Fractional Volterra Hierarchy And Its Correspondence To Cubic Hodge Hierarchy

Boris Dubrovin and Di Yang conjectured in their work that the generating function of cubic Hodge integrals satisfying local Calabi-Yau condition is the tau function of a new integrable hierarchy,which should be some reduction of the semi-discrete 2D Toda hierarchy.We construct this hierarchy,which we call fractional Volterra hierarchy,in terms of Lax equations.Using the method of 2D Toda hierarchy,we research on the Hamiltonian structure,tau structure,bilinear identities and Virasoro symmetries of fractional Volterra hierarchy.On the other hand,as a certain deformation of KdV hierarchy,cubic Hodge hierarchy also has Virasoro constraints,which can be naturally induced from those of KdV hierarchy.By comparing the Virasoro operators of fractional Volterra hierarhy and cubic Hodge hierarchy,we obtain the transformation between the two hierarchies.We prove the Dubrovin-Yang conjecture with the help of the uniqueness of solutions of Virasoro constraints.