Supersymmetric covariant derivative ($D_{\alpha },{\bar{D}}_{\dot{\alpha }}$)

Supersymmetric covariant derivatives anticommute with the supersymmetry generators. Their antichiral member defines a chiral superfield through ${\bar{D}}_{\dot{\alpha }}\Phi =0$.

In flat four-dimensional N=1 supersymmetry, let ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$, contracting the two spinor indices with the antisymmetric epsilon invariant tensor. The supercovariant derivative algebra with left derivatives has $\{{\bar{D}}_{\dot{\alpha }},{\bar{D}}_{\dot{\beta }}\}=0$. There are only two dotted components, so every product of three barred derivatives repeats one component and vanishes. Therefore ${\bar{D}}_{\dot{\alpha }}{\bar{D}}^{2}X=0$ for any superfield $X$: ${\bar{D}}^{2}X$ is a chiral superfield. This map is called chiral projection, although ${\bar{D}}^2$ is not an idempotent projector.

If $\Phi$ is a chiral superfield, holomorphic closure of chiral superfields also makes $W^{\prime }(\Phi )$ chiral. Thus an equation ${\bar{D}}^2{\Phi }^{†}=4W^{\prime }(\Phi )$ is consistent with the chiral constraint: applying any ${\bar{D}}_{\dot{\alpha }}$ gives zero on both sides. Both sides are scalar superfields, and the squared supersymmetric covariant derivative commutes with the supercharges, so the equation is Lorentz invariant and supersymmetry covariant.

For left Grassmann derivatives and $D_{\alpha }={\partial }_{\alpha }+i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{\mu }$, ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}{\partial }_{\mu }$, the coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ gives $D_{\alpha }={\partial }_{\alpha }{|}_y+2i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{y^{\mu }}$ and ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}{|}_y$. The odd chain rule gives ${\bar{\partial }}_{\dot{\alpha }}(\theta {\sigma }^{\mu }\bar{\theta })=-(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}$. A chiral superfield is consequently independent of $\bar{\theta }$ at fixed $y$.

For left Grassmann derivatives, choose $D_{\alpha }={\partial }_{\alpha }+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{D}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}+i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Differentiating the other operator's odd coefficient gives two equal contributions to the mixed anticommutator; coefficient-coefficient terms cancel by the Grassmann algebra. Thus the mixed anticommutator is $2i{\sigma }^{\mu }{\partial }_{\mu }$, while equal-chirality ones vanish. Also $\bar{D}(x+i\theta \sigma \bar{\theta })=0$, giving the coordinates used in the chiral-superfield component expansion. Negating the barred derivative negates the mixed algebra.