  We find a new obstruction to the existence of solutions of the
Hull-Strominger system, which goes beyond the balanced property of the
Calabi-Yau manifold $(X,\Omega)$ and the Mumford-Takemoto slope stability of
the bundle over it. The basic principle is the construction of a (possibly
indefinite) Hermitian-Einstein metric on the holomorphic string algebroid
associated to a solution of the system, provided that the connection $\nabla$
on the tangent bundle is Hermitian-Yang Mills. Using this, we define a family
of Futaki invariants obstructing the existence of solutions in a given balanced
class. Our results are motivated by a strong version of a conjecture by Yau on
the existence problem for these equations.
