• 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 $G$.

Is Schubert variety, a basis of homology or cohomology? I think it should be homology.

But, what does the algebraic model of affine grassmannian mean? And why when we turn on matter, we suddenly need the algebraic model?

Now, we consider the ravioli model. We consider the algebraic $G$ bundle over it. We have construction data (an element in $G(K)$), and isomorphism data (two copies of $G(O)$).

This is the moduli stack that we care about, right? Given two elements in $G(K)$, we can multiply them, but we cannot do so with two equivalences classes.

That's an interesting phenomenon. many operations cannot be defined on the level of equivalence classes. like, Lagrangians, we can define A-infinity operation on each individual Lagrangians, but we cannot