如何在loop space上定义纤维丛、Chern示性类呢？如果可以在上面建立纤维丛理论，不知道是否能够用来描述量子规范场理论？大家能不能讨论一下？

I think so. It will be about the cohomology of the loop space.

我想可能Cohomology没有清楚的表达出他本来的意思。纤维丛、陈类可以在任何拓扑空间上定义。

I don't think Chern class or Chern characteristic can be defined in all topological spaces. In general, Chern class is defined for a complex vector bundle. For real bundles, one can define Pontryagin class.

Geometric quantization has been proposed decades ago, I don't think they fits the need of physists. In mathematics, this becomes the area of symplectic geometry and topology.

Now after the boom of string theory, mathematicians are workng on string topology trying to build the foundation of string theory mathematically. The geometry and topology of loop space theory becomes more popular than ever under this project.

先搞清楚loop space的切丛吧.

It is 100% possible to consider vector bundles on loop space, although I am not sure when do complex / sympletic bundles exist for the loop space of some topological space X. You know, there is some obstructions you need to check first. I am not familiar with loop space.

如果可以在上面建立纤维丛理论，不知道是否能够用来描述量子规范场理论？大家能不能讨论一下？
～～～～～～～～～～～～～～

可以看看 Witten 的名著 supersymmetry and morse theory