Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies

Abstract

For any generalized Frobenius manifold with non-flat unity, we construct a bihamiltonian integrable hierarchy of hydrodynamic type which is an analogue of the Principal Hierarchy of a Frobenius manifold. We show that such an integrable hierarchy, which we also call the Principal Hierarchy, possesses Virasoro symmetries and a tau structure, and the Virasoro symmetries can be lifted to symmetries of the tau-cover of the integrable hierarchy. We derive the loop equation from the condition of linearization of actions of the Virasoro symmetries on the tau function, and construct the topological deformation of the Principal Hierarchy of a semisimple generalized Frobenius manifold with non-flat unity. We also give two examples of generalized Frobenius manifolds with non-flat unity and show that they are closely related to the well-known integrable hierarchies: the Volterra hierarchy, the q-deformed KdV hierarchy and the Ablowitz–Ladik hierarchy.