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We consider the topological string partition function, including the Nekrasovdeformation, for type IIB geometries with an A_{n-1} singularity over a Riemannsurface. These models realize the N=2 SU(n) superconformal gauge systemsrecently studied by Gaiotto and collaborators. Employing large N dualities weshow why the partition function of topological strings in these backgrounds iscaptured by the chiral blocks of A_{n-1} Toda systems and derive the dictionaryrecently proposed by Alday, Gaiotto and Tachikawa. For the case of genus zeroRiemann surfaces, we show how these systems can also be realized by Penner-likematrix models with logarithmic potentials. The Seiberg-Witten curve can beunderstood as the spectral curve of these matrix models which arisesholographically at large N. In this context the Nekrasov deformation maps tothe beta-ensemble of generalized matrix models, that in turn maps to the Todasystem with general background charge. We also point out the notion of a doubleholography for this system, when both n and N are large.

Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems