Abstract:Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of “theta functions” on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of “contractible” tropical curves when showing that the relevant log curves are torically transverse.

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中文翻译：

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簇变体的 Theta 基和 log Gromov-Witten 不变量

摘要：使用镜像对称的启发式方法，Gross、Hacking、Keel、Kontsevich 和 Siebert 的组合在各种 log Calabi-Yau 空间的坐标环上给出了“theta 函数”的规范基的组合构造，包括簇变体。我们证明簇变体的 theta 基由镜像/朗兰兹双簇变体的辛叶的某些后代对数 Gromov-Witten 不变量决定，正如 Gross-Hacking-Keel 的 Frobenius 结构猜想所预测的那样。我们进一步表明，这些 Gromov-Witten 计数通常由满足某些几何条件的有理曲线的朴素计数给出。作为一项关键的新技术工具，我们在显示相关对数曲线是环横向的时引入了“可收缩”热带曲线的概念。

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