Compactifying M-Theory to Four Dimensions
  We consider compactifications of ${\cal M}$-theory to four-dimensional Minkowski space on seven-dimensional non-compact manifolds. These compactifications include a warp factor which is non-constant due to the presence of sources coming from fivebranes wrapping two-dimensional submanifolds of the internal seven-dimensional space. We derive the expression for the field strengths and consider an explicit example of this general class of solutions. 
Katrin Becker and Melanie Becker