On counting special Lagrangian homology 3-spheres
On counting special Lagrangian homology 3-spheres
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计算特殊拉格朗日同调 3 球面
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发表时间：
1999
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通讯作者：
D. joyce
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作者：
D. joyce
本文试图通过在每个3-同调类中计算M中的特殊Lagrange有理同调3-球面N，定义（几乎）Calabi-Yau 3-folds M的一个新的不变量I，其权w（N）依赖于N的拓扑.这是由辛流形的Gromov-Witten不变量所激发的，该不变量计算每个2-同调类中的J-全纯曲线。
为了使这个不变量变得有趣，它应该不被基本的（几乎）卡-丘结构的变形所改变，或者当几乎卡-丘结构的周期通过M的上同调中的一些拓扑确定的超曲面时，根据一些严格的规则进行变换。
当我们变形基本的几乎卡-丘3-折叠时，特殊拉格朗日同调3-球面的集合只有在它们变得奇异时才会改变。因此，为了确定变形下的不变量的稳定性，我们需要知道特殊拉格朗日3-折叠的奇异行为，这是不太清楚的。
本文描述了两种特殊的拉格朗日3-折叠的奇异性，并导出了权函数w（N）的恒等式，使I在它们下保持不变或变换良好.权重函数w（N）=| H_1（N，Z）|满足这些身份。我们猜想，用这个权定义的不变量I与Kahler类无关，并且当全纯3-形式通过H ^3（M，C）中的某些真实的超曲面时，它以某种方式变化。
最后，我们考虑与弦论的联系。我们认为，我们的不变量I计数孤立的3-膜，它应该在镜像对称的故事卡拉比-丘3-折叠。
We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class.
In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M.
As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood.
We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C).
Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds.