# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] D. Abramovich, T. Graber, and A. Vistoli, Gromov-Witten theory for Deligne-Mumford stacks . · Zbl 1193.14070 · arxiv.org [2] J. Bryan and R. Pandharipande, Curves in Calabi-Yau threefolds and topological quantum field theory , Duke Math. J. 126 (2005), 369–396. · Zbl 1084.14053 · doi:10.1215/S0012-7094-04-12626-0 [3] -, Local Gromov-Witten theory of curves , J. Amer. Math. Soc. 21 (2008), 101–136. · Zbl 1126.14062 · doi:10.1090/S0894-0347-06-00545-5 [4] W. Chen and Y. Ruan, “Orbifold Gromov-Witten theory” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001) , Contemp. Math. 310 , Amer. Math. Soc., Providence, 2002, 25–85. · Zbl 1091.53058 [5] A. Gathmann, Relative Gromov-Witten invariants and the mirror formula , Math. Ann. 325 (2003), 393–412. · Zbl 1043.14016 · doi:10.1007/s00208-002-0345-1 [6] A. Gathmann and H. Markwig, The numbers of tropical plane curves through points in general position , J. Reine Angew. Math. 602 (2007), 155–177. · Zbl 1115.14049 · doi:10.1515/CRELLE.2007.006 [7] R. Gopakumar and C. Vafa, M-theory and topological strings—I ,\arxivhep-th/9809187v1$$\!\!$$. · Zbl 0922.32015 [8] -, M-theory and topological strings—II ,\arxivhep-th/9812127v1$$\!\!$$. [9] T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves , Duke Math. J. 30 (2005), 1–37. · Zbl 1088.14007 · doi:10.1215/S0012-7094-05-13011-3 [10] M. Gross and B. Siebert, From real affine geometry to complex geometry . · Zbl 1266.53074 · arxiv.org [11] E.-N. Ionel and T. H. Parker, Relative Gromov-Witten invariants , Ann. of Math. (2) 157 (2003), 45–96. JSTOR: · Zbl 1039.53101 · doi:10.4007/annals.2003.157.45 · links.jstor.org [12] P. Johnson, R. Pandharipande, and H.-H. Tseng, Abelian Hurwitz-Hodge integrals . · arxiv.org [13] M. Kontsevich and Y. Soibelman, “Affine structures and non-Archimedean analytic spaces” in The Unity of Mathematics , Progr. Math. 244 , Birkhäuser, Boston, 2006, 321–385. · Zbl 1114.14027 · doi:10.1007/0-8176-4467-9_9 [14] -, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations . · arxiv.org [15] A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds , Invent. Math. 145 (2001), 151–218. · Zbl 1062.53073 · doi:10.1007/s002220100146 [16] J. Li, Stable morphisms to singular schemes and relative stable morphisms , J. Differential Geom. 57 (2001), 509–578. · Zbl 1076.14540 [17] -, A degeneration formula for G-W invariants , J. Differential Geom. 60 (2002), 199–293. · Zbl 1063.14069 [18] D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory . · Zbl 1317.14126 · arxiv.org [19] G. Mikhalkin, Enumerative tropical algebraic geometry in $$\RR^2$$ , J. Amer. Math. Soc. 18 (2005), 313–377. · Zbl 1092.14068 · doi:10.1090/S0894-0347-05-00477-7 [20] T. Nishinou and B. Siebert, Toric degenerations of toric varieties and tropical curves , Duke Math. J. 135 (2006), 1–51. · Zbl 1105.14073 · doi:10.1215/S0012-7094-06-13511-1 [21] R. Pandharipande, Hodge integrals and degenerate contributions , Comm. Math. Phys. 208 (1999), 489–506. · Zbl 0953.14036 · doi:10.1007/s002200050766 [22] -, “Three questions in Gromov-Witten theory” in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) , Higher Ed. Press, Beijing, 2002, 503–512. · Zbl 1047.14043 [23] M. Reineke, Poisson automorphisms and quiver moduli . · Zbl 1232.53072 · arxiv.org [24] -, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants . · Zbl 1266.16013 · arxiv.org [25] N. Takahashi, Log mirror symmetry and local mirror symmetry . · Zbl 1066.14048 · arxiv.org [26] A. Zinger, A comparison theorem for Gromov-Witten invariants in the symplectic category . · Zbl 1225.14046 · doi:10.1016/j.aim.2011.05.021 · arxiv.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.