Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 48 / 2 / a / Solution

A four-dimensional $\mathcal{N}=1$ chiral superfield is a complex scalar superfield satisfyingThe supersymmetric covariant derivatives anticommute with the supercharges, so the constraint is preserved by supersymmetry. Use left Grassmann derivatives and the printed convention ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. The conjugate antichiral superfield satisfies $D_{\alpha }{\Phi }^{†}=0$.