# Why is the mirror of rigid Calabi-Yau threefold singularity theory?

Mirror symmetry relates two Calabi-Yau threefolds with mirrored Hodge diamonds. Since Calabi-Yau threefold is Kahler, this naive correspondence does not hold for rigid Calabi-Yau threefolds. Here Calabi-Yau theefold is called rigid if it has no complex deformation, i.e. $h^{2,1}=0$.

I heard that a "mirror manifold" of a rigid Calabi-Yau threefold is singularity theory, or Landau-Ginzburg theory. Are there any good explanation for this? Or can someone suggest a good reference for this (I am a math grad student with little physics background)?

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There is a toric complete intersection example, rather ad hoc, in arXiv:alg-geom/9402002.

This particular example is related to the situation when a decomposition of anticanonical class on toric Fano into sum of nef Cartier divisor classes can not be realized as nef-partition (decomposition of infinity divisor into sum of effective nef Cartier divisors).

I would imagine that there are other examples where the same SCFT can be obtained by different means, one coming from rigid CY, but I am not familiar with them.

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If you want to know why there is a natural reason for mathematicians to think some mirror symmetry may relate complex and symplectic manifolds which don't come from superconformal sigma models, see arXiv:math/0209319

The basic idea is that such pairs can be obtained from "mirror" smoothings of singular limits of Calabi-Yaus which are mirror. (One can make sense of this physically too, but not in a way that is accessible to known sigma model techniques).

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It is better to say that CY is mirror to the certain LG model. Namely if one considers quasi-homogeneous polynomial $W(x_1, \dots, x_n)$ defining isolated singularity at zero then the mirror pair of it is given by: $$\mathcal X := [\{W = 0\}/G],$$ where $G$ is group of symmetries of $W$. Then mirror symmetry is the relation between total ancestor potential of $W$ and GW potential of $\mathcal X$ or between Frobenius structures of $W$ and $\mathcal X$.

This is actually one very particular face of the mirror symmetry in singularity theory. The best reference is probably Chiodo-Ruan A global mirror symmetry framework for the Landau-Ginzburg/Calabi-Yau correspondence

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