As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, not only the fields?) for quantization. But moreover, their features also seem to turn up in questions regarding the defomration of one two-dimensional world sheet quantum field field theory to another. I see the introduction of operators in the Lagrangian etc. and it's reminiscent of renomralization (keyword: moduli space?). Then I read something about the beta function being zero, because it would be proportional to the Ricci curvature of the target manifold, which is zero for Calabi-Yau manifolds. And that's another argument, which speak for their consideration. (Remark: Maybe there is a broader spectrum of theories for which these questions don't make sense. I guess I'm talking about non-linear sigma models here.)
In string theory, do you use the renormalization techniques as representations of the geometrical deformations?
We get different theories for neightboring metrics right? How closely is the number of possible vacuum states is related to the number of Calabi-Yau metrics?