In string theory, could the space we live in be a general Calabi-Yau fibration over Minkowski space?

Question

Assuming that we live in a 10 dimensional spacetime, it is often informally argued that according to string theory we live in a product of Minkowski space and a 6-manifold, which because of the required compatibility with general relativity and with supersymmetry actually has to be a Calabi-Yau threefold (I hope what I say is more or less correct). Slightly more generally, it could be a fiber bundle over Minskowski space whose fiber is a Calabi-Yau threefold.

However, from these informal arguments it is not clear why or if we need to have the same Calabi-Yau variety at every point. Could it more generally be a smoothly varying family of (generically) Calabi-Yau manifolds over Minkowski space, or is there a physical reason why it most be locally trivial?

Comments

This question is interesting.  It is possible to think about the reciprocal picture : Minkowski fibration on a Calabi-Yau threefold.  More generally:  a K3 fibration on a Calabi-Yau threefold.  From other side in the case of a "Calabi-Yau fibration over Minkowski space", two Calabi-Yau threefolds corresponding respectively to two different Minkowski points, must be related via Mirror Symmetry of the Batyrev-Borisov`s kind.  Do you agree?

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@juancho I am not able to give an informed opinion w.r.t. your remark about the relation between the Calabi-Yau varieties at different points, but in its spirit, if more general Calabi-Yau fibrations are indeed possible, an interesting follow-up question would be to know in what ways such manifolds would have to be related.

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Would having a different Calabi-Yau threefold above each spacetime point not als mean that the laws of physics, particle spectrum etc vary in the same way from point to point?

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@Dilaton I guess so, right? To be honest I only thought about the mathematics (because I know very little of the physics), but since many Calabi-Yau varieties could give us a string theory that is compatible with what we know, this might not be a problem. We would however lose the homogeneity of spacetime. Could that be the reason that we prefer to assume having a fiber bundle? Please let me know if it doesn't make sense what I say.

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@doetoe the question is interesting. Yes, having varying Calabi-Yau compactifications inside the observable universe (would that be sheafs instead of fibre bundles?) would certainly break homogeneity of spacetime and probably have other bad consequences.

However, maybe such more general mathematical structures could be useful to describe internal inflation, which is a scenario where inflation starts at different regions (separated by domain walls) at different times leading to multiple universes that can have different laws of physics (?).

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There are many papers in string theory on situations where a modulus (such as size of a cycle in a CY) evolves, or where the actual topology of the CY changes. However, I think neither process is quite rigorously understood, even at the level of perturbative string theory.