Harmonic metrics for the Hull-Strominger system and stability

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 hyperK\"ahler 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.