Three dimensional tropical correspondence formula

TL;DR

This paper introduces a simplified tropical correspondence formula for computing Gromov-Witten invariants in three dimensions, incorporating genus information via a generating function and extending to Donaldson-Thomas invariants for toric manifolds.

Contribution

It presents a novel modification of the tropical correspondence formula for 3D curves, replacing vertex contributions with sine functions and applying it to various invariants.

Findings

Derived a new formula for tropical contributions involving sine functions.

Applied the formula to compute invariants of $ ext{CP}^3$ and other toric manifolds.

Extended the correspondence to Donaldson-Thomas invariants with a new substitution.

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 $\mathbb CP^{3}$. The tropical correspondence formula counting Donaldson-Thomas invariants replaces $n$ by $i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}$.