We prove a conjecture of Bousseau, van Garrel and the first-named author relating, under suitable positivity conditions, the higher genus maximal contact $\log$ Gromov–Witten invariants of Looijenga pairs to other curve counting invariants of Gromov–Witten/Gopakumar–Vafa type. The proof consists of a closed-form $q$-hypergeometric resummation of the quantum tropical vertex calculation of the $\log$ invariants in presence of infinite scattering. The resulting identity of $q$-series appears to be new and of independent combinatorial interest.

中文翻译：

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关于准驯服的 Looijenga 对

我们证明了 Bousseau、van Garrel 和第一作者的猜想，在适当的正性条件下，Looijenga 对的较高属最大接触 $\log$ Gromov–Witten 不变量与 Gromov–Witten/Gopakumar– 的其他曲线计数不变量有关。瓦法类型。证明包括在存在无限散射的情况下 $\log$ 不变量的量子热带顶点计算的封闭形式 $q$-超几何恢复。$q$ 系列的结果身份似乎是新的并且具有独立的组合兴趣。