The new exciting thing today is, the raising and lowering operator.

Let $E$ be the functor for adding strands, and $F$ the functor for removing strand, then we should

• define the functor $F$
• check that $F$ and $E$ are adjoint
• test it on T-branes.
• understand Rouquier's condition geometrically.

The setup is the following: $\gdef\lcal{\mathcal L}$

• we have a smooth closed surface $S$ with boundary, and there are some stops on the boundary.
• We can consider $k$ disjoint Lagrangians $L_1, \cdots, L_k$, which defines an object in $Sym^k(S) \RM \Delta$
• 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.

consider the simplest matrix factorization example on toric 3-folds. They are both gluing two copies of $\C^3$ together.

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, $Tot[O(-1) \oplus O(-1)]$.

There is another way to resolve. Let's not go there.

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 $a=(0, -1, 0), b=(1, 1, -1), c=(-1,1,-1)$. The problem with this set of generator is that, $b,c$ span a unsaturated sublattice, we have $d=(b+c)/2 = (0, 1, -1)$, which should be added. so the relation is $2d = b+c$. Then the coordinates are $A,B,C,D$ with $D^2 = BC$, and we do $\P^1$ coordinate like $u=B/D = D/C$ and $v = 1/u = D/B = C/D$. So $v^2 = C/B$, the local charts are

• $(A, B, v)$ and $(A, C, u)$, with change of coordinates like $u = 1/v$ and $C = v^2 B$. We have $Tot[O(0) \oplus O(-2)]$

Now, for the superpotential, we know we want the coordinate $(0,0,-1)$.

• In the first case, we have function $ABu = AB(C/B) = AC = AB(D/A)=BD$, similarly, we have $CDv=CD(B/C)=BD..$, so $u = c-b=(1,1,-1)$ and $a+b+u= a+c=(0,0,-1)$ hmm, it works.
• For the other case, we have $a+d = (0,0,-1)$, so the function is $AD = ABv = ACu$.

OK, great. Now, how do we compute its matrix factorization? We are no longer on an affine scheme, although, the function $W$ still is a well-defined function on the affinization.