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In the 90-91 pager "A PAIR OF CALABI-YAU MANIFOLDS AS AN EXACTLY SOLUBLE SUPERCONFORMAL THEORY", Candelas, de la Ossa, Green, and Parkes, brought up a family of Calabi-Yau 3-folds, canonically constructed from a sub-family of quintic CY 3folds, as a "mirror" to quintic an did some calculations on the mirror family to extract the GW invariants of the quintic.

I was having a discussion with a group of physicist on what argument lead them to take that particular family as the mirror, whatever mirror means; and apparently, the original point of view among physicists has been changed in past two decades and they did not know the answer either.

Does any one know what (may be physical) original recipe lead them to that particular mirror family?

This question should have been asked before, but a brief search did not lead me anywhere. Let me know if thats the case and I would remove this post.

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The history of this is as follows. In the paper by Candelas, Lynker and Schimmrigk there are two weighted hypersurfaces whose cohomology is mirror to that of the quintic. These therefore are two potential candidates for the mirror quintic. The question then was how to decide whether they provide mirror partners to the quintic or not. This was addressed in a paper by Lynker and Schimmrigk (http://inspirehep.net/record/27957) by transporting the Greene-Plesser construction of quotients of conformal field theories to the level of Landau-Ginzburg theories and hence weighted hypersurfaces. This established at level of physics that the two weighted hypersurfaces in the list of Candelas-Lynker-Schimmrigk are isomorphic and that at the level of physics they are both mirrors of the 1-parameter quintic family.

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    $\begingroup$ very interesting; I have no first-hand knowledge of the discovery, but I'm a bit surprised that Candelas et al. give all the credit for the discovery of the quintic mirror to Greene and Plesser (on page 23, top, and on page 24, footnote), while Lynker and Schimmrigk are quoted in passing with "For a more general discussion see ..." (page 26, bottom). I would think this is the type of citation you would give to a paper that did not play a key role in your own work. $\endgroup$ – Carlo Beenakker Nov 7 '14 at 19:24
  • $\begingroup$ The date of Lynker and Schimmrigk is very close to that of COGP! $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 7 '14 at 19:24
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The account Brian Greene gives in The Elegant Universe (a surprisingly useful reference for the early history of mirror symmetry) is that he and Plesser were studying techniques to create new Calabi-Yau manifolds using orbifolds. The mirror of the Fermat quintic arises naturally in this context. Nearly simultaneously, Candelas, Lynker, and Schimmrigk generated a large number of examples of probable mirror pairs by studying Calabi-Yau hypersurfaces in weighted projective spaces. (Math Reviews was highly enthusiastic: http://www.ams.org/mathscinet-getitem?mr=1067295 ).

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  • $\begingroup$ You mean they randomly picked that as the possible mirror; i.e. they had a family with the right hodge numbers and they tried that! $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 7 '14 at 18:04
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The history is described here: The quintic mirror was constructed by Greene and Plesser as one of a few hundred mirror manifold pairs. Candelas et al. acknowledge in their article that they got the quintic mirror from Greene and Plesser.

There is an interesting quote by Brian Greene why he did not himself pursue the enumeration problem solved by Candelas et al. using the quintic duality that he and Plesser had discovered:

You can have an equation that you know is abstractly correct, but it can nevertheless be a major challenge to evaluate that equation with adequate precision to extract numbers from it. We had the equation but we didn't have the tools to leverage it into the determination of numbers. Candelas and his collaborators developed the tools to do that, which was a huge accomplishment.

The Greene and Plesser construction as it is summarized here:

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  • $\begingroup$ Still does not answer my question; What was the recipe for getting that particular mirror. $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 7 '14 at 18:02
  • $\begingroup$ How did they know that they have an equation that is abstractly correct? $\endgroup$ – Mohammad Farajzadeh-Tehrani Nov 7 '14 at 18:09

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