Almost Calabi-Yau with torsion 6-manifolds and the instanton condition

Abstract: It is observed that on a compact almost complex Calabi-Yau with torsion (ACYT) 6-manifold with co-closed Lee form the curvature of the torsion connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the torsion connection. The same conclusion holds for any (non necessarily compact) balanced ACYT 6-manifold. In particular, on a CYT 6-manifold the Strominger-Bismut connection is an $SU(3)$-instanton if and only if the torsion is parallel with respect to the Strominger-Bismut connection provided either the CYT 6-manifold is compact with co-closed Lee form or it is a balanced CYT 6-manifold.