AI 中文总结
本文利用双束几何构造轨道间映射空间的无限维轨道结构，建立恰当平展李群胚表示，证明Morita等价类不依赖选择，局部为有限群作用群胚，为Floer理论等应用提供框架。
AI 中文摘要
轨道之间映射空间上轨道结构的存在性在先前的工作中已确立，特别是由W. Chen和B. Chen-Du-Liao完成。本文中，我们以更几何的方式，基于Lerman对轨道态射的描述，利用双束（或Hilsum-Skandalis态射）在映射空间上构造了一个无限维轨道结构。我们在$C^k$、$C^\infty$和$W^{k,p}$设定下构造了一个恰当平展（或Fréchet）李群胚来表示映射空间，并证明其Morita等价类独立于群胚表示的选择。局部上，映射群胚同构于一个有限群的作用群胚，其中有限群被识别为相应双束的自同构群。这为轨道上映射空间和模空间的进一步研究提供了具体的几何框架，并应用于辛轨道上的Floer理论。
英文摘要
The existence of orbifold structures on mapping spaces between orbifolds was established in previous work, notably by W. Chen and B. Chen-Du-Liao. In this paper, we construct an infinite-dimensional orbifold structure on the mapping space in a more geometric way, using bibundles (or Hilsum-Skandalis morphisms), based on Lerman's description of orbifold morphisms. We construct a proper étale (or Fréchet) Lie groupoid representing the mapping space in the $C^k$, $C^\infty$, and $W^{k,p}$ settings, and show that its Morita equivalence class is independent of the choices of groupoid presentations. Locally, the mapping groupoid is isomorphic to the action groupoid of a finite group, where the finite group is identified with the automorphism group of the corresponding bibundle. This provides a concrete geometric framework for further studies of mapping spaces and moduli spaces on orbifolds, with applications to Floer theory on symplectic orbifolds.
