[Paper Review] KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions

A. Alexandrov|arXiv (Cornell University)|Aug 23, 2021

Algebraic Geometry and Number Theory38 references4 citations

TL;DR

This paper establishes the KP integrability of triple Hodge integrals under the Calabi–Yau condition by deriving complete Heisenberg–Virasoro constraints and constructing cut-and-join operators that realize algebraic topological recursion. It proves that for KdV-reduced parameters, these tau-functions coincide with generating functions of intersection numbers of ψ and κ classes, confirming symplectic invariance of the Chekhov–Eynard–Orantin recursion in the general Θ-case.

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.

Motivation & Objective

• To establish KP integrability of triple Hodge integrals satisfying the Calabi–Yau condition.
• To derive complete families of Heisenberg–Virasoro constraints for the associated tau-functions.
• To construct cut-and-join operators that realize algebraic topological recursion for these invariants.
• To prove that KdV-reduced triple Hodge integrals match generating functions of ψ and κ class intersection numbers.
• To confirm symplectic invariance of the Chekhov–Eynard–Orantin topological recursion in the general Θ-case.

Proposed method

• Derives Heisenberg–Virasoro constraints from the symmetries of the KP hierarchy.
• Constructs multiple equivalent versions of cut-and-join operators that encode the algebraic topological recursion.
• Uses the cut-and-join operators to recursively generate tau-functions via a differential equation with the Euler operator as the leading term.
• Applies KdV reduction by translating parameters to κ-class relations, linking triple Hodge integrals to standard intersection theory.
• Identifies the KdV-reduced tau-functions with generating functions of ψ and κ class intersection numbers.
• Proves topological recursion for triple Hodge integrals and their Θ-deformations using the cut-and-join framework.

Experimental results

Research questions

• RQ1 Can the full set of Heisenberg–Virasoro constraints be derived for tau-functions of triple Hodge integrals under the Calabi–Yau condition?
• RQ2 How can cut-and-join operators be constructed to describe the algebraic topological recursion for these invariants?
• RQ3 Is there a precise correspondence between KdV-reduced triple Hodge integrals and intersection numbers of ψ and κ classes?
• RQ4 Does the Chekhov–Eynard–Orantin topological recursion hold for the general Θ-case of Hodge integrals?
• RQ5 Can symplectic invariance of the topological recursion be proven for the Θ-Hodge integral case?

Key findings

• The tau-functions of triple Hodge integrals under the Calabi–Yau condition satisfy a complete set of Heisenberg–Virasoro constraints.
• Cut-and-join operators are constructed in multiple equivalent forms, providing an algebraic realization of topological recursion.
• For KdV-reduced parameters, the tau-functions are proven to equal the generating functions of intersection numbers of ψ and κ classes.
• The KdV reduction maps the triple Hodge integral tau-functions to the Kontsevich–Witten and Brézin–Gross–Witten tau-functions.
• The paper proves that the Chekhov–Eynard–Orantin topological recursion holds for the general Θ-case, confirming symplectic invariance.
• Explicit topological expansions of the tau-functions are derived, with coefficients expressed in terms of rational functions in parameters p and q.