• still working on Teleman's shift operation

Let $G$ be compact Lie group. the pure gauge theory case is fine, just equivariant homology of the based loop space.

Now, you have a representation $V$, and $G$ acts on $V$. If you are BFN, you can consider a sphere, with a principal $G$ bundle over it, which is made from a $S^1 \to G$ a cluching function.

hmm, so topologically there is no interesting $G$ bundle on sphere. But, if you have a base space $B$ (maybe with interesting topology), and you want to consider a $S^2$ bundle over $B$, and then, you want to map the family of $S^1$ to $G$. I think this is about (based? free?) loop space from $S^1$ to $G$.