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Portugaliae Mathematica

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Volume 67, Issue 2, 2010, pp. 211–259
DOI: 10.4171/PM/1865

Published online: 2010-04-27

Quivers, curves, and the tropical vertex

Mark Gross[1] and Rahul Pandharipande[2]

(1) University of California, San Diego, La Jolla, USA
(2) 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.

Keywords: Quiver, Gromov–Witten, tropical

Gross Mark, Pandharipande Rahul: Quivers, curves, and the tropical vertex. Port. Math. 67 (2010), 211-259. doi: 10.4171/PM/1865