Solution
= Solution
Let $m_i=0$ and first take $\zeta>0$. The mass terms and <D-flatness> require $x=0$ and $\sum_i|\phi_i|^2-\sum_i|\widetilde\phi_i|^2=\zeta$, while <F-flatness> adds $\sum_i\widetilde\phi_i\phi_i=0$. Dividing by the U(1) gauge action gives the <Higgs branch>
$$
\boxed{\mathcal M_H=T^*\mathbb {CP}^{N-1}},
$$
with complex dimension $2N-2$. For $\zeta<0$, the same result follows after exchanging $\phi$ and $\widetilde\phi$.