compactifications to four dimensions supported by a flux of the self–dual five
form, where the string coupling is everywhere bounded. In [14] we shall
explore further these fluxed compactifications to four dimensions, addressing
in detail the key issue of perturbative stability, which has proved daunting
for non–supersymmetric $AdS$ vacua.

A common feature of non–supersymmetric vacua is the emergence of severe back-
reactions. This will be the case for all the examples discussed in this paper,
once quantum effects are taken into account. Back-reactions have long been
recognized as an important issue in the widely explored context of Scherk-
Schwarz compactifications [5], which have simple realizations in String Theory
[24], even when open string are present [25]. However, similar back-reactions
present themselves in a more basic context already in ten dimensions, in the
three ten–dimensional non–supersymmetric non–tachyonic models of [26], [27],
and of [28], where supersymmetry is non-linearly realized [29], and lead to
the emergence of a peculiar contribution to the low–energy effective action, a
“tadpole potential”

${}\Delta\,{\cal S}\ =\ -\ T\int d^{10}x\,\sqrt{-g}\ e^{\gamma\phi}\ ,$ (8.1)

where $\gamma=\gamma_{c}=\frac{3}{2}$ for the orientifold models and
$\gamma=\frac{5}{2}$ for the heterotic $SO(16)\times SO(16)$ model. The same
types of potentials are the leading contributions to generic string models
when quantum corrections due to broken supersymmetry are taken into account.
Their effects on vacuum solutions of the type considered here will lead to
generalizations of the Dudas–Mourad vacuum [11], of their counterparts for
generic values of $\gamma$ in [12], and of the climbing–scalar cosmologies
[13], with a number of novel features. These systems are the main topic of the
companion paper [9].

## Acknowledgments

We are grateful to E. Dudas for stimulating discussions, and also to G. Bogna
and Y. Tatsuta, who read an earlier version of the manuscript and made useful
comments. The work of AS was supported in part by Scuola Normale, by INFN (IS
GSS-Pi) and by the MIUR-PRIN contract 2017CC72MK_003. JM is grateful to Scuola
Normale Superiore for the kind hospitality extended to him while this work was
in progress. AS is grateful to U. Paris VII and DESY–Hamburg for the kind
hospitality, and to the Alexander von Humboldt foundation for the kind and
generous support, while this work was in progress.

##  Appendix A Some Technical Details

For the backgrounds of eq. (1.1), the Riemann tensor is computed most
conveniently starting from the one–form spin connection, with non–vanishing
components

${\omega^{\mu}}_{r}\ =\ -\ {\omega_{r}}^{\mu}\ =\
e^{A-B}\,A^{\prime}\,dx^{\mu}\ ,\qquad{\omega^{i}}_{r}\ =\ -\
{\omega_{r}}^{i}\ =\ e^{C-B}\,C^{\prime}\,dy^{i}\ .$ (A.1)

The non–vanishing contributions to the curvature two-form

${\Omega_{A}}^{B}\ \equiv\ \frac{1}{2}\,{{R_{MN}}_{A}}^{\ \ B}\ dx^{M}\wedge
dx^{N}\ =\ d\,{\omega_{A}}^{B}\ +\ {\omega_{A}}^{C}\wedge{\omega_{C}}^{B}\ ,$
(A.2)

when converted to curved indices, determine the background Riemann tensor,
where here we add also the contributions from the curvatures of the maximally
symmetric metrics $\gamma_{\mu\nu}$ and $\gamma_{mn}$:

$\displaystyle{\Omega}^{r\mu}$ $\displaystyle=$ $\displaystyle-\
\left(e^{A-B}\,A^{\prime}\right)^{\prime}\,dr\wedge dx^{\mu}\
,\qquad{\Omega}^{rm}\ =\ -\
\left(e^{C-B}\,C^{\prime}\right)^{\prime}\,dr\wedge dy^{m}\ ,$
$\displaystyle{\Omega}^{\mu\nu}$ $\displaystyle=$ $\displaystyle-\
\left[e^{2(A-B)}\left(A^{\prime}\right)^{2}\,-\,\frac{k}{\alpha^{\prime}}\right]dx^{\mu}\wedge
dx^{\nu}\ ,$ $\displaystyle{\Omega}^{m\nu}$ $\displaystyle=$ $\displaystyle-\
e^{A+C-2B}\,A^{\prime}\,C^{\prime}\,dy^{m}\wedge dx^{\nu}\ ,$
$\displaystyle{\Omega}^{mn}$ $\displaystyle=$ $\displaystyle-\
\left[e^{2(C-B)}\left(C^{\prime}\right)^{2}\,-\,\frac{k^{\prime}}{\alpha^{\prime}}\right]dy^{m}\wedge
dy^{n}\ .$ (A.3)

Consequently, the Riemann tensor with all curved indices has the non–vanishing
components

$\displaystyle{R_{r\mu r}}^{\nu}$ $\displaystyle=$ $\displaystyle-\
{\delta_{\mu}}^{\nu}\ e^{B-A}\left(e^{A-B}\,A^{\prime}\right)^{\prime}\
,\qquad{R_{rmr}}^{n}\ =\ -\ {\delta_{m}}^{n}\
e^{B-C}\left(e^{C-B}\,C^{\prime}\right)^{\prime}\ ,$
$\displaystyle{R_{\alpha\beta\mu}}^{\nu}$ $\displaystyle=$ $\displaystyle-\
\gamma_{\mu[\alpha}\,{\delta_{\beta]}}^{\nu}\left[e^{2(A-B)}\left(A^{\prime}\right)^{2}\,-\,\frac{k}{\alpha^{\prime}}\right]\
,\qquad{R_{\nu n\mu}}^{m}\ =\ -\
e^{2(A-B)}\,\gamma_{\mu\nu}\,{\delta_{n}}^{m}\,A^{\prime}\,C^{\prime}\ ,$
$\displaystyle{R_{mnp}}^{q}$ $\displaystyle=$ $\displaystyle-\
\gamma_{p[m}\,{\delta_{n]}}^{q}\left[e^{2(C-B)}\left(C^{\prime}\right)^{2}\,-\,\frac{k^{\prime}}{\alpha^{\prime}}\right]\
,\qquad{R_{\nu nm}}^{\mu}\ =\
e^{2(C-B)}\,{\delta_{\nu}}^{\mu}\,\gamma_{mn}\,A^{\prime}\,C^{\prime}\ ,$
(A.4)

together with others obtained from these by symmetry, and thus

$\displaystyle R_{\mu\nu}^{(0)}$ $\displaystyle=$ $\displaystyle-\
\gamma_{\mu\nu}e^{2A}\left[p(A^{\prime})^{2}\,e^{-2B}\ +\
(D-p-2)A^{\prime}\,C^{\prime}e^{-2B}\ +\
\left(A^{\prime}e^{A-B}\right)^{\prime}e^{-A-B}\right.$ $\displaystyle-$
$\displaystyle\left.\frac{p\,k}{\alpha^{\prime}}\,e^{-2A}\right]\ ,$
$\displaystyle R_{rr}^{(0)}$ $\displaystyle=$
$\displaystyle-\left[(p+1)\left(A^{\prime}e^{A-B}\right)^{\prime}e^{B-A}\ +\
(D-p-2)\left(C^{\prime}e^{C-B}\right)^{\prime}e^{B-C}\right]\ ,$
$\displaystyle R_{mn}^{(0)}$ $\displaystyle=$ $\displaystyle-\
\gamma_{mn}e^{2C}\left[(D-p-3)(C^{\prime})^{2}\,e^{-2B}\ +\
(p+1)A^{\prime}\,C^{\prime}e^{-2B}\ +\
\left(C^{\prime}e^{C-B}\right)^{\prime}e^{-B-C}\right.$ (A.5) $\displaystyle-$
$\displaystyle\left.\frac{(D-p-3)\,k^{\prime}}{\alpha^{\prime}}\,e^{-2C}\right]\
.$

In the harmonic gauge (2.14)

$\displaystyle{}{}R_{\mu\nu}^{(0)}$ $\displaystyle=$ $\displaystyle-\
\gamma_{\mu\nu}\left[e^{2(A-B)}\,A^{\prime\prime}\ -\
\frac{p\,k}{\alpha^{\prime}}\right]\ ,$ $\displaystyle R_{rr}^{(0)}$
$\displaystyle=$
$\displaystyle-\Big{[}\left(p+1\right)A^{\prime\prime}\,+\,\left(D-p-2\right)C^{\prime\prime}\,-\,p(p+1)\left(A^{\prime}\right)^{2}$
$\displaystyle-$
$\displaystyle(D-p-2)(D-p-3)\left(C^{\prime}\right)^{2}\,-\,2(p+1)(D-p-2)A^{\prime}C^{\prime}\Big{]}\,,$
$\displaystyle R_{mn}^{(0)}$ $\displaystyle=$ $\displaystyle-\
\gamma_{mn}\left[e^{2(C-B)}\,C^{\prime\prime}\ -\
\frac{(D-p-3)\,k^{\prime}}{\alpha^{\prime}}\right]\ .$ (A.6)

Consequently the scalar curvature is

$\displaystyle{}{}{}R$ $\displaystyle=$
$\displaystyle-2e^{-2B}\left[(p+1)A^{\prime\prime}\,+\,(D-p-2)C^{\prime\prime}\right]$
$\displaystyle+$
$\displaystyle\frac{p(p+1)k}{\alpha^{\prime}}\,e^{-2A}\,+\,\frac{(D-p-1)(D-p-2)k^{\prime}}{\alpha^{\prime}}\,e^{-2C}$
$\displaystyle+$ $\displaystyle
e^{-2B}\left[p(p+1)(A^{\prime})^{2}+2(p+1)(D-p-2)A^{\prime}C^{\prime}+(D-p-2)(D-p-3)(C^{\prime})^{2}\right]\,.$

and therefore

$\displaystyle{}{}{}G_{rr}$ $\displaystyle=$ $\displaystyle-\
\frac{1}{2}\,\frac{p(p+1)k}{\alpha^{\prime}}\,e^{2(B-A)}\,-\
\frac{1}{2}\,\frac{(D-p-1)(D-p-2)k^{\prime}}{\alpha^{\prime}}\,e^{2(B-C)}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[p(p+1)(A^{\prime})^{2}+2(p+1)(D-p-2)A^{\prime}C^{\prime}+(D-p-2)(D-p-3)(C^{\prime})^{2}\right]\,.$

In order to build the contributions of tensor fields strengths to the Einstein
equations one needs to take into account some properties of the field
strengths of eqs. (2.18) and (2.19),

$\displaystyle\left({{\cal H}_{p+2}}^{2}\right)_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\ (p+1)!\left(b^{\prime}{}_{p+1}(r)\right)^{2}e^{-2pA-2B}\
\gamma_{\mu\nu}\ ,$ $\displaystyle\left({{\cal H}_{p+2}}^{2}\right)_{rr}$
$\displaystyle=$ $\displaystyle-\
(p+1)!\left(b^{\prime}{}_{p+1}(r)\right)^{2}e^{-2(p+1)A}\ ,$ (A.9)

and

$\left({{\cal H}_{p+1}}^{2}\right)_{\mu\nu}\ =\ -\ p!\
h_{p+1}^{2}e^{-2pA}\gamma_{\mu\nu}\ .$ (A.10)

There are also two special cases related to self–dual forms. The first
concerns the self–dual profile of eq. (2.26), for which

$\displaystyle\left({{\cal H}_{5}}^{2}\right)_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\ 3\,H_{5}^{2}\ e^{2A-10C}\ \gamma_{\mu\nu}\ ,\quad\left({{\cal
H}_{5}}^{2}\right)_{rr}\,=\,-\ 3\,H_{5}^{2}\ e^{2B-10C}\ ,$
$\displaystyle\left({{\cal H}_{5}}^{2}\right)_{mn}$ $\displaystyle=$
$\displaystyle 3\,H_{5}^{2}\ e^{-8C}\ \gamma_{mn}\ ,$ (A.11)

consistently with

${\cal H}_{5}^{2}\ =\ 0\ ,$ (A.12)

which is implied by the self–duality. The second concerns the corresponding
$h$-profile of eq. (2.27), for which

$\displaystyle\left({{\cal H}_{5}}^{2}\right)_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\ 3\,h_{5}^{2}\ e^{-8A}\ \gamma_{\mu\nu}\ ,\quad\left({{\cal
H}_{5}}^{2}\right)_{rr}\,=\,3\,h_{5}^{2}\ e^{2B-10A}\ ,$
$\displaystyle\left({{\cal H}_{5}}^{2}\right)_{mn}$ $\displaystyle=$
$\displaystyle 3\,h_{5}^{2}\ e^{2C-10A}\ \gamma_{mn}\ .$ (A.13)

##  Appendix B A Useful Newtonian Model

In this Appendix we discuss the differential equation

$Z^{\prime\prime}\ =\ \epsilon\,\Delta^{2}\ e^{\,2\,Z}\ ,$ (B.1)

where $\epsilon=\pm 1$ and $\Delta$ is real and positive, which plays a role
in various parts of the paper. This equation has the first integral

$(Z^{\prime})^{2}\ =\ E\ +\ \epsilon\,\Delta^{2}\,e^{2\,Z}\ ,$ (B.2)

which has the form of an energy conservation condition for a Newtonian
particle, and in order to proceed further one must distinguish a few cases.

  1. 1.

If $\epsilon=1$ and $E=\frac{1}{\rho^{2}}>0$, letting

$x\ =\ \frac{r}{\rho}$ (B.3)

eq. (B.2) becomes

$\left(\frac{dZ}{dx}\right)^{2}\ =\ 1\ +\
\left(\Delta\,\rho\right)^{2}\,e^{2\,Z}\ .$ (B.4)

The redefinition

$Z=\ \tilde{Z}-\log(\Delta\,\rho)$ (B.5)

leads to the reduced equation

$\left(\frac{d\tilde{Z}}{dx}\right)^{2}\ =\ 1\ +\ e^{2\,\tilde{Z}}\ ,$ (B.6)

and therefore finally to the solutions

$Z\ =\ -\log\left[\Delta\,\rho\,\sinh\left(\frac{r-r_{0}}{\rho}\right)\right]\
,$ (B.7)

in the region $r>r_{0}$, or

$Z\ =\ -\log\left[\Delta\,\rho\,\sinh\left(\frac{r_{0}-r}{\rho}\right)\right]\
,$ (B.8)

in the region $r<r_{0}$.

  2. 2.

If $\epsilon=1$ and $E=-\frac{1}{\rho^{2}}<0$, letting

$x\ =\ \frac{r}{\rho}$ (B.9)

eq. (B.2) becomes

$\left(\frac{dZ}{dx}\right)^{2}\ =\ -\,1\ +\
\left(\Delta\,\rho\right)^{2}\,e^{2\,Z}\ .$ (B.10)

The redefinition

$Z=\ \tilde{Z}-\log(\Delta\,\rho)$ (B.11)

leads to the reduced equation

$\left(\frac{d\tilde{Z}}{dx}\right)^{2}\ =\ -\,1\ +\ e^{2\,\tilde{Z}}\ ,$
(B.12)

and therefore finally to the solution

$Z\ =\ -\log\left[\Delta\,\rho\,\cos\left(\frac{r-r_{0}}{\rho}\right)\right]\
,$ (B.13)

which is valid for $|r-r_{0}|<\frac{\pi\,\rho}{2}$, and one can conveniently
choose $r_{0}=0$. These solutions can be obtained from those of the preceding
case letting

$r-r_{0}\ \to\ i\left(r-r_{0}\ -\ \frac{\pi}{2}\right)\ ,\qquad\Delta\ \to\ -\
i\,\Delta\ .$ (B.14)

  3. 3.

If $\epsilon=1$ and $E=0$, eq. (B.2) reduces to

$Z^{\prime}\ =\ \pm\Delta\,e^{\,Z}\ ,$ (B.15)

which can be simply integrated and yields

$e^{-Z}\ =\ \mp\,\Delta\left(r-r_{0}\right)\ ,$ (B.16)

where $r_{0}$ is another integration constant. Here clearly the upper sign is
associated to the region $r<r_{0}$, where the real solution reads

$Z\ =\ -\ \log\Delta\left(r_{0}-r\right)\ .$ (B.17)

while the lower one is associated to the region $r>r_{0}$, where the real
solution reads

$Z\ =\ -\ \log\Delta\left(r-r_{0}\right)\ .$ (B.18)

These types of solutions capture the limiting behavior of the preceding cases
when $E$ is negligible with the respect to the exponential $e^{2Z}$, and thus
as $\rho\to\infty$.

  4. 4.

Finally, If $\epsilon=-1$ and $E=\frac{1}{\rho^{2}}>0$, letting

$x\ =\ \frac{r}{\rho}$ (B.19)

eq. (B.2) becomes

$\left(\frac{dZ}{dx}\right)^{2}\ =\ 1\ -\
\left(\Delta\,\rho\right)^{2}\,e^{2\,Z}\ .$ (B.20)

The redefinition

$Z=\ \tilde{Z}-\log(\Delta\,\rho)$ (B.21)

leads to the reduced equation

$\left(\frac{d\tilde{Z}}{dx}\right)^{2}\ =\ 1\ -\ e^{2\,\tilde{Z}}\ ,$ (B.22)

and therefore finally to the solution

$Z\ =\ -\log\left[\Delta\,\rho\,\cosh\left(\frac{r-r_{0}}{\rho}\right)\right]\
,$ (B.23)

for all real values of $r$. This can be continued to the first solution
combining the two transformations

$\frac{r}{\rho}\ \to\ \frac{r}{\rho}\ +\ i\frac{\pi}{2}\ ,\qquad\Delta\ \to\
-\ i\,\Delta\ .$ (B.24)

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