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Nolan Fitzpatrick: I understand that the extra dimensions of string theory are invisible to our low energy experiments, either the extra 6 of superstring theory and the extra 7 of M-theory. I also understand that these extra dimensions are either made invisible by compactification at the Planck length with Calabi-yau manifolds or our universes is confined to a 4-dimensional brane embedded in a bulk or hyperspace. However, the Calabi-yau manifold, as I have read, is typically taken to be 6-dimensional (3 complex dimensions). In M-theory, there are 7 extra dimensions made compact. Do these dimensions also take on the geometry of the Calabi-yau, some sort of 7-dimensional Calabi-yau or is there some other method of compactification?
John H. Schwarz:
In this case one considers 7-dimensional manifolds of G2 holonomy.
(Six-dimensional Calabi-Yau's have SU(3) holonomy.)
The math is very challenging. Also, there are lots of issues if one
wants to get realistic physics this way.
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