  We consider the topological string partition function, including the Nekrasov
deformation, for type IIB geometries with an A_{n-1} singularity over a Riemann
surface. These models realize the N=2 SU(n) superconformal gauge systems
recently studied by Gaiotto and collaborators. Employing large N dualities we
show why the partition function of topological strings in these backgrounds is
captured by the chiral blocks of A_{n-1} Toda systems and derive the dictionary
recently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zero
Riemann surfaces, we show how these systems can also be realized by Penner-like
matrix models with logarithmic potentials. The Seiberg-Witten curve can be
understood as the spectral curve of these matrix models which arises
holographically at large N. In this context the Nekrasov deformation maps to
the beta-ensemble of generalized matrix models, that in turn maps to the Toda
system with general background charge. We also point out the notion of a double
holography for this system, when both n and N are large.
