We reduce M theory on a $G_2$ 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 $\mathcal{N}=1$ supersymmetry with non-Abelian $\mathcal{N}=4$ 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 $G_2$ spaces. In addition, we consider the effects of switching on flux and Wilson lines on singular loci of the $G_2$ space, and we discuss the relation to $\mathcal{N}=4$ super-Yang-Mills theory.

• Received 20 September 2006

DOI:https://doi.org/10.1103/PhysRevD.74.086008