Communications in Mathematical Physics· 2026Q1
The Tropological Vertex
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- 2026year
Short summary
A new framework connects the topological vertex to relative Gromov–Witten invariants of log Calabi–Yau manifolds, revealing its inherent tropical symmetries.
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Key points
- The topological vertex is placed in the context of relative Gromov–Witten invariants of log Calabi–Yau manifolds.
- A gluing formula for tropical curves is used to compute these invariants.
- The topological vertex is shown to possess tropical symmetries.
- These symmetries are captured by a quantum torus Lie algebra related to the tropical vertex group.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mariño 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.
The authors' abstract, as published at the source. Communications in Mathematical Physics, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics