  We present solutions for the higher genus topological string amplitudes on
Calabi-Yau-manifolds, which are realized as complete intersections in
Grassmannians. We solve the B-model by direct integration of the holomorphic
anomaly equations using a finite basis of modular invariant generators, the gap
condition at the conifold and other local boundary conditions for the
amplitudes. Regularity of the latter at certain points in the moduli space
suggests a CFT description. The A-model amplitudes are evaluated using a mirror
conjecture for Grassmannian Calabi-Yau by Batyrev, Ciocan-Fontanine, Kim and
Van Straten. The integrality of the BPS states gives strong evidence for the
conjecture.
