In string theory there are physical reasons why the space we live in must locally be the product of Minkowski space with a Calabi-Yau manifold. The general theory doesn't say which Calabi-Yau manifold. However, different Calabi-Yau manifolds give rise to different physics. In fact, Gepner devised a way to obtain a fully determined conformal field theory from a given Calabi-Yau manifold. It can already be seen from his construction that some Calabi-Yau manifolds give rise to the same CFT.
In the other direction Greene-Vafa-Warner managed to create a Calabi-Yau manifold starting from a CFT. After applying some symmetry to the Lagrangian you obtain a different Calabi-Yau manifold giving rise to the same CFT. These manifolds are generally not even topologically equivalent, but they do reflect each other's geometrical structure, most notably by an interchange of the non-trivial hodge numbers, in a way that was not at all anticipated on purely mathematical grounds.
Is this more or less accurate? And can anyone say something more about the CFT's that are being considered? Are they the theories that are obtained when we follow the renormalization group flow to a fixed point in which only the massless states survive? I would greatly appreciate an answer that is precise but not overly technical, to give me an intuition that can help me read more technical accounts.