$R^4$ Couplings in M and Type II Theories on Calabi-Yau spaces

We discuss several implications of $R^4$ couplings in M theory when compactified on Calabi-Yau (CY) manifolds. In particular, these couplings can be predicted by supersymmetry from the mixed gauge-gravitational Chern-Simons couplings in five dimensions and are related to the one-loop holomorphic anomaly in four dimensional N=2 theories. We find a new contribution to the Einstein term in five dimensions proportional to the Euler number of the internal CY threefold, which corresponds to a one-loop correction of the hypermultiplet geometry. This correction is reproduced by a direct computation in type II string theories. Finally, we discuss a universal non-perturbative correction to the type IIB hyper-metric.