Three-Partition Hodge Integrals and the Topological Vertex

We give a new proof of the equivalence between the cubic Hodge integrals and the topological vertex in topological string theory. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters (w) over right arrow therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.