KP Integrability of Triple Hodge Integrals: III—Cut-and-Join Description, KdV Reduction, and Topological Recursions

In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive the complete families of the Heisenberg–Virasoro constraints. We also construct several equivalent versions of the cut-and-join operators. These operators describe the algebraic version of topological recursion. For the specific values of parameters associated with the KdV reduction, we prove that these tau-functions are equal to the generating functions of intersection numbers of \(\psi \) $\psi$ and \(\kappa \) $\kappa$ classes. We interpret this relation as a symplectic invariance of the Chekhov–Eynard–Orantin topological recursion and prove this recursion for the general \(\Theta \) $\mathrm{\Theta }$ -case.