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--- blog:2023-06-18 [2023/06/19 04:51] – pzhou
+++ blog:2023-06-18 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 17: / Line 17: @@
 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
+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