M-Theory on Manifolds with G_2 Holonomy

Abstract

We study M-theory on G_2 holonomy spaces that are constructed by dividing a seven-torus by some discrete symmetry group. We classify possible group elements that may be used in this construction and use them to find a set of possible orbifold groups that lead to co-dimension four singularities. We describe how to blow up such singularities, and then derive the moduli Kaehler potential for M-theory on the resulting class of G_2 manifolds. To consider the singular limit it is necessary to derive the supergravity action for M-theory on the orbifold C^2/Z_N. We do this by coupling 11-dimensional supergravity to a seven-dimensional Yang-Mills theory located on the orbifold fixed plane. We show that the resulting action is supersymmetric to leading non-trivial order in the 11-dimensional Newton constant. Obtaining this action enables us to then reduce M-theory on a toroidal G_2 orbifold with co-dimension four singularities, taking explicitly into account the additional gauge fields at the singularities. The four-dimensional effective theory has N=1 supersymmetry with non-Abelian N=4 gauge theory sub-sectors. We present explicit formulae for the Kaehler potential, gauge-kinetic function and superpotential. In the four-dimensional theory, blowing-up of the orbifold is described by continuation along D-flat directions. Using this interpretation, we demonstrate consistency of our results for singular G_2 spaces with 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 N=4 SYM theory.