Mirror of local Calabi-Yau - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T19:15:58Z http://mathoverflow.net/feeds/question/18631 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau Mirror of local Calabi-Yau Junwu Tu 2010-03-18T17:39:25Z 2010-04-15T23:35:49Z <p>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.</p> http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/18642#18642 Answer by Kevin Lin for Mirror of local Calabi-Yau Kevin Lin 2010-03-18T19:26:37Z 2010-03-19T03:37:30Z <p>Try Patrick Clarke's paper <a href="http://arxiv.org/abs/0803.0447" rel="nofollow">Duality for Toric Landau-Ginzburg Models</a>, which contains a general procedure for determining mirrors of (not necessarily compact) toric varieties (with superpotential, or without (i.e. with superpotential = 0)).</p> http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/18711#18711 Answer by Tony Pantev for Mirror of local Calabi-Yau Tony Pantev 2010-03-19T02:55:46Z 2010-03-30T20:27:09Z <p>The physicists (see e.g. <a href="http://arxiv.org/abs/hep-th/0012041" rel="nofollow"> this paper of Aganagic and Vafa</a>) 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 $$X : \ xy = u + v + a uv^{-1} + 1.$$</p> <p>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. </p> http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/19320#19320 Answer by Eric Zaslow for Mirror of local Calabi-Yau Eric Zaslow 2010-03-25T15:45:29Z 2010-03-25T15:45:29Z <p>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.</p> http://mathoverflow.net/questions/18631/mirror-of-local-calabi-yau/21517#21517 Answer by Mohammad F.Tehrani for Mirror of local Calabi-Yau Mohammad F.Tehrani 2010-04-15T23:35:49Z 2010-04-15T23:35:49Z <p>In some sense the answer is $T^*S^3$. Look at <a href="http://arxiv.org/pdf/hep-th/0211098v1" rel="nofollow">http://arxiv.org/pdf/hep-th/0211098v1</a></p>