Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 2 / ii / Solution

Suppose $\zeta >0$ and all $m_i$ are distinct. A nonzero ${\varphi }_j$ forces $x=m_j$. Distinctness then makes every other charged field vanish, and ${\sum }_i{\tilde{\varphi }}_i{\varphi }_i=0$ forces ${\tilde{\varphi }}_j=0$. The D-flatness equation fixes $|{\varphi }_j{|}^2=\zeta$, while its phase is removed by the gauge group. There is therefore one isolated supersymmetric vacuum for each flavor,for a total of $N$ vacua. Negative $\zeta$ gives the analogous vacua with ${\tilde{\varphi }}_j$ nonzero.