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--- blog:2023-06-15 [2023/06/15 20:34] – [add matter] pzhou
+++ blog:2023-06-15 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 46: / Line 46: @@
 ==== add matter: the original definition ====
-We follow BFN, and build the linear bundle $L_V$ over the algebraic model of loop to $G$, which is like the free loop quotient by equivalence $Gr_G = G((z) )/ G[[z ] ]$. Given a Laurent loop $\gamma \in G((z) )$, we can look for those $\gamma$-robust pairs, namely those nice global section of $V$ over the disk $D$, such that, even if we do a singular gauge transformation / multiply by the section to the group, we are still holomorphic. This $\gamma$ really looks like a transition function of a principal $G$ bundle on a raviolo $B$, and we take the associated $V$-bundle, and then $L_V|_\gamma$ by definition is the global section, $\Gamma(B, V_\gamma)$.
+We follow BFN, and build the linear bundle $L_V$ over the algebraic model of loop to $G$, which is like the free loop quotient by equivalence $Gr_G = G((z) )/ G[ [z ] ]$. Given a Laurent loop $\gamma \in G((z) )$, we can look for those $\gamma$-robust pairs, namely those nice global section of $V$ over the disk $D$, such that, even if we do a singular gauge transformation / multiply by the section to the group, we are still holomorphic. This $\gamma$ really looks like a transition function of a principal $G$ bundle on a raviolo $B$, and we take the associated $V$-bundle, and then $L_V|_\gamma$ by definition is the global section, $\Gamma(B, V_\gamma)$.
 aha, the massive version: why do we call that the massive version? Well, we have extra $S^1$ action, which rescales the fiber of $L_V$. So, we have additional equivariance parameter.
+Aha, you see, Teleman only introduced the matter / flavor group action on the side, instead of doing a big GIT thing, which you need to fix the base then quotient the fiber. There is no fiber to begin with.
+==== Euler Lagrangians ====
+Basically, we have the Toda integrable system, and we are defining a section of that integrable system.
+Example, $G = T = \C^*$, we have 3d Coulomb branch as $\mathfrak t \times T^\vee$, where $T^\vee$ is the fiber. Since the fiber is a group, there is a canonical section by picking out the identity element.
+If we have some representation, say $\C^*$ acts on $\C$ with of weight $\nu$, say $\nu=1$. We should define a section, deviating the 'identity section'. how so? First, we have a constant shift $\mu$. $\mu$ here just means a complex number, possibly zero. We take a point $\xi$ in the $\mathfrak t$, pair it with $\nu$, we get a number (an actual number! like, infinitesimal character turns a Lie algebra element into an actual number), add $\mu$ to it, still a number, then if that number is non-zero, we can use $\nu$ as a cocharacter to $T^\vee_\C$, and map that complex number to $T^\vee_\C$. If we do that for every section in the base, we should get a section. But, why is it a Lagrangian section? If you say, this is the graph of something, I would be much happier.
+OK, here we go. Consider the following function, then consider its $\Gamma_{df}$. In the additive case, given $\xi \in \mathfrak g_\C$ (wait, not even in a torus? so brave?) we consider
+$$ Tr_V [ (\xi+\mu) [(\log (\xi+\mu) - 1]] $$
+well, what are we doing $(x+c)(\log (x+c) - 1)$, that doesn't look like well-defined when $x=0$, but don't worry, let's take $d$, we get $(\log (x+c) - 1) dx + dx = \log (x+c) dx $, and then we exponentiate the fiber, we get $x+c$. Wait, why don't you directly say $x+c$, viewed as $\C^*$ nubmer?
+What are Theorem 1 and 2? They say, gluing up the two patches, matching $\epsilon_V$ with identity section of the other patch gives a good enough description of the coordinate ring.
+==== geometry, section 5 ====