General algebraic definition of mirror symmetry

Question

I'm trying to understand the following statement of Hori-Vafa from the algebraic perspective:

The mirror of the Hirzebruch surface $\mathbb{F}_{n}$ is the Landau-Ginzburg model $x+y+\frac{a}{x}+\frac{b}{x^{n}y}$, where $a,b \neq 0$.

However, from my search on literatures related to mirror symmetry in algebraic geometry, mirrors are usually only defined for Fano and Calabi-Yau varieties. Is there a general algebraic definition that explains what means "A is mirror to B"?

Answer 1

I don't know if there is a more up-to-date precise statement for $\mathbb{F}_n$ uniformly for all $n \in \mathbb{N}$, my knowledge is definitely not recent, but here is what I know.

Denote your LG-model / superpotential by $W \colon (\mathbb{C}^*)^2 \to \mathbb{C}$, $W = x+y+\frac{a}{x}+\frac{b}{x^{n}y}$ and let $\operatorname{Lag}_\mathrm{vc}(W)$ denote the corresponding category of Lagrangian vanishing cycles as defined by Seidel. Then:

• For $n = 0,1$ we have $D^b(\operatorname{Coh}(\mathbb{F}_n)) \cong D(\operatorname{Lag}_\mathrm{vc}(W))$;
• For $n \geq 2$ we have $D(\operatorname{Lag}_\mathrm{vc}(W)) \cong D^b(\operatorname{Coh}(\mathbb{CP}^2(n,1,1))$, where $\mathbb{CP}^2(n,1,1)$ is the weighted projective space $\mathbb{CP}^2$ with weights $(n,1,1)$. Furthermore, it is known that $D^b(\operatorname{Coh}(\mathbb{F}_2)) \cong D^b(\operatorname{Coh}(\mathbb{CP}^2(2,1,1))$, which gives you the correspondence in this case.
• For $n \geq 3$: $D^b(\operatorname{Coh}(\mathbb{F}_n))$ can be viewed as a full subcategory of $D^b(\operatorname{Coh}(\mathbb{CP}^2(n,1,1))$ generated by the exceptional collection $\langle \mathcal{O},\mathcal{O}(1),\mathcal{O}(n),\mathcal{O}(n+1) \rangle$, which suggests that replacing $D(\operatorname{Lag}_\mathrm{vc}(W))$ with a suitable modification should give the correct HMS equivalence (and indeed it does, but I don't recall the precise statement).