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--- blog:2023-09-12 [2023/09/13 14:46] – [MF example] pzhou
+++ blog:2023-09-12 [2023/09/14 00:10] (current) – [Challenge 2] pzhou
@@ Line 16: / Line 16: @@
   * We can consider $k$ disjoint Lagrangians $\lcal_1, \cdots, \lcal_k$ in $S \times \C^*$, which defines an object in $Sym^k(S \times \C^*) \RM \Delta$. We assume these $\lcal_i = L_i \times F_i \In S \times \C^*$ are of product type.
-===== MF example =====
+I immediately run into a trouble, which is, how to deal with the fact that, when I add another strand, the superpotential would be very different (there are interaction among many many terms).
-consider the simplest matrix factorization example on toric 3-folds. They are both gluing two copies of $\C^3$ together.
+I could say the following: either don't deal with the fiberwise non-compact space (no, I need T-brane).
+==== Challenge 1 ====
+define $E_L$. My Lagrangian is sitting in $Sym^k(B \times \C^*) \RM \Delta_B$. Then, I want to concatenate a strand, an object, near a specific stop. We can certainly do it for object, (hom from T to each strand you want to delete, you get a dg vector space as coef) and we can do it for morphism as well. it doesn't do much.
+define $F_R$. Suppose you want to subtract a strand. ok, this is really interesting. On the level of objects, you can wrap the $T$ brane, and count the intersections with the many strands. On the level of morphism, it works as well, by counting disks.
+Try see if we have $E_L F_R \to id$, and we have $id \to F_R E_L$. In the first case, you subtract, then you add. oh, it is then like tautology, because it is like $T \otimes hom(T, L_i) \to L_i$, sure. when you add and then substract, you always have that $id \to Hom(T,T)$.
+==== Challenge 2 ====
+See if you can recover $NH_k$? But this is already done. Let's double check.
+how to see $T^{k-1} \to (k) Hom(T, T) T^{k-1}$, this is like $(k) NH_{k-1} \C[x]$, OK, roughly speaking $\C[S_k]$ as $\C[S_{k-1}]$ module is $k$ dimensional. OK. fine.
+==== Challenge 3 ====
+Reproduce Rouquier.
-The first one is given by the fan with ray generator $(0,0,1), (1,0, 1), (0,1,1), (1,1,1)$. yes, a square in $x,y$ direction on level $z=1$. It is the total space of two line bundles on $\P^1$, but which two? Let's compute the face conormal, we have
-  * $a=(-1, 0, 0), b=(0, -1, 0) $ and $c=(1,0, -1), d=(0, 1, -1)$.
-one can see that $a+c =b+d$. That means the coordinate ring of the affine space is given by $AC = BD$, wher $A,B,C,D\in \C^4$. Now, we can do blow-up, I guess we can do two patches with coordinate $A, B, u = C/B=D/A$ and the other with $C, D, v = B/C = A/D$. OK, great! Now, how does the fiber coordinate tranform? We have $u B = C, uA = D$, so it is like, $Tot[O(-1) \oplus O(-1)]$.
-The second one is given by the fan on generators $(0, 0, 1), (1, 0, 1), (-1, 0, 1), (0, 1, 1)$. with dual cone generator being $a=(0, -1, 0), b=(1, 1, -1), c=(-1,1,-1)$.