Exact Results for Topological Strings on Resolved Yp,q Singularities

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-Yaus that we analyze are obtained as minimal resolution of cones over Y-p,Y-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,Y-q geometries.