seven-dimensional vectors ${\rm p}_i$ by writing $$F_i = 2e^{-{\bf p}\cdot\bbe}$$ where the new fields $\bbe = (\b_A)$ are defined by [^3] $${\rm Re}(T^A)=e^{\b_A}\; . \label{beta}$$ We expect this form of the Kähler potential to apply to a wide class of $G_2$ manifolds constructed by blowing up orbifolds, with the specifics of each example encoded in the vectors ${\bf p}_i$.

In the following we will, for concreteness, focus on the particular example of Ref. [@Lukas:2003dn], which is based on the orbifold $T^7/\ZM_2^3$. The blow-ups for this particular manifold can be labeled by a triple $(i)=(\t ,n,a)$ of indices where $\t =\a ,\b ,\g$ indicates the type, that is, under which of the three $\ZM_2$ orbifolding symmetries the associated fixed point remains invariant, $n=1,2,3,4$ labels the fixed points of equal type and $a=1,2,3$ is an index associated to each fixed point that describes the orientation of the blow-up relative to the bulk. Hence, there are $12$ fixed points, labeled by $(\t
,n)$, each with three associated blow-up moduli and, consequently, $36$ blow-up moduli in total. For simplicity of notation, we will use the single index $i$ to label the blow-ups whenever possible and only split into the triple $(\t ,n,a)$ when required. 