The tropical vertex

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.

57 pages, 1 figure; some typoes corrected and additional references given. Example 1.11 added

Cited by in corpus (38)

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• Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants
• Equations of tropical varieties
• 13/2 ways of counting curves
• Tropical refined curve counting from higher genera and lambda classes
• Vafa-Witten invariants from exceptional collections
• The quantum tropical vertex
• The canonical wall structure and intrinsic mirror symmetry
• Block-Göttsche invariants from wall-crossing
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• Stable maps to Looijenga pairs
• Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$
• Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting
• MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• Scattering diagrams, theta functions, and refined tropical curve counts
• A tropical view on Landau-Ginzburg models
• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Donaldson-Thomas invariants from tropical disks
• Refined floor diagrams from higher genera and lambda classes
• The local-orbifold correspondence for simple normal crossings pairs
• Stable maps to Looijenga pairs: orbifold examples
• Tropical Theta Functions and Log Calabi-Yau Surfaces
• Tangent curves to degenerating hypersurfaces
• Smoothing Toric Fano Surfaces Using the Gross-Siebert Algorithm
• Equations of mirrors to log Calabi--Yau pairs via the heart of canonical wall structures
• Dilogarithm identities in cluster scattering diagrams
• Rubber tori in the boundary of expanded stable maps
• Tropical counting from asymptotic analysis on Maurer-Cartan equations
• Quivers and curves in higher dimension
• Variations of BPS structure and a large rank limit
• Mirror symmetry for the Tate curve via tropical and log corals
• Scattering diagrams for generalized cluster algebras
• Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
• Relative Gromov-Witten and maximal contact conics
• Convergence of the mirror to a rational elliptic surface
• Enumerative geometry of surfaces and topological strings
• Tropical vertex and real enumerative geometry