论文标题
群体值的动量图，用于自动形态群体的行动
Group-valued momentum maps for actions of automorphism groups
论文作者
论文摘要
符号纤维束的平滑部分的空间具有自然的符号结构。我们提供了一个通用框架，以确定束束自动形态在该空间中的作用的动量图。由于总的来说，此操作不承认经典的动量图，因此我们介绍了更一般的组值势头图，这是受Poisson Lie设置启发的。在这种方法中，组值的动量图分配给与连接的主圆圈捆绑包的每个部分。在许多示例中说明了这种一般框架的力量：我们为具有整体螺旋性的流体构建了广义的Clebsch变量；事实证明，抗传统捆绑包是符合兼容复杂结构空间的动量图。 Teichmüller模量空间被实现为与$ \ mathrm {Slrm {sl}（2，\ Mathbb {r}）$相关的符号轨道缩小空间，并且与其他coadchoint Orbits相关的空间被识别和研究。此外，我们表明，除了曲率外，还在riemann表面编码的连接空间上的一组捆绑式自动形态的动量图，也是捆绑包的拓扑信息。
The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illustrated in many examples: we construct generalized Clebsch variables for fluids with integral helicity; the anti-canonical bundle turns out to be the momentum map for the action of the group of symplectomorphisms on the space of compatible complex structures; the Teichmüller moduli space is realized as a symplectic orbit reduced space associated to a coadjoint orbit of $\mathrm{SL}(2,\mathbb{R})$ and spaces related to the other coadjoint orbits are identified and studied. Moreover, we show that the momentum map for the group of bundle automorphisms on the space of connections over a Riemann surface encodes, besides the curvature, also topological information of the bundle.
