膜上的 skein 结构
Skeins on Branes
AI总结：
研究了具有Lagrangian边界的全纯曲线族，展示了期望的码元一现象可与量子拓扑的HOMFLYPT skein关系匹配，从而通过Lagrangian边界类在skein模块中计数全纯曲线得到变形不变结果。
摘要：
我们研究了具有Lagrangian边界的全纯曲线族的1参数族。我们证明期望的码元一现象可以组织以匹配量子拓扑的HOMFLYPT skein关系。由此可知，通过Lagrangian边界类在skein模块中计数全纯曲线可以得到变形不变结果。这是Witten断言的数学严格实现，即开拓拓扑弦的边界在Chern-Simons理论中创造线缺陷。利用这一理论，我们严格建立了Ooguri和Vafa的预测：链结在三维球体上的HOMFLYPT多项式系数计数了解决后的锥形中的全纯曲线，其边界在（链结共轭的）共轭上。
英文摘要：
We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein module of the Lagrangian gives a deformation invariant result. This is a mathematically rigorous incarnation of Witten's assertion that boundaries of open topological strings create line defects in Chern-Simons theory. Using this theory, we rigorously establish the following prediction of Ooguri and Vafa: the coefficients of the HOMFLYPT polynomial of a link in the three-sphere count the holomorphic curves in the resolved conifold, with boundary on (a push-off of) the link conormal.
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