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--- blog:2023-06-11 [2023/06/11 23:52] – created pzhou
+++ blog:2023-06-11 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 Here they defined a parabolic Coulomb branch. Recall what is a Coulomb branch, you first construct the BFN space. No? No. You first construct the $V(K)/G(K)$ stack, then you consider $V(O) / I_P$, where $I_P \In G(O)$ is such that at the center of the disk, we restrict the group to be in $P$. Why do we want that? Well, $G(O)$ is the gauge transformation group. So, $I_P$ is saying, you can do whatever gauge transformation you want on the disk, formal disk, except that I want to 'preserve a partial flag' in my fiberwise group. Just some constraint. You could even specify some further condition, not just the value of the group needs to be something, but also on the derivative.
-OK, so you have these sections, upto these equivalences, and then you pushforward to the base. What do you get? Does it make sense to ask for the fiber? Well, we can first push to $V(O)/G(O)$, then to $V(K) / G(K)$. Given a meromorphic sections' equivalence class, $V(K)/ G(K)$, we can ask whether there exist lifts.
+OK, so you have these sections, upto these equivalences, and then you pushforward to the base. What do you get? Does it make sense to ask for the fiber? Well, we can first push to $V(O)/G(O)$, then to $V(K) / G(K)$. Forget about convolution, we are just asking how to hom from one line operator to another.
-Somehow, there are D-modules on the stack.
+So, what does these parabolic Coulomb branch know? Do they know more stuff? Or less stuff? I believe that they know more.
+section 2.2 says, we have an inclusion of algebra $$ A(L, N) \to A^P(G, N). $$. What does that mean? The two are endomorphism algebra of different modules on different spaces. Concretely, both of them are convolution algebra, sheaf cohomology on two different convolution spaces. They are stratified bundles over the corresponding affine Grassmannian / affine flag variety.
+  * $R(L, N)$ the moduli space of $L$-bundle over the ravioli, with section in the associated $N$-bundle. Hmm, somehow I don't understand who is quotienting who. I thought, we had a fiber-product, and we have two quotients, one of the gauge group on the left, and one on the right slice. So, the resulting module should lives over the diagonal in the spec of $H^*(BL) \times H^*(BL)$. How do I know it is the diagonal? I just guessed. Otherwise, why we have the bimodule structure reducing to just a module structure? [need to check]
+  * $R^P(G, N)$ is the convolution space of $N(O)/I_P \times_{N(K)/G(K)} N(O)/I_P$. This lives over the Hecke modification space $pt/I_P \times_{pt/G(K)} pt/I_P$. This is like $I_P \ G(K) / I_P$. Since basically, we have a pair-of-flags.
+Try again. $T_{G,N}$ is the data of a section of $N \times D \to D$, and together with a meromorphic gauge transformation (modulo gauge transformation on the disk $D$).
+$R_{G,N}$ is the submoduli space of $T_{G, N}$, such that section in the trivial $N$-bundle, after doing that meromorphic section, extends from $N \times D^*$ to $N \times D$.
+The parabolic version $T^P_{G,N}$, is still a section of $N$ over $D$, together with a group element $G(K)$, modulo the equivalence relation that we can modify both the group action, and the section, by a gauge transformation. It is just, the gauge transformation group is restricted.
+The parabolic version $R^P_{G,N}$, is alsomostthe same as before, except we impose the regularity condition on the N-section, after the action.
+These are the BFN spaces. Then, we take equivariant $K$ theory or cohomology on the spaces. And then, we define convolutions. So, again, what is the parabolic induction?
+We first have a restriction, and