$SL(2,\mathbb{Z})$ Cosmological Attractors

We study cosmological theory where the kinetic term and potential have $SL(2,\mathbb{Z})$ symmetry. Potentials have a plateau at large values of the inflaton field, where the axion forms a flat direction. Due to the underlying hyperbolic geometry and special features of $SL(2,\mathbb{Z})$ potentials, the theory exhibits an $\alpha$-attractor behavior: its cosmological predictions are stable with respect to significant modifications of the $SL(2,\mathbb{Z})$ invariant potentials. We present a supersymmetric version of this theory in the framework of $\overline {D3}$ induced geometric inflation. The choice of $\alpha$ is determined by underlying string compactification. For example, in a CY compactification with $T^2$, one has $3\alpha=1$, the lowest discrete Poincar\'e disk target for LiteBIRD