Cubics, Integrable Systems, and Calabi-Yau Threefolds
Cubics, Integrable Systems, and Calabi-Yau Threefolds
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三次方程、可积系统和 Calabi-Yau 三重方程
DOI：
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
E. Markman
中科院分区：
文献类型：
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作者：
R. Donagi;E. Markman
在本文中，我们构造了一个解析完全可积哈密顿系统，它与任意Calabi-Yau三倍矩阵族具有正则关联。该系统的基是量规Calabi-Yaus族的模空间，光纤是三倍矩阵的Deligne上同群（或中间雅可比矩阵）。这个系统有几个有趣的性质：作为一族曲线的一般变种的Abel-Jacobi图像或“正规函数”得到的多值截面总是拉格朗日的；在镜像对应中使用的基上的自然仿射坐标作为可积系统的作用变量出现；汤川立方表示的是霍奇结构在族中的无限小变化，本质上等同于总空间上的辛结构。
In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds. The base of this system is a moduli space of gauged Calabi-Yaus in the family, and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the threefolds. This system has several interesting properties: the multivalued sections obtained as Abel-Jacobi images, or ``normal functions', of a family of curves on the generic variety of the family, are always Lagrangian; the natural affine coordinates on the base, which are used in the mirror correspondence, arise as action variables for the integrable system; and the Yukawa cubic, expressing the infinitesimal variation of Hodge structure in the family, is essentially equivalent to the symplectic structure on the total space.