## Abstract

A tropical curve in 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 λ 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 (nλ/ 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 CP^{3}. The tropical correspondence formula counting Donaldson–Thomas invariants replaces n by i^{-} ^{(} ^{1} ^{+} ^{n} ^{)}q^{n} ^{/} ^{2}+ i^{1} ^{+} ^{n}q^{-} ^{n} ^{/} ^{2}.

Original language | English |
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Pages (from-to) | 791-819 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 353 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 2017 |

Externally published | Yes |