effective theories, specifically the kinetic terms, for these fields were unknown. However, in a recent paper [@Lukas:2003dn], the four-dimensional moduli Kähler potential for M-theory on a $G_2$ manifold has been computed explicitly and our analysis will be based on these results.

The $G_2$ manifold considered in Ref. [@Lukas:2003dn] is based on the compact examples due to Joyce [@joyce1]–[@joyceb], constructed by blowing-up a seven-dimensional $G_2$ orbifold. The moduli for such a $G_2$ manifold split into two groups, namely the bulk moduli (related to the radii of the underlying seven-torus) and the blow-up moduli (related to the volume of the blow-ups). The Kähler potential of Ref. [@Lukas:2003dn] was obtained in the usual large-radius supergravity approximation and for blow-up volumes small compared to the total volume of the space. In the limit of a vanishing blow-up modulus the space develops a singularity (which corresponds to a singularity of the underlying orbifold) leading to new fields in the low-energy theory. It is singular limits such as this which are essential for a phenomenologically interesting low-energy theory.

In this paper, we will focus on the simple case of free moduli evolution based on the Kähler potential of Ref. [@Lukas:2003dn]. That is we will neglect the 