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The most general $\delta g_{mn}$ that preserve the Ricci-flatness on the original Calabi-Yau backgrounds is the sum of several components: the pure infinitesimal diffeomorphisms (I won't discuss those because they're physically vacuous and trivial); changes of the Kähler moduli; and changes of the complex structure. A Calabi-Yau three-fold has $h_{1,1}$ ...

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Yes, if all the dimensions are compact, well, we really mean that all spatial dimensions are compactified on a torus $T^9$, then (multiple) T-duality may map any simple D$p$-brane aligned with some dimensions to a D0-brane. Under T-duality, D$p$-brane is mapped either to a D$(p+1)$-brane or a D$(p-1)$-brane, so its dimension either increases or decreases. ...

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In some extra-dimensional models, such as brane cosmology, the fields (except gravity) are indeed confined to a lower-dimensional surface, which is sort of like "sharing almost the same coordinates in the extra dimensions". In Kaluza-Klein theory with compact extra dimensions, the fields are basically spread evenly across the entire size of the extra ...

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Your diagram looks like an illustration of the Ekpyrotic universe. In this model the extra dimensions are not compactified (i.e. curled up) so there is no uncurling of them. The reason we don't see the extra dimensions is because our universe is confined to a 3D brane, not because the extra dimensions are curled up. One well known theory for what determines ...

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