 Supersymmetric vacuum manifold of a bilinear hybrid superpotential

For nonzero $g$ and $\zeta$, these field values give $W=0$ and all Kähler covariant derivatives of a superpotential zero, hence zero supergravity auxiliary fields and zero vacuum energy. With only the supplied chiral fields, the product condition gives a complex vacuum manifold rather than one isolated point. A specified gauged opposite-charge symmetry can add a D-flatness condition and a gauge quotient, but no such gauge sector should be invented from the field labels alone.