ag.algebraic geometry - Mirror of local Calabi-Yau

The physicists (see e.g. this paper of Aganagic and Vafa) will write the mirror as a threefold $X$ which is an affine conic bundle over the holomorphic symplectic surface $mathbb{C}^{times}times mathbb{C}^{times}$ with discriminant a Seiberg-Witten curve $Sigma subset mathbb{C}^{times}times mathbb{C}^{times}$. In terms of the affine coordinates $(u,v)$ on $mathbb{C}^{times}times mathbb{C}^{times}$, the curve $Sigma$ is given by the equation
$$Sigma : u + v + a uv^{-1} + 1 = 0,$$
and so $X$ is the hypersurface in $mathbb{C}^{times}times mathbb{C}^{times} times mathbb{C}^2$ given by the equation
$$X : xy = u + v + a uv^{-1} + 1.$$

From geometric point of view it may be more natural to think of the mirror not as an affine conic fibration over a surface but as an affine fibration by two dimensional quadrics over a curve. The idea will be to start with the Landau-Ginzburg mirror of $mathbb{P}^{1}$, which is $mathbb{C}^{times}$ equipped with the superpotential $w = s + as^{-1}$ and to consider a bundle of affine two dimensional quadrics on $mathbb{C}^{times}$ which degenerates along a smooth fiber of the superpotential, e.g. the fiber $w^{-1}(0)$. In this setting the mirror will be a hypersurface in $mathbb{C}^{times}times mathbb{C}^{3}$ given by the equation
$$xy - z^2 = s + as^{-1}.$$
Up to change of variables this is equivalent to the previous picture but it also makes sense in non-toric situations. Presumably one can obtain this way the mirror of a Calabi-Yau which is the total space of a rank two (semistable) vector bundle of canonical determinant on a curve of higher genus.

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