Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 307 / 1 / v / Solution

With canonical normalization, the renormalizable supersymmetric gauge theory hasThe first term is the gauge-covariant Kähler potential and the second is the Supersymmetric Yang-Mills action. A holomorphic superpotential would also be integrated as $\int d^2\theta \mathcal{W}(\Phi )+h.c.$, but one commuting field in the fundamental representation of $SU(3)$ admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic ${ϵ}_{ijk}{\Phi }^i{\Phi }^j{\Phi }^k$ vanishes.