Chirality constraint on extended supersymmetry

In a four-dimensional theory with unbroken $\mathcal{N}\ge 2$ supersymmetry, a full charged hypermultiplet contains $\mathcal{N}=1$ chiral superfields in $R$ and $\overline{R}$. Their net complex chiral gauge spectrum vanishes. The vector multiplet is in the real adjoint gauge group representation. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex gauge chirality. Thus the chiral matter of the Standard Model can have at most unbroken $\mathcal{N}=1$ supersymmetry in four dimensions.