• Fukaya category of a singular space

Suppose you have a singular affine space $Y_0$, as the unique fiber of some fibration $W: Y \to \C$, where the singularity is not so bad. There are two ways to define the Fukaya category of the singular space. We can either do $$Fuk(Y_0) = Fuk(Y_1) / \text{vanishing cycle}$$ Or we can do $$Fuk(Y_0) = Fuk(Y \times \C_\eta, \eta W)$$

Example Consider $Y_0 = \{xy=0\}$, we know its mirror is $\C^* \RM \{1\}$. One way to realize this is quotient $\Coh(\C^*) / \langle O_1 \rangle$, another way is to do $Fuk(\C^3_{x,y,\eta}, \eta xy)$.

The last approach might be a bit democratic.

Consider another example Consider $Y_0 = \{x^2+y^2+z^2 - 1 = 0\}$ (well it is not singular), still we can check $$Fuk(Y_0) = Fuk( \C^4, \eta( x^2+y^2+z^2 - 1) )$$

Now, $Y_0 \cong T^*S^2$, we know its Fukaya category, which is $Loc(S^2)$, somehow is $\C[x]$-mod, where $|x|=-1$ (who know they can be of negative degree??). To test if my theory is correct, we compute the LG-model, we first compute what is $$Fuk( (\C^*)^4, \eta( x^2+y^2+z^2 - 1) ) \cong Coh((\C / \Z_2)^3 \times \C)$$ Then, you partially compactify on the A-side, and turn on superpotentialon the B-side.

$$W_B = z_1^2 + z_2^2 + z_3^2 + z_1 z_2 z_3 z_4$$

If we considering $W_A = \eta(x+y+z-1)$, so $m_1, \cdots, m_4$. They are basis of the $M_A$. we need to have $n_1, n_2, n_3, n_4$ for compactification, so that we get