On quaternionic complexes over unimodular quaternionic manifolds

Penrose's two-spinor notation for 4-dimensional Lorentzian manifolds is extended to two-component notation for quaternionic manifolds, which is a useful tool for calculation. We can construct a family of quaternionic complexes over unimodular quaternionic manifolds only by elementary calculation. On complex quaternionic manifolds as complexification of quaternionic Kähler manifolds, the existence of these complexes was established by Baston by using twistor transformations and spectral sequences. Unimodular quaternionic manifolds constitute a large nice class of quaternionic manifolds: there exists a very special curvature decomposition; the conformal change of a unimodular quaternionic structure is still unimodular quaternionic; the complexes over such manifolds are conformally invariant. This class of manifolds is the real version of torsion-free QCFs introduced by Bailey and Eastwood. These complexes are elliptic. We also obtain a Weitzenböck formula to establish vanishing of the cohomology groups of these complexes for quaternionic Kähler manifolds with negative scalar curvatures.