The tropological vertex

TL;DR

This paper reinterprets the topological vertex within the framework of relative Gromov-Witten invariants and tropical geometry, revealing symmetries and providing explicit computational methods.

Contribution

It connects the topological vertex to tropical and relative Gromov-Witten invariants, introducing a quantum algebraic symmetry structure.

Findings

Proves tropical symmetries of the topological vertex.

Develops a gluing formula for tropical curve enumeration.

Provides explicit descriptions of Gromov-Witten invariants.

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.