G2G2-manifolds with parallel characteristic torsion

We classify 7-dimensional cocalibrated G2G2-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection ∇c∇c with totally skew-symmetric torsion and a spinor field Ψ1Ψ1 solving the equations in the common sector of type II superstring theory. There exist G2G2-structures with parallel characteristic torsion that are not naturally reductive.