (N.O.T.)
Schmid:1973  states that the period vector is represented by the following
expansion:

$\mathbf{\Pi}(t^{i},\xi^{\bar{j}})=e^{\sum_{i=1}^{n}t^{i}N_{i}}\left(\mathbf{a}_{0}(\xi)+\mathbf{a}_{j}(\xi)e^{2\pi
it^{j}}+\mathbf{a}_{ij}(\xi)e^{2\pi i(t^{i}+t^{j})}+...\right),$ (2.21)

where the entries of the vectors $\mathbf{a}_{\bullet}(\xi)$ are holomorphic
functions, in general non-polynomial, of the non-singular coordinates
$\xi^{\bar{j}}$. This implies that, up to exponential corrections, the
nilpotent orbit:

$\mathbf{\Pi}_{nil}(t^{i},\xi^{\bar{j}})=e^{\sum_{i=1}^{n}t^{i}N_{i}}\mathbf{a}_{0}(\xi)$
(2.22)

is a good approximation of the period vector near the locus
$\Delta_{k_{1}...k_{n}}$.
Following this expansion, we can define near each asymptotic region the
limiting filtration:

$F^{p}_{lim}\left(\Delta_{k_{1}...k_{n}}\right)=\lim_{t_{k_{1}},...,t_{k_{n}}\mapsto+i\infty}e^{-\sum_{i=1}^{n}t^{i}N_{i}}F^{p}(t),$
(2.23)

which is related with the pure Hodge structure $F^{p}$ by the N.O.T. and it
has the property that is stays finite.

Just like the Hodge filtration $F^{p}$ produces the decomposition (2.7) of the
middle cohomology, the limiting filtration $F^{p}_{0}$, together with the
information about the monodromies $N_{1},...,N_{n}$, packaged in any element
$N$ of the cone
$\sigma(N_{1},N_{2},...,N_{n})=\left\\{\sum_{i=1}^{n}a^{i}N_{i}|a^{i}>0\right\\}$,
contains all the ingredients to construct a mixed Hodge decomposition where
the middle cohomology lifts into a finer splitting $\left\\{I^{p,q}\right\\}$,
with $0\leq p,q\leq 4$, known as Deligne splitting:

$H_{p}^{4}\left(Y_{4},\mathbb{C}\right)\quad\longrightarrow\quad\bigoplus_{0\leq
p,q\leq 4}I^{p,q}$ (2.24)

The significant ingredient that allows us to give a formal definition for this
splitting is encoded in the vector spaces:

$W_{i}(N)=\sum_{j\geq max(-1,l-4)}kerN^{j+1}\cap ImN^{j-l+4},$ (2.25)

producing a monodromy weight filtration under the action of $N$ such that
Cattani:1982cka :

$NW_{l}\subseteq W_{l-2}$ (2.26)

Using the vector spaces $W_{i}$ and $F^{p}_{lim}$ we can define the Deligne
splitting through the formula:

$I^{p,q}=F_{lim}^{p}\cap W_{p+q}\cap\left(\overline{F}^{q}_{lim}\cap
W_{p+q}+\sum_{j\geq 1}\overline{F}^{q-j}_{lim}\cap W_{p+q-j-1}\right).$ (2.27)

This is the unique definition such that the following three properties are
satisfied:

  * (i)

$F^{p}_{lim}=\bigoplus_{r\geq p}\bigoplus_{s}I^{r,s}$;

  * (ii)

$W_{l}=\bigoplus_{p+q\leq l}I^{p,q}$;

  * (iii)

$I^{p,q}=\overline{I^{q,p}}$ mod $\bigoplus_{r<p,s<q}I^{r,s}$.

Acting on the space $I^{p,q}$ with the Nilpotent matrix $N$, we have:

$NI^{p,q}\subset I^{p-1,q-1}.$ (2.28)

However, in general not the whole lower $(p,q)-$ spaces can be obtained
through this action. The subspaces $P^{p,q}\in I^{p,q}$ that cannot be
obtained acting with $N^{k}$ on $I^{p+k,q+k}$ form the primitive part of the
splitting, and we have:

$I^{p,q}=\oplus_{i\geq 0}N^{i}\left(P^{p+i,q+i}\right),$ (2.29)

The elements of $P^{p,q}$ satisfy the following polarization conditions:

$\begin{split}&S\left(P^{p,q},N^{l}P^{r,s}\right)=0\quad\quad\quad\text{for}\quad
p+q=r+s=l+4\quad\text{and}\quad\left(p,q\right)\neq\left(s,r\right)\\\
&i^{p-q}S\left(v,N^{p+q-4}\overline{v}\right)>0\quad\quad\quad\text{for}\quad
v\in P^{p,q},\quad v\neq 0,\\\ \end{split}$ (2.30)

which guarantees us that the elements belonging to the primitive part have
strictly-positive norm. These conditions will become important to study the
allowed Deligne splittings occurring at the enhanced singularities. It will be
crucial that they are correctly transmitted, and this will put severe
constraints on the form of the enhancement.

###  2.3 Strict Asymptotic regime and $SL(2)-$orbit theorem

In this work we are especially interested to construct the effective three-
dimensional actions for the complex structure moduli near any asymptotic
region of $\Delta$. This requires to have a prescription to approximate the
Hodge norm of four-forms in order to explicitly compute the kinetic metric
(2.3) and the leading behavior of the scalar potential (2.4). The prescription
is given by the Growth Hodge-norm theorem, proven in Cattani:1986cks , using
the information of the monodromy matrices $N_{i}$ and the vector
$\mathbf{a}_{0}$.
Another fundamental data we need when we treat with singular loci of
codimension higher than 1 is the specification of the growth sector we use to
reach the final singularity. This means to choose an order to send the moduli
to infinity in the asymptotic region around $\Delta_{k_{1}...k_{n}}$. Each of
these choice defines a growth sector:

$\mathcal{R}_{12...n}=\left\\{t^{i}=a^{i}+is^{i}\big{|}\frac{s^{1}}{s^{2}}>\gamma,\frac{s^{2}}{s^{3}}>\gamma,...\frac{s^{n-1}}{s^{n}}>\gamma,s^{n}>\gamma,\varphi^{i}<\delta\right\\},$
(2.31)

which specifies what we call a strict asymptotic regime (SAR) around
$\Delta_{k_{1}...k_{n}}$. For application to the growth theorem, the objects
that better encode the previous data are a set of $n$ $sl(2,\mathbb{C})$
commuting algebras:

$sl\left(2,\mathbb{C}\right)_{i}=\langle N_{i}^{-},N_{i}^{+},Y_{i}\rangle,$
(2.32)

associated to the $i-$th divisor $\Delta_{k_{i}}$ involved in the intersection
$\Delta_{k_{1}...k_{n}}$. The triplets
$\left(N_{i}^{-},N_{i}^{+},Y_{i}\right)$ can be computed following a recursive
method applied to the splittings $\left\\{\tilde{I}^{p,q}_{(i)}\right\\}$
which are associated with all the loci $\Delta_{k_{1}...k_{i}}$, with $i<n$,
traversed to reach the highest intersection $\Delta_{k_{1}...k_{n}}$ following
the order of (2.31). These refined splittings are $\mathbb{R}-$split333They
satisfy the property (iii) below (2.24) with zero modulo. Deligne splitting
computed using the filtration $\hat{F}^{p}$ related to $F^{p}$ up to the
action of two operators $\hat{\xi}$ and $\hat{\delta}$:

$\hat{F}^{p}=e^{\hat{\xi}}e^{-i\hat{\delta}}F^{p}.$ (2.33)

The $Sl(2)-$orbit theorem in Cattani:1986cks  prove this statement and provide
an explicit construction of the matrices $\hat{\xi}$ and $\hat{\delta}$.
Now, starting from the higher Deligne splitting
$\left\\{\tilde{I}^{p,q}_{(n)}\right\\}$ we can compute the operator
$Y_{(n)}=Y_{1}+...+Y_{n}$ such that:

$Y_{(n)}\tilde{I}_{(n)}^{p,q}(\Delta_{k_{1}...k_{n}})=(\underbrace{p+q}_{l_{n}}-4)\tilde{I}_{(n)}^{p,q}.$
(2.34)

Then, repeating the construction for all the subsequent splittings
$\left\\{I^{p,q}_{(i)}\right\\}$ until the operator $Y_{1}$ associated to the
divisor $\Delta_{1}$, we can reconstruct all the operators
$Y_{i}=Y_{(i)}-Y_{(i-1)}$. At this point, decomposing $N_{i}$ into the basis
of the eigenvectors of $Y_{(i-1)}$, namely
$N_{i}=\sum_{\alpha}N_{i}^{\alpha}$, we can define $N_{i}^{-}:=N_{i}^{0}$,
which is the only element in the decomposition that commutes with $Y_{(i-1)}$.
Finally, we can complete the triples looking for the elements $N^{+}_{i}$
satisfying the correct commutation relations and preserving the polarization.

The first important consequence of the $Sl(2)$-orbit theorem is the definition
of a further approximation of the nilpotent orbits in the strict asymptotic
regime, specified by the sector $R_{12...n}$, with the $Sl(2)-$orbits:

$\mathbf{\Pi}_{Sl(2)}=e^{i\sum_{i=1}^{n}s^{i}N^{-}_{i}}\tilde{\mathbf{a}}_{0}^{(n)}(\xi)$
(2.35)

related with $\mathbf{\Pi}_{nil}$ via:

$\mathbf{\Pi}_{nil}=e^{\sum_{i=1}^{n}t^{i}N_{i}}\mathbf{a}_{0}(\xi)=e^{\sum_{i=1}^{n}a^{i}N_{i}}\cdot
M(s)\cdot\mathbf{\Pi}_{Sl(2)},$ (2.36)

where $M(s)$ is a $s^{i}-$dependent matrix introduced in section $5$ of
Schmid:1973 .
This new approximation drops the subleading polynomial corrections in
$\mathbf{\Pi}_{nil}$.

The second relevant consequence for our aim is the fact that the
$sl\left(2,\mathbb{C}\right)$ triplets allow to decompose the cohomology group
as:

$H^{4}_{p}\left(Y_{4},\mathbb{C}\right)=\bigoplus_{\mathbf{l}\in\mathcal{E}}V_{\mathbf{l}}$
(2.37)

where the entries of the vector $\mathbf{l}=(l_{1},l_{2},...,l_{n})$ are
related to the eigenvalues of $\mathbf{v_{l}}$ with respect the operators
$Y_{(i)}$ as:

$Y_{(i)}v_{\mathbf{l}}=(l_{i}-4)v_{\mathbf{l}}.$ (2.38)

Note that this decomposition holds near the locus $\Delta_{k_{1}...k_{n}}$ and
strictly depends on the growth sector we use to reach that locus. Now, any
vector $w\in H_{p}^{4}\left(Y_{4},\mathbb{C}\right)$ can be written in an
unique way as:

$w=\sum_{\mathbf{l}\in\mathcal{E}}w_{\mathbf{l}},$ (2.39)

where $w_{\mathbf{l}}\in V_{\mathbf{l}}$. For each of these vectors the growth
Hodge-norm theorem ensures that the leading growing of its norm in the
$Sl(2)-$approximation is captured by a term of the form:

$||w||^{2}\sim\textit{const}\cdot\left(\frac{s^{1}}{s^{2}}\right)^{l_{1}-4}...\left(\frac{s^{n-1}}{s^{n}}\right)^{l_{n-1}-4}(s^{n})^{l_{n}-4},\quad\quad\quad\textit{const}>0,$
(2.40)

where the list of numbers $(l_{1},...,l_{n})$ identifies the location of $w$
in the intersection of monodromy filtrations:

$w\in W_{l_{1}}(N_{1})\cap W_{l_{2}}(N_{1}+N_{2})\cap...\cap
W_{l_{n}}(N_{1}+N_{2}+...+N_{n}).$ (2.41)

We will exploit this theorem to compute the kinetic metrics and the flux
potentials for the three-dimensional spacetime action in section 4.

###  2.4 Classification of singularities

In this section we make use of the mathematical tools explained above in order
to classify the singularities that can occur in the complex structure moduli
space $\mathcal{M}_{cs}$ of Calabi-Yau 4-folds compactifications. This implies
a classification of all the possible Deligne splittings (2.24) associated with
the points of $\Delta$. For the moment we consider to approach the
discriminant locus from the bulk of the moduli space towards one of its
singular divisors $\Delta_{k}$ in a region very far away from any higher
intersection.

In order to better visualize the discussion we associate to each splitting
$\left\\{I^{p,q}\right\\}$ a lattice representing the associated Hodge-Deligne
diamond, as in figure 2, where each lattice point $(p,q)$ is labeled with the
dimension $i^{p,q}$ of the corresponding space $I^{p,q}$.

Figure 2: Hodge-Deligne diamond encoding the complex dimensions of the Deligne
spaces $dim_{\mathbb{C}}I^{p,q}=i^{p,q}$

These complex dimensions satisfy the following properties:

$\begin{split}&\textit{(i)}\quad
i^{p,q}=i^{q,p};\quad\quad\quad\quad\quad\quad\textit{(iii)}\quad i^{p,q}\leq
i^{p+1,q+1},\text{for }p+q\leq 2\\\ &\textit{(ii)}\quad
i^{p,q}=i^{4-q,4-p};\quad\quad\quad\quad\textit{(iv)}\quad\sum_{q=0}^{4}i^{p,q}=h^{p,4-p}.\\\
\end{split}$

For Calabi-Yau 4-folds we have a unique holomorphic four form $\Omega$, then
$h^{4,0}=h^{0,4}=1$. Moreover the property (iv) tells us that we have only
five possibilities to spread this value in the new Hodge-Deligne
decomposition:

$1=h^{4,0}=i^{4,d}\quad\quad\quad\quad\quad\text{with}\quad\quad\quad\quad
d=0,1,2,3,4.$ (2.42)

According to kerr2019polarized , these possibilities fix the Roman label of
five large classes of singularities:

$\displaystyle d\quad=\quad$ $\displaystyle 0,$ $\displaystyle 1,$
$\displaystyle 2,$ $\displaystyle 3,$ $\displaystyle 4$
$\displaystyle\downarrow$ $\displaystyle\downarrow$ $\displaystyle\downarrow$
$\displaystyle\downarrow$ $\displaystyle\downarrow$ $\displaystyle I$
$\displaystyle II$ $\displaystyle III$ $\displaystyle IV$ $\displaystyle V$

Let us now introduce the notation to indicate the dimensions of the Deligne
spaces with the number of dots on the corresponding lattice points. Using the
properties (i)-(ii) we can associate to each Roman class a basic lattice over
which we can build the remaining unspecified components of the splitting.
These basic lattices are summarized in figure 3. Note that the Roman label of
the class uniquely determines the dots on the external perimeter of the
diamond. In order to fully specify the type of singularity, we still need to
fix the number of dots on the internal square.

Figure 3: Basic lattices classifying the five large classes of singularities
for Calabi-Yau 4-folds.

Before proceeding, we need to specify the class of Calabi-Yau 4-folds manifold
we are dealing with. In particular we have to fix the dimension
$h^{1,3}=h^{3,1}$ of the groups $H^{1,3}$ and $H^{3,1}$. Such information
specifies the dimension of $\mathcal{M}_{cs}$ and, with the property (iv), it
tells us that the total number of dots along the columns $(1,q),(3,q)$ and the
rows $(p,1),(p,3)$ has to be $h^{1,3}$.

To make the description more explicit, we will focus on the case $h^{1,3}=2$,
but the same procedure can be repeated analogously for other classes of
Calabi-Yau manifolds.
Let us also fix $h^{2,2}=\widehat{m}$, so that the sum of dots along the
central column and row in the diamonds must be $\widehat{m}$.

Figure 4: Hodge-Deligne diamonds associated with all the possible
singularities of the $h^{3,1}=2$ dimensional complex structure moduli space of
Calabi-Yau 4-folds with $h^{2,2}=\hat{m}$.

Starting with the five models depicted in figure 3, and using the properties
(i)-(iv), we can construct all the Hodge-Deligne diamonds representing all the
possible singularities Grimm:2019ixq . To distinguish between all of these
possibilities we add two subindices $(a,a^{\prime})$ to each Roman number: the
first index indicates the dimension $i^{3,3}$ and the second one the sum
$i^{3,3}+i^{3,2}$. All allowed diamonds are summarized in the figure 4. In the
next sections we will not consider type $I_{a,a^{\prime}}$ singularities
because they are located at finite distance in the moduli space as explained
in section 4.1.

###  2.5 Allowed enhancements

In the previous section we classified all the allowed simple singularities
that occur in $\mathcal{M}_{cs}$ approaching a singular divisor $\Delta_{k}$.
Now we want to consider enhanced singularities which occur when two singular
divisors intersect. In the case $h^{1,3}=2$ the intersection loci
$\Delta_{k_{1}k_{2}}=\Delta_{k_{1}}\cap\Delta_{k_{2}}$ are 0-dimensional loci
(points).

As explained in section 2.3 we can reach the locus $\Delta_{k_{1}k_{2}}$
following paths in $\mathcal{M}_{cs}$ belonging to different growth sectors.
From now on we will focus our attention on the sectors $R_{12}$, but the same
procedure can be implemented also for the sectors $R_{21}$. Starting from a
generic point in the bulk of the moduli space, we send the first modulus
$s^{1}\mapsto\infty$ to get the singularity
$\textit{Type}_{A}\left(\Delta_{1}\right)$. Then we send also the second
modulus $s^{2}\mapsto\infty$ to reach the intersection $\Delta_{12}$ with the
divisor $\Delta_{2}$. The singularity occurring at this point must be one of
those classified in the previous section. Let us call such singularity
$\textit{Type}_{B}\left(\Delta_{12}\right)$ and indicate the enhancement
through the sector $R_{12}$ with the following notation:

$\textit{Type}_{A}\left(\Delta_{1}\right)\quad\quad\longrightarrow\quad\quad\textit{Type}_{B}\left(\Delta_{12}\right).$
(2.43)

Roughly speaking we have to understand which possible Deligne-splitting
$\left\\{I^{p,q}\left(\Delta_{12}\right)\right\\}$ can enhance from the
Deligne splitting $\left\\{I^{p,q}\left(\Delta_{1}\right)\right\\}$. The
fundamental mathematical object containing this information is the primitive
part of $\left\\{I^{p,q}\left(\Delta_{k_{1}}\right)\right\\}$, which can be
written as:

$P^{p,q}=I^{p,q}\cap\ker N_{1}^{p+q-3}.$ (2.44)

Using this definition and the equation (2.29), we make explicit in figure 5
which spaces $I^{p,q}$ could contain a non-trivial primitive part. These
primitive parts can be grouped in the following vector spaces:

$\begin{split}&P^{8}=P^{4,4}\\\ &P^{7}=P^{3,4}\oplus P^{4,3}\\\
&P^{6}=P^{2,4}\oplus P^{3,3}\oplus P^{4,2}\\\ &P^{5}=P^{1,4}\oplus
P^{2,3}\oplus P^{3,2}\oplus P^{4,1}\\\ &P^{4}=P^{0,4}\oplus P^{1,3}\oplus
P^{2,2}\oplus P^{3,1}\oplus P^{4,0}\\\ \end{split}$ (2.45)

each of which can be used to construct a pure Hodge structure of respective
weight $8,7,6,5,4$, exactly as $H^{4}_{p}$ admits a pure Hodge structure of
weight $4$.

Figure 5: Principal parts of the Deligne splitting for the middle cohomology
of $CY$ $4-$folds.

These new Hodge structures will be the basis to construct the next Deligne
splitting $\left\\{I^{p,q}\left(\Delta_{12}\right)\right\\}$ occurring at the
intersection. They will play the role of
$H^{4}_{p}\left(Y_{4},\mathbb{C}\right)$ for simple singularities.

For each $P^{j}$ we can construct a Deligne-splitting
$\left\\{I^{p,q}\left(\Delta_{12}\right)\right\\}^{j}$, with $0\leq p+q\leq
2j$, following the standard procedure described in section 2.2. Then, there is
a systematic way to rearrange these set of splittings to construct the
Deligne-splitting $\left\\{I^{p,q}\left(\Delta_{12}\right)\right\\}$ occuring
at the enhancement. This systematic way contains in itself the rules to
determine whether a particular enhancement is admissible with respect to its
capability to correctly transmit the polarization conditions. We refer to
kerr2019polarized  for a complete description of the procedure, while in the
following we just summarize the result.

Given an Hodge-Deligne diamond associated to a $j-$weight Hodge structure
polarized by $S(\cdot,N\cdot)$, it is possible to define an integer-valued
function on its corresponding lattice:

$\diamondsuit\quad:\quad(p,q)\quad\longrightarrow\quad\mathbb{Z}$ (2.46)

such that the following properties are satisfied:

  * (i)

$\sum_{q=0}^{j}\diamondsuit(p,q)=h^{p,j-p}$ for all $p$;

  * (ii)

$\diamondsuit(p,q)=\diamondsuit(q,p)=\diamondsuit(j-q,j-p)$ for all $p,q$;

  * (iii)

$\diamondsuit(p-1,q-1)\leq\diamondsuit(p,q)$ for $p+q\leq j$.

A simple choice for such a function is $\diamondsuit(p,q)=i^{p,q}$. Using this
definition we can easily define new Hodge-Deligne diamonds taking the sum of
two diamonds:

$\diamondsuit(p,q)=\diamondsuit_{1}(p,q)+\diamondsuit_{2}(p,q)$ (2.47)

or shifting the entries of one initial diamond:

$\diamondsuit\left[a\right](p,q)=\diamondsuit(p+a,q+a).$ (2.48)

Now let $\diamondsuit\left(F_{1},N_{1}\right)$ be the Hodge-Deligne diamond
associated with the singularity $\textit{Type}_{A}\left(\Delta_{1}\right)$ and
$\diamondsuit\left(F_{2},N_{2}\right)$ the one associated with the singularity
$\textit{Type}_{B}\left(\Delta_{12}\right)$. From
$\left(F_{1},W(N_{1})\right)$ we can construct the primitive vector spaces
$P^{j}(N_{1})$, and we can associate to each of these a $j-$weight pure Hodge
structure polarized by $S(\cdot,N_{1}^{j}\cdot)$ with Hodge-Deligne diamond
$\diamondsuit\left(F^{\prime}_{j},N^{j}_{1}\right)$.
If the following decomposition is possible:

$\diamondsuit\left(F_{2},N_{2}\right)=\sum_{4\leq j\leq 8}\sum_{0\leq a\leq
j-4}\diamondsuit\left(F^{\prime}_{j},N^{j}_{1}\right)\left[a\right]$ (2.49)

then the enhancement (2.43) is allowed.

In order to classify all the allowed enhancements in the space
$\mathcal{M}_{cs}$ we have to check the condition (2.49) for all the
enhancements we can construct using the singularities in figure 4. The results
are summarized in table 1, in agreement with table $(5.2)$ of Grimm:2019ixq .

Enhancement | $(l_{1},l_{2})\in\mathcal{E}$
---|---
$\textit{II}_{0,0}\quad\longrightarrow\quad\left\\{\begin{matrix}&\quad\textit{II}_{0,1}\\\ &\quad\textit{II}_{1,1}\end{matrix}\right.\quad$ | $\begin{matrix}&(3,3),(4,3),(4,4),(4,5),(5,5)\\\ &(3,3),(4,2),(4,4),(4,6),(5,5)\end{matrix}$
$\textit{II}_{0,1}\quad\longrightarrow\quad\left\\{\begin{matrix}&\quad\textit{II}_{1,1}\\\ &\quad\textit{III}_{0,0}\\\ &\quad\textit{V}_{2,2}\end{matrix}\right.$ | $\begin{matrix}&(3,2),(3,3),(3,4),(4,4),(5,4),(5,5),(5,6)\\\ &(3,2),(3,4),(4,4),(5,4),(5,6)\\\ &(3,0),(3,2),(3,4),(3,6),(4,4),(5,2),(5,4),(5,6),(5,8)\end{matrix}$
$\textit{III}_{0,0}\quad\longrightarrow\quad\left\\{\begin{matrix}&\quad\textit{III}_{0,1}\\\ &\quad\textit{III}_{1,1}\end{matrix}\right.$ | $\begin{matrix}&(2,2),(4,3),(4,4),(4,5),(6,6)\\\ &(2,2),(4,2),(4,4),(4,6),(6,6)\end{matrix}$
$\textit{III}_{0,1}\quad\longrightarrow\quad\quad\quad\textit{III}_{1,1}$ | $(2,2),(3,2),(3,4),(4,4),(5,4),(5,6),(6,6)$
$\textit{III}_{1,1}\quad\longrightarrow\quad\quad\quad\textit{V}_{2,2}$ | $(2,0),(2,2),(4,2),(2,4),(4,4),(6,4),(4,6),(6,6),(6,8)$
$\textit{IV}_{0,1}\quad\longrightarrow\quad\quad\quad\textit{V}_{2,2}$ | $(1,0),(1,2),(3,2),(3,4),(4,4),(5,4),(5,6),(7,6),(7,8)$
$\textit{V}_{1,1}\quad\longrightarrow\quad\left\\{\begin{matrix}&\quad\textit{V}_{1,2}\\\ &\quad\textit{V}_{2,2}\end{matrix}\right.$ | $\begin{matrix}&(0,0),(2,2),(4,3),(4,4),(4,5),(6,6),(8,8)\\\ &(0,0),(2,2),(4,2),(4,4),(4,6),(6,6),(8,8)\end{matrix}$
$\textit{V}_{1,2}\quad\longrightarrow\quad\quad\quad\quad\textit{V}_{2,2}$ | $(0,0),(2,2),(3,2),(3,4),(4,4),(5,4),(5,6),(6,6),(8,8)$
Table 1: Allowed Enhancements for Calabi-Yau fourfolds with $h^{1,3}=2$.

Since we are considering flux compactifications and we need to compute the
leading behavior of the potential (2.4) near each possible enhancement using
the theorem (2.40), we have to know all the possible location of $G_{4}$ in
the intersection of the monodromy filtrations:

$G_{4}\in W_{l_{1}}\left(N_{1}\right)\cap W_{l_{2}}\left(N_{1}+N_{2}\right).$
(2.50)

This is equivalent to classify all the possible doublets $(l_{1},l_{2})$ that
can contain some component of $G_{4}$. The table 1 summarizes the results for
all available enhancements which corresponding expression of the flux
potential will be explicitly studied in section 4.2. As mentioned before, we
do not consider enhancements from type $I_{a,a^{\prime}}$ singularities
because they are located at finite distance in the moduli space.

##  3 Dynamical Cobordisms

Dynamical Cobordism techniques Buratti:2021fiv ; Buratti:2021yia ;
Angius:2022aeq ; Angius:2022mgh ; Angius:2023rma ; Angius:2023xtu ;
Huertas:2023syg ; Blumenhagen:2022mqw ; Blumenhagen:2023abk  (see Dudas:2000ff
; Blumenhagen:2000dc ; Dudas:2002dg ; Dudas:2004nd  for early related works
and Basile:2018irz ; Antonelli:2019nar ; Basile:2020xwi ; Basile:2021mkd ;
Mourad:2021mas ; Mourad:2021mas2 ; Mourad:2022loy ; Basile:2022ypo  for more
recent developments, and Charmousis:2010cke ; Kiritsis:2017knp  for
holographic applications) represent a powerful method to explore large field
regimes through dynamical solutions of generic spacetime actions. Moreover, in
the context of the Swampland program, they are able to provide effective
realizations of the configurations cobordant to nothing predicted at the
topological level by the Cobordism Conjecture McNamara:2019rup .

###  3.1 Codimension-1 ETW branes

In this first section we provide a quick review of the method, that
generalizes the construction in Angius:2022aeq  including also time-dependent
solutions as discussed in Angius:2022mgh  (see also Dudas:2002dg ;
Basile:2018irz ; Antonelli:2019nar ; Mourad:2021mas  for early works on time-
dependent solutions) in the simplest case of codimension-1 ETW branes.

Consider the $d-$dimensional action in $M_{P}=1$ units:

$S=\int
d^{d}x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}\left(\partial\phi\right)^{2}-V(\phi)\right]$
(3.1)

containing Einstein-gravity coupled to a real scalar with arbitrary potential.
We consider solutions for the equations of motion associated to this action
that run along one spacetime coordinate $y$ according with the ansatz:

$\begin{split}&ds^{2}_{d}=e^{-2\sigma(y)}ds^{2}_{d-1}\pm dy^{2}\\\
&\phi=\phi(y),\\\ \end{split}$ (3.2)

where the sign $\pm$ encompasses the possibilities to consider $y$ as a
spacelike or a timelike coordinate. The condition for such a solution to
realize a dynamical cobordism to nothing is the presence of a metric
singularity at finite distance in spacetime corresponding to a divergent
regime for the scalar. Imposing these requirements within the equations of
motion we obtain the following behavior for the fields:

$\begin{split}&\phi(y)\simeq-\frac{2}{\delta}\log y\\\
&\sigma(y)\simeq-\frac{4}{\delta^{2}(d-2)}\log y.\\\ \end{split}$ (3.3)

The real number $\delta$ parameterizes the class of solutions and controls the
growing of the scalar potential:

$V(\phi)=\mp ace^{\delta\phi},$ (3.4)

where $c$ is a free parameter, $a$ is related to $\delta$ by:

$\delta=2\sqrt{\frac{d-1}{d-2}(1-a)}$ (3.5)

and the overall sign depends on whether $y$ is a spacelike or timelike
coordinate. The equations of motion also impose the condition $a<1$ for this
family, which imposes some bounds on $\delta$ when we choose a spacelike or a
timelike running coordinate. In particular, if $V>0$ we have that, for $a<0$
(hence $\delta>2\sqrt{\frac{d-1}{d-2}}$) the coordinate $y$ is spacelike, and
for $0<a<1$ (hence $\delta<2\sqrt{\frac{d-1}{d-2}}$) the coordinate $y$ is
timelike; while if $V<0$, for $a<0$ (hence $\delta>2\sqrt{\frac{d-1}{d-2}}$)
the coordinate $y$ is timelike, and for $0<a<1$ (hence
$\delta<2\sqrt{\frac{d-1}{d-2}}$) the coordinate $y$ is spacelike.

Another interesting feature of these solutions is the existence of universal
scaling relations linking the spacetime and the field space distances $\Delta$
and $\mathcal{D}$ with the spacetime scalar curvature $|R|$ in the following
way:

$\Delta\sim e^{-\frac{\delta}{2}\mathcal{D}},\quad\quad|R|\sim
e^{\delta\mathcal{D}}.$ (3.6)

Such relations, controlled by the critical exponent $\delta$, encode the
defining properties for the realization of a dynamical cobordism in the
spacetime solutions.

###  3.2 Intersecting ETW branes

In this section we overview the local description of intersecting ETW branes
configuration done in Angius:2023rma . These solutions, given by a mere
superposition of branes, represent the simplest building block to probe
multiple infinite distance limits in the field space. However, as we will
discuss in section 4.2, these solutions are not adequate to probe infinite
distance limits involving multiple scalars in the Calabi-Yau moduli space, but
it will be necessary a generalization of the ansatz as described in section
3.4.

We consider the following $(n+2)-$dimensional action containing Einstein
gravity coupled to two real scalar fields constrained by the general potential
$V\left(\phi_{1},\phi_{2}\right)$:

$S=\int
d^{n+2}x\sqrt{-g}\left\\{\frac{1}{2}R-\frac{1}{2}\left(\partial\phi_{1}\right)^{2}-\frac{1}{2}\left(\partial\phi_{2}\right)^{2}-\frac{\alpha}{2}\partial_{\rho}\phi_{1}\partial^{\rho}\phi_{2}-V(\phi_{1},\phi_{2})\right\\}.$
(3.7)

Note that the presence of a mixed term in the kinetic sector of the action is
essential to solve the equations of motion for the class of solutions
considered in Angius:2023rma .

We consider the following structure for the spacetime metric:

$ds_{n+2}^{2}\,=\,e^{-2\sigma_{1}-2\sigma_{2}}ds_{n}^{2}+e^{-2\sigma_{2}}dy_{1}^{2}\pm
e^{-2\sigma_{1}}dy_{2}^{2}\,,$ (3.8)

which, by setting only one of the two running coordinates $y_{i}$ to a
constant, nicely reduces to a local codimension-1 ETW metric up to a constant.
This is equivalent to restricting our investigation to an $(n+1)$ dimensional
$y_{i}-$constant slice in the $(n+2)$dimensional spacetime. The sign $\pm$ in
the metric indicates the possibilities to consider the coordinate $y_{2}$ as a
spacelike or a timelike direction.
Moreover, we require the following coordinate dependence for the scalars:

$\phi_{1}=\phi_{1}(y_{1})\quad,\quad\quad\quad\phi_{2}=\phi_{2}(y_{2}).$ (3.9)

A picture of the spacetime configuration described in this structure is shown
in Figure 6, with two codimension-1 ETW branes intersecting at the
codimension-2 locus $y_{i}=0$.

Figure 6: Spacetime configuration representing two intersecting ETW branes of
type $\delta_{1}$ and $\delta_{2}$, as figure 2 of Angius:2023rma .

Imposing the equations of motion, the local description near the codimension-2
locus is given by logarithmic functions:

$\begin{split}\phi_{1}&=-\frac{2}{\delta_{1}}\log
y_{1}\quad,\quad\phi_{2}=-\frac{2}{\delta_{2}}\log y_{2}\\\
\sigma_{1}&=-\frac{4}{n\delta_{1}^{2}}\log
y_{1}\quad,\quad\sigma_{2}=-\frac{4}{n\delta_{2}^{2}}\log y_{2}\\\
\end{split}$ (3.10)

where the parameters $\delta_{i}$ specify the type of intersecting branes and
control the growing of the potential in the quarter circle area of figure 6:

$V=V_{1}+V_{2}=-c_{1}v_{1}\,e^{\delta_{1}\phi_{1}}e^{\frac{4}{n\delta_{2}}\phi_{2}}\mp
c_{2}v_{2}\,e^{\frac{4}{n\delta_{1}}\phi_{1}}e^{\delta_{2}\phi_{2}}$ (3.11)

with $c_{i}$ positive integration constants. The sign $\mp$ in the second term
of $V$ refers to the cases where $y_{2}$ is a spacelike or a timelike
coordinate, respectively. The coefficients $v_{i}$ are related to the
parameters $\delta_{i}$ as:

$\delta_{i}^{2}=\frac{8(n+1)}{n+\sqrt{n\left[n+8v_{i}(n+1)\right]}}.$ (3.12)

The equations of motion also constrain the coefficient $\alpha$, controlling
the mixed kinetic term in the action, to a specific value:

$\alpha=\frac{8}{n\delta_{1}\delta_{2}}.$ (3.13)

This highlights the fact that the solutions apply to a preferential basis of
the field space which metric contain a non-trivial mixed term weighed by the
parameter (3.13). As we will explain in section 4.2, this invalidates their
naive application in the Calabi-Yau moduli space, where each Calabi-Yau
modulus is a scalar field, because in the corresponding infinite distance loci
the asymptotic metrics are diagonal.

###  3.3 Intersecting configurations for decoupled scalar fields

Since the Calabi-Yau moduli space metrics are diagonal in the asymptotic
limit, as we will discuss in detail in section 4.2, we consider the following
$(n+2)-$dimensional action for gravity coupled with two decoupled real scalar
fields constraint by the general potential $V(\psi_{1},\psi_{2})$:

$S=\int
d^{n+2}x\sqrt{-g}\left\\{\frac{1}{2}R-\frac{1}{2}\left(\partial\psi_{1}\right)^{2}-\frac{1}{2}\left(\partial\psi_{2}\right)^{2}-V(\psi_{1},\psi_{2})\right\\}$
(3.14)

Note that in this new action the dynamics of the scalars is controlled by a
flat metric kinetic term.

To solve the equations of motion, we consider the following ansatz for the
fields:

$ds^{2}_{n+2}=e^{2A(y_{1},y_{2})}ds^{2}_{n}+e^{2B(y_{1},y_{2})}dy_{1}^{2}\pm
e^{2C(y_{1},y_{2})}dy^{2}_{2}$ (3.15)
$\psi_{1}=\psi_{1}(y_{1},y_{2})\quad,\quad\quad\psi_{2}=\psi_{2}(y_{2}).$
(3.16)

Note that, in order to find intersecting ETW brane solutions for scalars with
diagonal kinetic terms, we now allow one of the scalars to depend on two
coordinates (although the equations of motion make this dependence relatively
simple). Despite this could seem like a complication, we will actually show
that it leads to an interpretation very much in the spirit of enhancements of
singularities in CY moduli space in section 4.2. A linearly independent set of
combination of equations of motion for these functions is:

$\begin{split}(1)\quad&\mp
e^{-2C}n\left[A^{\prime}\left(nA^{\prime}+B^{\prime}-C^{\prime}\right)+A^{\prime\prime}\right]-e^{-2B}\left[\dot{A}\left(n\dot{A}-\dot{B}+\dot{C}\right)+\ddot{A}\right]=2V\\\
(2)\quad&e^{-2B}\left[-n\dot{A}^{2}+(n-2)\dot{A}\left(\dot{C}-\dot{B}\right)+(n-2)\ddot{A}+2\dot{C}\left(\dot{C}-\dot{B}\right)+2\ddot{C}+\left(\dot{\psi_{1}}\right)^{2}\right]+\\\
\pm
e^{-2C}&\left[-nA^{\prime\prime}+(n-2)A^{\prime}\left(B^{\prime}-C^{\prime}\right)+(n-2)A^{\prime\prime}+2B^{\prime}\left(B^{\prime}-C^{\prime}\right)+2B^{\prime\prime}+\left(\psi_{1}^{\prime}\right)^{2}+\left(\psi_{2}^{\prime}\right)^{2}\right]=0\\\
(3)\quad&n\left[\dot{A}B^{\prime}-\dot{A}A^{\prime}+\dot{C}B^{\prime}-\dot{A}^{\prime}\right]=\dot{\psi_{1}}\psi_{1}^{\prime}\\\
(4)\quad&e^{-2B}\left[\left(n\dot{A}-\dot{B}+\dot{C}\right)\dot{\psi_{1}}+\ddot{\psi_{1}}\right]\pm
e^{-2C}\left[\left(nA^{\prime}+B^{\prime}-C^{\prime}\right)\psi_{1}^{\prime}+\psi_{1}^{\prime\prime}\right]-\frac{\partial
V}{\partial\psi_{1}}=0\\\ (5)\quad&\pm
e^{-2C}\left[\left(nA^{\prime}+B^{\prime}-C^{\prime}\right)\psi_{2}^{\prime}+\psi_{2}^{\prime\prime}\right]-\frac{\partial
V}{\partial\psi_{2}}=0,\\\ \end{split}$ (3.17)

where we have used the notation $\dot{f}:=\partial_{y_{1}}f$ and
$f^{\prime}:=\partial_{y_{2}}f$. The upper signs refer to the case where
$y_{2}$ is a spacelike coordinate and the bottom signs to the case where
$y_{2}$ is a timelike coordinate.
As in the previous section, we are interested in studying classes of solutions
for these equations realizing a dynamical cobordism, with the scalars running
to infinity in a finite distance in spacetime. We can conventionally choose
that this happens as we approach the origin of the coordinates $y_{i}$.

Let us focus our attention in the simple class of solutions featuring an
additive structure for the warp factors of the metric:

$A(y_{1},y_{2})=-\sigma_{1}(y_{1})-\sigma_{2}(y_{2})\quad,\quad
B(y_{1},y_{2})=-\sigma_{2}(y_{2})\quad,\quad
C(y_{1},y_{2})=-\sigma_{1}(y_{1})$ (3.18)

and logarithmic profiles for the functions $\sigma_{i}$ and $\psi_{i}$:

$\begin{split}\sigma_{1}=-a_{1}\log y_{1}+\frac{1}{2}\log
c_{1}\quad&,\quad\sigma_{2}=-a_{2}\log y_{2}+\frac{1}{2}\log c_{2}\\\
\psi_{1}=b_{11}\log y_{1}+b_{12}\log y_{2}\quad&,\quad\psi_{2}=b_{2}\log
y_{2}\\\ \end{split}$ (3.19)

Note that this means that at $y_{1}=0$ the scalar $\psi_{1}$ diverges, whereas
at $y_{2}=0$ both scalars diverge. This will provide a nice match with the
picture of singularity enhancement in CY moduli space of section 2.5, as we
explain later on. The functions $B$ and $C$ do not have a respectively $y_{1}$
and $y_{2}$ dependence because such a dependence can be easily reabsorbed by a
change of variables in the metric. The real numbers $c_{1}$ and $c_{2}$ are
integration constants, giving only subleading contributions in the $y_{i}\to
0$ limits.
Replacing these profiles in (3.17) we get:

$\begin{split}&V=-\frac{1}{2}c_{1}na_{1}\left(na_{1}+a_{1}-1\right)y_{1}^{-2}y_{2}^{-2a_{2}}\mp\frac{1}{2}c_{2}na_{2}\left(na_{2}+a_{2}-1\right)y_{1}^{-2a_{1}}y_{2}^{-2}=\\\
&:=-c_{1}v_{1}y_{1}^{-2}y_{2}^{-2a_{2}}\mp
c_{2}v_{2}y_{1}^{-2a_{1}}y_{2}^{-2}\\\ &b_{11}^{2}=na_{1}\\\
&b_{12}^{2}=na_{1}a_{2}^{2}\\\ &b_{2}^{2}=na_{2}\left(1-a_{1}a_{2}\right)\\\
\end{split}$ (3.20)

For $a_{1},a_{2}\neq 1$ the scalar potential splits in two pieces with a
different dependence on $y_{1}$ and $y_{2}$, which means a different
dependence on $\psi_{1}$ and $\psi_{2}$. These terms encode the leading
contribution of the function $V$ in the asymptotic regime near the locus
$y_{i}=0$. The constant coefficients $v_{i}$ are related to the parameters
$a_{i}$ by:

$a_{i}=\frac{1\pm\sqrt{1+8v_{i}\left(1+\frac{1}{n}\right)}}{2(n+1)}$ (3.21)

In order to give to the reader a clear overview on the results, we summarize
below the spacetime solutions written in terms of the parameters $a_{i}$:

$\begin{split}&ds^{2}_{n+2}=y_{1}^{2a_{1}}y_{2}^{2a_{2}}ds^{2}_{n}+y_{2}^{2a_{2}}dy_{1}^{2}\pm
y_{1}^{2a_{1}}dy_{2}^{2}\\\ &\psi_{1}=-\sqrt{na_{1}}\log
y_{1}-a_{2}\sqrt{na_{1}}\log y_{2}\\\
&\psi_{2}=-\sqrt{na_{2}(1-a_{1}a_{2})}\log y_{2}.\\\ \end{split}$ (3.22)

The potential takes the form:

$V=-c_{1}v_{1}e^{\frac{2}{\sqrt{na_{1}}}\psi_{1}}\mp
c_{2}v_{2}e^{2\sqrt{\frac{a_{1}}{n}}\psi_{1}}e^{2\sqrt{\frac{1-a_{1}a_{2}}{na_{2}}}\psi_{2}}.$
(3.23)

Note that the spacetime metric in (3.22) is conformally flat: this is a
straightforward generalization of the metric (3.2) of the codimension-1 case.

Actually, although we have constructed the above solutions from scratch from
the action (3.14), they can be regarded as a reinterpretation of the solutions
in section 3.2. This is because the two solutions are related by a simple
redefinition of the fields:

$\begin{split}&\phi_{1}=\psi_{1}-\left(\frac{a_{1}a_{2}}{1-a_{1}a_{2}}\right)^{1/2}\psi_{2}\\\
&\phi_{2}=\left(\frac{1}{1-a_{1}a_{2}}\right)^{1/2}\psi_{2}\\\ \end{split}$
(3.24)

for which the spacetime action (3.14) takes the form (3.7) with

$\alpha=2\sqrt{a_{1}a_{2}}.$ (3.25)

Respect to these new fields we can read the solutions as the superposition of
two codimension 1 ETW branes located at the respective positions $y_{1}=0$ and
$y_{2}=0$, as depicted in figure 6, and characterized by the critical
exponents $\delta_{i}$ given in (3.12), which relation with the parameters
$a_{i}$ is:

$\delta_{i}^{2}=\frac{4}{na_{i}}.$ (3.26)

Note that when we approach the locus $y_{1}=0$ the new redefined scalar
$\phi_{1}$ runs to infinity such as the scalar $\psi_{1}$ of the decoupled
setup, while when we approach the locus $y_{2}=0$ only the scalar $\phi_{2}$
goes to infinity in the new fields redefinition but both the scalars
$\psi_{1}$ and $\psi_{2}$ diverge at the same time in the decoupled picture.
In section 4.2 we will take advantage of this feature to associate to the ETW
brane located at the position $y_{1}=0$, where only the scalar $\psi_{1}$
diverges, a codimension-1 divisor in the Calabi-Yau moduli space, and to the
other ETW brane, the one located at the position $y_{2}=0$ where both the
scalars $\psi_{1}$ and $\psi_{2}$ diverge, not a second divisor intersecting
with the first one but the intersection of the two divisors itself, i.e. the
enhancement.

Although this is a nice interpretation to solve the problem to have diagonal
asymptotic metrics, the structure of the potential that support this class of
solutions is not of the same form of the potentials that we get in the Calabi-
Yau moduli space near the enhanced singularities, as we will see in section
4.2. In the next section we will introduce a generalization of the above
solution, which is similar in spirit in the interpretation of the ETW branes,
but it is supported by scalar potentials having the same form of the flux
potentials computed using the growth theorem for the Hodge norm applied to the
$G_{4}$ fluxes supported by the asymptotic Hodge structures allowed at the
enhancements and summarized in table 1.

###  3.4 A new solution beyond the conformal flatness

In this section we present a generalization of the solutions described in the
previous section and associated with the same action (3.14). These new
solutions are governed by a scalar potential of the right form to match the
flux potential supported in the asymptotic regions of the Calabi-Yau moduli
space around the enhancements. The simple modification with respect to the
ansatz (3.18) that we will consider in this section consists to go beyond the
conformal flatness of the metric.

We consider a non-conformally flat ansatz for the spacetime metric (3.15):

$\begin{split}&A(y_{1},y_{2})=a_{1}\log y_{1}+a_{2}\log y_{2}\\\
&B(y_{1},y_{2})=a_{2}\log y_{2}-\frac{1}{2}\log c_{2}\\\
&C(y_{1},y_{2})=(1-a_{1}n)\log y_{1}-\frac{1}{2}\log c_{1}\\\ \end{split}$
(3.27)

and we keep the same logarithmic profiles for the scalar fields:

$\psi_{1}=b_{11}\log y_{1}+b_{12}\log y_{2}\quad,\quad\psi_{2}=b_{2}\log
y_{2}.$ (3.28)

The profile of the scalars is the same as in the previous section, so these
solutions also follow the interpretation that one ETW brane corresponds to a
singular divisor and the other ETW brane corresponds to the intersection of
divisors, i.e. to the enhancement of the singularity.

Replacing these functions in the equations of motion (3.17) we obtain the
following constraints for the parameters:

$\begin{split}&b_{11}=-\sqrt{a_{1}n(2-a_{1}-a_{1}n)}\\\
&b_{12}=-a_{2}(1-a_{1}n)\sqrt{\frac{n}{a_{1}(2-a_{1}-a_{1}n)}}\\\
&b_{2}=-\sqrt{a_{2}n\left[1+\frac{a_{2}(a_{1}n-1)^{2}}{a_{1}(-2+a_{1}+a_{1}n)}\right]}\\\
\end{split}$ (3.29)

and the following shape for the potential:

$V=\mp\frac{c_{2}}{2}a_{2}n\left(a_{2}n+a_{2}-1\right)y_{1}^{-2+2a_{1}n}y_{2}^{-2}=\mp
c_{2}v_{2}y_{1}^{-2+2a_{1}n}y_{2}^{-2}.$ (3.30)

As in the previous section, the upper signs refer to the case where $y_{2}$ is
a spacelike coordinate and the bottom ones to the case where $y_{2}$
represents a timelike direction. Note that if $y_{2}$ is timelike, and
$a_{2}>\frac{1}{1+n}$, the potential is positive. For positive $a_{i}$ the
relations (3.29) are well defined only if $a_{1}<\frac{2}{1+n}$.

For a clearer overall picture of the result, let us summarize the solutions by
making the dependence on the spacetime coordinates explicit:

$\begin{split}&ds^{2}_{n+2}=y_{1}^{2a_{1}}y_{2}^{2a_{2}}ds^{2}_{n}+y_{2}^{2a_{2}}dy_{1}^{2}-y_{1}^{2(1-a_{1}n)}dy_{2}^{2}\\\
&\psi_{1}=-\sqrt{a_{1}n(2-a_{1}-a_{1}n)}\log
y_{1}-a_{2}(1-a_{1}n)\sqrt{\frac{n}{a_{1}(2-a_{1}-a_{1}n)}}\log y_{2}\\\
&\psi_{2}=-\sqrt{a_{2}n\left[1+\frac{a_{2}(a_{1}n-1)^{2}}{a_{1}(-2+a_{1}+a_{1}n)}\right]}\log
y_{2}.\\\ \end{split}$ (3.31)

The potential written in terms of these fields takes the form:

$V=\mp
c_{2}v_{2}e^{\frac{2-2a_{1}n}{\sqrt{a_{1}n(2-a_{1}-a_{1}n)}}\psi_{1}}e^{2\sqrt{\frac{a_{2}+a_{1}(-2+a_{1}+a_{1}n-2a_{2}n+a_{1}a_{2}n^{2})}{a_{1}a_{2}n(-2+a_{1}+a_{1}n)}}\psi_{2}}.$
(3.32)

Note that with respect to the class of solutions treated in section 3.3, here
we have only a single term in the potential driving the dynamics of the fields
near the locus $y_{i}=0$. This feature makes these solutions appropriate for
the application to the infinite distance network of the Calabi-Yau moduli
space. In fact, the structure (3.32) of the potential is precisely the one
that matches the flux potentials obtained from the Hodge norm of $G_{4}$ for
all the admissible enhancements in the space $\mathcal{M}_{cs}$ with
$h^{3,1}=2$.

In order to make more manifest the interpretation of this solution to describe
the intersection of two codimension-1 ETW branes of the kind in section 3.1,
we make use of the following redefinition of the fields:

$\begin{split}&\phi_{1}=\psi_{1}-(1-a_{1}n)\sqrt{\frac{a_{2}}{a_{1}(2-a_{1}-a_{1}n)-a_{2}(a_{1}n-1)^{2}}}\psi_{2}\\\
&\phi_{2}=\left(1+\frac{a_{2}(a_{1}n-1)^{2}}{a_{1}(-2+a_{1}+a_{1}n)}\right)^{-1/2}\psi_{2}\\\
\end{split}$ (3.33)

to write the spacetime action (3.14) in the form (3.7), with

$\alpha=2(1-a_{1}n)\sqrt{\frac{a_{2}}{a_{1}(2-a_{1}-a_{1}n)}}$ (3.34)

and

$V=\mp
c_{2}v_{2}e^{\frac{2-2a_{1}n}{\sqrt{a_{1}n(2-a_{1}-a_{1}n)}}\phi_{1}}e^{\frac{2}{\sqrt{a_{2}n}}\phi_{2}},$
(3.35)

Importantly, one has to remember that the fields $\phi_{1}$ and $\phi_{2}$
should not be regarded as independent Calabi-Yau moduli, but rather that
$\phi_{1}$ is a Calabi-Yau modulus and $\phi_{2}$ is a combination of
$\phi_{1}$ and another orthogonal modulus. In this interpretation, the non-
trivial mixed term (3.34) emphasizes the fact that $\phi_{2}$ is not
orthogonal to $\phi_{1}$.

With respect to these new fields we can read the solutions as the intersection
of two codimension 1 ETW branes located at the respective positions $y_{1}=0$
and $y_{2}=0$. And we characterize these branes using the critical exponents:

$\begin{split}&\delta_{1}=\frac{2-2a_{1}n}{\sqrt{a_{1}n(2-a_{1}-a_{1}n)}}\\\
&\delta_{2}=\frac{2}{\sqrt{a_{2}n}}\\\ \end{split}$ (3.36)

controlling the growing of the potential as we approach the codimension-2
intersection between the two codimension-1 branes.

As anticipated, the new key feature of these solutions is that they go beyond
the conformally flat ansatz of sections 3.2 and 3.3. Indeed, using a
convenient redefinition of the spacetime coordinates we can write the metric
(3.31) in the following form:

$\begin{split}ds^{2}_{n+2}=&y_{1}^{2(1-a_{1}n)}y_{2}^{2a_{2}}\left[y_{1}^{4a_{1}n-2}ds^{2}_{n}+y_{1}^{-2(1-a_{1}n)}dy_{1}^{2}-y_{2}^{-2a_{2}}dy_{2}^{2}\right]=\\\
&\propto\tilde{y}_{1}^{2\frac{(1-a_{1}n)}{a_{1}n}}\tilde{y}_{2}^{\frac{2a_{2}}{1-a_{2}n}}\left[\tilde{y}_{1}^{4-\frac{2}{a_{1}n}}ds^{2}_{n}+d\tilde{y}_{1}^{2}-d\tilde{y}_{2}^{2}\right],\\\
\end{split}$ (3.37)

which is not conformally flat due to the presence of the extra factor in front
of the $n$ dimensional metric $ds^{2}_{n}$.

The conformally flat solutions in section 3.2 were shown in Angius:2023rma  to
correspond to the backreaction of the superposition of two codimension-1
source terms, with no codimension-2 sources. In our solutions the extra factor
beyond the conformally flat ansatz suggests the presence of an additional
codimension-2 source localized at the intersection between the two ETW
branes.. It would be interesting to explore this further.

####  3.4.1 Scaling Relations

As we point out above, one interesting feature of the Dynamical Cobordism
solutions is the existence of specific scaling relations linking the spacetime
and the field space distances $\Delta$ and $\mathcal{D}$ with the spacetime
scalar curvature $|R|$. Such relations are summarized in (3.6) for the case of
codimension-1 singularities and they are controlled by the critical exponent
$\delta$ characterizing the type of ETW brane dressing the singularity. In
Angius:2023xtu  these scaling relations were extended to the case when we
approach the codimension-2 singularity located at the intersection between two
distinct codimension-1 ETW branes. In these setups, the parameter controlling
the scaling is a path-dependent combination of the critical exponents
associated with the two individual intersecting branes. In this section we
discuss the same relations for the new class of non-conformally flat
solutions.

Consider the following parameterization for the spacetime paths in the region
$y_{i}<1$ that approach the intersecting locus $y_{i}=0$ as $t\mapsto 0$:

$\Gamma(\gamma_{1},\gamma_{2}):\quad\begin{cases}&y_{1}=t^{\gamma_{1}}\\\
&y_{2}=t^{\gamma_{2}}\end{cases},$ (3.38)

where $\gamma_{i}$ are two positive real numbers.
The spacetime distance to the origin for the path $\Gamma$ identified by the
pair of numbers $(\gamma_{1},\gamma_{2})$, is:

$\Delta=\int_{\Gamma}\big{|}y_{2}^{2a_{2}}\left(\partial_{t}y_{1}\right)^{2}-y_{1}^{2(1-a_{1}n)}\left(\partial_{t}y_{2}\right)^{2}\big{|}^{1/2}dt=\int_{\Gamma}\big{|}\gamma_{1}^{2}t^{2r_{1}}-\gamma_{2}^{2}t^{2r_{2}}\big{|}^{1/2},$
(3.39)

with:

$\begin{split}&r_{1}=\gamma_{1}-1+a_{2}\gamma_{2}\\\
&r_{2}=\gamma_{2}-1+(1-a_{1}n)\gamma_{1}.\\\ \end{split}$ (3.40)

The two contributions to the integral are comparable in the case
$r_{1}=r_{2}$:

$\frac{\gamma_{1}}{\gamma_{2}}=\frac{1-a_{2}}{a_{1}n},$ (3.41)

at which the distance $\Delta$ is minimized. In all the other cases one of the
two terms dominates over the other and the spacetime distance behaves as:

$\Delta=\int_{\Gamma}\gamma_{i}^{2}t^{r_{i}}dt=\frac{\gamma_{i}}{r_{i}+1}t^{r_{i}+1},$
(3.42)

with $i=1$ for the paths above the tangent line to the path (3.38) at
$r_{1}=r_{2}$, and $i=2$ for the paths below that line.

Since we know the profiles of the scalars $\phi_{i}(y_{i})$ in terms of the
spacetime coordinates, we can translate each spacetime path in (3.38) into a
path in the field space $\phi_{i}\left(y(t)\right)$ and compute the
corresponding distance using the kinetic metric appearing in the spacetime
action:

$\mathcal{D}=\int\big{|}d\phi_{1}^{2}+d\phi_{2}^{2}+\alpha
d\phi_{1}d\phi_{2}\big{|}^{1/2}$ (3.43)

with $\alpha$ given in (3.34). For each of the two regimes in (3.42) we can
write the distance $\mathcal{D}$ in the field space in terms of the distance
in the spacetime:

$\mathcal{D}\sim\frac{\sqrt{n\left[a_{1}\left(2-a_{1}-a_{1}n\right)\gamma_{1}^{2}+2a_{2}(1-a_{1}n)\gamma_{1}\gamma_{2}+a_{2}\gamma_{2}^{2}\right]}}{r_{i}+1}\log\Delta.$
(3.44)

Using the parameterization $\gamma_{1}=\sqrt{\gamma}$ and
$\gamma_{2}=1/\sqrt{\gamma}$ such that the ratio
$\gamma_{1}/\gamma_{2}=\gamma$ and the value of $\gamma$ separating the two
regimes is $\gamma^{\ast}=(1-a_{2})/a_{1}n$ we obtain the scaling relation:

$\Delta\sim e^{-\frac{1}{2}\delta_{int}\mathcal{D}}$ (3.45)

with

$\delta_{int}=\begin{cases}&\frac{2\left(\frac{a_{2}}{\sqrt{\gamma}}+\sqrt{\gamma}\right)}{\sqrt{n\left[\frac{a_{2}}{\gamma}+a_{1}(2-a_{1}-a_{1}n)\gamma+2a_{2}(1-a_{1}n)\right]}}\quad\quad\text{for}\quad\gamma>\gamma^{\ast},\\\
&\frac{2\left(\frac{1}{\sqrt{\gamma}}+\sqrt{\gamma}(1-a_{1}n)\right)}{\sqrt{n\left[\frac{a_{2}}{\gamma}+a_{1}(2-a_{1}-a_{1}n)\gamma+2a_{2}(1-a_{1}n)\right]}}\quad\quad\text{for}\quad\gamma<\gamma^{\ast}.\\\
\end{cases}$ (3.46)

This result is different with respect to the one discussed in section 3.3 of
Angius:2023rma  due to the modified relation between the parameter $a_{1}$ and
the critical exponent $\delta_{1}$ in (3.36). Nevertheless, the generalization
to non-conformally flat metrics preserves the fact that one still gets nice
scaling relations encoding the fundamental properties defining the realization
of a dynamical cobordism in the spacetime.

Notice that in the limits $\gamma\mapsto 0$ and $\gamma\mapsto\infty$ we
recover respectively the critical exponent $\delta_{2}$ and the combination
$\delta_{1}/(1-a_{1}n)$ controlling the asymptotic behavior of the fields
(3.33).

##  4 ETW networks for infinite distance limits in CY moduli space

In this section we are finally ready to explore the network of singular
divisors located at infinite distance in the complex structure sector of the
Calabi-Yau moduli space using ETW brane solutions in spacetime.

The key tool in this dictionary between infinite distance limits in moduli
space and cobordisms to nothing in spacetime is the translation, for each
specific infinite distance limit, of the information encoded in the Hodge-
Deligne structure and in the flux with that encoded in the critical exponents
for ETW branes. As already anticipated, each ETW brane explores a singular
divisor, while intersecting configurations explore intersections of divisors
in a different way with respect to the naive expectation.

###  4.1 ETW branes for simple singularities

In this section we exploit the classification of singularities that can occur
in a two-moduli family of Calabi-Yau fourfolds with primitive Hodge number
$\hat{m}\geq 4$, done in Grimm:2019ixq  and reviewed in section 2.4, to write
for each of these boundaries an effective spacetime action and to build
codimension-1 Dynamical Cobordism solutions, characterized by their
appropriate critical exponent, which explore the infinite distance limit for
the corresponding divergent scalar.

####  4.1.1 Generalities

We consider the three-dimensional effective action obtained compactifying
M-theory on the above mentioned Calabi-Yau fourfold. The resulting theory is
completely specified by the Kähler potential, which determines the Weil-
Petersson metric $G_{WP}$, and the three-dimensional scalar potential induced
by turning on four-form fluxes $G_{4}$ on certain four-cycles of the internal
space.

In the regime we are interested, namely the asymptotic regime near the
singular divisor $\Delta_{1}$ and very far away from any intersection, we can
choose the local set of coordinate in $\mathcal{M}_{cs}$ introduced above
(2.16), such that the putative divisor is parameterized by the condition
$z_{1}=0$. The leading term of the Kähler potential in the strict asymptotic
regime near the divisor $\Delta_{1}$ is computed applying the growth theorem
(2.40) to the holomorphic form $\Omega$:

$\mathcal{K}^{cs}_{Sl(2)}\simeq-\log\left[(s_{1})^{d_{1}}f(\xi)\right],$ (4.1)

where $d_{1}$ identifies the location of the form in the monodromy filtration
$W\left(N_{1}\right)$. Note that $d_{1}$ is exactly the number used in section
2.4 to classify the five classes of singularities.

Using this result we can compute the following expansion for the Weil-
Petersson metric:

$G_{s_{1}s_{1}}=\partial_{t_{1}}\partial_{\bar{t}_{1}}\mathcal{K}^{cs}=\frac{1}{4}\frac{d_{1}}{(s_{1})^{2}}+\frac{\sharp}{(s_{1})^{3}}+...+\mathcal{O}\left(e^{2\pi
it_{1}}\right)$ (4.2)

and keep only the leading contribution. Note that the constant coefficient of
the leading term is completely determined by the integer $d_{1}$ which
contains the information about the specific type of singularity. Applying this
computation to each infinite distance singularity in $\mathcal{M}_{cs}$, we
can obtain, as done in Grimm:2019ixq , the following few possibilities to
construct the kinetic sector of the spacetime action:

Singularity | $d_{1}$ | Kähler potential | $G_{s_{1}s_{1}}$
---|---|---|---
II | $1$ | $K^{cs}\sim-\log\left[(s_{1})^{1}f(\xi)\right]$ | $\sim\frac{1}{4}\frac{1}{(s_{1})^{2}}$
III | $2$ | $K^{cs}\sim-\log\left[(s_{1})^{2}f(\xi)\right]$ | $\sim\frac{1}{4}\frac{2}{(s_{1})^{2}}$
IV | $3$ | $K^{cs}\sim-\log\left[(s_{1})^{3}f(\xi)\right]$ | $\sim\frac{1}{4}\frac{3}{(s_{1})^{2}}$
V | $4$ | $K^{cs}\sim-\log\left[(s_{1})^{4}f(\xi)\right]$ | $\sim\frac{1}{4}\frac{4}{(s_{1})^{2}}$

The introduction of a primitive $G_{4}$ four-form flux in some internal cycle
of the compactification generates an effective potential (2.4) for the three
dimensional action in the Einstein frame. According to section 2.3, we have
that in the strict asymptotic regimes of the moduli space the middle
cohomology split in eigenspaces of the $sl(2)$ operators $Y_{i}$, (2.37), and
the $G_{4}$ flux decomposes in this splitting as:

$G_{4}=\sum_{\mathbf{l}\in\mathcal{E}}G_{4}^{\mathbf{l}}$ (4.3)

This sum tells us that approaching the putative singular locus, $G_{4}$
spreads in the corresponding Hodge-Deligne diamond admitting components only
in the admissible sectors $V_{\mathbf{l}}$ of $H_{p}^{4}(Y_{4},\mathbb{C})$.
All this information allows to use the growth theorem for the Hodge norm
(2.40) to extract the leading complex structure dependence of the potential
$V_{M}$ via:

$V_{M}=\frac{1}{\mathcal{V}^{3}_{4}}||G_{4}||^{2}\sim\frac{1}{\mathcal{V}^{3}_{4}}||G_{4}||^{2}_{sl(2)}=\sum_{l_{1}\in\mathcal{E}}\left(s_{1}\right)^{l_{1}-4}||\rho_{l_{1}}(G_{4},a)||^{2}$
(4.4)

in the asymptotic regime near the singular divisor $\Delta_{1}$ parameterized
by the divergent coordinate $s_{1}\mapsto\infty$. The positive coefficient
$||\rho_{l_{1}}(G_{4},a)||^{2}$ is constraint by the flux quantization
conditions and it carries within it the dependence on the real part of the
coordinate $t_{1}$ as in (2.17).

Since in this section we are considering asymptotic regimes near to a specific
singular divisor $\Delta_{1}$ but very far away from any higher intersection,
in the previous formula we have a single divergent scalar $s_{1}$ and one
single component $l_{1}$ for the vector $\mathbf{l}$. Depending on the value
of $l_{1}$ we can have power divergent potentials, whenever $4<l_{1}\leq 8$,
or power vanishing potentials, when $0\leq l_{1}<4$. The leading contribution
of the three-dimensional spacetime action for the scalar $s_{1}$ is:

$S=\int
d^{3}x\sqrt{-g}\left[\frac{1}{2}R-\frac{d_{1}}{4(s_{1})^{2}}\left(\partial
s_{1}\right)^{2}-||\rho_{l_{1}}||^{2}s_{1}^{l_{1}-4}\right].$ (4.5)

In order to put it in the form (3.1) we redefine the field as:

$\phi=\sqrt{\frac{d_{1}}{2}}\log s_{1},$ (4.6)

and the action (4.5) becomes:

$S=\int
d^{3}x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}\left(\partial\phi\right)^{2}-||\rho_{l_{1}}||^{2}e^{\sqrt{\frac{2}{d_{1}}}(l_{1}-4)\phi}\right]$
(4.7)

For our analysis, we consider in the above action only power divergent
potentials, with $4<l_{1}\leq 8$. This is because for decreasing potentials
the Dynamical Cobordism solutions are all equivalent to the case with no
potential, and the value of $\delta$ is fixed to a specific number for all the
different types of singularities. This excludes the possibility to distinguish
between the different cases.

As shown in Angius:2022aeq  and revisited in section 3.1, the equations of
motion associated with this action admit a special class of solutions (3.3)
realizing a codimension-1 Dynamical Cobordism to nothing. It is remarkable
that the well-established setup of M-theory flux compactification leads to
solutions of this type requiring a non-trivial time dependence. It would be
interesting to explore the possible cosmological implications of this result
and potential relations with phenomena of nucleation of bubbles of something
Friedrich:2024cob . The important point is that these solutions provide a way
to explore the strict asymptotic regime of the corresponding boundary
$\Delta_{1}$ in $\mathcal{M}_{cs}$ through its spacetime realizations as an
ETW brane configuration. From equation (4.3) we have that the $G_{4}$ flux of
our compactification can have several components turned on that live in
different vector spaces $W_{l_{1}}\left(N_{1}\right)$. The relevant term in
this expansion for our analysis is the one that grows faster than the others
in the $s_{1}\mapsto\infty$ limit. Considering the label $l_{1}$ associates
with this leading term we compute the critical exponent identifying the
specific solution in the family through the formula:

$\delta=\sqrt{\frac{2}{d_{1}}}\left(l_{1}-4\right).$ (4.8)

Let us emphasize that $\delta$ completely determines the behavior of the
spacetime fields because it encodes the information about the growth of the
potential as we approach to the ETW boundary. From a geometric point of view,
such information is completely encoded in the Hodge-Deligne splitting of the
middle cohomology near the corresponding singular divisor plus the information
about the flux.

####  4.1.2 Examples

Let us consider the example of the type $II_{0,0}$ singularity in the complex
structure moduli space of Calabi-Yau fourfold with $h^{3,1}=2$. Its Hodge-
Deligne diamond is depicted in Figure 4 and it encodes the information about
how the Hodge structure of $H^{4}_{p}(Y_{4},\mathbb{C})$ is spread in the
Hodge-Deligne splitting as we approach the singularity. We have the following
non trivial groups:

$\begin{split}&(i^{0,3},i^{1,2},i^{2,1},i^{3,0})=(1,0,0,1)\\\
&(i^{0,4},i^{1,3},i^{2,2},i^{3,1},i^{4,0})=(0,1,\widehat{m},1,0)\\\
&(i^{1,4},i^{2,3},i^{3,2},i^{4,1})=(1,0,0,1)\\\ \end{split}$ (4.9)

which implies three possibilities for the location of $G_{4}$ in the
splitting: $l_{1}=3,4,5$. We neglect the cases $l_{1}=4,3$ because, according
to (4.7), they produce vanishing potential, while for the case $l_{k}=5$ the
equation (4.8) gives:

$\text{Type }II_{0,0}\quad\quad\text{with
}l_{k}=5\quad\quad\longrightarrow\quad\quad\delta=\sqrt{2}.$ (4.10)

Following the same procedure for all the singularities classified in section
2.4, with a $G_{4}$ flux specified by its location $l_{1}$ in the splitting,
we are able to associate to each of these possibilities a specific spacetime
solution characterized by the corresponding critical exponent. The results are
summarized in Table 2

Type | $d_{k}$ | $l_{k}$ | $\delta$ | Type | $d_{k}$ | $l_{k}$ | $\delta$
---|---|---|---|---|---|---|---
$II_{0,0}$ | $1$ | $5$ | $+\sqrt{2}$ | $II_{0,1}$ | $1$ | $5$ | $+\sqrt{2}$
$II_{1,1}$ | $1$ | $5$ | $+\sqrt{2}$ | $II_{1,1}$ | $1$ | $6$ | $+2\sqrt{2}$
$III_{0,0}$ | $2$ | $6$ | $+2$ | $III_{1,1}$ | $2$ | $6$ | $+2$
$III_{0,1}$ | $2$ | $5$ | $+1$ | $III_{0,1}$ | $2$ | $6$ | $+2$
$IV_{0,1}$ | $3$ | $5$ | $+\sqrt{6}/3$ | $IV_{0,1}$ | $3$ | $7$ | $+\sqrt{6}$
$V_{1,1}$ | $4$ | $6$ | $+\sqrt{2}$ | $V_{1,1}$ | $4$ | $8$ | $+2\sqrt{2}$
$V_{1,2}$ | $4$ | $5$ | $+\sqrt{2}/2$ | $V_{1,2}$ | $4$ | $6$ | $+\sqrt{2}$
$V_{1,2}$ | $4$ | $8$ | $+2\sqrt{2}$ | $V_{2,2}$ | $4$ | $6$ | $+\sqrt{2}$
$V_{2,2}$ | $4$ | $8$ | $+2\sqrt{2}$ |  |  |  |
Table 2: Critical exponents characterizing the ETW brane solutions in
spacetime for each type of singularity that can occur in the complex structure
moduli space of Calabi-Yau fourfold with $h^{1,3}=2$.

Notice that all the critical exponents classified in the table stay inside the
upper bound $\delta\leq 2\sqrt{\frac{d-1}{d-2}}$, that for three spacetime
dimensions means $\delta\leq 2\sqrt{2}$. Since we are dealing with positive
potentials, this implies that the spacetime solutions of the form (3.3)
realizing these infinite distance limits involve a timelike running
coordinate. The cases $II_{1,1}$ with a non-trivial flux component in
$l_{1}=6$ and $V_{1,1},V_{1,2}$ and $V_{2,2}$ with $l_{1}=8$ are special
because they saturate the bound for the critical exponent: they correspond in
regimes where the scalar potential is subleading with respect to the kinetic
term, so they behave as in the case of zero potential.

###  4.2 ETW networks for the enhancements

The main claim of the previous section is the capability to explore any
singular divisor of the Calabi-Yau moduli space with an ETW brane solution in
spacetime. In this section we propose the use of real codimension-2
intersecting ETW brane solutions to explore the network of intersecting
divisors in CY moduli space with flux potential.

As anticipated in section 3.3, the naive expectation is that individual ETW
branes explore singular divisors in the moduli space, according to the
dictionary of the previous section, and their intersection would explore the
intersection between the corresponding divisors. However the actual
interpretation of the spacetime solutions turns out to be more subtle, and
much closer to the mathematical description of the network of divisors in
terms of a structure of singularity enhancements in specific growth sectors,
as we now explain.

####  4.2.1 Generalities

The regime we are interested is the strict asymptotic regime that approaches
the codimension-2 singularity $\Delta_{12}=\Delta_{1}\cap\Delta_{2}$ through
the growth sector $R_{12}$, staying very far away from any higher
intersection. As explained in section 2.2, in this region we can fix a local
set of coordinates $\left\\{t_{1},t_{2},\xi^{\bar{j}}\right\\}$, such that the
putative locus $\Delta_{12}$ is parameterized by the conditions $s_{1}\succ
s_{2}\mapsto\infty$. Applying the growth theorem (2.40) to the Hodge norm of
the holomorphic form $\Omega$, we can extract the leading term of the Kähler
potential in the sector $R_{12}$:

$\mathcal{K}^{cs}_{Sl(2)}\simeq-\log\left[(s_{1})^{d_{1}}(s_{2})^{(d_{e}-d_{1})}f(\xi)\right],$
(4.11)

where $d_{1}$ labels the type of singularity associated with the divisor
$\Delta_{1}$, according with the classification of section 2.4, and $d_{e}$
labels the type of singularity occurring at the intersection $\Delta_{12}$,
according with the allowed enhancements listed in the table 1. Using the
equation (4.11), we compute for all the possible enhancements of
$\mathcal{M}_{cs}$, listed in table 1, the corresponding asymptotic Weil-
Petersson metric via (2.3) and we summarize the results in table 3.

Enhancement | $d_{1}$ | $d_{e}$ | Kähler Potential | WP Metric
---|---|---|---|---
$\textit{II}_{0,0}\mapsto\textit{II}_{0,1}$ | $1$ | $1$ | $\mathcal{K}^{cs}\sim-\log\left[s_{1}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{II}_{0,0}\mapsto\textit{II}_{1,1}$ | $1$ | $1$ | $\mathcal{K}^{cs}\sim-\log\left[s_{1}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{II}_{0,1}\mapsto\textit{II}_{1,1}$ | $1$ | $1$ | $\mathcal{K}^{cs}\sim-\log\left[s_{1}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{II}_{0,1}\mapsto\textit{III}_{0,0}$ | $1$ | $2$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})(s_{2})f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&\frac{1}{4(s_{2})^{2}}\end{matrix}\right)$
$\textit{II}_{0,1}\mapsto\textit{V}_{2,2}$ | $1$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})(s_{2})^{3}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&\frac{3}{4(s_{2})^{2}}\end{matrix}\right)$
$\textit{III}_{1,1}\mapsto\textit{V}_{2,2}$ | $2$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{2}(s_{2})^{2}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{2(s_{1})^{2}}&0\\\ 0&\frac{1}{2(s_{2})^{2}}\end{matrix}\right)$
$\textit{III}_{0,0}\mapsto\textit{III}_{0,1}$ | $2$ | $2$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{2}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{III}_{0,0}\mapsto\textit{III}_{1,1}$ | $2$ | $2$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{2}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{III}_{0,1}\mapsto\textit{III}_{1,1}$ | $2$ | $2$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{2}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{4(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{IV}_{0,1}\mapsto\textit{V}_{2,2}$ | $3$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{3}(s_{2})f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{3}{4(s_{1})^{2}}&0\\\ 0&\frac{1}{4(s_{2})^{2}}\end{matrix}\right)$
$\textit{V}_{1,1}\mapsto\textit{V}_{2,2}$ | $4$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{4}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{V}_{1,2}\mapsto\textit{V}_{2,2}$ | $4$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{4}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
$\textit{V}_{1,1}\mapsto\textit{V}_{1,2}$ | $4$ | $4$ | $\mathcal{K}^{cs}\sim-\log\left[(s_{1})^{4}f(\xi)\right]$ | $G_{s_{i}s_{j}}\sim\left(\begin{matrix}\frac{1}{(s_{1})^{2}}&0\\\ 0&0\end{matrix}\right)$
Table 3: Leading behavior of the infinite distances in the moduli space from a
double intersection singularity.

Note that, except for the enhancements:

$\begin{split}\textit{(i)}\quad\textit{II}_{0,1}\mapsto\textit{III}_{0,0}\quad\quad&\quad\quad\textit{(ii)}\quad\textit{II}_{0,1}\mapsto\textit{V}_{2,2}\\\
\textit{(iii)}\quad\textit{III}_{1,1}\mapsto\textit{V}_{2,2}\quad\quad&\quad\quad\textit{(iv)}\quad\textit{IV}_{0,1}\mapsto\textit{V}_{2,2}\\\
\end{split}$

whose corresponding Weil-Petersson metrics show a diagonal form, for the rest
of the enhancements we obtain degenerate metrics. This implies that these
cases cannot be studied with the methods we developed in section 3.

As explained in the previous section, turning on a primitive $G_{4}$-flux in
some internal cycles of the compactification we generate an effective three
dimensional potential for the Calabi-Yau moduli according with the definition
(2.4). The $sl(2)$ decomposition (2.37) of the middle cohomology near the
asymptotic locus $\Delta_{12}$ allows to write the $G_{4}-$flux as in (4.3),
where the allowed doublets $\mathbf{l}=\left(l_{1},l_{e}\right)$ appearing in
the expansion are the one classified in table 1. Using these allowed doublets
in the growth theorem for the Hodge norm, the authors of Grimm:2019ixq
compute the complex structure dependence of the flux potential for all the
allowed enhancements in $\mathcal{M}_{cs}$ with $h^{3,1}=2$ through the
formula:

$V_{M}=\frac{1}{\mathcal{V}^{3}_{4}}||G_{4}||^{2}\sim\frac{1}{\mathcal{V}^{3}_{4}}||G_{4}||^{2}_{sl(2)}=\sum_{(l_{1},l_{e})\in\mathcal{E}}\left(s_{1}\right)^{l_{1}-4}\left(s_{2}\right)^{l_{e}-l_{1}}||\rho_{(l_{1},l_{e})}(G_{4},a)||^{2}$
(4.12)

Since Dynamical Cobordism solutions are not able to distinguish decreasing
potentials from the case with zero potential, we consider only the doublets
$(l_{1},l_{e})$ leading to divergent flux-potentials. In the table 4 we
summarize the results only for the four enhancements equipped with a non-
degenerate metric.

Enhancement | Doublets | Potential
---|---|---
$\textit{II}_{0,1}\mapsto\textit{III}_{0,0}$ |  $(5,6)$, $(5,4)$ | $\sim c_{1}(s_{1})(s_{2})+c_{2}\frac{s_{1}}{s_{2}}$
$\textit{II}_{0,1}\mapsto\textit{V}_{2,2}$ |  $(5,8)$, $(5,6)$, $(5,4)$, $(5,2)$ | $\sim c_{1}(s_{1})(s_{2})^{3}+c_{2}(s_{1})(s_{2})+c_{3}\frac{s_{1}}{s_{2}}+c_{4}\frac{s_{1}}{(s_{2})^{3}}$
$\textit{III}_{1,1}\mapsto\textit{V}_{2,2}$ |  $(6,8)$, $(6,6)$, $(6,4)$ | $\sim c_{1}(s_{1})^{2}(s_{2})^{2}+c_{2}(s_{1})^{2}+c_{3}(s_{2})^{2}$
$\textit{IV}_{0,1}\mapsto\textit{V}_{2,2}$ |  $(7,8)$, $(7,6)$, $(5,6)$, $(5,4)$ | $\sim c_{1}(s_{1})^{3}(s_{2})+c_{2}\frac{(s_{1})^{3}}{(s_{2})}+c_{3}(s_{1})(s_{2})+c_{4}\frac{s_{1}}{s_{2}}$
Table 4: Divergent potentials associated to the enhancements in
$\mathcal{M}_{cs}$ with $h^{3,1}=2$ equipped with a non-degenerate metric. The
parameters $c_{i}$ are positive constant coefficients constrained by the flux
quantization conditions.

####  4.2.2 Spacetime solutions and interpretation

Using the Weil-Petersson metric and the dominant contribution of the scalar
potential for all the possible enhancements reviewed in the previous section
and all the possible components for the $G_{4}-$flux, we can write the leading
contribution of the spacetime action for the involved moduli, valid in the
growth sector $R_{12}$, near the intersection locus
$\Delta_{12}=\Delta_{1}\cap\Delta_{2}$:

$S=\int
d^{3}x\sqrt{-g}\left[\frac{1}{2}R-\frac{d_{1}}{4(s_{1})^{2}}\left(\partial
s_{1}\right)^{2}-\frac{d_{e}-d_{1}}{4(s_{2})^{2}}\left(\partial
s_{2}\right)^{2}-||\rho_{\mathbf{l}}||^{2}s_{1}^{l_{1}-4}s_{2}^{l_{e}-l_{1}}\right].$
(4.13)

In order to put the action in the form (3.14), we fix the canonical
normalization for the scalars through the following redefinition of the
fields:

$\psi_{1}=\sqrt{\frac{d_{1}}{2}}\log
s_{1}\quad\quad,\quad\quad\psi_{2}=\sqrt{\frac{d_{e}-d_{1}}{2}}\log s_{2},$
(4.14)

and the action (4.13) takes the form:

$S=\int
d^{3}x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}\left(\partial\psi_{1}\right)^{2}-\frac{1}{2}\left(\partial\psi_{2}\right)^{2}-||\rho_{\mathbf{l}}||^{2}e^{(l_{1}-4)\sqrt{\frac{2}{d_{1}}}\psi_{1}}e^{(l_{e}-l_{1})\sqrt{\frac{2}{d_{e}-d_{1}}}\psi_{2}}\right].$
(4.15)

As pointed out above, the diagonal kinetic metric appearing in this spacetime
action prevents the naive interpretation of the two scalars as two independent
moduli. This is due to the fact that intersecting ETW brane configurations
contain a mixed kinetic term in their spacetime action (3.7) which is always
non-zero, (3.13), as imposed by the equations of motion. The key idea to solve
this drawback is encoded in the redefinition of the fields (3.24) and (3.33)
relating the decoupled scalars $\psi_{i}$ with the scalars $\phi_{i}$
satisfying the equations of motion associated to the action (3.7) with a
specific non-trivial mixing term as in section 3.2. The solutions presented in
section 3.3 for decoupled scalars have the nice property that when we approach
the locus $y_{1}=0$ only the scalar $\psi_{1}$ diverges, while when we
approach the locus $y_{2}=0$ both the scalars $\psi_{1}$ and $\psi_{2}$
diverge at the same time. On the other hand, the solutions with respect to the
fields $\phi_{i}$ have a clear interpretation for these loci as two
intersecting ETW branes. In terms of Calabi-Yau moduli, the ETW brane located
at the position $y_{1}=0$, at which the modulus $\psi_{1}$ diverge, is
interpreted as the spacetime realization of the divisor $\Delta_{1}$ involved
in the intersection $\Delta_{12}$. The brane located at position $y_{2}=0$, at
which both the Calabi-Yau moduli diverge, is then naturally interpreted as the
spacetime realization of the intersection $\Delta_{12}$ of the two divisors,
i.e. the enhancement. This new interpretation matches very well with the
description of the network of divisors provided in section 2 where the
asymptotic Hodge theory strongly constrains how to jump from a singular
divisor $\Delta_{1}$ to the enhanced singularity at the intersection
$\Delta_{12}$ through the growth sector $R_{12}$. Here we have something very
similar in the spacetime: the solution describes in the same configuration
what happens near the singular divisor $\Delta_{1}$, corresponding to the ETW
brane located at $y_{1}=0$, and near its enhanced singularity in
$\Delta_{12}$, corresponding to the ETW brane located at $y_{2}=0$.

Although the interpretation of the intersecting ETW configurations as
spacetime realizations of the enhancements in the moduli space along specific
growth sectors nicely solve the problem to have asymptotic diagonal metrics,
the structure of the potential that supports the class of solutions in section
3.3 is not of the same form of the potential that we have in the Calabi-Yau
moduli space near the enhanced singularities. The potential (3.23) contains
two different contributions: the first one dominating in a cone region near
the ETW brane located at the position $y_{1}=0$, and the second one dominating
in the complementary region near the ETW brane located at $y_{2}=0$. On the
other hand, in the spacetime action for the Calabi-Yau moduli the potential is
given by a single term: the leading contribution in the sum (4.12) when we
approach the intersection between the two different divisors. Then, there is
no straight direct way to match the two expressions.

One possibility is to focus on trajectories in the moduli space where one term
of the potential (3.23) is subleading with respect to the other and match with
(4.12) only the dominant term of (3.23). However, this approach does not
provide a solution in the full spacetime region between the two intersecting
ETW branes, but only in the sector in which the trajectories selecting the
dominant term in the potential are supported.

Happily, a fully satisfactory realization can be provided using the new class
of solutions described in section 3.4, driven by a single potential term
(3.32), and performing a special redefinition of the fields (3.33) which leads
to the interpretation of these multiple infinite distance limits as
configurations of intersecting ETW branes.

Note that for each particular enhancement $\Delta_{1}\rightarrow\Delta_{12}$,
characterized by its corresponding variation of the mixed Hodge-Deligne
structure and equipped with the flux information, we have a specific spacetime
action (4.15) defined in terms of the pairs of parameters
$\left(d_{1},d_{e}\right)$ and $\left(l_{1},l_{e}\right)$. The main advantage
of the solutions computed in section 3.4, whose potential (3.32) matches
perfectly the structure of the Calabi-Yau flux potential given by the
$sl(2)-$approximation of the Hodge norm of $G_{4}$, is that we can associate
to each of these actions a specific solution of the form (3.31) and interpret
each growth sector $R_{12}$ as an intersecting configuration of branes in
spacetime.

####  4.2.3 Examples

In this section we will perform the explicit matching between the potential
(4.12), appearing in the action (4.15), and the Dynamical Cobordism potential
(3.32), with $n=1$, supporting the class of solutions (3.31), for each
enhancement of table 4 and for each possible leading term of the corresponding
potential. The matching produces for each of these enhancements the exact
values of the parameters $a_{i}$, which uniquely determine the spacetime
solutions via (3.31). The list of values of $a_{i}$ obtained for all the
allowed enhancements of table 4 which are equipped with a non-trivial diagonal
kinetic metric, computed in table 3, and with a divergent flux potential
showing an explicit dependence from both the moduli is summarized in table 5.

Enhancement | $(l_{2},l_{e})$ | $a_{1}$ | $a_{2}$ | $\delta_{1}$ | $\delta_{2}$
---|---|---|---|---|---
$II_{0,1}\mapsto III_{0,0}$ | $(5,6)$ | $1/2$ | $1$ | $\sqrt{2}$ | $2$
$II_{0,1}\mapsto III_{0,0}$ | $(5,4)$ | $1/2$ | $1$ | $\sqrt{2}$ | $2$
$II_{0,1}\mapsto V_{2,2}$ | $(5,8)$ | $1/2$ | $1/2$ | $\sqrt{2}$ | $2\sqrt{2}$
$II_{0,1}\mapsto V_{2,2}$ | $(5,6)$ | $1/2$ | $3/2$ | $\sqrt{2}$ | $2\sqrt{6}/3$
$II_{0,1}\mapsto V_{2,2}$ | $(5,4)$ | $1/2$ | $3/2$ | $\sqrt{2}$ | $2\sqrt{6}/3$
$II_{0,1}\mapsto V_{2,2}$ | $(5,2)$ | $1/2$ | $1/2$ | $\sqrt{2}$ | $2\sqrt{2}$
$III_{1,1}\mapsto V_{2,2}$ | $(6,8)$ | $1/3$ | $1/4$ | $2$ | $4$
$III_{1,1}\mapsto V_{2,2}$ | $(6,6)$ | $1/3$ | $1$ | $2$ | $2$
$III_{1,1}\mapsto V_{2,2}$ | $(6,4)$ | $1/2$ | $1/4$ | $2$ | $4$
$IV_{0,1}\mapsto V_{2,2}$ | $(7,8)$ | $1/4$ | $1/2$ | $\sqrt{6}$ | $2\sqrt{2}$
$IV_{0,1}\mapsto V_{2,2}$ | $(7,6)$ | $1/4$ | $1/2$ | $\sqrt{6}$ | $2\sqrt{2}$
$IV_{0,1}\mapsto V_{2,2}$ | $(5,6)$ | $3/4$ | $3/2$ | $\sqrt{6}/3$ | $2\sqrt{6}/3$
$IV_{0,1}\mapsto V_{2,2}$ | $(5,4)$ | $3/4$ | $3/2$ | $\sqrt{6}$ | $2\sqrt{6}/3$
Table 5: Table showing the parameters $a_{i}$ appearing in the spacetime
solutions (3.31) for each possible enhancement and the corresponding values of
the critical exponents characterizing the nature of the intersecting ETW
branes.

In order to obtain the correct sign in the potential we are requiring the
direction $y_{2}$ to be timelike. The corresponding solutions physically
describe a configuration of intersecting branes with a boundary in the
spacelike direction $y_{1}$ and an origin in time at the location $y_{2}=0$.
It would be interesting to explore the role of such solutions in cosmological
applications. A special configuration with a boundary defining the beginning
of time and a boundary with respect to a spacelike coordinate has been
explored in Angius:2022mgh  in a supercritical bosonic string theory setup
with tachyon condensation along a lightlike direction. However, in that case
there is an interpretation of the two boundaries as two different phases of a
same recombinating ETW brane. In the present case of Calabi-Yau moduli we have
an interpretation of the two boundaries in terms of an intersecting
configuration of two distinct ETW branes.

Using the definition of critical exponents given in (3.36) we can characterize
the two intersecting branes appearing in each of these spacetime realizations
in the language of the Dynamical Cobordism. The values of the critical
exponents for all the accounted enhancements are summarized in the last two
columns of table 5. Notice that the list of values obtained for $\delta_{1}$
nicely agree with the dictionary between singular divisors and codimension 1
ETW branes provided in the table 2 of the previous subsection. As explained
above, the ETW brane at $y_{2}=0$ is interpreted as the spacetime
representation of the enhanced singularity occurring at the intersection locus
$\Delta_{12}$ in $\mathcal{M}_{cs}$. For that reason the corresponding
critical exponent $\delta_{2}$ does not have to match with the values
appearing in table 2: the corresponding ETW brane does not represent a
singular divisor in the moduli space but an enhanced singularity.

###  4.3 Application to Swampland Distance Conjecture

One of the most explored swampland conjecture is the Distance Conjecture
Ooguri:2006in , which predicts for any infinite distance limit in moduli space
the emergence of an infinite tower of states becoming exponentially massless
with the field space distance, therefore the cutoff of the corresponding
effective theory is lowered as:

$\Lambda\sim e^{-\lambda\mathcal{D}}$ (4.16)

where $\lambda$ is an $\mathcal{O}(1)$ parameter.

On the other hand, Dynamical Cobordism solutions probes infinite distance
limits in field spaces, then it is natural to ask for their interplay. Since
the Dynamical Cobordism analysis provides scaling relations linking the
spacetime and the field space distances, we can offer a spacetime version of
(4.16) expressing the lowering of the cutoff in terms of paths in the
spacetime:

$\Lambda\sim\Delta^{\frac{2\lambda}{\delta_{int}}}$ (4.17)

This spacetime rephrasing of the Distance Conjecture was proposed in
Angius:2022aeq  for codimension 1 infinite distance limits explored with
codimension-1 ETW brane. In Angius:2023rma , it was shown that the same
relation is still valid for codimension-2 infinite distance limits in the
field space probed in spacetime through configurations of intersecting ETW
branes. The only difference in this second case, with respect to the case of
single ETW brane, is that the relation (4.17) is controlled by a path-
dependent parameter $\delta_{int}$ which takes different values for each
specific spacetime direction traveled to reach the intersection between the
two ETW branes. Due to the knowledge of the profiles of the divergent scalars
that parameterize the codimension-2 infinite distance locus in the field space
in terms of the spacetime coordinates, it is possible to map each of these
spacetime paths to a specific path in field space that reaches the multiple
infinite distance limit, and to obtain the decay rate of $\Lambda$ for each of
these paths.

For the class of solutions reviewed in section 3.2, considering all the
spacetime paths reaching the brane intersection along all the possible
different directions, we are able to reproduce all the possible paths in the
field space reaching the corresponding codimension-2 infinite distance limit.
However, this is not true for the new class of solutions studied in section
3.4. The new insight lies in the interpretation of these solutions as the
intersection of two ETW branes, where one brane represents a codimension-1
infinite distance locus in field space, corresponding to the singular divisor
$\Delta_{1}$, and the other intersecting brane does not represent one of its
intersecting divisors $\Delta_{2}$, but the enhanced singularity arising at
the codimension-2 locus $\Delta_{12}=\Delta_{1}\cap\Delta_{2}$. This new
interpretation implies that none of the paths parameterized in (3.38)
corresponds to a path in field space able to reach the divisor $\Delta_{2}$.
Roughly speaking, using the profiles of the fields given in (3.31) to
translate the spacetime paths parameterized in (3.38) to the corresponding
ones in the field space, we are able to reproduce only the field space sector
containing paths that achieve the locus $\Delta_{12}$ passing close to the
divisor $\Delta_{1}$. In particular, the field space path
$\tilde{\Gamma}(\gamma_{1},\gamma_{2})$, corresponding to the path
$\Gamma(\gamma_{1},\gamma_{2})$ in spacetime, is:

$\tilde{\Gamma}(\gamma_{1},\gamma_{2}):\quad\begin{cases}&\psi_{1}=-\left[\gamma_{1}\sqrt{a_{1}n(2-a_{1}-a_{1}n)}+\gamma_{2}a_{2}(1-a_{1}n)\sqrt{\frac{n}{a_{1}(2-a_{1}-a_{1}n)}}\right]\log
t\\\
&\psi_{2}=-\gamma_{2}\sqrt{a_{2}n\left[1+\frac{a_{2}(a_{1}n-1)^{2}}{a_{1}(-2+a_{1}-a_{1}n)}\right]}\log
t\\\ \end{cases}$ (4.18)

Since the choice $\gamma_{1}=0$ in (3.38) corresponds to approach the ETW
brane located at $y_{2}=0$ following a transversal path to the brane, this
direction corresponds to the furthest path from $\Delta_{1}$ that we can have
in the field space. Therefore the parameterization (4.18) includes all the
field space paths that satisfy the condition:

$\frac{\psi_{1}}{\psi_{2}}>\frac{\sqrt{a_{2}}(1-a_{1}n)}{\sqrt{a_{1}(-2+a_{1}+a_{1}n)+a_{2}(a_{1}n-1)^{2}}}.$
(4.19)

This interpretation nicely reproduces the growth sectors $R_{12}$ of the
Calabi-Yau moduli space.

##  5 Conclusions

M-theory compactifications on Calabi-Yau fourfolds with four-form flux produce
a rich landscape of vacua reproducing interesting phenomenology. The infinite
distance limits of the corresponding moduli/field spaces offer an intriguing
arena to test many of the Swampland conjectures and get information about the
mechanism breaking the effective field theory description by quantum gravity
effect. Interestingly, the structure of the infinite distance limits in the
moduli space displays a rich network of intersecting divisors, characterized
by the rich mathematical tools of asymptotic Hodge theory Schmid:1973 ;
Cattani:1982cka ; kerr2019polarized ; Grimm:2019ixq ; Grimm:2018cpv ;
Grimm:2018ohb ; Grimm:2019grh ; Bastian:2020bgh . In this paper we have
initiated the study of this network of normal crossing divisors using
Dynamical Cobordism solutions in the resulting three dimensional effective
field theories. Some of our results in this context are:

  * •

We have provided a dictionary associating to each singular divisor in the
network its spacetime realization as a codimension-1 ETW brane characterized
by a specific critical exponent;

  * •

We have shown that, due to the properties of the flux potential, this match
requires the corresponding Dynamical Cobordism to describe a time-dependent
solution;

  * •

We have extended this dictionary to codimension-2 loci in the network of
divisors at infinity in the Calabi-Yau moduli space by using codimension-2
Dynamical Cobordisms describing intersecting ETW branes;

  * •

We have shown that the two intersecting ETW branes in the spacetime picture,
rather than describe two intersecting divisors in the moduli space, describe
the singularity enhancement of a divisor as it approaches a singularity in a
specific growth sector. It is remarkable that the spacetime picture reproduces
the spirit of the mathematical approach to the study of intersections of
divisors;

  * •

We have shown that, in order to match the leading behaviour of the flux
potential given by the asymptotic growth of the Hodge norm, the required
spacetime solutions for intersecting ETW branes are more general than those
considered hitherto and we have provided the explicit construction of such
generalization, by relaxing the constraint of conformally flat ansatz in the
solutions in Angius:2023rma ;

  * •

We studied the scaling relations between the spacetime distance and the field
space distance along general paths in the new intersecting ETW brane solution,
generalizing those in the literature, and we explored its relation to the
cutoff implied by the Swampland Distance Conjecture, thus providing a
generalization valid for non-trivial scalar potentials in Calabi-Yau flux
compactifications.

Some related interesting open questions are the following:

  * •

It would be interesting to further study the class of non-conformally flat
solutions involving two divergent scalar fields we constructed in this paper.
For instance to display the source terms supporting it and to characterize the
properties of the possible codimension-2 source term localized at the
intersection of the ETW branes, and study possible connections with other
setups including such codimension-2 sources Blumenhagen:2022mqw .

  * •

We performed explicit computations of the critical exponents characterizing
the ETW branes associated to each possible singular divisor and some of the
enhanced singularities in the complex structure moduli space of Calabi-Yau
fourfolds with $h^{1,3}=2$. It would be interesting to reproduce the same
analysis in higher dimensional moduli space, thus allowing for the exploration
of higher codimension intersections in the Calabi-Yau moduli space.

  * •

In our dictionary between singular divisors in the moduli space and ETW branes
in spacetime, we translate the information about the Hodge-Deligne splittings
characterizing the moduli space singularities in terms of the critical
exponents characterizing the branes. However the inverse is not true: the
known class of Dynamical Cobordism solutions does not reproduce all the
singularity enhancements in the Calabi-Yau moduli space. In particular, those
involving degenerate metrics at infinity in field space may require, from the
Calabi-Yau side, a better characterization of the subleading contributions to
the Calabi-Yau moduli metric, and from the Dynamical Cobordism side, a
generalization of the solutions for scalars with non-trivial metrics for the
running scalars. It would be interesting to fully understand the spacetime
realizations of the enhancements in order to investigate if the complete
picture contains more details allowing a 1-1 correspondence between the two
approaches.

Our work has provided an important step in the exploration of the intricate
structure of the asymptotic Calabi-Yau moduli space in the presence of general
flux potential. We hope to come back to these exciting open questions in the
future.

## Acknowledgments

We would like to acknowledge Angel Uranga for his fundamental contribution to
this manuscript through meticulous comments and extensive discussions. We are
also pleased to thank José Calderon-Infante, Matilda Delgado, Jesús Huertas
and Andriana Makridou for useful discussions and for collaborations on related
topics. We also thank Ivano Basile, Stefano Giaccari, Alvaro Herráez, Elias
Kiritsis, Fernando Marchesano, Anthony Massidda, Irene Valenzuela and Roberto
Volpato for helpful discussions. R.A. wishes to acknowledge the hospitality of
the Max Planck Institute for Physics in Munich during the last stages of this
work. The research of R.A. has been supported by the grants CEX2020001007-S
and PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/501100011033 and by ERDF
A way of making Europe.

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