Understanding Compactified Dimensions in Superstring and M-Theory

[Kevin_Axion]

In superstring theory there is a 10 dimensional space-time with 6 compactified spatial dimensions in Calabi-Yau manifolds and 4 expanded dimensions with three being space and one temporal. Now with m-theory there is one more dimension so does that mean there are 7 compactified dimensions (I know it appears illogical to ask this question but I always here String Theorists saying there are 6 compactified dimensions, most likely because they aren't discussing m-theory). If so what is the shape of these compactified dimensions?

 

[negru]

Well you can get usual 10D string theory by compactifying M-theory on a circle.

 

[suprised]

If one wants to get down from d=11 to d=4 and preserve N=1 supersymmetry, then the 7-dimensional compact manifold needs to have G2 holonomy. This group is the analog of SU(3) for a six-dimensional Calai-Yau manifold.

Such "G2-manifolds" are generically not equivalent to a direct product of a circle times a Calabi-Yau manifold, so a priori yield a different class of theories in d=4.
However some of such compactifications are dual to "ordinary" compactifications of d=10 strings on Calabi-Yau spaces. This shows again that there is no absolute notion of what the dimension of the compact space is.