Differences

This shows you the differences between two versions of the page.

--- blog:2023-06-14 [2023/06/14 18:18] – created pzhou
+++ blog:2023-06-14 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
 ====== 2023-06-14 ======
   * Reading Teleman
+  * chatting with Spencer
 ===== The role of Coulomb branches in 2D gauge theory =====
@@ Line 6: / Line 7: @@
 I don't know what's the difference between $g^{reg}/G$ and $g/G$ (adjoint action). OK, well, if we just do $g/G$, there might be too many orbits (with the same eigenvalues), and picking out $g^{reg}$ is saying, I am looking at the most generic (?) guy with this eigenvalues.
+Why we call this integrable structure "Toda integrable system". I thought it is like Hitchin integrable system, because we are taking eigenvalues of a matrix. Who is Toda, and what did he do?  What's 'inverse scattering method'?
+OK, let's not be distracted.
+The 'reconstruction' result, you mean the gluing two copies method? , is from the 2d gauge theory interpretation. (what do you mean by that 2d gauge theory moduli space?)
+===== Teleman's ICM =====
+The hook question: if you know Floer homology with Lagrangians is categorifying the intersection of middle dimensional homology cycles (quantum homology?)
+wait, do we have a notion of product in the decategorified case? Wow, previously, you have access to each individual intersection points, so you can say who talks to who. if you only remember the intersection number, you don't have enough structure!
+why do we care about chain complex, rather than just its cohomology? because it is interesting to have a local system family of chain complexes over $S^2$ (simply connected). However,if you pass to cohomology, you lose too much information, and you cannot detect the variation. Passing to fiberwise cohomology, and taking global section, gives the wrong answer.
+Why does Teleman says, if $G$ is simply connected, (then $\pi_2(BG)=0$), then we don't have interesting 'local system of categories on BG'?
+What does he mean, when he says representation admits a character theory? Do you mean restrict to an abelian subgroup, then the representation split as direct sum of representations of the subgroup? No, he doesn't mean that. Say $G$ is compact Lie group. Given a representation, $\rho: G \to End(V)$, and there is the trace map $End(V) \to k$, so composing them, we can get a function from the conjugacy classes of $G$ to $k$. That is the character function, living on $G/G$. (what does it look like? examples? intersting questions about it?)
+So, are you suggesting, we look at coherent sheaves on $G/G$? well, what does that even mean?
+Wait wait wait. So, you are saying, (categorical) representation of $G$ is related to the BFM space of the $G^v$ (tada, Langlands dual group is here). So, what do you mean by this $BFM(G^v)$? what's your convention?
+This is related to $T^*(G^\vee / conj)$. Now, I am confused. why we change so much? Why we go to the Langldans dual?
+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.
+===== Adding and subtracting strands in usual Heegard Floer =====
+Read Auroux, Perutz, Weirheim