The tropical vertex

TL;DR

This paper establishes a deep connection between the algebraic structure of the tropical vertex group and genus 0 relative Gromov-Witten invariants of toric surfaces, using advanced techniques like scattering diagrams and tropical curve counts.

Contribution

It proves that ordered product factorizations in the tropical vertex group correspond to calculations of specific genus 0 relative Gromov-Witten invariants, revealing new links between algebraic and enumerative geometry.

Findings

Ordered product factorizations are equivalent to genus 0 relative Gromov-Witten invariants calculations.

The invariants involve full tangency to a toric divisor at an unspecified point.

Uses scattering diagrams, tropical curve counts, and degeneration formulas.

Abstract

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.