• 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.

but why does this construction have anything to do with 3d N=4 gauge theory? Well, beat me. The $\Omega G$ is a perfectly nice group: for any two elements on it, we can do honest multiplications. But affine Grassmannian is not a group, unless you take the homology of it. So, it is homotopic to a group? like you can deformation retract to a group, then away from setwise construction, you are a group.

We have parameter from $H^*(B GL(1)(K) )$. That should be the Lie algebra of it