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

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.