Hodge–FVH correspondence for special cubic Hodge integrals

Let $\mathcal{H}({\bf t};p,q,r;\epsilon)$ be the cubic Hodge free energy, let $w=\epsilon^2\partial_{t_0}^2\mathcal{H}({\bf t};p,q,r;\epsilon)$, and let $Z_{\rm cubic}({\bf t};p,q,r;\epsilon)=\exp(\mathcal{H}({\bf t};p,q,r;\epsilon))$ be the corresponding cubic Hodge partition function. The cubic Hodge hierarchy is the integrable hierarchy satisfied by $w$, deforming the Riemann hierarchy; the fractional Volterra hierarchy (FVH) is the hierarchy appearing in the conjectured correspondence. Hodge–FVH correspondence. The Hodge hierarchy for the special cubic Hodge integrals is equivalent, under a certain Miura-type transformation, to the fractional Volterra hierarchy (FVH). Furthermore, the corresponding cubic Hodge partition function gives a tau function of the FVH. This conjecture proposes an integrable-systems description of special cubic Hodge integrals, identifying their Hodge hierarchy with the fractional Volterra hierarchy and interpreting the cubic Hodge partition function as a tau function; the source does not provide evidence resolving the conjecture.