Conormal bundles to knots and the Gopakumar-Vafa conjecture
Conormal bundles to knots and the Gopakumar-Vafa conjecture
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共正规束到结和 Gopakumar-Vafa 猜想
DOI：
10.4310/atmp.2007.v11.n4.a3
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发表时间：
2005
影响因子：
1.5
通讯作者：
Sergiy Koshkin
中科院分区：
文献类型：
--
作者：
Sergiy Koshkin
我们给出了Gopakumar-Vafa猜想的一种新的拉格朗日子流形的构造，该猜想将三维球面上的Chern-Simons理论和可分解锥上的Gromov-Witten理论联系起来。给出3-球面上的一个纽结，它的余法丛被扰动以使其与零点分离，然后拉过锥形过渡。这种构造产生了分解的锥形的全实子流形，这些子流形在扰动的辛结构中是拉格朗日的，并且以自然和显式的方式对应于纽结。证明了可分解的锥形和其中的纽结拉格朗日函数都有有界几何，并且全纯曲线的模空间在Gromov拓扑中是紧的。
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in it have bounded geometry, and that the moduli spaces of holomorphic curves ending on the Lagrangians are compact in the Gromov topology.