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arXiv · 2401.12726

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

Zhiyuan Wang·Chenglang Yang·Jian Zhou
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Abstract

We present a mathematical proof of the Aganagic-Dijkgraaf-Klemm-Mari\~no-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.

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Zhiyuan Wang, Chenglang Yang, Jian Zhou. 2024-01-23. On a Proof of the ADKMV Conjecture -- $3$-KP Integrability of the Topological Vertex. https://arxiv.org/abs/2401.12726

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