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I just finished learning M(atrix) theory and the basics of the compactification of extra dimensions.

The extra 6 dimensions of superstring theory can be compactified on 3 Calabi-Yau manifolds (because 6 real dimensions means 3 complex dimensions).

However, when it comes to M-theory, one cannot compactify on 3.5 Calabi-Yau manifolds, so after compactifying 6 dimensions, where does the extra 1 dimension go? Is it just compactified on a circle, or something like that?

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For clarity, there are CY's of complex dimension two, three, etc., not just one. – Vibert May 26 '13 at 14:16
Keyword in this context: $G_2$-manifolds. – Qmechanic May 26 '13 at 16:00
@Vibert: That was what I meant. When I said 3 Calabi Yau manifolds, I meant Calabi Yau manifolds of complex dimension 3. – centralcharge May 27 '13 at 13:50
@Qmechanic: Thanks a lot! – centralcharge May 27 '13 at 13:51
See on the nLab at – Urs Schreiber Sep 2 '13 at 13:16
up vote 0 down vote accepted

$$\newcommand{\holonomy}{[\mathcal{H\mathbb{O} \ell}]}$$

This answer is an expansion of Qmechanic's comment/. *****************


Holonomy can be imagined as the integral, or global version, of the Riemann Curvature Tensor. The Riemann Curvature tensor, indeed is


Where $\mathcal{\holonomy}$ is the Holonomy.

Holonomy Groups

Now, this holonomy is the group action of the Holonomy group of the manifold. So, in other words, the holonomy of the identity of the holonomy group (not doing any sort of a transport) doesn't do anything to a point on the manifold, and that holonomies are hand - wavily, sort - of "associative" (use this statement with caution!), i.e., instead of writerighteing $\phi(g,x)$ or something, if we choose to write something like, say, $g\dagger x$, then:

$$g\dagger\left(h\dagger x\right)=\left(gh\right)\dagger x$$

Oh, and the first statpement becomes, :

$$e\dagger x =x $$

Now, this is not as trivial as it looks. $e$ is the identity of the holonomy group, NOT of the manifold! .

So, where does $G(2)$ come in?

Now, where in the world does $G(2)$ come from? $G(2)$ is a holonomy group of $\bf{\mathbf{\it{7}}}$-dimensional manifolds, called $G(2)$ manifolds. This means that it is possible to use this as a compactification manifold for M-theory. M-theory has a supersymmetry of $\mathcal N=8$. But, if we waNt a supersymmetry of $\mathcal{N}=1$ (accessible at lower energies), n the compactificaqtion manifoldk must get rid of $\frac78$ of the supersymmetry, i.e. retain only $\frac18$.

It so happens to be that $G(2)$ manifolds do indeed satisfy this criteriaon.

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