• 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 define operation on the equivalence classes.

Then, what is BFN space with matter representation over it? You have some vector bundle on one patch, and another vector bundle on another patch. You are looking for compatible pairs. Then $G(O)_L, G(O)_R$ acts on this data.

Compared with the case without matter, we have the same group, but more things to be acted upon. The convolution structure is also clear.

Then, we need to consider homology of the space. We cannot just handwave, otherwise it is the same as saying 'path-integral'.

What does homology mean? The Schubert cell is a nice cell. Given a cocharacter $\C^* \to G$, means given an element in $G(K)$. We may consider the $G(O)-G(O)$ double coset. That might be what we mean when we say $G(O)$-equivariant cohomology. I guess, we can do $G(O)$-conjugation action's equivariance. Then, it would be compatible with composition.

What's the most naive thing? just do set, and union. pointwise operation, take the image of the map.

Then, what does monopole operator mean? And what does a BM equivariant homology cycle mean? What does equivariant mean? If we consider $S^2$ mod $U(1)$, what do we get? I would take the Borel construction.

Let's blackbox a bit. BFN and Teleman deals with matter differently. In BFN, we use the same indexing set for basis. The multiplication rule for the 'monopole' operators are different. The monopole operators are the global coordinate, the global fiber coordinates. If you compose two fundamental cycle of Schubert varieties, you can get a lot of stuff. When you encounter folding, in the sense that two input arrows