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--- blog:2023-06-14 [2023/06/14 19:38] – [Teleman's ICM] pzhou
+++ blog:2023-06-14 [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ====== 2023-06-14 ======
   * Reading Teleman
+  * chatting with Spencer
 ===== The role of Coulomb branches in 2D gauge theory =====
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 Multiplicity (vector) space, of two G-representations; is replaced by the multiplicity categories.
+==== What's BFM space? ====
+. well, we can say, the convolution homology of $H_*(pt/G(O) \times_{pt/G(K)} pt/G(O))$, the spherical Coulomb branch algebra. This is Morita equivalent to $H_*(pt/I \times_{pt/G(K)} pt/I)$, which is the Webster(?) cylindrical Nil-Hecke.
+Why such constructible sheaf endomorphism algebra, can be computed effectively, using Fukaya categories? Why?
+What if the group is $G=\C^*$? Well, we have $Gr_G = \Z$, so homology is like $\C[\Z] = \C[z, z^{-1}]$, that's the cylindrical wrapping part.
+Hold on, I don't understand, people usually say, $G(K)/G(O)$ has T-fixed points, labelled by $\C^* \to G$, why? If I give you such a map (doesn't have to be a group homomorphism), can you give me an element in $G[t,t^{-1}]$? An element in $G(K)$ is just a section of $G$ over the annuli (formal). so $G(K)/G(O)$ is just the 'singular' part of the loop. In the case $G$ itself is $\C^*$, everything is topological. So, that $\Z$ is just the winding number.
+But, for $G = GL(2)$, that $G(K)/G(O)$ is more intereting. topologically, we have connected component being $\Z$. The question is, what is the possible behavior, as $t \to 0$ for $G$? I mean, different entries have different divergent speed.
+No, that's the red herring. We really should be doing Iwahori, but there is no difference with $GL(1)$. So, can you translate $GL(1)$? Don't think about the B-side, just think about the convolution algebra. The algebra has a trivial $H^*_{\C^*}(pt)$ part, that is $\C[u]$, cohomology on $BG$. oh, that's super easy. then, we have the matrix part, the covolution part, the concatenation of loop part. That is super important. concatenation of loop. concatenation of Reeb chord, both are concatenations.
+Now, let's be super careful, which one is which. We start our life by doing gauge theory. Let $G = U(1)$ be our starting point. Then, we should do $\Omega G$, convolution homology for loop space here, that is $BFM(G^\vee)$'s coordinate ring.
+  * $H_*(\Omega K)$ is the endomorphism algebra of wrapped cotangent fiber in $T^*K$, which is commutative (since $K$ is a group).
+  * Let $T$ be the maximal torus. $H^T_*(\Omega K)$, this should be the $T$-equivariant Fukaya category of $T^*K$, where $T$ acts by conjugation, where I think the only fixed loci is $T$ itself. Can I do $Fuk(T^*K)^T$? I take the fiber over $e$, although $e$ itself is invariant, but the cotangent fiber $\mathfrak k^*$ suffers from $T$-action. The action seems ok. except the diagonal action acts trivially. I don't show, should I put some local system on the $T$-orbits? Say, we have some equivariant $U(1)$ local system over the Lagrangian.
+  * The endomorphism algebra should be an algebra over $H^*(pt/K)$. So, spec of that base algebra should be $t/W$.
+Let's face it. I don't know where does the upstairs space come from, and the upstairs superpotential.
+===== Chat with Spencer =====
+==== points on a circle ====
+What does $n$ points on a circle mean? Can you get $cNH_n$? There must be some flag variety in this game. How do you get the usual $NH_n$, where does the usual polynomial come in? So, I guess, just guessing, the dots are equivariant variables in the usual equivariant cohomology, like cohomology of BG, which is $G$-invariant function on $\mathfrak g$, the generators are in degree two.
+But, precisely, why we use flag variety? Are we using $pt / G(K)$, and what do we push down? $pt/I$? Then, the fiber is huge, $G(K)/I$. This sounds right, since we have no matter. What does hom mean?
+wait wait, why did BFN reduces to BFM in the pure gauge theory case?
+How about this: conjugation action of $U(n)$ by $U(n)$, the result is $Sym^n U(1)$. We have $U(n) \to Sym^n U(1)$, the fiber is $U(n) / \prod U(n_i)$, which should be partial flag. I just don't know how to use this data.