  A conjecture on the relation between the cubic Hodge integrals and the
topological vertex in topological string theory is resolved. 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 therein is shown to be a tau function of the
generalized KdV (aka Gelfand-Dickey) hierarchies.
