The tropical vertex
type: Item
instance of: scholarly article
title: The tropical vertex
zbMATH Open document ID: 1205.14069
DOI: 10.1215/00127094-2010-025
author: Mark Gross
author: Bernd Siebert
author: Rahul Pandharipande
published in: Duke Mathematical Journal
publication date: 2010-07-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.0779
review text: The tropical vertex group consists of formal 1-parameter families of symplectomorphisms of the 2-dimensional algebraic torus, and in this article, the authors establish a formula for an ordered product factorization of the commutator of generators in the tropical vertex group in terms of certain relative genus zero Gromov-Witten invariants of toric surfaces. More generally, they prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of relative genus zero Gromov-Witten invariants. The proof begins by using scattering diagram expansions to connect commutators to tropical curve counts, which can be then related to holomorphic curve counts. The commutator formulas are then established via degeneration and exact Gromov-Witten calculations. An interesting open question raised by this work is whether and how the higher genus Gromov-Witten invariants are related to the tropical vertex group.
reviewed by: Hans U. Boden
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 53D45
zbMATH DE Number: 5731760
zbMATH Keywords: tropical vertex group
zbMATH Keywords: relative Gromov-Witten invariants
MaRDI profile type: Publication
OpenAlex ID: W3100215086
arXiv ID: 0902.0779
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description: scientific article