  We study the Nekrasov-Shatashvili limit of the N=2 supersymmetric gauge
theory and topological string theory on certain local toric Calabi-Yau
manifolds. In this limit one of the two deformation parameters \epsilon_{1,2}
of the Omega background is set to zero and we study the perturbative expansion
of the topological amplitudes around the remaining parameter. We derive
differential equations from Seiberg-Witten curves and mirror geometries, which
determine the higher genus topological amplitudes up to a constant. We show
that the higher genus formulae previously obtained from holomorphic anomaly
equations and boundary conditions satisfy these differential equations. We also
provide a derivation of the holomorphic anomaly equations in the
Nekrasov-Shatashvili limit from these differential equations.
