Peng Zhou

It is useful to recap what I did today, or these days.

I have been thinking about Koszul duality a lot these days.

• where does the mysterious grading come from. What is mixed category? The terminology is weird that, mixed constructible sheaf does not form a mixed category. you need to take a (not full) subcategory.
• why the two spaces, Coulomb branch and Higgs branch, are related? I don't see the relation at all. You can say, categorically, one is about $Map(S^1, V/G)_{dR}$ the other is $Map(S^1_{dR}, V/G)$. But concretely, on a low brow way, what are we talking about?

You want to say, for the same theory, the Coulomb branch and Higgs branch, are just two shadows of the same object, the theory. Therefore, the category of line operators in the two branches are the same. Just some equivalence of categories (but where are the generators?) The BDGH paper. let me shut-up and read.

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I read Tom Braden's paper on mixed category for toric varieties. At least one idea is useful, namely where does the extra grading come from.

The idea is that, given a vector space with a unipotent operator, then you can split the vector space into Jordan blocks. Each Jordan block has a Z-indexed-filtration, with the index well-defined upto a global shift.

Suppose you have T^*P^n, with the stratification given by P^0, P^1, ... , like in category O. Then we want to equip the wrapped Fukaya category's cocore's endomorphism with extra grading.

Combinatorially, we can try to match the $End(T)$ algebra with the combinatorially defined $A(V)$ algebra. $A(V)$ algebra has extra grading. You can say that this is like $T$-equivariant cohomology, where the generator has cohomologically grading $2$. Then, we forget the cohomological grading and only keep the extra grading. Then the question is, why the endomorphism of $T$ brane is the same as $A(V)$. You can say that Gammage-Mcbreen-Webster solved the problem, but not quite. Local system is naturally identified with coherent sheaves module over torus; Only unipotent local system is identified with module near the identity of the torus; and only unipotent system with an explicit Frobenius action allows you to have a $\Z$-index filtration. So, the Fukaya category needs a graded lift.

There should be a version of the A-side, whose endomorphism of co-core is isomorphic to $A(V)$ as an ungraded algebra, then we can choose some graded lift. Well, the crossing over (between strands of different color) should have grading $1$, dot having grading $2$. That's the grading convention of $A(V)$.

Now, what's the relation between

• Frobenius on a variety $X$ defined over $\F_q$. we get, mixed constructible sheaf.
• algebraic Lagrangian skeleton in a algebraic symplectic variety, over $\F_q$. we should get mixed microlocal sheaf.
• holomorphic Lagrangian skeleton in a holomorphic symplectic variety, over $\C$. we should get mixed microlocal sheaf.
• Ask Saito, what's the analog in $\C$ for the mixed constructible sheaf?

In principle, all these question should be answered by these Koszul duality people. Koszul duality is about, two holomorphic skeletons, and a duality correspondence that sends cocore to core. It has been proven in the realization of mixed DQ module; but not yet in the relatization of Fukaya category.

First of all, Reeb chord, is not like Morse critical point. So, we don't necessarily have a canonical grading on it. It depends on a choice of holomorphic symplectic structure. There is some Conley-Zehnder index that I don't quite know. Maybe only $\Z/2$-grading is canonical. But that is Morse grading.

Let's dream a bit. Consider the 3d mirror between $T^*\P^2$ and $A_2$ surface, the smoothing of $\Z^2/\mu_3$ (we say $A_1$ surface is the smooth $T^*\P^1$).

What's the classical story of Koszul duality? Quadratic duality.

• symmetric algebra vs. anti-symmetric algebra.
• symmetric algebra but with a bit more quadratic relations; less anti-symmetric algebra. I don't have a geometric intuition for what does that mean. I don't understand why quadratic plays a big role here.

There is another way to think about this. Say, you have some semi-simple ring $R_0 = \oplus k e_\alpha$, and then some graded ring $R = R_0 \oplus R_1 \cdots $. So we have some 'augmentation', $R \to R_0$, most likely sending all the $R_{>0} \to 0$. Then, we should have some $R^! = End_{R-mod}(R_0)$. That might serve as the dual algebra. maybe graded? diagonally graded?

OK fine.

How about the algebra $A(V)$, it is positively graded, with degree $0$ part semi-simple. When the conormal intersects with the 'diagonal Lagrangian' (ok, I don't really know what does that mean? what is this diagonal Lagrangian in $T^*\R^n$? There is no good looking one. We can say

Suppose we have a region $\Delta_\alpha \In \eta + V$ and dual region $\Delta^\vee_\alpha \In V^\vee_\xi$, They don't necessarily intersect. Projectively, they might intersect.

Consider the following construction. Starting from $T^*\C^N$, with Hamiltonian $(\C^*)^N$-action, with alg moment map to $\C^N$ with singularity hyperplanes. OK, take a slice $\eta + V_\C \In \C^N$ (taking complex moment map), then take a quotient.

Consider SES $$ V_\Z \to \Z^N \to (V^\perp)^*_\Z $$ and $$ V^\perp \to (\Z^N)^* \to V^*_\Z $$ We want to say that $(V^\perp_\Z)^* \otimes \C$ is like the dual Lie algebra $Lie(L)^*$ of $L \In T=(\C^*)^N$. Then, $Lie(T)^* \cong \Z^N$.

• We are alg symp reduction by $L = (V^\perp_\Z)\otimes \C^*$ to get $M_H(V)$.
• We are alg symp reduction by $(V_\Z)\otimes \C^*$ to get $M_H(V)$.

It is sort of wrong to say $(V_\Z)\otimes \C^*$ and $(V^\perp_\Z)\otimes \C^*$ are two different subgroups of the same torus. They are different. Although the dual torus of $\C^*$ is $\C^*$, it is not the same one. So, we should not say that they are reduction from the same torus.

However, maybe we can take the product $M_H(V) \times M_H(V^\vee)$. We can say that, we have $T^*(\C^{2N})$, and do Hamiltonian reduction by a Lagrangian torus $V_{\C^*} \times V^\perp_{\C^*}$, and is left with still a product space.

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So I talked with YX a bit on 'what is the weight grading', in mixed sheaves. It is just the eigenvalue of the Frobenius action, on etale sheaves.

Then, why the endomorphism of the big tilting has that weight grading.

Let's think about hypertoric variety, Gale duality.

Given a vector space in $\R^N$, we have $$ V \to \R^N \to (V^\perp)^* $$ $$ V^\perp \to (\R^N)^* \to V^* $$ Great. Given $\eta \in (V^\perp)^*$ and $\xi \in V^*$, we look at the fiber $V_\eta$ and $(V^\perp)_\xi$, they are partitioned by the restriction of the sign partitions in $\R^N$ and $(\R^N)^*$.

Feasible is dual to bounded. So the two sides has the same collection of feasible and bounded.

Consider the Lagrangian $(V \oplus V^\perp)_{\eta, \xi}$ intersecting with the diagonal sign blocks in $T^*\R^N$.

I still don't know why we have this correspondence of bounded and feasible chambers. But we do, and the two linear spaces $V$ and $V^\perp$ can be really different.

• 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.

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• obsidian?

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• abelian with matter is the trouble

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• what to say next

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Declaration: I was lost in abstract definitions for too long. I will now compute examples, examples, and examples. Not only because abstraction makes me sleepy, but also examples makes my hands dirty and my mind happy.

• Reading Ginzburg's paper
• Heisenberg algebra and Weyl algebra (just names)
• Writing my own paper

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