doi:10.1016/j.geomphys.2023.105014engOtal, AntonioUgarte, LuisSix dimensional homogeneous spaces with holomorphically trivial canonical bundleART-2023-135852We classify all the 6-dimensional unimodular Lie algebras gadmitting a complex structure with non-zero closed (3, 0)-form. This gives rise to 6-dimensional compact homogeneous spaces M= \G, where is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being HermitianYang-Mills, we prove that if a compact non-Kähler homogeneous space M= \Gadmits an invariant solution with respect to some non-flat connection ∇in the family ∇ε,ρ, then Mis a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton.2023http://zaguan.unizar.es/record/12966910.1016/j.geomphys.2023.105014http://zaguan.unizar.es/record/129669oai:zaguan.unizar.es:129669info:eu-repo/grantAgreement/ES/AEI/PID2020-115652GB-I00JOURNAL OF GEOMETRY AND PHYSICS 194 (2023), 105014 [27 pp.]by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess