KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
Alexander Alexandrov
TL;DR
This work develops an algebraic approach to the topological recursion for triple Hodge integrals under Calabi–Yau constraints by embedding the generating functions into the KP/KdV hierarchy and exploiting Heisenberg–Virasoro symmetries. It derives complete Heisenberg–Virasoro constraints and cut-and-join descriptions, enabling a concrete algebraic topological recursion that generalizes KW/BGW models. The paper proves a KdV reduction identifying these triplet Hodge tau-functions with shifted KW/BGW intersection numbers and interprets this through symplectic invariance of Chekhov–Eynard–Orantin topological recursion, providing topological recursion for both α = 1 and α = 0 cases. It also develops a detailed factorization and translation framework for Virasoro group elements, yielding explicit cut-and-join operators and recursive structures for the associated tau-functions, with explicit computations for the low-genus, low-point terms.
Abstract
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 $ψ$ and $κ$ classes. We interpret this relation as a symplectic invariance of the Chekhov--Eynard--Orantin topological recursion and prove this recursion for the general $Θ$-case.