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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.

Thanks!

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  • $\begingroup$ The CFTs are two-dimensional and describe the embedding of the string in the CY, i.e. the bosonic fields of the CFT correspond to the different directions in the CY geometry that the string can oscillate. $\endgroup$ – Mitchell Porter Aug 17 '18 at 14:07
  • $\begingroup$ @MitchellPorter Thanks. Does that mean that the CFT is a field theory on these surfaces (which I suppose are world sheets of strings)? $\endgroup$ – doetoe Aug 17 '18 at 14:10
  • $\begingroup$ "the different directions in the CY geometry that the string can oscillate": does that mean that they take their values in the normal bundle to the tangent bundle of the embedded surface in the tangent bundle of the CY? $\endgroup$ – doetoe Aug 17 '18 at 14:15
  • $\begingroup$ The bosonic part of the CFT is basically a position vector for the string worldsheet in the CY... But the CFT is more fundamental than the CY. Perturbative string theory is the sum over Riemann surfaces for this CFT, and the CY emerges from that. $\endgroup$ – Mitchell Porter Aug 17 '18 at 21:09
  • $\begingroup$ @MitchellPorter Does this also mean that rather than a single CFT, we have a CFT on each possible world sheet embedded in the CY (or in the full 10D space)? $\endgroup$ – doetoe Aug 18 '18 at 14:11

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