论文标题
旨在完善紧凑型卡拉比（Calabi-Yau）上的拓扑字符串3倍
Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds
论文作者
论文摘要
我们提出了一项建议，以根据精制的全体形态异常方程来计算椭圆纤维纤维纤维的Calabi-yau上的精制gopakumar-vafa数（GVN）。关键的例子是（几乎）Fano表面上平滑的椭圆纤维。我们包括对现有的数学方法的详细审查，以使用稳定的吊杆模量3倍稳定的sheaves稳定的sheaves来定义和计算（未精制的）gopakumar-vafa不变式（GVI）和GVN，该语言应使用物理学家可以访问的语言。特别是，我们讨论了GVN对复杂结构模量的依赖性以及方向的选择。我们在许多情况下计算GVN，并将B模型预测与几何计算进行比较。我们还通过分析传播剂的准模块化特性，从骨膜异常方程中得出模块化异常方程。我们推测可以为方向做出的数学选择的物理相关性。
We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces. We include a detailed review of existing mathematical methods towards defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a language that should be accessible to physicists. In particular, we discuss the dependence of the GVNs on the complex structure moduli and on the choice of an orientation. We calculate the GVNs in many instances and compare the B-model predictions with the geometric calculations. We also derive the modular anomaly equations from the holomorphic anomaly equations by analyzing the quasi-modular properties of the propagators. We speculate about the physical relevance of the mathematical choices that can be made for the orientation.
