Quivers, curves, and the tropical vertex

  • Mark Gross

    University of California, San Diego, La Jolla, USA
  • Rahul Pandharipande

    ETH Zürich, Switzerland


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.

Cite this article

Mark Gross, Rahul Pandharipande, Quivers, curves, and the tropical vertex. Port. Math. 67 (2010), no. 2, pp. 211–259

DOI 10.4171/PM/1865