xi/ Y_2$ breaking term grows more rapidly than $\xi/Y_1$. 
The other possibility which avoids the asymmetric destabilization is by enforcing the diagonal direction via suitable geometrical flux terms. 

 
\section{Conclusions}
\label{conclu}
 
 
 
 
 The first part of our work is motivated by the  observation that  the pure ${\cal R}^2$ theory is the only scale invariant gravity theory 
without ghost. As a first step we show that this theory is conformally equivalent to a conventional Einstein  gravity with an extra 
scalar degree of freedom $\phi$ without potential thanks to the underlying  classical scale invariance.  At the classical level there are three and only three possible vacua according to the sign of the coupling $\mu^2$ of the  ${\cal R}^2$ theory. The de Sitter vacuum ($dS$) arises, if  $\mu^2$ is positive,  the anti de Sitter vacuum ($AdS$), if is $\mu^2$ is negative  and the flat Minkowski space (${\cal M}$), if $\mu^2$ is zero. Although this is an interesting  observation it is not useful  in the absence of at least the observed Standard Model matter with the gauge, Yukawa  and self scalar interactions. This forces us to generalize the ${\cal R}^2$ theory by coupling it to conformal matter including all possible interactions. The resulting classical theory has the desired scale symmetry as a remnant of the conformal invariance of the matter sector.  
Any mass breaking terms in both the bosonic and fermionic sectors are forbidden classically. We know however how the quantum corrections associated to the matter interactions violate the scale invariance inducing non-trivial scales due to renormalization effects, namely, introducing Coleman-Weinberg mass terms, anomalous dimensions in the fields and couplings which break ``softly" the classical scale invariance. This kind of scale violation effects are under control via the renormalizability of the matter sector around a weakly curved gravitational background (in the approximation where the gravitational background is treated classically). 

Disregarding at this step the well known field theory mass-scale hierarchy problem (which will not be present in the supersymmetric extension of the theory), we present in section 2 the classical ${\cal R}^2$ theory coupled to matter.  We show that in the Einstein frame the conformally coupled scalars and the dual field $\phi$ (the no scale modulus), associated to the extra degrees of freedom of the ${\cal R}^2$ theory, form together a non-trivial scalar manifold in the conventional Einstein frame: ${\cal M}_0 =H^{n+1}\equiv SO(1,1+n)/SO(1+n)$. This is an important result, since any extension (supersymmetric or not) of the scale invariant ${\cal R}^2$ theory has to contain ${\cal M}_0 $ as a submanifold. This universality  property of  ${\cal M}_0 $ let us to examine in more detail the vacuum structure of the universal sector based on $SO(1,1+n)$ theory in the presence of scaling violating terms, like for instance the undressed Einstein term ${\cal R}$ in the Jordan frame.  We have shown explicitly that the ${\cal R}$ term is the origin of an inflationary slow roll transition from the de Sitter to the flat space, in a very similar way as in the minimal Starobinsky model.  More importantly, we claim that this transition can be treated semi-classically once the fluctuation of the scalars fields are protected by the induced de Sitter (mass$)^2$. 
The structure of the Minkowski vacuum is generically different from that of the minimal Starobinsky model.
The classical Minkowski vacuum can be degenerate in certain cases so that the quantum mass terms are becoming relevant,
since they will be needed to determine the correct quantum vacuum of the theory. 

The third step of this work is devoted to the supersymmetric extension of the minimal $SO(1,1+n)$ model. This was achieved in section $3$ in the framework of $N=1$ supergravity where we show that the $SO(1,1+n)/SO(1+n)$ manifold is minimally extended to a   K\"ahlerian manifold ${\cal M}_1=SU(1,1+n)/ U(1)\times SU(1+n)$.  
The observation here is that ${\cal M}_1$ is nothing but the
no-scale manifold which controls the kinetic part of the scalars and absorbs in a ``magic" way the negative contribution of the supergravity auxiliary fields, namely the well known $-3m^2_{3/2}$ term of the supergravity potential. This ``magic'' property of ${\cal M}_1 $ leads to the positivity properties the $F$ part of the potential {\it independently} if supersymmetry is broken or not, provided that the no-scale modulus does not appear in the superpotential. As a result, the supergravity $F$ part of the potential looks like the global supersymmetric potential dressed by the no-scale modulus $\phi$ in a multiplicative sense. This positivity property is explained in section $3$ and is further extended in more general no-scale manifolds coming from strings in section 5. In fact, this is the main reason for the failure of the majority of the models proposed in the literature when trying to find metastable de Sitter vacua by modifying in an ad-hoc way the $T$-dependent superpotential. We show by presenting explicit examples, both 
in supergravity in section 3 but also in superstring induced supergravity models in section 5,
 that the only natural way to resolve the instabilities of a $T$-dependent superpotential can be achieved when a non trivial Fayet-Iliopoulos term is created by a gauging of a $U(1)_R$ symmetry. 

 However, this is still not yet an achievement, since all possible $U(1)_R$ gaugings, able to create Fayet-Iliopoulos terms, are anomalous! This means that at the quantum level the theory is inconsistent. It implies that the gauging of $U(1)_R$ alone is not enough and has to be extended in a consistent way by gauging simultaneously some of the isometries of the scalar manifold. This leads to two generic extensions of $U(1)_R$, namely to the $U(1)_d$ and $U(1)_t$ gaugings, involving the dilatation symmetry or the axionic symmetry of the scalar manifold.  Both of them cure the anomaly by screening the $D$-term contribution in a scale invariant manner 
or, respectively, in a non-scale invariant manner. In both cases the de Sitter vacuum is destabilized either in a rapid or slow way 
giving in both cases inflationary behavior during the transition to the flat Minkowski vacuum.  

Concerning the applications in cosmology, for the case of the $U(1)_d$ resolution of the anomaly, the de Sitter instability is relatively fast and is given in terms of a polynomial potential of degree two in terms of $B$ and degree four in terms of $\Phi$. This kind of inflationary potentials has to be treated as two component chaotic inflationary model.  Generically in this
 kind of inflationary potentials the predictions for the ratio $r$ of the tensor to scalar fluctuations is relatively large $r$=0.2 to 0.01. These models will be relevant if the BICEP2 preliminary results turn out to be confirmed.  

In section 5 we investigated the superstring vacuum structure in the generic case where three anomalous 
$U(1)^A_R$ are involved. The resolution of the anomaly via  $U(1)^A_t$  induces a slow transition from de Sitter to flat Minkowski space via three, two or one
 inflaton components. We postulate that in superstring theory the resolution of the various anomalous $U(1)^A_R$ is generic in  all string compactifications.  

Namely we show that $N=1$ superstring constructions provide metastable de Sitter vacua supported by Fayet-Iliopoulos $D_R$-terms associated to the several anomalous $U(1)^A_R$  gauge symmetries. 
The superstring resolution of these anomalies is achieved via 
local axion shifts, which promote the several $U(1)_R$ symmetries to $ U(1)_{t}$ or $U(1)_d$ symmetries.
  
 \section{Acknowledgement*}
We are grateful to S. Ferrara for fruitful discussions, especially for some clarifications concerning the Fayet-Iliopoulos terms in supergravity and also about some of 
the restrictions concerning the equivalence of the  ``minimal'' versus ``non-minimal'' formulation of $N=1$ supergravity. 
We also appreciated fruitful discussions with V. Mukhanov, H. Partouche, K. Stelle, J. Troost and  B.~de~ Wit.
 C. K. acknowledges the theoretical physics group of Ludwig Maximilians University and 
Max-Planck-Institute in Munich for hospitality and he
likes to thank the University of Cyprus for hospitality, where part of the work was done.
C.K. and D.L. like to thank the theory department of CERN, where part of the work was done.
N.T.  acknowledge the Laboratoire de Physique Th\'eorique of Ecole Normale Sup\'erieure for hospitality. 
The work of C.K. and N.T. is also supported by the CEFIPRA