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For $\zeta =0$ and distinct masses, any nonzero charged field can occur only at one $x=m_j$. At such a point, D-flatness gives $|{\varphi }_j|=|{\tilde{\varphi }}_j|$, whereas F-flatness gives ${\tilde{\varphi }}_j{\varphi }_j=0$; together they force both fields to vanish. Thus only the Coulomb branch remains,of complex dimension one. The points $x=m_i$ are special because the corresponding charged multiplets become massless there.

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.

Let $m_i=0$ and first take $\zeta >0$. The mass terms and D-flatness require $x=0$ and ${\sum }_i|{\varphi }_i{|}^2-{\sum }_i|{\tilde{\varphi }}_i{|}^2=\zeta$, while F-flatness adds ${\sum }_i{\tilde{\varphi }}_i{\varphi }_i=0$. Dividing by the U(1) gauge action gives the Higgs branchwith complex dimension $2N-2$. For $\zeta <0$, the same result follows after exchanging $\varphi$ and $\tilde{\varphi }$.