  Motivated by the programme on mirror symmetry for non-K\"ahler manifolds, we
construct representations of the $N=2$ superconformal vertex algebra associated
to solutions of the Hull-Strominger system. The construction is via embeddings
of the $N=2$ superconformal vertex algebra in the chiral de Rham complex of a
string Courant algebroid. Our results require that the connection $\nabla$, one
of the unknowns of the system, is Hermitian-Yang-Mills. Our main theorem proves
that any solution of the Hull-Strominger system satisfying this condition has
an associated $N=2$ embedding.
