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The tropical vertex. (English) Zbl 1205.14069
The tropical vertex group consists of formal 1-parameter families of symplectomorphisms of the 2-dimensional algebraic torus, and in this article, the authors establish a formula for an ordered product factorization of the commutator of generators in the tropical vertex group in terms of certain relative genus zero Gromov-Witten invariants of toric surfaces. More generally, they prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of relative genus zero Gromov-Witten invariants. The proof begins by using scattering diagram expansions to connect commutators to tropical curve counts, which can be then related to holomorphic curve counts. The commutator formulas are then established via degeneration and exact Gromov-Witten calculations. An interesting open question raised by this work is whether and how the higher genus Gromov-Witten invariants are related to the tropical vertex group.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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