# 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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