Four-dimensional effective M theory on a singular G2 manifold

We reduce M theory on a G2 orbifold with codimension four singularities, taking explicitly into account the additional gauge fields at the singularities. As a starting point, we use 11-dimensional supergravity coupled to seven-dimensional super-Yang-Mills theory, as derived in a previous paper. The resulting four-dimensional theory has N=1 supersymmetry with non-Abelian N=4 gauge theory subsectors. We present explicit formulas for the Kähler potential, gauge-kinetic function and superpotential. In the four-dimensional theory, blowing up of the orbifold is described by a Higgs effect induced by continuation along D-flat directions. Using this interpretation, we show that our results are consistent with the corresponding ones obtained for smooth G2 spaces. In addition, we consider the effects of switching on flux and Wilson lines on singular loci of the G2 space, and we discuss the relation to N=4 super-Yang-Mills theory.