  We consider the topological B-model on local Calabi-Yau geometries. We show
how one can solve for the amplitudes by using W-algebra symmetries which
encodes the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the
highly effective fermionic/brane formulation this leads to a free fermion
description of the amplitudes. Furthermore we argue that topological strings on
Calabi-Yau geometries provide a unifying picture connecting non-critical
(super)strings, integrable hierarchies, and various matrix models. In
particular we show how the ordinary matrix model, the double scaling limit of
matrix models, and Kontsevich-like matrix model are all related and arise from
studying branes in specific local Calabi-Yau three-folds. We also show how
A-model topological string on P^1 and local toric threefolds (and in particular
the topological vertex) can be realized and solved as B-model topological
string amplitudes on a Calabi-Yau manifold.
