${\cal R}^2$ inflation from scale invariant supergravity and anomaly free superstrings with fluxes

The ${\cal R}^2$ scale invariant gravity theory coupled to conformally invariant matter is investigated. We show that in the non-supersymmetric case the conformally coupled scalars belong to an $SO(1, 1+n)/SO(1+n)$ manifold, while in the supersymmetric case the scalar manifold becomes isomorphic to the K\"ahlerian space ${\cal M}_n$=$SU(1, 1+n)/ U(1)\times SU(1+n)$. In both cases when the underlying scale symmetry is preserved the vacuum corresponds to de Sitter space. Once the scale symmetry is broken by quantum effects, a transition to flat space becomes possible. We argue that the scale violating terms are induced by anomalies related to a $U(1)_R$ symmetry. The anomaly is resolved via the gauging of a Peccei-Quinn axion shift symmetry. The theory describes an inflationary transition from de Sitter to flat Minkowski space, very similar to the Starobinsky inflationary model. The extension to metastable de Sitter superstring vacua is also investigated. The scalar manifold is extended to a much richer manifold, but it contains always ${\cal M}_n$ as a sub-manifold. In superstrings the metastability is induced by axions that cure the anomalies in chiral $N=1$ (or even $N=0$) supersymmetric vacua via a Green-Schwarz/Peccei-Quinn mechanism generalized to four dimensions. We present some typical superstring models and discuss the possible stabilization of the no-scale modulus.