Harmonic complex structures and special Hermitian metrics on products of Sasakian manifolds

It is well known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures $(J_{a,b},g_{a,b})$. We show in this article that the complex structure $J_{a,b}$ is harmonic with respect to $g_{a,b}$, i.e. it is a critical point of the Dirichlet energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or $k$-Gauduchon ($k\geq 2$). Finally, we study the Bismut connection associated to $(J_{a,b}, g_{a,b})$ and we provide formulas for the Bismut-Ricci tensor $\operatorname{Ric}^B$ and the Bismut-Ricci form $\rho^B$. We show that these tensors vanish if and only if each Sasakian factor is $\eta$-Einstein with appropriate constants and we also exhibit some examples fulfilling these conditions, thus providing new examples of Calabi-Yau with torsion manifolds.