Dubrovin-Zhang Type Loop Equations for Hodge Integrals with Target Varieties

For any smooth projective variety with semisimple quantum cohomology, in 2014, Dubrovin, Liu, Yang and Zhang proposed an integrable hierarchy for its associated Hodge integrals, but the loop equation is yet to be established. In this paper, we explicitly derive Dubrovin-Zhang type loop equation for general Hodge integrals of any smooth projective variety (not necessarily with semisimple quantum cohomology) using Virasoro constraints. As applications, we study the loop equations and their solutions for Hodge integrals of a point and \ (CP¹\).