What happens when you try to do Fukaya-Seidel category with the function $f = xy^2$ on $\C^2$?

Well, you would say first, let's compute the regular fiber, which is $(\C^*)$, parameterized by $y$. Then, you ask, what is the monodromy. I don't think there is anything special, so let's say, the monodromy is trivial.

But, is it really trivial? hmmm, what if we set the fiberwise superpotential like $W = x+y$, we get some interesting fiberwise stop and wrapping. But I have no idea why we want these fiberwise wrapping. Anyway, on the fiber over $c$, we get $W = c/y^2 + y$.

Suppose I build a U brane on the base, and compute its endomorphism.

Let me try something else. $xy(y-a)$. How about this function? Is it better in terms of being more generic? partial $x$ gives $y(y-a)=0$, and partial y gives $x (2y-1)=0$, so we get either $(0,0)$ or $(0,a)$ as singular point, all with singular value $0$. OK, so we have two critical points over the same critical value, it seems they don't talk to each other.

Is that what happens with $f=xy^2$? Not quite, here is the trouble, the generic fiber was $\C_y \RM \{y=0, y=a\}$, And it is quite different from $a=0$. Maybe you can kill something, so that it is ok.

What's the story for $f=xy$? Say, we are on the boundary $|x|^2+|y|^2=1$, and we have something stupid, like $xy > 0$ intersecting with it. This says, we have $|x|, |y|$ free, but $\theta_x = -\theta_y$. So, the stop on the boundary is like, an open cylinder in $S^3$, which contracts to $S^1$, and which gives the stop (or the thimble as cone over the stop).

Now, we do $2 \theta_y + \theta_x = 0$. Good, suppose we now cone over this Legendrian knot, what can I say?