We compute the moduli Kähler potential for M theory on a compact manifold of $G_2$ holonomy in a large radius approximation. Our method relies on an explicit $G_2$ structure with small torsion, its periods and the calculation of the approximate volume of the manifold. As a verification of our result, some of the components of the Kähler metric are computed directly by integration over harmonic forms. We also discuss the modification of our result in the presence of codimension-four singularities and derive the gauge-kinetic functions for the massless gauge fields that arise in this case.

• Received 6 June 2003

DOI:https://doi.org/10.1103/PhysRevD.69.066003