The Aeppli Parameter for the Heterotic Moduli

In this paper, we construct a family of solutions to the $3$-fold Hull-Strominger system using the Aeppli class, without introducing the auxiliary gauge connection on the tangent bundle. In particular, we deform the conformally balanced metric along an Aeppli class off-shell and then tune it by a hermitian $(1,1)$-form dependent on this Aeppli class on-shell to satisfy the anomaly cancellation condition. The existence of the family of solutions is then obtained by the implicit function theorem. This refines the previous work by not introducing auxiliary gauge connection, thereby matching the expected dimension with the Bott-Chern parameter. This construction of the Aeppli parameter also extends to the $n$-fold Hull-Strominger system.