The extended Toda hierarchy

Using construction of logarithm of a difference operator, we present the Lax pair formalism for certain extension of the continuous version of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the ${\mathbb C}P^1$ topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrödinger hierarchy.