We present the Lax pair formalism for certain extension of the continuous limit 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 CP 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.

Abstract. We generalize the Toda lattice hierarchy by considering N + M dependent variables.We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ǫ-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [4]. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groupsW (N) (AN+M−1) of the A series, defined in [9].

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.

Abstract. We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.

We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold C k,m with a tau function of the bigraded Toda hierarchy.Date: 4/4/2013.