Hyper-Kähler Hierarchies and Their Twistor Theory
Hyper-Kähler Hierarchies and Their Twistor Theory
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Hyper-Kähler 层次结构及其扭曲理论
DOI：
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发表时间：
2000
期刊：
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通讯作者：
Lionel J. Mason
中科院分区：
文献类型：
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作者：
M. Dunajski;Lionel J. Mason
摘要：给出了与度量上的超凯勒方程（四维反自对偶爱因斯坦真空方程，ASDVE）相关的层次结构的扭量构造。构造递归算子 R 并用于构建无限维对称代数，特别是超凯勒方程的更高流。结果表明，R 通过与有理函数相乘来作用于扭量数据。这些结构通过斯帕林-托德（江口-汉森）解决方案的例子进行说明。扩展时空？其额外维度对应于层次结构的更高流。表明 ?是扭量空间中具有正态束 ?(n)⊕?(n) 的有理曲线模空间，并且规范地配备了 ASDVE 层次结构的 Lax 分布。空间？被证明是由四维超凯勒切片组成的。给出了天文方程形式的 ASDVE 的拉格朗日、哈密顿和双哈密顿公式。推导了天方程组解的模空间上的辛形式，并证明了它与递归算子的兼容性。
Abstract: A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-Kähler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling–Tod (Eguchi–Hansen) solution.An extended space-time ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? is shown to be foliated by four dimensional hyper-Kähler slices.The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator.