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In a previous question (Calabi-Yau manifolds and compactification of extra dimensions in M-theory), I was told that the $G(2)$ lattice can be used to compactify the extra 7 dimensions of M-theory and preserve exactly $\mathcal N=1$ supersymmetry.

However, since there is only 1 $G(2)$ lattice, there should be only 1 4-dimensional M-theory. Then, why is there such a huge fuss about the M-theory landscape?

Thanks!

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up vote 5 down vote accepted

It's not a "$G(2)$ lattice" one has to compactify the M-theoretical dimensions upon (after all, the $G_2$ lattice is 2-dimensional); it's the $G_2$ holonomy manifolds. There are lots of different topologies of these seven-dimensional manifolds. They're analogous to the Calabi-Yau manifolds but don't allow one to use the machinery of complex numbers.

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Thanks. Now that I get it. – centralcharge Jun 2 '13 at 9:22

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