Abstract
A tropical curve in \({\mathbb R^{3}}\) contributes to Gromov–Witten invariants in all genus. Nevertheless, we present a simple formula for how a given tropical curve contributes to Gromov–Witten invariants when we encode these invariants in a generating function with exponents of \({\lambda}\) recording Euler characteristic. Our main modification from the known tropical correspondence formula for rational curves is as follows: a trivalent vertex, which before contributed a factor of n to the count of zero-genus holomorphic curves, contributes a factor of \({2\sin(n\lambda/2)}\). We explain how to calculate relative Gromov–Witten invariants using this tropical correspondence formula, and how to obtain the absolute Gromov–Witten and Donaldson–Thomas invariants of some 3-dimensional toric manifolds including \({\mathbb{C}P^{3}}\). The tropical correspondence formula counting Donaldson–Thomas invariants replaces n by \({i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich D., Chen Q.: Stable logarithmic maps to Deligne–Faltings pairs II. Asian J. Math. 18(3), 465–488 (2014)
Abramovich, D., Wise, J.: Invariance in Logarithmic Gromov–Witten Theory (2013). arxiv:1306.1222
Chen Q.: Stable logarithmic maps to Deligne–Faltings pairs I. Ann. Math. (2) 180(2), 455–521 (2014)
Gross M., Siebert B.: Logarithmic Gromov–Witten invariants. J. Am. Math. Soc. 26(2), 451–510 (2013)
Ionel E.-N.: GW invariants relative to normal crossing divisors. Adv. Math. 281, 40–141 (2015)
Markwig H., Rau J.: Tropical descendant Gromov–Witten invariants. Manuscr. Math. 129(3), 293–335 (2009)
Maulik D., Oblomkov A., Okounkov A., Pandharipande R.: Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds. Invent. Math. 186(2), 435–479 (2011)
Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}^{2}}\). J. Am. Math. Soc. 18(2), 313–377 (2005) (electronic)
Nishinou T., Siebert B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)
Pandharipande R.: Hodge integrals and degenerate contributions. Commun. Math. Phys. 208(2), 489–506 (1999)
Parker, B.: Notes on Exploded Manifolds and a Tropical Gluing Formula for Gromov–Witten Invariants (2016). arXiv:1605.00577
Parker, B.: Tropical Enumeration of Curves in Blowups of the Projective Plane (2014). arXiv:1411.5722
Parker, B.: De Rham Theory of Exploded Manifolds (2011). arXiv:1003.1977
Parker, B.: Gromov–Witten Invariants of Exploded Manifolds (2011). arXiv:1102.0158
Parker, B.: Exploded Manifolds. Adv. Math. 229, 3256–3319 (2012). arXiv:0910.4201
Parker, B.: Log geometry and exploded manifolds. Abh. Math. Sem. Hamburg 82, 43–81 (2012). arxiv:1108.3713
Parker, B.: Holomorphic Curves in Exploded Manifolds: Kuranishi Structure (2013). arXiv:1301.4748
Parker, B.: Universal Tropical Structures for Curves in Exploded Manifolds (2013). arXiv:1301.4745
Parker, B.: Holomorphic Curves in Exploded Manifolds: Virtual Fundamental Class (2015). arXiv:1512.05823
Speyer D.E.: Parameterizing tropical curves I: curves of genus zero and one. Algebra Number Theory 8(4), 963–998 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by N. A. Nekrasov
Funded by ARC Grant DP140100296.
Rights and permissions
About this article
Cite this article
Parker, B. Three Dimensional Tropical Correspondence Formula. Commun. Math. Phys. 353, 791–819 (2017). https://doi.org/10.1007/s00220-017-2874-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-017-2874-1