  The invariant balanced Hermitian geometry of nilmanifolds of dimension 6 is
described. We prove that the holonomy group of the associated Bismut connection
reduces to a proper subgroup of SU(3) if and only if the complex structure is
abelian. As an application we show that if J is abelian then any invariant
balanced J-Hermitian structure provides solutions of the Strominger system.

Differential Geometry