Differences

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

--- blog:2023-08-05 [2023/08/05 22:01] – pzhou
+++ blog:2023-08-05 [2023/08/06 04:10] (current) – pzhou
@@ Line 2: / Line 2: @@
   * write up the notes that are useful for myself.
   * why quiver gauge theory has anything to do with Kac-Moody algebra?
+The stuff that I typed up below, are so incoherent and dreamy, that I don't know what am I talking about.
+So they should be either cleaned up or deleted.
+I also cleaned up some to read papers.
+I don't think I want to write up the example computation of the spaces.
 ===== Statements,  Examples about Coulomb branches =====
@@ Line 33: / Line 40: @@
 In the $K$-theory story, when you have $pt \in \C$, with $\C^*$ action in the standard way, you get $\iota^* \iota_*([0]) = [0] (1 - 1/t)$, where $1/t$ is the representation of the conormal (representation of the linear coordinates).
-In the homoloyg story, when we pushforward, we have a $dim=0$ class, so it is $deg=codim=2$ cohomology class, then we pull-back, it is a degree $2$ guy. So, it might be the 'equivarint euler class' of the conormal bundle, which is $-t$.
+In the homoloyg story, when we pushforward, we have a $dim=0$ class, so it is $deg=codim=2$ cohomology class, then we pull-back, it is a degree $2$ guy. So, it might be the 'equivarint euler class' of the normal or conormal bundle. I guess it is normal bundle, but I am not sure why. (https://arxiv.org/pdf/1305.4293.pdf) they uses $e_T(\nu_p)$, the normal bundle.
+In the example of $\P^1$, at point $[1;0]$, where we use $T_2 / T_1$ as local coordinates. Let me say, the normal bundle is cashed in for $Y_2 - Y_1$. Then, we say
+$$ [\P^1] = \frac{[0]}{Y_2 - Y_1} + \frac{[\infty]}{Y_1 - Y_2}. $$
+If we understand both sides as cohomology, then both are in degree $0$, $deg([0])=2$ but $deg(Y_i)=2$, so they cancel.
+Why does localization to fixed point works? Why does it play well with convolution? Isn't convolution very complicated?