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

This answer is an expansion of Qmechanic's comment.

# Holonomy

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

$$R_{\mu\nu\rho}^\sigma=\mbox{d}\holonomy$$

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 criterion.