Peng Zhou

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====== 2023-07-16 ======
  * planning for the Coulomb branch

===== planning =====

Let me think about my desired statement first. What do I want to state in this section. Yes, I need to discuss about the geometry of the Coulomb branch in this section. What is my point? Don't be bind up by the setup, do your own setup. If I were to rewrite this paper, I would do this: immediately after the introduction, I will do the most important computation, 2 strands nilhecke, and the interaction ones. They can be stated without any reference to the Coulomb branches. 

Then, I will recall the background on the general setup of the Coulomb branch. It is all you need to know about Coulomb branch in order to understand this paper, and a first year grad student in math or in physics should be able to follow. Should I split the old results and new results? Yes. I think the geometry and the localization and abelianizations are the old results. 
  * Definition. 
  * Example: pure abelian case, abelian case with matter, pure gauge theory case. 
  * Reducing to Model case: In general, without any deformation, one can obtain the local description of the affine Coulomb branches, over the descriminant loci. 
  * Deformation and Resolution. One could build a bigger Coulomb branch with a bigger group, that corresponds to split the box node into many small circles nodes. 


==== Aside: do we care about codimension 2 guys? ====
Question: suppose you have a Kahler manifold, that you only know very well up to complex codimension two loci, is that good enough to define a Fukaya category? Hmm, what does codimension two mean? It means that you start with a space, singular maybe, and you deleted something from it. Why it should not matter? Because when we define things, because a (complex) codimension 2 guy intersect a dimension 1 guy is a complex codimension 1 condition. And, when we build things in a family, like, checking associativity condition, one only need to build real one-dimensional disk moduli spaces, so it is OK. You will not see it. 

Well, if you say so, how do you define the Fukaya category of $T^*\P^1$, which is a resolution of the nilpotent cone of $sl_2$? The singularity downstairs is codimension 2. It is the tip of a cone. In this case, you can certainly imagine a disk bounded by Lagrangian that run through the exceptional divisor. So, it would be a 'sin' to delete this seemingly harmless singularity part, unless we know that apriori, the original space is smooth. 

Answer: If the ambient space is smooth, then we don't care about codim 2 guy, no matter if it is smooth or not. 

==== Just write it. in 1 hour ====
10:53am now.