Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

中文翻译：

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Log Calabi-Yau 表面的日志 BPS 数

令 $(S,E)$ 是一个对数 Calabi-Yau 曲面对，其中 $E$ 是一个平滑除数。我们在 $(S,E)$ 中定义了 $\mathbb{A}^1$-curves 的新的推测整数值计数。这些 log BPS 数是通过类似于磁盘计数的多重覆盖公式的公式从沿 $E$ 的最大切线的 0 类 log Gromov-Witten 不变量推导出来的。描述和验证了算术属的 del Pezzo 曲面和曲线类与属 0 局部 BPS 数的猜想关系。我们陈述了一些猜想并提供了计算证据。