What is the mirror manifold of the total space of the bundle $O(-1)\oplus O(-1)$ over $P^1$? I have tried to find the answer on the web but failed. Is there a good reference for this? Thanks.

## 4 Answers

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.

I think this is a stubborn case which does not fit into the general picture. For example, if you use the standard toric procedure to try to construct a differential equation for the log periods of the mirror, then try to find the GW invariants from the solutions (in this case, to recover the famous 1/d^3 formula), then it won't quite work. There are various adjustments you can make, based on knowing the answer ahead of time, and I once saw a paper (sorry, I forget where or by whom) which tried to make sense of all this, but I don't think there is a good general picture of this case. I would check Klemm's written record for some guidance.

Try Patrick Clarke's paper Duality for Toric Landau-Ginzburg Models, which contains a general procedure for determining mirrors of (not necessarily compact) toric varieties (with superpotential, or without (i.e. with superpotential = 0)).

In some sense the answer is $T^*S^3$. Look at http://arxiv.org/pdf/hep-th/0211098v1