量子几何、稳定性与模性
Quantum geometry, stability and modularity
AI总结：
本文利用 PT 不变量与秩 0 DT 不变量间的新数学关系，计算了超几何型紧致 Calabi-Yau 三维流形的 Abel D4-D2-D0 指标生成级数，证实其模性并预测无穷级指标，同时为直接积分法计算更高亏格 GV 不变量提供了新约束。
摘要：
通过利用 Pandharipande-Thomas (PT) 不变量（与 Gopakumar-Vafa (GV) 不变量密切相关）和计数 D4-D2-D0 BPS 束缚态的秩 0 Donaldson-Thomas (DT) 不变量之间的新数学关系，我们严格计算了超几何类型的紧单参数 Calabi-Yau 三维流形的 Abel D4-D2-D0 指标生成级数的前几项。在所有能将 GV 不变量计算到足够高亏格的情况下，我们发现了生成级数具有模性的惊人证据，并预测了无穷级数的 Abel D4-D2-D0 指标。反之，我们利用这些结果为直接积分法提供了新约束，从而能将 GV 不变量（进而拓扑弦配分函数）计算到比以往更高的亏格。GV/PT/DT 不变量之间的三角关系由一个关联 PT 和秩 0 DT 不变量的新显式公式支撑，该公式在附录中由第二作者证明。作为推论，我们得到了 Picard 秩为 1 的 CY 三维流形 PT 和 GV 不变量的严格 Castelnuovo 型界。
英文摘要：
By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
