Let K be a compact Lie group and fix an invariant inner product on its Lie algebra. Given a Hamiltonian action of K on a compact symplectic manifold X, the normsquare of the moment map defines a Morse stratification of X by locally closed symplectic submanifolds such that the stratum to which any x in X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow with respect to a suitably compatible Riemannian metric on X. The open stratum indexed by 0 retracts K-equivariantly via this gradient flow to the minimum which is the zero-locus of the moment map (if this is not empty). The usual 'symplectic quotient' for the action of K on any other stratum is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural 'symplectic quotients' for the unstable strata labelled by nonzero indices. There is an analogous infinite-dimensional picture for the Yang--Mills functional over a Riemann surface with strata determined by Harder-Narasimhan type.

中文翻译：

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矩图范数平方的不稳定莫尔斯层的辛商

设 K 是一个紧李群并在其李代数上固定一个不变的内积。给定 K 在紧辛流形 X 上的哈密顿作用，矩图的范数平方通过局部闭合辛子流形定义 X 的莫尔斯分层，使得 X 中任何 x 所属的层由其向下的极限行为决定梯度流下的轨迹相对于 X 上适当兼容的黎曼度量。 由 0 索引的开放层通过此梯度流将 K 等变收缩到最小值，即矩图的零轨迹（如果这不是空的） . K 对任何其他层的作用的通常“辛商”是空的。尽管如此，受非还原几何不变理论的最新结果的启发，我们发现辛商构造可以修改为由非零指数标记的不稳定层提供自然的“辛商”。对于具有由 Harder-Narasimhan 类型确定的地层的黎曼曲面上的 Yang-Mills 泛函，有一个类似的无限维图片。