related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants countingD4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.

中文翻译：

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量子几何、稳定性和模块化


与 Gopakumar-Vafa (GV) 不变量相关，并且排名 0 Donaldson-Thomas (DT) 不变量计数 D4-D2-D0 BPS 束缚态，我们严格计算紧凑型阿贝尔 D4-D2-D0 索引生成序列中的前几项一参数 Calabi-Yau 超几何类型的三重。在所有可以将 GV 不变量计算到足够高的属的情况下，我们发现生成级数是模的，并预测阿贝尔 D4-D2-D0 索引的无限级数。相反，我们使用这些结果为直接积分方法提供新的约束，该方法允许将 GV 不变量（以及拓扑串配分函数）计算到比迄今为止更高的属数。 GV/PT/DT 不变量之间的三角关系由与 PT 和 0 级 DT 不变量相关的新显式公式提供支持，该公式由第二位作者在附录中进行了证明。作为推论，我们获得了 CY 三倍的 PT 和 GV 不变量的严格 Castelnuovo 型界限，皮卡德排名第一。