Peng Zhou

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====== 2024-12-20 ======

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: 
  - $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 $NH_{k_1, k_2}$. 
  - This parabolic subgroup $P_{k_1, k_2}$ is the automorphism subgroup that preserves the partial flag $\C^{k_1} \In \C^k$. 
  - 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 $P_{k_1, k_2}$. 
  - 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 $Sh(pt \to pt)$?