Peng Zhou

It's a good place to work and study, the Joshua Tree Field Station (very cool hotel).

I am still trapped by the disk with 3 stops, no punctures.

But do you understand the disk with 2 stops, how the gluing works? one disk has $k_1$ strands; another disk has $k_2$ strands, both with $2$ stops, put them together, get $k=k_1+k_2$ strands.

It is about going from $BGL(k_1)$ and $BGL(k_2)$ to $BGL(k)$. I just know $T^{k_1}$ and $T^{k_2}$ merges to $T^{k_1+k_2}$. Let's try parabolic induction: we have $$ G_1\times G_2 = GL(k_1) \times GL(k_2) \gets P=P_{k_1, k_2} \to GL(k_1+k_2)=G $$

Assume everything is fine, then we have $$ BP \to B(G_1 \times G_2), \quad BP \to BG $$ The first one is saying, if you have a principle $P$ bundle $E$ over something $M$, then you can build the associated bundle of $E \otimes_P L$.

I guess the corresponding construction on sheaf is just doing pull-push along the parabolic induction.

Is there classifying space for two step flags? well, one needs to say the ranks, so it would be the $BP_{k_1, k_2}$.

Let's keep on thinking about disk with three stops.

There are a few things in common with disk with two stops and one feature in the middle, like a hole, a puncture. One need to decide if the T-branes are on the left or on the right. In the case of punctures, or several punctures, we are looking at how the standard flag in $\C^k$ map to the target flag $0 \In \C$.

I really don't know where is that target space come from?

What is the disk with three stops?

Disk with two stops, $k$ strands T-brane, has endomorphism algebra $NH_k$, with $q$ grading for crossing $q^{-2}$. Correspondingly, we have $(\pi: BB \to BG)_* \C_{BB}$, whose endomorphism involves $\pi^!$. Recall that for sheaves (not coherent sheaves), $\pi^! = \pi^* [\dim_\R fiber]$. This explains why we have those negative cohomological degrees.

Now we add another stop, say at the top. We will have $(k,0), (k-1,1), \cdots, (0,k)$ different types of $T$-branes.

We have a few observations:

1. $End(T_{(k_1, k_2)}) = NH_{k_1} \otimes NH_{k_2} =: NH_{k_1, k_2}. $. On the other side, we can consider $(BB \to BP_{k_1, k_2})_* \C$ whose endomorphism is also $NH_{k_1, k_2}$.
2. This parabolic subgroup $P_{k_1, k_2}$ is the automorphism subgroup that preserves the partial flag $\C^{k_1} \In \C^k$.
3. We have different kinds of flags there. They are different quiver representations of $ \bullet \to \bullet$. For example, consider the injection $j: \C^{k_1}\into \C^k$ as an object in the quiver rep. The moduli stack for this object is $Gr(k_1, k) / GL(k) = pt/P_{k_1, k_2}$, this is because $GL(k)$ acts on $Gr(k_1, k)$ transitively, with stabilizer of a point $P_{k_1, k_2}$.
4. How to think about $T_{2,3} \to T_{1,4}$? We have flag $Fl_1:=( \C^3 \to \C^5)$ and $Fl_2 := (\C^4 \to \C^5)$. We do have map from $Fl_1 \to Fl_2$, just like $T_{2,3} \to T_{1,4}$ admits maps.

Our goal is to build a category living over the Higgs side space. But I don't know what is the ambient space.

One guess is that, the space is still and always is $BGL_k$, and the T-brane sheaf is still $BB \to BG$ pushing forward. But this is like after doing stop removal. How to remember?

Another way of thinking is, we don't have to use $[pt/GL_k]$, but $[(pt \to pt)/GL_k]$, and then do sheaves on it. What does it mean? A sheaf of vector spaces on a space $X$, is a functor from the open category to $dgVect$. A sheaf on a quotient stack $[X/G]$, or the category of sheaves on $[X/G]$ is the limit (equalizer in the infinite version) $$ Sh(X) \to Sh(X \times G) \to Sh(X \times G \times G) \to \cdots $$

But what is $Sh(pt \to pt)$?

So Mauricio asks, what is the 'A-side' window? I thought I know the answer, a cheap one, not in the form of a generator, but just in terms of a subcategory generated by certain classes of objects. And I am not sure if it is right.

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What is that hemisphere partition function? Consider a gauged linear sigma model with superpotential, namely a reductive complex Lie group $G$ acting on $V$ preserving volume form and a superpotential $W: V \to \C$. To get $\Z$-graded MF category, we choose certain non-negative $R$-charge (could be in $\Q$) so that $W$ has weight $2$.

We consider the $G \times \R$-equivariant MF, call it $M$. In different GIT chamber, we get different contour deformation $Z(M;t)$.

• anything supported on the unstable loci has $Z=0$
• deep in the GIT chamber, anything converges.

When migrating the server, I forgot to backup the yiye website, lost a lot of memories, sad. I also dumped a lot of my phd notes to dumpster anyway when I move away from Chicago, it is inevitable to let go.

If I consider coherent sheaf category as the category of B-branes, then how does it depends on the complexified Kahler parameters? One can either do it in the physical way, reading Witten, Hori etc; or one can do it in the math way, thinking about deformation of the Coh category by gerbes.

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reading the classic papers, Khovanov-Lauda, Rouquier, Varagnolo-Vasserot, to dig out how to setup the convolution algebra (in general)

I don't even understand why these things can be realized using Fukaya category and Floer theory, besides 'physics motivation'. What's the math motivation?

Question 1: where does convolution algebra come from? Like Hecke algebra? What is the physical setup? you can say, some correspondence and push-pull.

Let's summarize what can be written.

1. HMS for K-theoretic Coulomb branches

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