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On a Proof of the ADKMV Conjecture -- $3$-KP Integrability of the Topological Vertex

Zhiyuan Wang, Chenglang Yang, Jian Zhou

TL;DR

The work addresses the connection between open Gromov-Witten invariants of toric Calabi–Yau threefolds and integrable systems by proving the ADKMV conjecture in the framed, three-leg case. It introduces a novel boson-fermionic field assignment that yields a fermionic expression for the framed topological vertex as a Bogoliubov transform, and then derives deterministic matrix formulas via Wick’s theorem. The main contributions are (i) the validation of the framed ADKMV Conjecture for arbitrary partitions and framings, (ii) the demonstration that the generating function of the open Gromov-Witten invariants is a tau-function of the $3$-component KP hierarchy, and (iii) the establishment that gluing of topological vertices preserves multi-component KP integrability, thereby controlling open GW theory of all smooth toric Calabi–Yau threefolds. This work provides a rigorous mathematical foundation for the integrable structure underlying topological vertex methods and local mirror symmetry in toric geometry.

Abstract

We present a mathematical proof of the Aganagic-Dijkgraaf-Klemm-Mariño-Vafa Conjecture proposed in 2006, which states that the generating function of the topological vertex, i.e., the generating function of the open Gromov-Witten invariants of $\mathbb{C}^3$, satisfies the $3$-component KP hierarchy. In our proof we introduce a boson-fermionic field assignment which generalizes the well-known boson-fermion correspondence. The proof also works for the generalization to the framed topological vertex case conjectured by Deng and Zhou. As a consequence, open Gromov-Witten theory of all smooth toric Calabi-Yau threefolds are controlled by the multi-component KP hierarchy.

On a Proof of the ADKMV Conjecture -- $3$-KP Integrability of the Topological Vertex

TL;DR

The work addresses the connection between open Gromov-Witten invariants of toric Calabi–Yau threefolds and integrable systems by proving the ADKMV conjecture in the framed, three-leg case. It introduces a novel boson-fermionic field assignment that yields a fermionic expression for the framed topological vertex as a Bogoliubov transform, and then derives deterministic matrix formulas via Wick’s theorem. The main contributions are (i) the validation of the framed ADKMV Conjecture for arbitrary partitions and framings, (ii) the demonstration that the generating function of the open Gromov-Witten invariants is a tau-function of the -component KP hierarchy, and (iii) the establishment that gluing of topological vertices preserves multi-component KP integrability, thereby controlling open GW theory of all smooth toric Calabi–Yau threefolds. This work provides a rigorous mathematical foundation for the integrable structure underlying topological vertex methods and local mirror symmetry in toric geometry.

Abstract

We present a mathematical proof of the Aganagic-Dijkgraaf-Klemm-Mariño-Vafa Conjecture proposed in 2006, which states that the generating function of the topological vertex, i.e., the generating function of the open Gromov-Witten invariants of , satisfies the -component KP hierarchy. In our proof we introduce a boson-fermionic field assignment which generalizes the well-known boson-fermion correspondence. The proof also works for the generalization to the framed topological vertex case conjectured by Deng and Zhou. As a consequence, open Gromov-Witten theory of all smooth toric Calabi-Yau threefolds are controlled by the multi-component KP hierarchy.
Paper Structure (28 sections, 13 theorems, 218 equations)

This paper contains 28 sections, 13 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Preliminaries of Boson-Fermion Correspondence
  3. Schur functions and skew Schur functions
  4. Fermionic Fock space
  5. Cut-and-join operator
  6. Boson-fermion correspondence
  7. Properties of the vertex operators $\Gamma_\pm$
  8. KP hierarchy, Bogoliubov transform, and multi-component generalization
  9. Framed Topological Vertex and Framed ADKMV Conjecture
  10. Partition function of open Gromov-Witten invariants of $\mathbb{C}^3$
  11. Topological vertex and framed topological vertex
  12. ADKMV Conjecture and framed ADKMV Conjecture
  13. Main result
  14. Step 1. An alternative expression of the framed topological vertex as transfer matrix elements
  15. Step 2. Determinantal expression for $C_{\mu^1,\mu^2,\mu^3}^{(a_1,a_2,a_3)} (q)$
  16. ...and 13 more sections

Key Result

Theorem 1.2

For a partition $\mu$ and an integer $a\in {\mathbb Z}$, define an operator $\Psi_\mu^{(a)}$ on the fermionic Fock space as in Definition def-boson-ferm. Then the framed topological vertex is given by: where $K$ is the cut-and-join operator on the fermionic Fock space.

Theorems & Definitions (29)

  • Conjecture 1.1: ADKMV Conjecture
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Conjecture 3.1: ADKMV Conjecture adkmvakmv
  • Conjecture 3.2: framed ADKMV Conjecture dz1
  • Theorem 3.3: Theorem 5.2 in dz2
  • Theorem 3.4: = Theorem \ref{['thm:Main2']}
  • ...and 19 more