 Chiral spinor curvature operator

The spinor covariant derivative commutator is a differential two-form in its derivative indices, so the two-component spinor calculus decomposition defines symmetric curvature operators ${\Delta }_{AB}$ and ${\Delta }_{A^{\prime }{B}^{\prime }}$. With ${ϵ}^{AB}{ϵ}_{AB}=2$, the unprimed projection is ${\Delta }_{AB}={ϵ}^{A^{\prime }{B}^{\prime }}[{\nabla }_{A{A}^{\prime }},{\nabla }_{B{B}^{\prime }}]/2$, and the primed projection is analogous. For a torsion-free connection these are algebraic curvature derivations: they annihilate scalar functions and obey the Leibniz rule on tensor products.