order terms in $\e_{(\t ,n)}$.

A simple form of the Lagrangian
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The Kähler potential described in the previous section depends on $43$ chiral superfields. We, therefore, have $43$ real moduli fields from the real parts of theses superfields, which are associated to the geometry of the $G_2$ manifold, plus $43$ axions from the imaginary parts. It is clear that the axions can be consistently set to constants and for simplicity this is what we will do in the following. We are, hence, left with the real scalar fields $\b_A$ and $u_i$.

However, their kinetic terms, as computed from Eq. , are still fairly complicated. A substantial simplification can be achieved by introducing a new set of fields $\f_A$ and $z_i$ defined by the Ansatz $$\begin{aligned}
 \b_A &=& \f_A+\sum_i c_{iA}e^{{\bf q}_i\cdot\bph}z_i^2 \\
 u_i &=& e^{{\bf s}_i\cdot\bph}\; .\end{aligned}$$ Here, ${\bf c}_i=(c_{iA})$, ${\bf q}_i=(q_{Ai})$ and ${\bf
s}_i=(s_{Ai})$ are constants to be determined shortly. The kinetic terms for $\b_A$ and $u_i$ can now be rewritten using this field reparametrisation. For the particular choice of constants $$\begin{aligned}
 {\bf c}_i &=& -4\m_i{\bf p}_i\\
 {\bf s}_i &=& \frac{1}{2}(\m_i+1){\bf p}_i\\
 {\bf q}_i &=& \m_i{\bf p}_i\; ,\end{aligned}$$ where $\m_i\in\{ -1,+1\}$ are arbitrary signs, we find that the Lagrangian 