We investigate stability conditions related to the existence of solutions of the Hull-Strominger system with prescribed balanced class. We build on recent work by the authors, where the Hull-Strominger system is recasted using non-Hermitian Yang-Mills connections and holomorphic Courant algebroids. Our main development is a notion of harmonic metric for the Hull-Strominger system, motivated by an infinite-dimensional hyperKähler moment map and related to a numerical stability condition, which we expect to exist generically for families of solutions. We illustrate our theory with an infinite number of continuous families of examples on the Iwasawa manifold.

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, Ω) 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 ∇ 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.