G2-manifolds with Parallel Characteristic Torsion

We classify 7-dimensional cocalibrated G2-manifolds with parallel characteristic torsion and non-abelian holonomy. All these spaces admit a metric connection ∇ with totally skew-symmetric torsion and a spinor field Ψ1 solving the equations in the common sector of type II superstring theory. There exist G2-structures with parallel characteristic torsion that are not naturally reductive. 1. Metric connections with parallel torsion Consider a Riemannian manifold and denote by ∇g its Levi-Civita connection. Any 3-form T defines via the formula ∇XY := ∇gXY + 1 2 T(X,Y, ∗) a metric connection ∇ with totally skew-symmetric torsion T. We are interested in the case that the torsion form is parallel, ∇T = 0. Then T is coclosed, δ(T) = 0, and the differential dT depends only on the algebraic type of T (see [13]),