The Loop Equation for Special Cubic Hodge Integrals

As the first step of proving the Hodge-FVH correspondence recently proposed in [19], we derive the Virasoro constraints and the Dubrovin–Zhang loop equation for special cubic Hodge integrals. We show that this loop equation has a unique solution, and provide a new algorithm for the computation of these Hodge integrals. We also observe the gap phenomenon for certain special cubic Hodge free energies.