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We investigate stability conditions related to the existence of solutions ofthe Hull-Strominger system with prescribed balanced class. We build on recentwork by the authors, where the Hull-Strominger system is recasted usingnon-Hermitian Yang-Mills connections and holomorphic Courant algebroids. Ourmain development is a notion of harmonic metric for the Hull-Strominger system,motivated by an infinite-dimensional hyperK\"ahler moment map and related to anumerical stability condition, which we expect to exist generically forfamilies of solutions. We illustrate our theory with an infinite number ofcontinuous families of examples on the Iwasawa manifold.

Harmonic metrics for the Hull-Strominger system and stability