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We classify all the $6$-dimensional unimodular Lie algebras $\mathfrak{g}$admitting a complex structure with non-zero closed $(3,0)$-form. This givesrise to $6$-dimensional compact homogeneous spaces $M=\Gamma\backslash G$,where $\Gamma$ is a lattice, admitting an invariant complex structure withholomorphically trivial canonical bundle. As an application, in the balancedHermitian case, we study the instanton condition for any metric connection$\nabla^{\varepsilon,\rho}$ in the plane generated by the Levi-Civitaconnection and the Gauduchon line of Hermitian connections. In the setting ofthe Hull-Strominger system with connection on the tangent bundle beingHermitian-Yang-Mills, we prove that if a compact non-K\"ahler homogeneous space$M=\Gamma\backslash G$ admits an invariant solution with respect to somenon-flat connection $\nabla$ in the family $\nabla^{\varepsilon,\rho}$, then$M$ is a nilmanifold with underlying Lie algebra $\mathfrak{h}_3$, asolvmanifold with underlying algebra $\mathfrak{g}_7$, or a quotient of thesemisimple group SL(2,$\mathbb{C}$). Since it is known that the system can besolved on these spaces, our result implies that they are the unique compactnon-K\"ahler balanced homogeneous spaces admitting such invariant solutions. Asanother application, on the compact solvmanifold underlying the Nakamuramanifold, we construct solutions, on any given balanced Bott-Chern class, tothe heterotic equations of motion taking the Chern connection as (flat)instanton.

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle