Quivers, curves, and the tropical vertex

2009/09/28 by M. Gross, Mark Gross, Gross, M. +3 · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT

paper · pdf · doi:10.48550/arxiv.0909.5153

43 pages

arxiv created 2009/09/28 · openalex publication_date 2009/09/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the Gromov-Witten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by Gross-Siebert and Kontsevich). Where possible, we present both the quiver and Gromov-Witten perspectives.

Citations

• Poisson automorphisms and quiver moduli
• Moduli of representations of quivers
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Cohomology of quiver moduli, functional equations, and integrality of Donaldson–Thomas type invariants

Cited by

• Lectures on stabilized ellipsoid embeddings
• Computations in equivariant Gromov-Witten theory of GKM spaces
• Wild wall crossing and BPS giants

Related

• Refined GW/Kronecker correspondence
• The extended tropical vertex group
• Universal covers and the GW/Kronecker correspondence
• Quivers and curves in higher dimension
• Quiver representations in toric geometry