The tropological vertex
Abstract
The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mari\~no and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants.
Cite
@article{arxiv.2205.02555,
title = {The tropological vertex},
author = {Norman Do and Brett Parker},
journal= {arXiv preprint arXiv:2205.02555},
year = {2025}
}
Comments
39 pages, 24 figures. v3: Incorrect sign in previous version's equation (18) now appears correctly in equation (19), necessitating further sign changes later in the paper. References updated. v4: note added about the contribution of constant curves