Peng Zhou

reading Etingof on quantum group

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What's new?

1. What is the central charge formula on the B-side? What's the relationship with stability condition? SOD?
2. What's Xin Jin's story? How does the open torus embed?

a category with a notion of 'equal', or 'isomorphism', or 'quasi-isomorphism'.

1. category of set? then isomorphism, bijection
2. category of vector space? isomorphism
3. of chain complex of abelian groups? Well, quasi-isomorphism may not be.
4. the category of dg categories. we really just want to do equivalence of category, not isomorphism of categories. that requires that we have two ways functors.

Let's go simple. Suppose you have a sheaf on $\R$, and you want to test whether it is locally constant, so you do restriction. what's the condition? to test $(x,\xi)$ is not in the SS, it means there is an open set of $U$, such that $F(x-\epsilon, x+\epsilon) \to F(x-\epsilon, x)$ is a nice arrow.

Are we working in the homotopy category of dg categories?

Suppose the category $C$ have the notion of isomorphisms. Like set, or homotopy category of dg categories.

Then, we need to say, what does locally constant mean, indeed restriction to smaller open induces isomorphism in $C$.

Does Ho(dg-cat) admits arbitrary limit and colimit? Should be. The cat dg-cat embed into Ho(dg-cat), since it is a localization.

In the simplest setting, we have mirror symmetry for $Coh(\C^* \times \C^2)$.

Next, we are going to take symmetric power.

Do you remember what happens when two eigenvalues collide? No, don't do Hermitian matrices, that will never be nilpotent. What if you have a matrix that looks like $( (1,1), (0, 1+x) )$. What is the eigenvector for e.v. $1+x$? How about $(1,x)$? Eigenvector for $\lambda = 1$, is $(1,0)$. So you see, when $x \to 0$, the two eigenspaces also collide!

The key question is: what is the superKLRW algebra?

From the A-side, let me guess, we have 'fiber product' of two MC. So, when y1 = y2, in the fiber, we need to have x1 = x2, and z1 = z2 (the new pair of fiber coord). And we need to remember the ratio of (x1-x2)/(y1-y2), and (z1-z2)/(y1-y2).

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going to revise the VGIT paper.

Here, the base is one dimensional, fiber is easy. We can take the easy way out, just prove enough for this case. Or, we can do full proof.

I don't want to be super general, but just for this case. I want to consider skeleton on the total space, and skeleton on a sub-level set.

let's just glue category, that is pretty cool, and useful.

So, we have a constructible sheaf of categories. I think, we can define a sheaf of stable categories. We don't have the notion a microlocal stalk, but we do have the notion of equivalences. In that sense, it is not very satisfactory.

I need to say, non-characteristic deformation lemma. what is 2.7.2?

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well, two months passed.

what did I learn today? with Alexei's talk yesterday and the discussion today.

what I

fk, a month has passed.

what do I want? I want raising and lowering operator, which is adding and removing strand operator, which is a special case of gluing an extra guy and put some extra strands operator.