  The generating function of cubic Hodge integrals satisfying the local
Calabi-Yau condition is conjectured to be a tau function of a new integrable
system which can be regarded as a fractional generalization of the Volterra
lattice hierarchy, so we name it the fractional Volterra hierarchy. In this
paper, we give the definition of this integrable hierarchy in terms of Lax pair
and Hamiltonian formalisms, construct its tau functions, and present its
multi-soliton solutions.
