On Projective Gravity and the vanishing of the Cosmological Constant

We generalize Einstein's Lagrangian in a non-polynomial (in R) way. The usual Lagrangian (linear in R) is the zero $\alpha'$ limit of our theory, where $\alpha'$ is a parameter that is interpreted as the inverse cosmological costant before the Planck time. The theory space of this lagrangian admits a ${\bf Z_{2}}$ modular group, namely $R \leftrightarrow 1/R$. Independence of the modular invariant expectation values from the number of `Big Bangs' enforces a quantization condition for the cosmological constant. At the semiclassical approximation we obtain $\Lambda =0$, and a vacuum equation which is equivalent to inflation cosmology. D=4 and D=1 universes are obtained as unique (and topologically separated by the D=2 semiclassical barrier) integer dimension solutions. They correspond to the first excited level and the ground state respectively of our projective gravity.