  We obtain exact results in \alpha' for open and closed A-model topological
string amplitudes on a large class of toric Calabi-Yau threefolds by using
their correspondence with five dimensional gauge theories. The toric
Calabi-Yau's that we analyze are obtained as minimal resolution of cones over
Y(p,q) manifolds and give rise via M-theory compactification to SU(p) gauge
theories on R^4 x S^1. As an application we present a detailed study of the
local F_2 case and compute open and closed genus zero Gromov-Witten invariants
of the C^3/Z_4 orbifold. We also display the modular structure of the
topological wave function and give predictions for higher genus amplitudes.The
mirror curve in this case is the spectral curve of the relativistic A_1 Toda
chain. Our results also indicate the existence of a wider class of relativistic
integrable systems associated to generic Y(p,q) geometries.
