# Connecting flux vacua through scalar field excursions

Gary Shiu<EMAIL_ADDRESS>Department of Physics, University of
Wisconsin-Madison, 1150 University Avenue, Madison, WI 53706, USA  Flavio
Tonioni<EMAIL_ADDRESS>Instituut voor Theoretische Fysica, K.U.
Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium  Vincent Van Hemelryck
<EMAIL_ADDRESS>Institutionen för Fysik och Astronomi,
Uppsala Universitet, Box 803, SE-751 08 Uppsala, Sweden  Thomas Van Riet
<EMAIL_ADDRESS>Instituut voor Theoretische Fysica, K.U. Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium

###### Abstract

We show how flux vacua that differ from each other in flux quanta can be seen
as different vacua in a single scalar potential of an enlarged field space,
which resolves the separation by thin domain walls. This observation, which is
motivated by the AdS Distance Conjecture, allows one to compute distances
between different vacua using the usual field space metric. We verify for
explicit examples such as scale-seperated IIA flux vacua and the IIB Freund-
Rubin vacua that the Distance Conjecture (for scalar fields) is satisfied and
that the asymptotic directions in the enlarged field space are indeed
hyperbolic. This enlarged field space contains the tachyon fields on the
unstable $\tilde{\mathrm{D}}p$-branes of type II string theory, which can
induce the brane charges of the stable D-branes. We suggest that requiring
continuous interpolations refines the Cobordism Conjecture and postdicts the
existence of unstable $\tilde{\mathrm{D}}p$-branes.

††preprint: UUITP-32/23

##  I Introduction

The rich structure of the vacuum manifold of string theory, a.k.a. the
Landscape, is often depicted by an energy functional (scalar potential) with
many local minima. While this picture is useful for conveying to the public,
it is not commonly used in quantitative investigations of the Landscape. In
particular, we do not tend to think of flux vacua which differ by flux quantum
numbers or compactification manifold as different vacua in a single potential.
Instead, vacua that differ in flux numbers or choices of compactification
manifold can be thought of as being separated by domain walls. The Swampland
Cobordism Conjecture [1] formalises this picture by postulating that such
domain walls have to exist.

However, domain walls can be thin or thick;the latter can be described as a
kink solution with fields gradually moving from one vacuum to another. When
the thickness of the domain wall is much smaller than the short-distance
cutoff of the effective field theory (EFT), we can regard the wall as thin.

Consider for instance what happens in the absence of moduli stabilisation for
the landscape of 4d supersymmetric (SUSY) $\mathcal{N}=2$ vacua from type II
string theory on Calabi-Yau 3-folds. According to Reid’s conjecture [2], all
Calabi-Yau spaces can be connected to each other through geometric transitions
which can change the topology. From a 4d viewpoint, these transitions can be
seen as motion in the space of (potentially massive) scalars from one vacuum
to another.

In this letter we continue on our previous work [3] and argue for a similar
picture in the presence of moduli stabilisation through fluxes. In other
words: thin domain walls separating flux quanta can be seen as thick domain
walls by integrating in certain open-string degrees of freedom. Hence, in the
larger field space containing (possibly very heavy) open-string fields, we
arrive at a picture in which different flux vacua are different local minima
inside a single scalar potential. In ref. [3], we demonstrated this for the
original scale-separated flux vacua of type IIA orientifold compactifications
of refs. [4, 5, 6, 7]. This was partially motivated by an ongoing debate on
the consistency of these vacua as string-theoretic backgrounds. The open-
string fields in this case come from unstable D4-branes wrapping trivial
1-cycles inside non-trivial 2-cycles Poincaré dual to the four-form fluxes of
the would-be solution. The positions of these branes correspond to 4d scalars
whose scalar potential variations are smaller than the size of the actual
vacuum energy.

The present work aims to generalise this picture to any setup. Let us consider
some flux vacuum characterised by a specific $n$-dimensional internal manifold
and a set of integers describing flux quanta over various submanifolds. Using
Hodge duality, one can describe any flux as some internal $k$-form $F_{k}$,
with $k\leq n$, threading compact dimensions only. If this form is for
instance an RR-flux, then a D$(8-k)$-brane wrapping the dual $(n-k)$-cycle
would appear as a domain wall in the non-compact $(10-n)$-dimensional
spacetime separating vacua with RR-flux quanta differing by the D$(8-k)$-brane
charge. A universal way to think of this domain wall as a thick wall is by
regarding the D$(8-k)$-brane as a tachyon kink solution of an unstable
$\tilde{\mathrm{D}}(9-k)$-brane [8]. In other words, one integrates in open-
string tachyons and, in the enlarged field space of compactification moduli
and tachyons, the different vacua reside in one single scalar potential.

Besides the general conceptual picture, another motivation for our work
originates in the so-called Anti-de Sitter (AdS) Distance Conjecture [9],
which states that, for any $d$-dimensional AdS flux vacuum with cosmological
constant $\Lambda_{\mathrm{AdS}}$, there exists a tower of states with a mass
scale $m$ that behaves as

$\dfrac{m}{m_{\mathrm{P},d}}\overset{\Lambda_{\mathrm{AdS}}\sim
0}{\sim}|m_{\mathrm{P},d}^{-2}\Lambda_{\mathrm{AdS}}|^{\alpha},$ (I.1)

for a positive constant $\alpha$, where $m_{\mathrm{P},d}$ is the Planck mass.
The strong form of this conjecture states that $\alpha=1/2$ for SUSY vacua.
Whereas the general conjecture is supported by every controlled vacuum
solution ever constructed, the strong form is more controversial and even more
far-reaching as it would imply that SUSY AdS vacua do not admit a separation
of scales and cannot be regarded as vacua inside a lower-dimensional EFT as
the tower of particles does not decouple. Since SUSY AdS vacua are known to
have CFT duals, the implications for holography are equally far-reaching.

The argument for the general form of the conjecture relies on an extension of
the ordinary Distance Conjecture [10], which quantifies how a tower of states
becomes light when the geodesic distance travelled in moduli space becomes
large in Planck units. The generalisation that leads to the AdS Distance
Conjecture of ref. [11] relied on distances in metric space instead of
(scalar) moduli space. Unfortunately, there is no independent support for the
interesting suggestion that the Distance Conjecture can be applied to fields
of all spin.

In light of this, it is instructive to connect flux vacua through trajectories
in scalar field space since then one can use the kinetic term of the scalar
fields to compute the distance between different vacua. This was done in ref.
[3] for the scale-separated IIA flux vacua of refs. [4, 5, 6, 7], and it was
found that the Distance Conjecture is obeyed by the scalars that provide the
interpolation of flux quanta. We refer to refs. [12, 13, 14] for extensions of
these observations to other vacua and to refs. [15, 16] for alternative
measures of distances between vacua.

In this paper, using unstable $\tilde{\mathrm{D}}p$-branes, we can demonstrate
in general circumstances that flux vacua can be seen as critical points in a
single potential and how this realises the exponential dependence on the
distance of the tower mass-scale. It furthermore verifies the suggestion [10]
that the field space metric becomes hyperbolic in the large-distance limit.
However, since the field trajectories involve massive fields, strictly
speaking, we cannot see this as a manifestation of the original Distance
Conjecture for moduli spaces but its stronger and more speculative extension
to fields which are lifted in a scalar potential.

##  II Non-BPS branes as domain walls

In addition to the RR-charged stable BPS D-branes, type II theories also
feature non-BPS $\tilde{\mathrm{D}}$-branes, which are unstable D$p$-branes
for $p$ odd/even in type IIA/IIB theories; for a review, see e.g. refs. [8,
17, 18]. As a manifestation of the instability of such non-BPS branes, the
spectrum of open strings ending on the latter is non-supersymmetric and
contains a tachyon along with a gauge field.

At small string coupling, unstable non-BPS $\tilde{\mathrm{D}}p$-branes are
described by a worldvolume action that resembles in many aspects the action of
stable BPS D-branes. First, they have a string-frame Dirac-Born-Infeld term
[18]

$\displaystyle
S_{\mathrm{DBI}}^{\tilde{\mathrm{D}}p}=-\tilde{\mu}_{p}\int_{\Sigma_{1,p}}\mathrm{d}^{1,p}\xi\;\mathrm{e}^{-\Phi}\,V(T)\,\sqrt{-\mathrm{det}\,\bigl{[}\gamma_{\alpha\beta}\bigr{]}},$
(II.1)

where
$\smash{\gamma_{\alpha\beta}=G_{\alpha\beta}+\partial_{\alpha}T\partial_{\beta}T+(B+2\pi\ell_{s}F)_{\alpha\beta}}$.
Here, $\Phi$ is the 10-dimensional dilaton, while $G_{\alpha\beta}$ and
$B_{\alpha\beta}$ are the metric and Kalb-Ramond field. Furthermore,
$F=\mathrm{d}A$ is the field-strength of the open-string gauge field
$A_{\alpha}$ and $T$ is the open-string tachyon, while
$\tilde{\mu}_{p}=\sqrt{2}\,\mu_{p}=\sqrt{2}\,2\pi/(2\pi\ell_{s})^{p+1}$ is the
tension of the brane. Finally, $V(T)$ is the function

$\displaystyle
V(T)=\dfrac{1}{\displaystyle\mathrm{cosh}\,\biggl{(}\dfrac{T}{\sqrt{2}\ell_{s}}\biggr{)}},$
(II.2)

which represents the tachyon potential. Second, the Wess-Zumino term, in poly-
form notation, is [19, 18]

$S_{\mathrm{WZ}}^{\tilde{\mathrm{D}}p}=\tilde{\mu}_{p}\int_{\Sigma_{p+1}}C\wedge\mathrm{tr}\,\bigl{[}W(T)\,\mathrm{d}T\wedge\mathrm{e}^{B+2\pi\ell_{s}F}\bigr{]}\,,$
(II.3)

where $W(T)$ is a function that behaves like

$W(T)\overset{T\sim\pm\infty}{\sim}\dfrac{1}{2}\,\mathrm{e}^{\mp\frac{T}{\sqrt{2}\ell_{s}}}$
(II.4)

A kink solution of the theory interpolates between the vacua at $T=\pm\infty$;
such a kink solution, once plugged back into the action, represents a lower-
dimensional stable D$(p-1)$-brane since
$\smash{\tilde{\mu}_{p}\int_{-\infty}^{\infty}\mathrm{d}\xi\,V(T(\xi))=\mu_{p-1}}$,
and in the WZ-term we have
$\tilde{\mu}_{p}\int_{-\infty}^{\infty}W(T)\,\mathrm{d}T=\mu_{p-1}$.

Given this quick summary, let us consider a type II compactification of the
form $\mathcal{M}_{d}\times\mathrm{X}_{n}$, with $d+n=10$, and imagine that
there is some (unbounded) RR-flux $F_{k}$ winding a $k$-cycle $\Sigma_{k}$
inside the space $\mathrm{X}_{n}$. Now imagine a spacetime-filling non-BPS
$\tilde{\mathrm{D}}(9-k)$-brane wrapping the $(n-k)$-cycle dual to
$\Sigma_{k}$. Then, changing the tachyonic worldvolume scalar field from one
tachyon vacuum to another induces a change in $F_{k}$ flux by one unit.

This is motivated due to the fact that the non-BPS
$\tilde{\mathrm{D}}(9-k)$-brane sources the Bianchi identity for $F_{k}$,
being

$\mathrm{d}F_{k}=2\kappa_{10}^{2}\,\tilde{\mu}_{p}\,W(T)\,\mathrm{d}T\wedge\delta(\Sigma_{k}),$
(II.5)

where we ignored the term $H_{3}\wedge F_{k-2}$ as such a term is absent in
the examples treated in this paper. The Bianchi identity can be integrated to
become

$\dfrac{1}{(2\pi\ell_{s})^{k-1}}F_{k}=U(T)\,\delta(\Sigma_{k})+N\epsilon_{k},$
(II.6)

where we defined
$U(T)=\sqrt{2}\int_{-\infty}^{T}W(S)\mathrm{d}S/(2\pi\ell_{s})$ and $N$ is the
constant background flux on the background-volume form $\epsilon_{k}$. By
integrating over the $k$-cycle $\Sigma_{k}$, we obtain

$\dfrac{1}{(2\pi\ell_{s})^{k-1}}\int_{\Sigma_{k}}F_{k}=U(T)+N.$ (II.7)

Since $U(T=-\infty)=0$, the quantised flux is just $N$, but when
$U(T=+\infty)=1$, there are $N+1$ units of flux.

One might worry that the flux vacuum we end up with is not a genuine vacuum
due to the presence of the non-BPS brane. However, following Sen’s conjecture
[20], a non-BPS brane with the tachyon condensed in its tachyon vacuum
corresponds to not having any brane at all. This means that the non-BPS brane
only exists during the transition, and one truly starts and ends with a flux
vacuum without extra local objects.

For all practical purposes, one can think of the non-BPS
$\tilde{\mathrm{D}}(9-k)$-brane as a dimensional extension of a D$(8-k)$-brane
domain wall in spacetime, wrapping the same $(n-k)$-cycle $\Sigma_{n-k}$. Such
a brane is the usual domain wall that is responsible for interpolations
between $F_{k}$-flux vacua.

As an example, let us have a look at the Freund-Rubin solution
$\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ in type IIB supergravity. The solution
is parameterised by the integer $N$, measuring the quantised self-dual
$F_{5}$-flux. Additionally, there is the massless 10d dilaton. The AdS scale
$L_{\mathrm{AdS}}$ and the radius $L$ of the $\mathrm{S}^{5}$ are related and
both depend on the flux as $L_{\mathrm{AdS}}\sim N^{1/4}\ell_{s}$ and $L\sim
N^{1/4}\ell_{s}$. According to our proposal, the tachyon of a spacetime
filling non-BPS $\tilde{\mathrm{D}}4$-brane does the trick of controlling the
$F_{5}$-flux on the dual 5-cycle, i.e. the $\mathrm{S}^{5}$.

##  III Field-space metrics and distances

As we argued above, moving in tachyon space allows us to change the RR-fluxes
in the compactification. However, changing the flux alone does not guarantee
one to end up in a new flux vacuum. Indeed, the scalar fields in the
compactification need to adjust accordingly. For example, in
$\mathrm{AdS}_{5}\times\mathrm{S}^{5}$, the volume field of the
$\mathrm{S}^{5}$, or its radius $L\sim N^{1/4}\ell_{s}$, also shifts when the
flux is changed. The goal of this section is to identify what the scalar field
space, enlarged with the tachyons, looks like. We argue that the field space
takes a hyperbolic structure for unbounded fluxes. It is then natural to
contemplate distances between the different flux vacua. However, there is a
potential involved, hence the field space is not a moduli space. For the
latter, a well-defined notion of distance exists as the geodesic distance in
the moduli space: for a theory of scalar fields $\phi^{a}$, when moving along
a trajectory in the moduli space, parametrised by $s$, the dimensionless
field-space distance is

$\Delta=\int_{0}^{1}\mathrm{d}s\;\kappa_{d}\sqrt{g_{ab}\,\dfrac{\mathrm{d}\phi^{a}}{\mathrm{d}s}\dfrac{\mathrm{d}\phi^{b}}{\mathrm{d}s}},$
(III.1)

where $g_{ab}$ is the field-space metric and $\kappa_{d}$ is the
$d$-dimensional gravitational coupling. This definition has been extensively
used in the context of the Swampland Distance Conjecture. It is less clear
what a good definition of distance is in field spaces with potentials (see
ref. [21] where it was argued that the same formulas should apply; see also
refs. [15, 16] for alternatives).

We want to emphasise that we keep an open mind to what the precise definition
of field distances should be. Nevertheless, if we take the geodesic distance
in the field space, we find agreement with the Swampland Distance Conjecture,
due to the hyperbolic structure of the field-space geometry.

An expansion of the DBI-action of a non-BPS $\tilde{\mathrm{D}}(9-k)$-brane
shows that the field-space metric111In this note, we follow the standard
convention in which the field-space metric $g_{ab}$ of a set of scalar fields
$\phi^{a}$ can be read off from the kinetic action  $\displaystyle
S=-\int_{M_{1,d-1}}\mathrm{d}^{1,d-1}x\sqrt{-\tilde{g}_{1,d-1}}\;\dfrac{1}{2}\,g_{ab}(\phi)\,\tilde{g}^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}.$
for the tachyon $T$ reads

$\displaystyle
g_{TT}=\dfrac{2\pi\sqrt{2}}{(2\pi\ell_{s})^{d}}\,\dfrac{\mathcal{V}_{n-k}}{\sqrt{\mathcal{V}_{n}}}\,\mathrm{e}^{\Phi_{d}}\,V(T).$
(III.2)

Here, $\Phi_{d}$ is the $d$-dimensional dilaton and $\mathcal{V}_{n-k}$ and
$\mathcal{V}_{n}$ are the dimensionless string-frame volumes of the wrapped
cycle and of the total internal space, respectively. Below, we redefine the
tachyon as $\smash{X\equiv\gamma\int\mathrm{d}T\,\sqrt{V(T)}}$, where the
factor $\gamma$ absorbs all constant factors and the appropriate dimensionful
terms, except for a factor of $\sqrt{2}$ for convenience. If, for instance,
the internal volume is factorised such that two string-frame radions
$\sigma_{1}$ and $\sigma_{2}$ control such volumes as
$\smash{\mathcal{V}_{n-k}=\mathrm{e}^{(n-k)\sigma_{1}}}$ and
$\smash{\mathcal{V}_{n}=\mathrm{e}^{(n-k)\sigma_{1}+k\sigma_{2}}}$,222The
total volume can be controlled by several radions, but here a simple
factorised structure of the internal space has been considered for simplicity;
in the explicit examples, we will be precise. the tachyon metric then becomes

$\displaystyle
g_{XX}=2\,\mathrm{e}^{\frac{\sqrt{d-2}}{2}\,\tilde{\Phi}_{d}+\frac{\sqrt{n-k}}{2}\,\tilde{\sigma}_{1}-\frac{\sqrt{k}}{2}\,\tilde{\sigma}_{2}},$
(III.3)

where we also canonically normalised the closed-string scalars.333A review of
how to canonically normalise the dilaton and the radions is in appendix A. For
brevity, in the main text, we will drop the factors of $\kappa_{d}$. Now, the
tachyon vacua at $\smash{T=\pm\infty}$ correspond to vacua for $X$ at finite
values, which after fixing the integration constant in the definition of $X$,
can be taken at $\smash{X=0,X_{0}}$. Furthermore, we can imagine extending the
range of $X$ from this interval to the real line $\smash{\mathbb{R}}$, where
the points $\smash{X=NX_{0}}$, for $\smash{N=0,1,\dots}$, correspond to vacua
where the associated flux is $\smash{N}$.

In the case of a single unbounded flux, the field space clearly contains a
hyperbolic factor $\mathbb{H}^{2}$ since there is one open-string tachyon
which is of course coupled to a single linear combination of closed-string
fields (dilaton and radions). This is actually a more general structure. If
there are $m$ fluxes that are unbounded, and which are not related to each
other by tadpole conditions, then the field space contains a hyperbolic factor
$\smash{(\mathbb{H}^{2})^{m}}$: this is because the kink solutions for each of
the tachyons controlling one of the $m$ flux quanta would allow one to make a
jump in this flux without affecting the others. Therefore, the tachyon kinetic
couplings must be orthogonal. In other words, the field-space metric is always
of the form

$\mathrm{d}s_{\text{field}}^{2}=2\sum_{r=1}^{m}\mathrm{e}^{\sum_{a}\alpha_{ra}\varphi^{a}}\,(\mathrm{d}X_{r})^{2}+\sum_{a}(\mathrm{d}\varphi^{a})^{2},$
(III.4)

where $\smash{\varphi^{a}=(\tilde{\Phi}_{d},(\tilde{\sigma}_{i})_{i})}$
represents the canonically-normalised closed-string moduli, and is such that

$\sum_{a}\alpha_{ra}\alpha_{s}{}^{a}=\delta_{rs}.$ (III.5)

In the remainder of this section, we verify this behaviour for some examples.
We illustrate it here for the simplest scale-separated vacua of massive type
IIA string theory and for $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ vacua of type
IIB string theory. We also compute the scalar field distances in the large
flux regime. In app. B, we explain how these distances can be calculated, and
we show that they are always logarithmic in the flux quantum $N$.
Additionally, we performed an analogous analysis for the scale-separated vacua
in massless type IIA of ref. [22] and for
$\mathrm{AdS}_{3}\times\mathrm{S}^{3}\times\mathbb{T}^{4}$ vacua of type IIB
in app. C. In all of these examples, we only consider the scalars whose vacuum
expectation values shift under flux changes.

####  $\mathrm{AdS}_{4}\times\mathrm{CY}_{3}$ vacua in massive type IIA

Among our main examples of interest are the flux solutions on a toroidal
orientifold (or more generally on Calabi-Yau orientifolds) of massive type IIA
string theory found in refs. [4, 5, 6, 7]. There are three unbounded
$F_{4}$-fluxes on three different 4-cycles. Following the prescription
outlined above, we can wrap non-BPS $\tilde{\mathrm{D}}5$-branes on the
2-cycles dual to the four-cycles. Therefore, we have three different tachyons
$X_{i}$ in the problem, with $i$ corresponding to the $i$-th 2-cycle that the
brane is wrapping, whose string-frame radions are denoted as
$\tilde{\sigma}_{i}$. The field-space metric is

$\mathrm{d}s_{\text{field}}^{2}=2\sum_{i}\mathrm{e}^{\frac{\tilde{\Phi}_{4}+\sum_{j}\epsilon_{ij}\tilde{\sigma}_{j}}{\sqrt{2}}}\mathrm{d}X_{i}^{2}+\mathrm{d}\tilde{\Phi}_{4}^{2}+\sum_{i}\mathrm{d}\tilde{\sigma}_{i}^{2},$
(III.6)

where $\epsilon_{ij}=-1+2\delta_{ij}$, and by the $\mathrm{SO(4)}$-rotation

$\displaystyle P_{i}$
$\displaystyle\equiv\frac{1}{2}\,\bigl{(}\tilde{\Phi}_{4}+\sum_{j}\epsilon_{ij}\tilde{\sigma}^{j}\bigr{)},$
(III.7a) $\displaystyle Q$
$\displaystyle\equiv\frac{1}{2}\,\bigr{(}\tilde{\Phi}_{4}+\tilde{\sigma}_{1}+\tilde{\sigma}_{2}+\tilde{\sigma}_{3}\bigr{)}\,,$
(III.7b)

and after defining $P_{i}=-\sqrt{2}\,\mathrm{log}Z_{i}$, it becomes

$\mathrm{d}s_{\text{field}}^{2}=2\sum_{i}\frac{\mathrm{d}X_{i}^{2}+\mathrm{d}Z_{i}^{2}}{Z_{i}^{2}}+\mathrm{d}Q^{2}.$
(III.8)

As expected, the field space has the form
$(\mathbb{H}^{2})^{3}\times\mathbb{R}$. Therefore, the geodesic distance can
be calculated with the techniques illustrated in app. B. As
$\tilde{\sigma}_{i}\sim(\sqrt{2}/4)\,\log N$ and
$\tilde{\Phi}_{4}\sim-(3\sqrt{2}/2)\,\log N$, we find that $Z_{i}\sim N^{7/8}$
and $Q\sim(3/8)\,\log N$ and still $X_{i}\sim N$, such that the distance
becomes $\Delta=\sqrt{63/8}\,\log N$. Notice that the distance is not only
logarithmic in $N$, but also smaller than what was found in ref. [3].

This logarithmic dependence ensures that the Distance Conjecture is satisfied.
Indeed, the Distance Conjecture applied to the Kaluza-Klein (KK) tower, taking
the form
$\smash{m_{\mathrm{KK}}(N)/m_{\mathrm{KK},0}=\mathrm{e}^{-\alpha\Delta}}$,
agrees with the power-law scaling of $\smash{m_{\mathrm{KK}}(N)\sim
N^{1/4}\ell_{s}}$. Here $m_{\mathrm{KK},0}$ is the KK mass one starts from.
Furthermore, we find that $\smash{\alpha=\sqrt{1/126}<1/\sqrt{2}}$, violating
the sharpened Distance Conjecture of ref. [23] proposed for exact moduli
spaces.

####  $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ in type IIB

In the $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ solutions of type IIB strings,
it will be a non-BPS $\tilde{\mathrm{D}}4$-brane that has the right couplings
for changing the $F_{5}$ flux. This $\tilde{\mathrm{D}}4$-brane fills
spacetime and has no cycle to wrap in the internal space. The relevant field
space is made up of the 5d volume, the dilaton and the tachyon and it reads

$\mathrm{d}s_{\text{field}}^{2}=2\,\mathrm{e}^{\frac{\sqrt{3}}{2}\tilde{\Phi}_{5}-\frac{\sqrt{5}}{2}\tilde{\sigma}}\mathrm{d}X^{2}+\mathrm{d}\tilde{\Phi}_{5}^{2}+\mathrm{d}\tilde{\sigma}^{2}.$
(III.9)

After the $\mathrm{SO}(2)$-rotation

$\left(\\!\\!\begin{array}[]{c}P\\\
Q\end{array}\\!\\!\right)\equiv\dfrac{1}{2\sqrt{2}}\left(\\!\begin{array}[]{cc}\sqrt{3}&\,-\sqrt{5}\\\
\sqrt{5}&\,\sqrt{3}\end{array}\\!\right)\left(\\!\\!\begin{array}[]{c}\tilde{\Phi}_{5}\\\
\tilde{\sigma}\end{array}\\!\\!\right),$ (III.10)

and redefining $P=-\sqrt{2}\,\mathrm{log}\,Z$, we find

$\mathrm{d}s_{\text{field}}^{2}=2\,\dfrac{\mathrm{d}X^{2}+\mathrm{d}Z^{2}}{Z^{2}}+\mathrm{d}Q^{2}\,.$
(III.11)

Again, the field space is the product of the hyperbolic plane and the real
line, i.e. $\mathbb{H}^{2}\times\mathbb{R}$. From the proper scalings,
$\tilde{\sigma}\sim(\sqrt{5}/4)\,\log N$ and
$\tilde{\Phi}_{5}\sim-(5/4\sqrt{3})\,\log N$ one finds that $Z\sim N^{5/8}$
and $Q\sim(1/4)\,\log N$ and $X\sim N$, from which it follows that the
distance is $\Delta=\sqrt{23/6}\,\log N$. Here the logarithmic behaviour also
ensures that the Distance Conjecture is satisfied. Using that
$\smash{m_{\mathrm{KK}}\sim N^{1/4}\ell_{s}}$, we find that
$\smash{\alpha=\sqrt{3/184}}$ also violating the sharpened Distance Conjecture
of ref. [23].

##  IV Discussion

We have argued on general grounds, and shown with concrete examples, that
vacua that differ in flux quanta can be seen as different critical points of a
single-valued scalar potential, extending ideas of ref. [3]. This allows us to
define, in a natural way, the distance between flux vacua as a distance
travelled in scalar field space. This can be done by integrating in open
string fields. In this paper, we took them to be tachyon fields of unstable
$\tilde{\mathrm{D}}p$-branes, but this choice is not unique. In our previous
work [3], we took those open-string fields to be the positions of
$\mathrm{D}p$-branes wrapping cycles trivial in homology. A benefit of using
tachyon fields of unstable $\tilde{\mathrm{D}}p$-branes is that one can argue
more model-independently how flux vacua arise as critical points in a
universal single-valued scalar potential.

We suggest that requiring interpolations between vacua to be continuous
postdicts the existence of these non-BPS branes, in the spirit of the
Swampland Cobordism Conjecture [1]. However, string theory has more to offer
than vacua where the moduli are stabilised with RR-fluxes alone. There exist
scenarios where unbounded NSNS-fluxes do stabilise the moduli, such as e.g. in
the non-SUSY vacua of ref. [24], and M-theory has vacuum solutions with
$G_{7}$\- and $G_{4}$-fluxes. If a continuous version of the Cobordism
Conjecture is to be taken seriously, then our proposal implies the existence
of extended _non-BPS_ objects with NS5-charge and non-BPS objects with
$\mathrm{M}2$\- or $\mathrm{M}5$-charges in M-theory. The idea is then again
that BPS thin domain walls are described as thick domain walls that lift to
space-filling non-BPS branes.

For the examples that we have studied, this formalism confirms that the field
space connecting asymptotic vacua in the large-flux regime is hyperbolic and
the distances are logarithmic in the flux quantum $N$. Since the KK-masses
scale with a negative power of $N$ this agrees with the ordinary Swampland
Distance Conjecture [10], stating that the mass-scale of the (KK-)tower is
exponential in the distance, and not just the AdS Distance Conjecture [11].
Note that we do not touch upon the _strong_ AdS Distance Conjecture of ref.
[11], which rules out supersymmetric scale-separated vacua.

One could argue that there is not as much evidence for a Distance Conjecture
applied to scalar fields that are not pure moduli but are lifted by a scalar
potential instead. In such cases, we tend to use the Distance Conjecture only
when the potential is exponentially suppressed in asymptotic regimes. Below we
argue why, despite the naive high mass scale of the tachyon fields, there is a
notion in which they behave as pure moduli in the large-$N$ limit, which could
be the explanation of why the ordinary Distance Conjecture is obeyed.

The scalar potential of the tachyon in the $d$-dimensional Einstein frame
scales as

$V_{\text{tachyon}}\sim\dfrac{2\pi\sqrt{2}}{(2\pi\ell_{s})^{d}}\frac{\mathcal{V}_{n-k}}{\sqrt{\mathcal{V}_{n}}}\,\mathrm{e}^{\frac{d+2}{2\sqrt{d-2}}\tilde{\Phi}_{d}}\,V(T).$
(IV.1)

Therefore, because $\smash{\partial^{2}V_{\text{tachyon}}/\partial T^{2}\sim
V_{\text{tachyon}}/(2\ell_{s}^{2})}$, we can estimate the scaling of the
tachyon mass in $d$-dimensional Planck units as

$m_{\text{tachyon}}^{2}\sim\dfrac{1}{2}\,g_{TT}^{-1}\dfrac{\partial^{2}}{\partial
T^{2}}V_{\text{tachyon}}\sim\ell_{s}^{-2}\,\mathrm{e}^{\frac{2\,\tilde{\Phi}_{d}}{\sqrt{d-2}}},$
(IV.2)

as we should have expected since the mass scale of the tachyon is the string
scale. On the other hand, the KK-mass in $d$-dimensional Planck units scales
as

$m_{\mathrm{KK}}^{2}\sim{L_{\mathrm{KK}}^{-2}}\,\mathrm{e}^{\frac{2\,\tilde{\Phi}_{d}}{\sqrt{d-2}}},$
(IV.3)

where $L_{\mathrm{KK}}$ is the radius in string units of the largest cycle in
the compact manifold. Since we require large volumes, we see that the
effective tachyon mass is higher than the KK-scale. One might worry that this
should invalidate the whole effective field theory, as the KK-scale provides
an upper bound for the cut-off scale of the theory. However, we will argue
that this is not necessarily true. Indeed, we should also compare the tachyon
energy with the energy coming from the flux potential. How this scales with
the flux parameter $N$ depends on the compactification. For instance, for the
simple scale-separated flux vacua in type IIA string theory [4, 5, 6, 7], the
tachyon potential is parametrically smaller than the flux potential, being

$\frac{V_{\text{tachyon}}}{V_{\mathrm{AdS}_{4}\times\mathrm{CY}_{3}}}\sim
N^{-1/4}.$ (IV.4)

For the $\mathrm{AdS}_{5}\times\mathrm{S}^{5}$ vacua of type IIB strings, we
have a similar pattern, being

$\frac{V_{\text{tachyon}}}{V_{\mathrm{AdS}_{5}\times\mathrm{S}^{5}}}\sim
N^{-3/4}.$ (IV.5)

This means that, in the tachyon direction $X$, both examples have tachyon
vacua with three very specific features:  (i) they are equidistant, when we
are considering just the $X$-direction while keeping the $Z$-field constant;
(ii) they are separated by hills in the potential that become smaller in size;
(iii) they become sharper with respect to the KK-scale, as $N$ becomes
larger.444This specific wiggly behaviour does not happen for all examples, as
shown in app. C. This is illustrated qualitatively in fig. 1.

$X$$V_{\text{tachyon}}$$N\to\infty$ Figure 1: The figure shows the main
qualitative features of the potentials that we consider in this paper. The
different local minima correspond to AdS vacua corresponding to different flux
units $N$, while the wiggles represent the contribution to the total potential
coming from the potential along the tachyonic direction $X$. As $N$ grows, the
tachyon potential that lets the theory hop between vacua with different flux
units has three main features: the local minima are equidistant, the convexity
of the minima increases (with respect to the KK-scale), and the height of the
wiggles decreases.

In this particular sense the tachyon field behaves as a modulus at large $N$:
the wiggles (i.e. energy differences between the vacua) become arbitrarily
small in AdS units. Hence, the tachyon field can be given a kinetic energy
well below the AdS-scale so that it will fly through the potential as if the
wiggles were absent. This is not different from our usual notion of asymptotic
field variations in compactifications and the usual Distance Conjecture should
apply. Yet, one could object that the second derivative of the canonically
normalised tachyon field remains of order of the string scale in every vacuum.
This is small in Planck units since the string scale is suppressed by inverse
powers of $N$ with respect to the Planck scale. Yet, this is large in KK
units, which is our EFT cut-off. Since the tachyon is not part of the
$d$-dimensional supergravity multiplet we consider it possible that its mass
renormalisation is strong enough for it to obtain a mass below the cut-off,
but this is merely a suggestion at this point.

Finally, our arguments should apply beyond vacua with unbounded fluxes. One
could for instance consider the type IIB Landscape of 4d vacua from
compactifications on conformal Calabi-Yau three-folds. The complex structure
moduli are stabilised by three-form fluxes which are bounded by tadpole
constraints [25, 26]. One could interpolate between such flux vacua by using
non-BPS branes, or by space-filling 5-branes as in KPV [27]. It would be
interesting to investigate how scalar field distances in this context relate
to obtaining a small on-shell flux superpotential $W_{0}$ for KKLT
constructions as in ref. [28], but we leave this to further research.

###### Acknowledgements.

### Acknowledgments

GS is supported in part by the DOE grant DE-SC0017647. FT and TVR are
supported by the FWO Odysseus grant GCD-D0989-G0F9516N. VVH is supported by
the Olle Engkvists Stiftelse.

##  Appendix A Canonical normalization of closed-string scalars

Let $\ell_{s}=\sqrt{\alpha^{\prime}}$ be the string length. We can define the
$d$-dimensional Einstein-frame metric $\smash{\tilde{g}_{\mu\nu}}$, which
determines a line element expressed as
$\smash{d\tilde{s}_{1,d-1}^{2}=\tilde{g}_{\mu\nu}\,\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}}$,
by parameterising the string-frame metric, defined through the line element
$ds_{1,9}^{2}=G_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}$, as

$\begin{split}ds_{1,9}^{2}=\mathrm{e}^{\frac{4\Phi_{d}}{d-2}}\,\chi\,\tilde{g}_{\mu\nu}\,\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\mathrm{e}^{2\sigma}\,\breve{g}_{mn}\,\mathrm{d}y^{m}\mathrm{d}y^{n},\end{split}$
(A.1)

where $\Phi_{d}$ is the $d$-dimensional dilaton and $\sigma$ is the string-
frame radion. Here, the internal metrics are assumed to be normalised in such
a way that
$\smash{\int_{\mathrm{K}_{n}}\mathrm{d}^{n}y\,\sqrt{\breve{g}_{n}}=(2\pi\ell_{s})^{n}}$,
with $n=10-d$. An arbitrary constant $\chi$ has also been inserted: fixing it
as $\chi=\mathrm{e}^{-4\langle\Phi_{d}\rangle/(d-2)}$ makes the
$d$-dimensional metric components correspond in both frames, in the vacuum.
Another possible choice is

$ds_{1,9}^{2}=\mathrm{e}^{\frac{\phi}{2}}\,\biggl{[}\dfrac{\chi\,\tilde{g}_{\mu\nu}\,\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}}{\mathrm{e}^{2\omega\,\frac{10-d}{d-2}}}+\mathrm{e}^{2\omega}\,\breve{g}_{mn}\,\mathrm{d}y^{m}\mathrm{d}y^{n}\biggr{]},$
(A.2)

which involves the shifted 10-dimensional dilaton
$\phi=\Phi-\mathrm{ln}\,g_{s}$, where $g_{s}=\mathrm{e}^{\langle\Phi\rangle}$
is the string-coupling vacuum expectation value, and the Einstein-frame radion
$\omega$. The $(\Phi_{d},\sigma)$\- and $(\phi,\omega)$-bases can be related
via the linear transformations
$\smash{\Phi_{d}=[(d-2)/8]\,\phi-[(10-d)/2]\,\omega}$ and
$\smash{\sigma=\phi/4+\omega}$. For an isotropic compactification, both bases
are diagonal and the canonically normalised fields e.g. in the
$(\Phi_{d},\sigma)$-basis read

$\displaystyle\tilde{\Phi}_{d}$
$\displaystyle=\dfrac{2}{\sqrt{d-2}}\,\dfrac{\Phi_{d}}{\kappa_{d}},$
$\displaystyle\tilde{\sigma}$
$\displaystyle=\sqrt{10-d}\,\dfrac{\sigma}{\kappa_{d}}.$

Here, given the string-frame gravitational coupling
$\smash{2\kappa_{10}^{2}=(2\pi\ell_{s})^{8}/2\pi}$ (which is not proper, due
to the extra dilaton factor), the reduced $d$-dimensional gravitational
coupling is

$2\kappa_{d}^{2}=\dfrac{1}{2\pi}\dfrac{g_{s}^{2}\,\mathrm{e}^{2\langle\Phi_{d}\rangle}}{(2\pi\ell_{s})^{2-d}}=\dfrac{g_{s}^{2}\,(2\pi\ell_{s})^{d-2}}{2\pi\,\mathrm{e}^{(10-d)\langle\omega\rangle}}.$
(A.3)

This is related to the $d$-dimensional Planck mass $\smash{m_{\mathrm{P},d}}$
as $\smash{m_{\mathrm{P},d}=\kappa_{d}^{2/{d-2}}}$. Although the
10-dimensional dilaton and the Einstein-frame radions are more intuitive
quantities to work with, in an anisotropic compactification there are multiple
radions and they are kinetically-mixed in the $d$-dimensional Einstein frame;
the $d$-dimensional dilaton and the string-frame radions are instead always
diagonal. In sec. III, for instance, we consider internal metrics decomposed
as
$\smash{ds_{n}^{2}=\mathrm{e}^{2\sigma_{1}}d\breve{s}_{n-k}^{2}+\mathrm{e}^{2\sigma_{2}}d\breve{s}_{k}^{2}}$.
For the details of anisotropic compactifications, see e.g. ref. [29, app. B].

##  Appendix B Computing distances in hyperbolic space

Suppose we have a hyperbolic field space of the form

$\mathrm{d}s_{\text{field}}^{2}=\frac{1}{Z^{2}}\bigl{(}\mathrm{d}X^{2}+\mathrm{d}Z^{2}\bigr{)}.$
(B.1)

Let us suppose that we want to compute a distance in this field space along a
path $(X(s),Z(s))$ where $X(0)\sim 0$, $X(1)\sim N^{f}$ and $Z(0)\sim 1$,
$Z(1)\sim N^{g}$. Here, $N\gg 1$ is a large positive number, while $f$ and $g$
are constant powers one can specify for any given model of interest. Of
course, $N$ has to be interpreted as a large flux at the end of the trajectory
at $s=1$, whereas the beginning at $s=0$ would correspond to a low flux, say
$N=1$. The geodesics are arcs of semicircles, and the solution can be
parameterised as

$X(s)=l\,\tanh(d_{1}s+d_{2})+X_{c},\quad Z(s)=l\,\mathrm{sech}(d_{1}s+d_{2}).$
(B.2)

In particular, the geodesics are such that

$[X_{c}-X(s)]^{2}+[Z(s)]^{2}=l^{2}$ (B.3)

and the geodesic distance is simply

$\Delta=\int_{0}^{1}\mathrm{d}s\,\sqrt{\dfrac{1}{Z^{2}}\biggl{[}\biggl{(}\dfrac{\mathrm{d}X}{\mathrm{d}s}\biggr{)}^{2}+\biggl{(}\dfrac{\mathrm{d}Z}{\mathrm{d}s}\biggr{)}^{2}\biggr{]}}=|d_{1}|.$
(B.4)

A picture of the geodesic path is in fig. 2.

$X$$Z$$l$$s=0$$s=1$ Figure 2: A sketch of the geodesic path in hyperbolic
plane.

All the integration constants $l$, $d_{1}$, $d_{2}$ and $X_{c}$ are fixed by
the boundary conditions, as we explain below. To do this, let us assume that
one of the endpoints of the semicircle is (close to) the origin, and therefore
$X_{c}\simeq l$, which is justified in the large $N$-limit, since we also have
$l\gg 1$ as we show below. We can estimate the radius of the semicircle by
noticing that $\smash{[l-X(1)]^{2}+[Z(1)]^{2}=l^{2}}$ is solved for

$l=\frac{(X(1))^{2}+(Z(1))^{2}}{2X(1)}$ (B.5)

Let us look now at the case $f>g$, for which $X$ becomes much larger than $Z$.
It means that the trajectory goes over the top of the semicircle. In this
limit, the radius $l$ behaves as $X(1)$, and thus $l\sim N^{f}$. Let us now
look at the equation for $Z$, evaluated at $s=0$. We know that this should not
scale with $N$ at leading order. We find

$Z(0)=l\,\text{sech}\,d_{2}\sim N^{f}\>\text{sech}\,d_{2},$ (B.6)

We see that $d_{2}$ must scale as $d_{2}=\pm f\log N$ in the large-$N$ limit.
Without loss of generality, we take the minus sign. We now evaluate $Z$ at
$s=1$, and from the requirement that $Z(1)\sim N^{g}$, we deduce that there
are two options, either

$d_{1}=(2f-g)\log N\quad\text{or}\quad d_{1}=g\log N.$ (B.7)

It turns out that if $Z$ is to increase and decrease over the interval
$s\in[0,1]$, one needs to choose the first option in eq. (B.7). This is
because the $\mathrm{sech}$-function requires an argument, here
$d_{1}s+d_{2}$, that is first negative and then positive with increasing $s$
for that behaviour. Hence the geodesic distance in the large $N$-limit is

$\Delta=(2f-g)\log N.$ (B.8)

Suppose now that the opposite is true: $g>f$. This means that the
$Z$-coordinate becomes much larger than $X$ in the large-$N$ limit. It means
the endpoint is not yet reaching the top of the semicircle. The same formula
for the radius $l$ applies, and we now see from this that $l=N^{2g-f}$. From
$Z(0)\sim 1$, it then follows that

$d_{2}\sim\pm(2g-f)\log N$ (B.9)

and again we take the minus sign without loss of generality. Consequentially,
from $Z(1)\sim N^{g}$ we get that either

$d_{1}=(3g-2f)\log N\quad\text{or}\quad d_{1}=g\log N.$ (B.10)

This time, $Z$ is to increase but not decrease over the interval $s\in[0,1]$,
for which the first option in eq. (B.10) needs to be taken, as the argument of
the $\mathrm{sech}$-function needs to remain negative. Hence the distance is
now

$\Delta=g\log N.$ (B.11)

Suppose now that the field space is a direct product of hyperbolic planes and
the real line, with metric

$\mathrm{d}s_{\text{field}}^{2}=\sum_{i}\frac{a_{i}}{Z_{i}^{2}}\left(\mathrm{d}X_{i}^{2}+\mathrm{d}Z_{i}^{2}\right)+b\,\mathrm{d}Q^{2}$
(B.12)

In that case, the solution to the geodesic equation is for all the $X_{i}$’s
and $Z_{i}$’s the same as eq. (B.2), with associated integration constants
$d_{1,i}$, $d_{2,i}$, $l_{i}$ and $X_{0,i}$. The integration constants can be
found analogously to the above. Furthermore, $Q$ is solved as
$Q(s)=d_{3}s+d_{4}$. Finally, the distance becomes

$\Delta^{2}=\sum_{i}a_{i}\,d_{1,i}^{2}+b\,d_{3}^{2}\,.$ (B.13)

##  Appendix C Field spaces and distances for scale-separated vacua of
massless type IIA strings and
$\boldsymbol{\mathrm{AdS}_{3}\times\mathrm{S}^{3}\times\mathbb{T}^{4}}$ of
type IIB strings

In this appendix, we discuss the field space and compute the distances for the
scale-separated vacua of massless type IIA string theory and the
$\mathrm{AdS}_{3}\times\mathrm{S}^{3}\times\mathbb{T}^{4}$ vacua of type IIB
string theory.

#### Scale-separated $\mathrm{AdS}_{4}$ vacua in massless type IIA

The scale-separated solutions of ref. [22] are set in massless type IIA string
theory, compactified on a SU(3)-structure nilmanifold (the Iwasawa manifold).
For generalisations, see ref. [30]. In the simplest example of ref. [22],
there are three unbounded fluxes: two $F_{2}$-fluxes and the $F_{6}$-flux.
Following our prescription, we can use two non-BPS
$\tilde{\mathrm{D}}7$-branes wrapping the two 4-cycles dual to the 2-cycles
that the $F_{2}$-fluxes are winding to change them. Similarly, we can use a
spacetime-filling non-BPS $\tilde{\mathrm{D}}3$-brane to change the
$F_{6}$-flux. The field space metric is

$\displaystyle\begin{split}\mathrm{d}s_{\text{field}}^{2}=2\,\mathrm{e}^{\frac{\tilde{\Phi}_{4}-\tilde{\sigma}_{1}-\tilde{\sigma}_{2}-\tilde{\sigma}_{3}}{\sqrt{2}}}\mathrm{d}X_{1}^{2}+2\,\mathrm{e}^{\frac{\tilde{\Phi}_{4}+\tilde{\sigma}_{1}+\tilde{\sigma}_{2}-\tilde{\sigma}_{3}}{\sqrt{2}}}\mathrm{d}X_{2}^{2}\\\
+2\,\mathrm{e}^{\frac{\tilde{\Phi}_{4}+\tilde{\sigma}_{1}-\tilde{\sigma}_{2}+\tilde{\sigma}_{3}}{\sqrt{2}}}\mathrm{d}X_{3}^{2}+\mathrm{d}\tilde{\Phi}_{4}^{2}+\sum_{i}\mathrm{d}\tilde{\sigma}_{i}^{2}&,\end{split}$
(C.1)

where $X_{1}$ represents the D7 tachyon and $X_{2}$ and $X_{3}$ the D3
tachyons. After an $\mathrm{SO}(4)$-rotation of the fields, i.e.

$\displaystyle P_{1}$
$\displaystyle\equiv-\dfrac{1}{2}\bigl{(}\tilde{\Phi}_{4}-\tilde{\sigma}_{1}-\tilde{\sigma}_{2}-\tilde{\sigma}_{3}\bigr{)},$
(C.2a) $\displaystyle P_{2}$
$\displaystyle\equiv-\dfrac{1}{2}\bigl{(}\tilde{\Phi}_{4}+\tilde{\sigma}_{1}+\tilde{\sigma}_{2}-\tilde{\sigma}_{3}\bigr{)},$
(C.2b) $\displaystyle P_{3}$
$\displaystyle\equiv-\dfrac{1}{2}\bigl{(}\tilde{\Phi}_{4}+\tilde{\sigma}_{1}-\tilde{\sigma}_{2}+\tilde{\sigma}_{3}\bigr{)},$
(C.2c) $\displaystyle Q$
$\displaystyle\equiv-\dfrac{1}{2}\bigl{(}\tilde{\Phi}_{4}-\tilde{\sigma}_{1}+\tilde{\sigma}_{2}+\tilde{\sigma}_{3}\bigr{)},$
(C.2d)

and defining $P_{i}=-\sqrt{2}\,\mathrm{log}Z_{i}$, the field space metric
becomes

$\mathrm{d}s_{\text{field}}^{2}=2\sum_{i}\frac{\mathrm{d}X_{i}^{2}+\mathrm{d}Z_{i}^{2}}{Z_{i}^{2}}+\mathrm{d}Q^{2},$
(C.3)

which means that the field space is again of the form
$(\mathbb{H}^{2})^{3}\times\mathbb{R}$. According to ref. [22], the scalars
scale as $\tilde{\Phi}_{4}\sim-[\sqrt{2}(a+b+c)/4]\,\log N$,
$\tilde{\sigma}_{1}\sim[\sqrt{2}(a-b-c)/4]\,\log N$,
$\tilde{\sigma}_{2}\sim[\sqrt{2}(a-b+c)/4]\,\log N$ and
$\tilde{\sigma}_{3}\sim[\sqrt{2}(a+b-c)/4]\,\log N$. Consequentially,
$Z_{1}\sim N^{(5a+b+c)/8}$, $Z_{2}\sim N^{(a+5b+c)/8}$, $Z_{3}\sim
N^{(a+b+5c)/8}$ and $Q\sim[(a+b+c)/8]\,\log N$. The tachyons, playing the role
of the fluxes, have to scale as $X_{1}\sim N^{a}$, $X_{2}\sim N^{b}$ and
$X_{3}\sim N^{c}$. The expression for the distance now depends on the relative
size of $a$, $b$ and $c$. They all need to be positive. Without loss of
generality, we take $c\geq b$ and it follows from growing volumes that
$a>c\geq b$ and also $a>2b$. From that, it can be seen that $X_{1}$ always
scales higher than $Z_{1}$, whereas $X_{2}$ always scales lower than $Z_{2}$.
If $a+b>3c$ , then $X_{3}$ scales lower than $Z_{3}$ too, and the distance
becomes

$\Delta_{1}=\frac{\sqrt{31a^{2}-2a(b+c)+7b^{2}+6bc+7c^{2}}}{2\sqrt{2}}\,\log
N.$ (C.4)

In the other case, when $a+b<3c$, we find

$\Delta_{2}=\frac{\sqrt{31a^{2}-2a(b+5c)+7b^{2}-2bc+31c^{2}}}{2\sqrt{2}}\,\log
N.$ (C.5)

These expressions are compatible with each other when $a+b=3c$.

Looking at the energy scales, whether the tachyon potentials are lower than
the flux potential depends on the relative size of the fluxes. Indeed, we have

$\frac{V_{X_{i}}}{V_{\text{AdS}_{4}\times\text{Iw}_{6}}}\sim
N^{(a+b+c)/4-a\delta_{i1}-b\delta_{i2}-c\delta_{i3}}$ (C.6)

With constraints on the scaling exponents $a$, $b$ and $c$ discussed above, we
notice that the potential for the D3-tachyon $X_{1}$ is always lower than the
flux potential, whereas the D7-tachyon $X_{2}$ always scales higher. For the
D7-tachyon $X_{3}$ it depends: if $a+b>3c$, its potential scales lower than
the flux potential and higher otherwise. This is reminiscent of the distance
computations above, where we needed to investigate when the $X_{i}$ scale
higher or lower than the $Z_{i}$.

#### $\mathrm{AdS}_{3}\times\mathrm{S}^{3}\times\mathbb{T}^{4}$

We can also take a look at type IIB string theory compactified to
AdS${}_{3}\times\mathrm{S}^{3}\times\mathbb{T}^{4}$ with an $F_{3}$-flux along
the $\mathrm{S}^{3}$ and an $F_{7}$-flux along the whole compact space. Here
the relevant scalar fields are the 3d dilaton $\tilde{\Phi}_{3}$, and the
volume moduli of the $\mathrm{S}^{3}$ and the $\mathbb{T}^{4}$, which we call
$\tilde{\sigma}$ and $\tilde{\tau}$ respectively. Our proposal suggests we
need a non-BPS $\tilde{\mathrm{D}}2$-brane and D6-brane inducing changes in
the $F_{7}$-flux and $F_{3}$-flux respectively. We call their worldvolume
tachyons $X_{1}$ and $X_{2}$ respectively. The relevant field space metric is

$\begin{split}\mathrm{d}s_{\text{field}}^{2}=2\,\mathrm{e}^{\frac{\tilde{\Phi}_{3}}{2}-\frac{\sqrt{3}}{2}\tilde{\sigma}-\tilde{\tau}}\mathrm{d}X_{1}^{2}+2\,\mathrm{e}^{\frac{\tilde{\Phi}_{3}}{2}-\frac{\sqrt{3}}{2}\tilde{\sigma}+\tilde{\tau}}\mathrm{d}X_{2}^{2}\\\
+\mathrm{d}\tilde{\Phi}_{3}^{2}+\mathrm{d}\tilde{\sigma}_{2}+\mathrm{d}\tilde{\tau}^{2}&.\end{split}$
(C.7)

After the field rotation

$\displaystyle P_{i}$
$\displaystyle\equiv-\dfrac{1}{2}\biggl{(}\dfrac{\tilde{\Phi}_{3}}{2}-\dfrac{\sqrt{3}}{2}\tilde{\sigma}+(-1)^{i}\,\tilde{\tau}\biggr{)},$
(C.8a) $\displaystyle Q$
$\displaystyle\equiv-\dfrac{1}{2}\biggl{(}\tilde{\Phi}_{3}+\dfrac{\tilde{\sigma}}{\sqrt{3}}\biggr{)},$
(C.8b)

and defining $P_{i}=\log(Z_{i})$, we find

$\mathrm{d}s_{\text{field}}^{2}=2\sum_{i}\frac{\mathrm{d}X_{i}^{2}+\mathrm{d}Z_{i}^{2}}{Z_{i}^{2}}+\mathrm{d}Q^{2},$
(C.9)

i.e. a field space of the form $(\mathbb{H}^{2})^{2}\times\mathbb{R}$. Then
again, one should find how the solutions scale. One can infer that if the
$F_{7}$-flux is given by $N^{a}$ and the $F_{3}$-flux by $N^{b}$, then
$\tilde{\Phi}_{3}\sim-[3(a+b)/4]\,\log N$,
$\tilde{\sigma}\sim[\sqrt{3}(a+b)/4]\,\log N$ and
$\tilde{\tau}\sim[(a-b)/2]\,\log N$. With that, the new fields scale as
$Z_{1}\sim N^{(5a+b)/8}$, $Z_{2}\sim N^{(a+5b)/8}$ and $Q\sim[(a+b)/4]\,\log
N$. In order to have large volume for the four-torus, we need $a>b$. This
guarantees that $X_{1}\sim N^{a}$ scales higher than $Z_{1}$ in the large
$N$-limit. The same happens for $Z_{2}$ and $X_{2}\sim N^{b}$ when $3b>a>b$,
otherwise $X_{2}$ scales lower than $Z_{2}$. When $3b>a$, the distance is

$\Delta_{1}=\sqrt{\frac{112a^{2}-13ab+32b^{2}}{8}}\log N\,.$ (C.10)

In the other case, when $a>3b$, we have

$\Delta_{1}=\sqrt{\frac{112a^{2}-5ab+8b^{2}}{8}}\log N\,.$ (C.11)

Comparing the tachyon energy densities with the flux potential, we notice that

$\frac{V_{X_{1}}}{V_{\mathrm{AdS}_{\times}\mathrm{S}^{3}\times\mathbb{T}^{4}}}\sim
N^{(b-3a)/4},\quad\frac{V_{X_{2}}}{V_{\mathrm{AdS}_{\times}\mathrm{S}^{3}\times\mathbb{T}^{4}}}\sim
N^{(a-3b)/4}.$ (C.12)

Here the potential energy of the D2-tachyon $X_{1}$ is parametrically smaller
than the flux provided by the fluxes (as $a>b$ for the large volume limit).
Whether the energy of the D6-tachyon $X_{2}$ is lower than the flux potential,
depends again on whether $a<3b$ or not, similar to how whether $X_{2}$ scales
higher than $Z_{2}$ and how it influences the distance computation.

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Few common failure cases in mobile robots

Ramviyas Parasuraman 111The author was affiliated with European Organization
for Nuclear Research (CERN), Switzerland during this work. Contact:
<EMAIL_ADDRESS>CVAP, Royal Institute of Technology (KTH), Stockholm, Sweden.

Abstract
A mobile robot deployed for remote inspection, surveying or rescue missions
can fail due to various possibilities and can be hardware or software related.
These failure scenarios necessitate manual recovery (self-rescue) of the robot
from the environment. It would bring unforeseen challenges to recover the
mobile robot if the environment where it was deployed had hazardous or harmful
conditions (e.g. ionizing radiations). While it is not fully possible to
predict all the failures in the robot, failures can be reduced by employing
certain design/usage considerations. Few example failure cases based on real
experiences are presented in this short article along with generic suggestions
on overcoming the illustrated failure situations.

##  1 Introduction

Mobile robots are increasingly considered in applications in hostile or
hazardous environments where humans cannot perform some tasks due to safety
issues such as high radiation levels or because of challenges in the
environments [1, 2]. Furthermore, there exist a special category called rescue
robots, which are meant to help the rescuers during disaster situation and
conditions where human may not be fully able to perform the tasks due to
harmful or hazardous conditions. Robots failing in such environments brings
unfortunate situations such as the robot getting abandoned in the site [1].

In this article, some possible failure scenarios based on the experiences
learned with the robots at CERN as well as from other researchers meet-up
during the IEEE SSRR summer school 2012 are presented. The experiences are
based on (at least) the following robots: TIM [3] \- a robotic train used for
remote inspection and radiation survey for the Large Hadron Collider (LHC),
Telemax - an Explosive Ordinance Disposal (EOD) robot used for remote
inspection at CERN, and a KUKA youBot mobile platform used for research in
robotic and manipulation applications.

Limited suggestions for design considerations aiming to reduce the chances of
failures are also discussed from a simplistic perspective. For a detailed
analysis on failure scenarios (how and when a mobile robot could fail), refer
[4]. Additionally, observing rescue robot design standards (in progress) from
NIST and ASTM [5] could also be beneficial in avoiding the failure scenarios.

##  2 Hardware and software failures

It should be considered that mechanical or electronic component failures may
occur at any time during the usage of a mobile robot. We explain in this
section several cases of failure or unfortunate scenarios and explain a
possible pointers to avoid these problems.

  1. 1.

Chronic: When a robotic device is left switched on and was not used for a long
time, they might have a tendency to become mobile due to friction loss or
inertia effects. (E.g. A forklift could lose its pressure on the fork carrying
a heavy load for quite a while). Therefore it should be made sure that the
robot is immobile (by using a normally closed mechanical brake or the motion
power) when the robot is static.

  2. 2.

Calibration: In the TIM robot [3], we observed that the robot consumed twice
as much power when moving forward relative to backward movement. This was
later discovered to be caused by a motor failure. The lesson here is that,
before using a robot, it should be calibrated or verified for proper
functioning of movements in all directions. This also applies to other
important functional elements for e.g. sensing.

  3. 3.

Thermal: There are high chances of minor failures in robots due to thermal
influences. For instance, once when we were about to give a specific
demonstration with the youBot to a large audience, its onboard computer got
shutdown due to overheating in a warm conference room which was unexpected.
This bad experience could have been avoided if we had designed for proper heat
dissipation/ventilation in the robot.

  4. 4.

Over-current: Even though short circuit problems are usually negligible in
most robots as they employ professional electronics components, over current
and circuit related problem may occur anytime. For instance, a USB powered
WiFi card consumed too much of power from the USB port (exceeded its limit)
resulting in damage to the the computer motherboard and eventual failure of
the robot operation. Thus ensuring proper power availability (by using current
limiter or a relay) to every devices and sensors connected to the robot could
help avoiding such issues.

  5. 5.

Memory: In one of the experiences, a sudden crash in the robot’s computer hard
disk resulted in instant robot failure and prevented its mission of autonomous
localization and mapping of the environment. This brings up the need to have
(and execute) the robot programs in redundant efficient manner, use solid
state drives and also include a recovery routine in the RAM memory to switch a
defective hard disk containing the main routine or atleast to shut-down the
robot safely.

  6. 6.

Frequency: Over-clocking of a robots computer could result in serious damages
to the robot and its performance as well. It could also trigger a spark or
fire (there was a reported case that a high frequency switch got fired but
then the system was immediately shut-out to avoid further damage) and
therefore the robot systems and functions should carefully be designed not to
be overcooked or over-clocked.

  7. 7.

Faults: If the robot is used in high energy particle physics or ionizing
radiation facilities, the probability that the electronics getting damaged due
to single event upset or cumulative effects of radiation are high (there were
several reported incidents for such cases and for instance there exist a
special group at CERN to study the effects of radiation to electronics).
Hence, for such applications, we emphasize the importance of having redundant
fault-tolerant hardware and software modules for important functionality as
well as having fail-safe algorithms on-board the robot

  8. 8.

Data verification and validation: Autonomous robots fully depend on the data
obtained from its sensors or sent to its actuators. Data related issues might
occur if algorithms are not devised properly. In one instance, the laser range
scanner provided false values in improper (black painted/water surface)
surfaces as expected but was not accounted in the algorithm. This data was
used for autonomous path planning and resulted in undesired actions/behaviors.
Therefore, sensor data should always be validated for proper values even if
one use a safety sensing device.

In another incident, a faulty Joystick provided error (or fault) value to the
algorithm which coincided with the reverse motor actuation value, and thus
resulted in robot behaving strangely by moving in the opposite of the intended
direction when the joystick abruptly failed (may be due to its low battery
level). A lesson here is that the range of sensor error values and the
actuators input values should be strictly verified for non-coincidences,
properly tuned and essentially utilized.

##  3 Energy and Communication

Managing a mobile robot’s energy and communication (wired or wireless) system
form the key priority in predicting and avoiding most of the failure
situations. We have observed several energy and communication failure
situations of the TIM robot either when the robot ran out of energy or when it
was not possible to communicate to the ground staff operating the robot. Thus,
ensuring the energy and communication requirement for a robotic mission can be
fulfilled before starting the mission is highly important.

A basic energy management system consists of an energy autonomy prediction
system (estimating the time and distance autonomy) will be helpful in making
sure that the robot meets energy expectations. This can be done by simulating
the energy consumption behavior before a mission or online prediction of
energy autonomy during a mission (e.g. [6]).

Similarly, a (wireless) communication autonomy prediction system could be
employed to estimate achievable distance (range), message latency, data
transfer rate, etc in a given environment. This can be included in the mission
planning stage or in offline mode (not during the mission) as exemplified in
[7] where offline prediction of wireless capabilities was examined. For
example, we faced several challenges in wireless communication with robots
(such as low operational ranges, frequent disconnections, abrupt quality
changes, etc.) at CERN underground tunnels. This is because the nature of the
environment itself is challenging for radio signal propagation due to the
effects of reflections from large metallic objects or deep multipath fading.
Therefore, for a successful robotic operation, monitoring of the connectivity
(connection quality) and deciding what action to take in the event of loss of
communication is certainly required if cases such as loss of robots in high
radiation zones (e.g. abandoned Quince robot in Fukushima nuclear reactor
building [1]) should not be repeated in the future.

Finally, a vital consideration for robot practitioners is to develop/implement
a functionality in robots to alert/inform the operator before the robot
running out of energy or loosing wireless connectivity. This could also be
extended to include situations such as when the remaining energy in robot is
not enough to get the robot back to its home station or when the predicted
communication capability/reach in the robot’s path is not adequate.

## References

  * [1] K. Nagatani, S. Kiribayashi, Y. Okada, S. Tadokoro, T. Nishimura, T. Yoshida, E. Koyanagi, and Y. Hada, “Redesign of rescue mobile robot Quince,” in _IEEE International Symposium on Safety Security and Rescue Robotics_ , 2011, pp. 13–18.
  * [2] R. Murphy, J. Kravitz, S. Stover, and R. Shoureshi, “Mobile robots in mine rescue and recovery,” _Robotics Automation Magazine, IEEE_ , vol. 16, no. 2, pp. 91–103, June 2009.
  * [3] K. Kershaw, F. Chapron, A. Coin, F. Delsaux, T. Feniet, J.-L. Grenard, and R. Valbuena, “Remote inspection, measurement and handling for LHC,” in _Particle Accelerator Conference, 2007. PAC. IEEE_ , June 2007, pp. 332 –334.
  * [4] J. Carlson and R. Murphy, “How UGVs physically fail in the field,” _IEEE Transactions on Robotics_ , vol. 21, no. 3, pp. 423–437, 2005.
  * [5] R. R. Murphy, “Proposals for New UGV, UMV, UAV, and HRI Standards for Rescue Robots,” in _Proceedings of the 10th Performance Metrics for Intelligent Systems Workshop_. ACM, 2010, pp. 9–13.
  * [6] R. Parasuraman, K. Kershaw, P. Pagala, and M. Ferre, “Model based on-line energy prediction system for semi-autonomous mobile robots,” in _5th International Conference on Intelligent Systems, Modelling and Simulation (ISMS)_ , 2014.
  * [7] R. Parasuraman, K. Kershaw, M. Ferre Perez, _et al._ , “Experimental investigation of radio signal propagation in scientific facilities for telerobotic applications,” _International Journal of Advanced Robotic Systems_ , vol. 10, pp. 1–11, 2013.


# Optimization approach for the Monge-Ampère equation

###### Abstract.

This paper studies the numerical approximation of solution of the Dirichlet
problem for the fully nonlinear Monge-Ampère equation. In this approach, we
take the advantage of reformulation the Monge-Ampère problem as an
optimization problem, to which we associate a well defined functional whose
minimum provides us with the solution to the Monge-Ampère problem after
resolving a Poisson problem by the finite element Galerkin method. We present
some numerical examples, for which a good approximation is obtained in 68
iterations.

Key words. elliptic Monge-Ampère equation, gradient conjugate method, finite
element Galerkin method.

AMS subject classifications. 35J60, 65K10, 65N30.

fethi ben belgacem 111 Department of Mathematics, Higher Institute of Computer
Sciences and Mathematics of Monastir (ISIMM), Avenue de la Korniche - BP 223 -
Monastir - 5000, TUNISIA. (fethi.benbelgacem@isimm.rnu.tn).

##  1\. Introduction

In this paper, we give a numerical solution for the following Monge-Ampère
problem

(1.1) $\left\\{\begin{array}[]{cc}\mbox{det}[D^{2}u]=f(x)&x\in\Omega,\\\
u_{|\Gamma}=0,&u\>\textrm{convex},\end{array}\right.$

where $\Omega$ is a smooth convex and bounded domain in $\mathbb{R}^{2},$
$\left[D^{2}u\right]$ is the Hessian of $u$ and $f\in
C^{\infty}(\overline{\Omega}),\textrm{ }f>0.$

Equation (1.1) belongs to the class of fully nonlinear elliptic equation. The
mathematical analysis of real Monge-Ampère and related equations has been a
source of intense investigations in the last decades; let us mention the
following references ( among many others and in addition to [7], [9], [15]):
[10], [8], [17, chapter 4], [2], [28], [11]-[14]. Applications to Mechanics
and Physics can be found in [27], [4], [5], [18], [24], [26],[31], (see also
the references therein).

The numerical approximations of the Monge-Ampère equation as well as related
equations have recently been reported in the literature. Let us mention the
references [4], [29], [39], [26 ], [11], [32], [25],[28], [33]; the method
discussed in [11], [32],[25] is very geometrical in nature. In contrast with
the method introduced by Dean and Glowinski in [19 ] [20] [21], which is of
the variational type.

On the existence of smooth solution for (1.1), we recall that if $f\in
C^{\infty}(\overline{\Omega})$ equation (1.1) has a unique strictly convex
solution $u\in C^{\infty}(\overline{\Omega})$ (see [14]).

To obtain a numerical solution for (1.1), we propose a least-square
formulation of (1.1). In this approach, we take the advantage of reformulation
of the Monge-Ampère problem as a well defined optimization problem, to which
we associate a well functional whose minimum provides us with the solution to
the Monge-Ampère problem after resolving a Poisson problem by the finite
element Galerkin method. The minimum is computed by the conjugate gradient
method.

The remainder of this article is organized as follows. In section 2, We
introduce the optimization problem. In section 3, we discuss a conjugate
gradient algorithm for the resolution of the optimization problem. The finite
element implementation of the above algorithm is discussed in section 4.
Finally, in section 5, we show some numerical results.

##  2\. Formulation of the Dirichlet problem for the elliptic Monge-Ampère
equation

Let $u_{I}$ be the solution of (1.1). Let $\lambda_{1}$ and $\lambda_{2}$ be
the eigenvalues of the matrix $[D^{2}u_{I}].$ We have

$\left\\{\begin{array}[]{ccc}\lambda_{1}+\lambda_{2}&=&\Delta u_{I},\\\
\lambda_{1}\lambda_{2}&=&\textrm{det}[D^{2}u_{I}]=f.\end{array}\right.$

Then $\lambda_{1}$ and $\lambda_{2}$ are the solutions of the equation

$X^{2}-\Delta u_{I}X+f=0.$

So

$(\Delta u_{I})^{2}-4f\geq 0.$

Then

$\Delta u_{I}-2\sqrt{f}\geq 0.$

Let us set

$\Delta u_{I}-2\sqrt{f}=\widetilde{g}\in C^{\infty}(\overline{\Omega}).$

We conclude that $u_{I}$ is solution of the following Dirichlet Poisson
problem

$\mathcal{P}_{\widetilde{g}}\left\\{\begin{array}[]{c}\Delta
u=2\sqrt{f}+\widetilde{g},\\\ u_{|\Gamma}=0.\end{array}\right.$

To compute $\widetilde{g},$ we consider the least-squares functional $J$
defined on

$E=\left\\{\varphi\in C^{\infty}(\overline{\Omega}),\>\varphi\geq 0\right\\},$

as follows:

$J(g)=\frac{1}{2}\int_{\Omega}\left(\textrm{det}\left(D^{2}u^{g}\right)-f\right)^{2}dx,$

where $u^{g}$ is the solution of the Dirichlet Poisson problem

$\mathcal{P}^{g}\left\\{\begin{array}[]{c}\Delta u=2\sqrt{f}+g,\\\
u_{|\Gamma}=0.\end{array}\right.$

The minimization problem

(2.1) $\left\\{\begin{array}[]{cc}\widetilde{g}\in E,\\\ J(\widetilde{g})\leq
J(g)&\forall g\in E,\end{array}\right.$

is thus a least-squares formulation of (1.1).

######  Theorem 1.

$u_{I}$ is the strictly convex solution of (1.1) if and only if there exist a
unique solution $\widetilde{g}$ of (2.1) such that $u_{I}=u^{\tilde{g}}.$

###### Proof.

Since $u_{I}$ is solution of $(\mathcal{P}^{\tilde{g}}),$ we have
$u_{I}=u^{\tilde{g}}.$ So $J(\widetilde{g})=0$ and $\tilde{g}$ is a unique
solution of (2.1).

Conversely, let $\overline{g}$ be a solution of (2.1). Since (1.1) has a
solution $u_{I},$ we can deduce immediatly that $J(\widetilde{g})=0$ and so,
$J(\overline{g})=0.$ It follows that

$\left\\{\begin{array}[]{c}\textrm{det}[D^{2}u^{\bar{g}}]=f,\\\
u{}_{|\Gamma}^{\bar{g}}=0.\end{array}\right.$

We have $\Delta u^{\bar{g}}=2f+\overline{g}>0$ and
$\textrm{det}[D^{2}u^{\bar{g}}]>0,$ we can deduce that $u^{\bar{g}}$ is
strictly convex and from the uniqueness of solution for (1.1) we get
$u^{\bar{g}}=u_{I}.$ ∎

##  3\. Iterative solution for the minimisation problem

###  3.1. Description of the algorithm.

The algorithm we consider to solve the problem (2.1) which is based on the PRP
(Polak-Ribière-Polyak [36,37]) conjugate gradient method reads:

Given $g^{0}\in E;$

then, for _ $k\geq 0,$ $g^{k}$_ being known in $E$, solve __

$\mathcal{P}^{g^{k}}\left\\{\begin{array}[]{c}\Delta u=2\sqrt{f}+g^{k},\\\
u_{|\Gamma}=0.\end{array}\right.$

_Compute, $\nabla J(g^{k})$, _

> _If $k\geq 1,$ $\beta^{k}=\dfrac{\nabla J(g^{k})^{T}(\nabla J(g^{k})-\nabla
> J(g^{k-1})}{\left\|\nabla J(g^{k-1})\right\|_{2}^{2}};$_
>
> __
>
> $d^{k}=\left\\{\begin{array}[]{c}-\nabla J(g^{0})\quad\quad\quad\qquad\qquad
> if\,k=0\\\ -\nabla J(g^{k})+\beta^{k}d^{k-1}\quad\quad\quad\quad if\,k\geq
> 1;\end{array}\right.$
>
> _and update $g^{k}$_ _by_
>
> $g^{k+1}=g^{k}+\alpha^{k}d^{k}.$

Where $\alpha_{k}$ is computed with the Armijo-type line search.

###  3.2. Solution of sub-problem $(\mathcal{P}^{g}).$

We consider first the variational formulation of $(\mathcal{P}^{g})$

(3.1) $\left\\{\begin{array}[]{ccc}\textrm{Find}&u^{g}\in
H_{0}^{1}(\Omega),&\textrm{such that,}\\\ a(u^{g},v)=L(v),&\forall v\in
H_{0}^{1}(\Omega),\end{array}\right.$

where

(3.2) $a(u,v)=\int_{\Omega}\nabla u\nabla vdx$

and

(3.3) $L(v)=-\int_{\Omega}\left(2\sqrt{f}+g\right)vdx,$

$a$ in (3.2) is coercive on $H_{0}^{1}(\Omega).$ For $f\in L^{2}(\Omega)$we
have $\sqrt{f}\in L^{2}(\Omega).$ Since $\Omega$ is bounded and for $g\in
L^{2}(\Omega),$ $L$ in (3.3) is continuous, then by the Lax-Milgram theorem
$(\mathcal{P}^{g})_{V}$ has a unique solution $u^{g}.$

##  4\. Finite element approximation of the minimization problem

For simplicity, we assume that $\Omega$ is a bounded polygonal domain of
$\mathbb{R}^{2}.$ Let $\mathcal{T}_{h}$ a finite triangulation of $\Omega$
(like those discussed in e.g, [16]).

We introduce a

with $P_{1}$ the space of the two-variable polynomials of degree $\leq 1.$ A
function $\varphi$ being given in $H^{2}(\Omega)$ we denote
$\frac{\partial^{2}\varphi}{\partial x_{i}x_{j}}$ by

(4.1) $\int_{\Omega}\frac{\partial^{2}\varphi}{\partial
x_{i}^{2}}vdx=-\int_{\Omega}\frac{\partial\varphi}{\partial
x_{i}}\frac{\partial v}{\partial x_{i}}dx,\>\forall v\in
H_{0}^{1}(\Omega),\>\forall i=1,2,$ (4.2)
$\int_{\Omega}\frac{\partial^{2}\varphi}{\partial
x_{1}x_{2}}vdx=-\frac{1}{2}\int_{\Omega}\left[\frac{\partial\varphi}{\partial
x_{1}}\frac{\partial v}{\partial x_{2}}+\frac{\partial\varphi}{\partial
x_{2}}\frac{\partial v}{\partial x_{1}}\right]dx,\>\forall v\in
H_{0}^{1}(\Omega).$

Let $\varphi\in V_{h};$ taking advantage of relations (4.1) and (4.2) we
define the discrete analogues of the differential operators $D_{ij}^{2}$ by

(4.3) $\left\\{\begin{array}[]{c}\forall i=1,2,\>D_{hii}^{2}(\varphi)\in
V_{0h},\\\
\int_{\Omega}D_{hii}^{2}(\varphi)vdx=-{\displaystyle\int_{\Omega}\frac{\partial\varphi}{\partial
x_{i}}\frac{\partial v}{\partial x_{i}}dx,\>\forall v\in
V_{0h},}\end{array}\right.$ (4.4)
$\left\\{\begin{array}[]{c}D_{h12}^{2}(\varphi)\in V_{0h},\\\
\int_{\Omega}D_{h12}^{2}(\varphi)vdx=-{\displaystyle\frac{1}{2}\int_{\Omega}\left[\frac{\partial\varphi}{\partial
x_{1}}\frac{\partial v}{\partial x_{2}}+\frac{\partial\varphi}{\partial
x_{2}}\frac{\partial v}{\partial x_{1}}\right]dx,\>\forall v\in
V_{0h}.}\end{array}\right.$

To compute the above discrete second order partial derivatives we will use the
trapezoidal rule to evaluate the inegrals in the left hand sides of (4.3) and
(4.4). We consider the set ${\scriptstyle\sum}_{h}$ of the vertices of
$\mathcal{T}_{h}$ and
${\scriptstyle{\textstyle}\sum}_{0h}$=$\left\\{P\left|\right.P\in{\scriptstyle\sum}_{h},P\notin\Gamma\right\\}.$
We define the integers $N_{h}$ and $N_{0h}$ by
$N_{h}=\mbox{Card}({\scriptstyle\sum}_{h})$ and
$N_{0h}=\mbox{Card}({\scriptstyle\sum}_{0h}).$ So dim$V_{h}=N_{h}$ and
dim$V_{0h}=N_{0h}.$

For $P_{k}\in{\scriptstyle\sum}_{h}$ we associate the function $w_{k}$
uniquely defined by

$w_{k}\in
V_{h},\,w_{k}(P_{k})=1,\>w_{k}(P_{l})=0,\>\mbox{if}\>l=1,...N_{h,}\>l\neq k.$

It is well known (e.g., [16]) that the sets
$\mathfrak{B}_{h}=\left\\{w_{k}\right\\}_{k=1}^{N_{h}}$ and
$\mathfrak{B}_{0h}=\left\\{w_{k}\right\\}_{k=1}^{N_{0h}}$ are vector bases of
$V_{h}$ and $V_{0h},$ respectively.

We denote by $A_{k}$ the area of the polygonal which is the union of those
triangles of $\mathcal{T}_{h}$ which have $P_{k}$ as a common vertex. By
applying the trapezoidal rule to the integrals in the left hand side of
relations (4.3) and (4.4) we obtain:

(4.5) $\left\\{\begin{array}[]{c}\forall i=1,2,\>D_{hii}^{2}(\varphi)\in
V_{0h},\\\
D_{hii}^{2}(\varphi)(P_{k})=-{\displaystyle\frac{3}{A_{k}}\int_{\Omega}\frac{\partial\varphi}{\partial
x_{i}}\frac{\partial w_{k}}{\partial x_{i}}dx,\>\forall
k=1,2,...,N_{0h},}\end{array}\right.$ (4.6)
$\left\\{\begin{array}[]{c}D_{h12}^{2}(\varphi)\left(=D_{h21}^{2}(\varphi)\right)\in
V_{0h},\\\
D_{h12}^{2}(\varphi)(P_{k})=-{\displaystyle\frac{3}{2A_{k}}\int_{\Omega}\left[\frac{\partial\varphi}{\partial
x_{1}}\frac{\partial w_{k}}{\partial x_{2}}+\frac{\partial\varphi}{\partial
x_{2}}\frac{\partial w_{k}}{\partial x_{1}}\right]dx,\>\forall
k=1,2,...,N_{0h}.}\end{array}\right.$

Computing the integrals in the right hand sides of (4.5) and (4.6) is quite
simple since the first order derivatives of $\varphi$ and $w_{k}$ are
piecewise constant.

Taking the above relations into account. We approximate the space $E$ by

$E_{h}=\left\\{\varphi\in V_{h},\>\varphi\geq 0\right\\},$

and then the minimization problem (2.1) by

$\left\\{\begin{array}[]{cc}\widetilde{g}_{h}\in E_{h},\\\
J_{h}(\widetilde{g}_{h})\leq J_{h}(g_{h})&\forall g_{h}\in
E_{h},\end{array}\right.$

Where

$J_{h}(g_{h})=\frac{1}{6}{\displaystyle\sum_{k=1}^{N_{h0}}A_{k}\left|D_{h11}^{2}(u_{h}^{g_{h}})(P_{k})D_{h22}^{2}(u_{h}^{g_{h}})(P_{k})-\left(D_{h12}^{2}(u_{h}^{g_{h}}(P_{k})\right)^{2}-f_{h}(P_{k})\right|^{2}},\>\forall
g_{h}\in E_{h},$

and $f_{h},$ $\widetilde{g}_{h}$, $g_{h}$ are respectively a continuous
approximations of functions $f,$ $\widetilde{g},$ $g$ and $u_{h}^{g_{h}}$ is
the solution of the discret variant of the Dirichlet Poisson problem
$(\mathcal{P}^{g})$.

###  4.1. Discrete variant of the algorithm

We will discuss now the solution of (2.1) by a discrete variant of algorithm
3.1.

_Given $g_{h}^{0}\in E_{h};$ _

then, for _ $k\geq 0,$ $g_{h}^{k}$ being known in $E_{h}$, solve
$\mathcal{P}^{g_{h}^{k}},$ _

_Compute, $\nabla J(g_{h}^{k})$, $\gamma_{h}^{k}=\left\|\nabla
J(g_{h}^{k})\right\|_{2}^{2};$_

> _If $k\geq 1,$ $\beta_{h}^{k}=\gamma_{h}^{k}/\gamma_{h}^{k-1};$_
>
> __
>
> $d_{h}^{k}=\left\\{\begin{array}[]{c}-\nabla
> J(g^{0})\quad\quad\quad\qquad\qquad if\,k=0\\\ -\nabla
> J(g_{h}^{k})+\beta_{h}^{k}d_{h}^{k-1}\;if\,k\geq 1;\end{array}\right.$
>
> and update $g_{h}^{k}$_by_
>
> $g_{h}^{k+1}=g_{h}^{k}+\alpha_{h}^{k}d_{h}^{k}.$

######  Remark 2.

There are many approches for finding an avaible step size $\alpha_{h}^{k}$.
Among them the exact line search is an ideal one, but is cost-consuming or
even impossible to use to find the step size. Some inexact line searches are
sometimes useful and effective in practical computation, such as Armijo line
search [1], Goldstein line search and Wolfe line search [24,38].

The Armijo line search is commonly used and easy to implement in practical
computation.

Armijo line search

Let $s>0$ be a constant, $\rho\in\left(0,1\right)$ and
$\mu\in\left(0,1\right).$ Choose $\alpha_{k}$ to be the largest $\alpha$ in
$\left\\{s,\>s\rho,\>s\rho^{2},...,\right\\}$ such that

$J_{h}(g_{h}^{k})-J_{h}(x_{k}+\alpha d_{h}^{k})\geq-\alpha\mu\nabla
J_{h}(g_{h}^{k})^{T}d_{h}^{k}.$

However, this line search cannot guarantee the global convergence of the PRP
method and even cannot guarantee $d_{k}$ to be descent direction of $J$ at
$g^{k}.$

####  4.1.1. Solution of the sub problem _$\mathcal{P}_{g_{h}^{k}}$ _

Any sub-problem _$(\mathcal{P}^{g_{h}^{k}}),$ _is equivalent to a finite
dimensional variational linear problem which reads as follows: Find
$u_{h}^{g_{h}}\in V_{0h}$ such that

(4.7) $a(u_{h}^{g_{h}},v_{h})=L(v_{h}),\forall v_{h}\in V_{0h}.$

By the Lax-milgram theorem we can easily show that (4.7) has a unique solution
$u_{h}^{g_{h}}\in V_{0h}.$

##  5\. Numerical experiments

In this section we are going to apply the method discussed in the previous
section to the solution of some test problems. For all these test problems we
shall assume that $\Omega$ is the unit disk. We first approximate $\Omega$ by
a polygonal domain $\Omega_{h}.$ We consider $\mathcal{T}_{h}$ a finite
triangulation of $\Omega_{h}.$

The first test problem is expressed as follows

(5.1)
$\left\\{\begin{array}[]{cc}\mbox{det}[D^{2}u]=4\left(1+2\left(\left|x\right|^{2}\right)\right)e^{2\left(\left|x\right|^{2}-1\right)}&x\in\Omega,\\\
u_{|\Gamma}=0,&u\>\textrm{convex}.\end{array}\right.$

with $\left|x\right|^{2}=x_{1}^{2}+x_{2}^{2}.$ The exact solution _$u\in
C^{\infty}(\bar{\Omega})$_ to problem (5.1) is given by

$u\left(x\right)=e^{\left(\left|x\right|^{2}-1\right)}-1.$

######  Remark 3.

When computing the approximate solutions of these problems, we stopped the
iterations of the algorithm as soon as $\left|J_{h}(g_{h})\right|\leq
10^{-6}.$

We have discretized the optimization problem associated to the problem (5.1).
We solved the Poisson problem encountred at each iteration of the algorithm by
a fast Poisson solvers.

We have used as initial guess three different constant values for $g_{h}^{0}.$
The results obtained after 68 iterations are summarized in Table 1 (where
$u_{h}^{c}$ denotes the computed approximate solution and
$\left\|.\right\|_{0,\Omega}=\left\|.\right\|_{L^{2}(\Omega)}$).

The graph of $u_{h}^{c}$ and its contour plot obtained, for ${\displaystyle
h=\nicefrac{{1}}{{128}}}$ has been respectively visualized on Figure 2 and
Figure 3.

Table 1. First test problem : Convergence of the approximate solution. $h$ | $g_{h}^{0}$ | $\left\|u-u_{h}^{c}\right\|_{0,\Omega}$
---|---|---
$\begin{array}[]{c}\nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.8861\times 10^{-4}\\\ 0.5497\times 10^{-4}\\\ 0.3720\times 10^{-4}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.3416\times 10^{-4}\\\ 0.9121\times 10^{-5}\\\ 0.7554\times 10^{-5}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.6305\times 10^{-5}\\\ 0.4981\times 10^{-5}\\\ 0.7203\times 10^{-6}\end{array}$
Figure 5.1. First test problem : Graph of $u_{h}^{c}.$ Figure 5.2. First test
problem : Contour plot of $u_{h}^{c}.$

We conclude from the results in Table 1 that the value $g^{0}=0.3$ is optimal
and quite accurate approximations of the exact solutions are obtained.

######  Remark 4.

We did not try to find the optimal value of $g^{0}$(it seems that is a
difficult problem).

In the second test problem we take

$f(x)={\displaystyle\left(\frac{4}{5}\right)^{2}\pi^{2}\left[\mbox{cos}^{2}\left(\frac{\pi}{2}\left(1-\left|x\right|^{2}\right)\right)+\frac{\pi}{2}\left(\left|x\right|^{2}\right)\mbox{sin}\left(\pi\left(1-\left|x\right|^{2}\right)\right)\right].}$

The solution to the corresponding Monge-Ampère problem is the function _$u\in
C^{\infty}(\bar{\Omega})$_ defined by

$u\left(x\right)=-\frac{4}{5}\mbox{sin}\left(\frac{\pi}{2}\left(1-\left|x\right|^{2}\right)\right).$

The method provides after 64 iterations the results summurized in Table 2.

The value $g^{0}=0.3$ is again optimal.

Table 2. second test problem : Convergence of the approximate solution. $h$ | $g_{h}^{0}$ | $\left\|u-u_{h}^{c}\right\|_{0,\Omega}$
---|---|---
$\begin{array}[]{c}\nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.6466\times 10^{-4}\\\ 0.4510\times 10^{-4}\\\ 0.2983\times 10^{-4}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.1749\times 10^{-4}\\\ 0.8507\times 10^{-5}\\\ 0.6221\times 10^{-5}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.3743\times 10^{-5}\\\ 0.1180\times 10^{-5}\\\ 0.5591\times 10^{-6}\end{array}$
Figure 5.3. Second test problem : Graph of $u_{h}^{c}.$ Figure 5.4. Second
test problem : Contour plot of $u_{h}^{c}.$

The third test problem is defined as follows

(5.2) $\left\\{\begin{array}[]{cc}\mbox{det}[D^{2}u]=1&x\in\Omega,\\\
u_{|\Gamma}=0,&u\>\textrm{convex}.\end{array}\right.$

The function $u$ given by

$u(x)=\frac{1}{2}\left(\left|x\right|^{2}-1\right)$

is the solution of (5.2 _) and $u\in C^{\infty}(\bar{\Omega}).$_

Table 3. Third test problem : Convergence of the approximate solution. $h$ | $g_{h}^{0}$ | $\left\|u-u_{h}^{c}\right\|_{0,\Omega}$
---|---|---
$\begin{array}[]{c}\nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\\\ \nicefrac{{1}}{{32}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.3830\times 10^{-3}\\\ 0.2564\times 10^{-3}\\\ 0.2971\times 10^{-3}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\\\ \nicefrac{{1}}{{64}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.7448\times 10^{-4}\\\ 0.8529\times 10^{-6}\\\ 0.9193\times 10^{-5}\end{array}$
$\begin{array}[]{c}\nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\\\ \nicefrac{{1}}{{128}}\end{array}$ | $\begin{array}[]{c}0.1\\\ 0.2\\\ 0.3\end{array}$ | $\begin{array}[]{c}0.6215\times 10^{-5}\\\ 0.5837\times 10^{-6}\\\ 0.3806\times 10^{-5}\end{array}$
Figure 5.5. Third test problem : Graph of $u_{h}^{c}.$ Figure 5.6. Third test
problem : Contour plot of $u_{h}^{c}.$

We deduce from Table 3 that $g_{0}=0.2$ is an optimal value.

Unfortunaly I did not find any other initial value that gives more accurate
results. Even for $g_{0}=0.4$ the results are not satisfied.

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