三维热带对应公式
Three dimensional tropical correspondence formula
摘要：
$\mathbb R^{3}$中的热带曲线对所有属中的Gromov-Witten不变量都有贡献。然而，我们提出了一个简单的公式，一个给定的热带曲线如何有助于Gromov-Witten不变量时，我们编码这些不变量的生成函数与指数的$λ$记录欧拉示性数。我们对已知的有理曲线的热带对应公式的主要修改如下：一个三价顶点，它以前贡献了一个因子$n$到零亏格全纯曲线的计数，贡献了一个因子$2\sin(nλ/2)$。我们解释了如何利用这个热带对应公式计算相对Gromov-Witten不变量，以及如何获得包括$\mathbb CP^{3}$在内的$3$维环面流形的绝对Gromov-Witten和Donaldson-Thomas不变量.计算唐纳森-托马斯不变量的热带对应公式用$i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}$代替$n$。
英文摘要：
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 $λ$ 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 $\mathbb CP^{3}$. The tropical correspondence formula counting Donaldson-Thomas invariants replaces $n$ by $i^{-(1+n)}q^{n/2}+i^{1+n}q^{-n/2}$.
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