Peng Zhou

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blog:2023-07-16 [2023/07/16 17:12] – [2023-07-16] pzhoublog:2023-07-16 [2023/07/16 17:53] (current) – pzhou
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 ====== 2023-07-16 ====== ====== 2023-07-16 ======
   * planning for the Coulomb branch   * 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.  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. 
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 +==== 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.  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.  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. 
 +
  
  
blog/2023-07-16.1689527544.txt.gz · Last modified: 2023/07/16 17:12 by pzhou