= Solution
Vary $F_i$ and $F_i^*$ independently. The <auxiliary fields> have algebraic, rather than differential, <Euler-Lagrange equations>:
$$
F_i^*+W_i=0,\qquad F_i+\overline{W_i}=0.
$$
Alternatively complete the square:
$$
\mathcal L_F=\sum_i|F_i+\overline{W_i}|^2-\sum_i|W_i|^2.
$$
Substitution therefore yields \b[the on-shell auxiliary fields and nonnegative scalar potential]
$$
\boxed{F_i=-\overline{W_i},\qquad\mathcal L_F^{\rm on\ shell}=-V_F,\qquad V_F=\sum_i|W_i|^2.}
$$
The sign in the <Lagrangian> is minus the <F-term scalar potential>; changing the metric convention does not change this algebraic result. A <supersymmetric vacuum> requires all $W_i=0$, namely <F-flatness>. For a noncanonical positive <Kähler metric> $K_{i\bar j}$, the purely bosonic elimination instead gives $V_F=K^{i\bar j}W_i\overline{W_j}$; fermionic connection terms must also be included in a complete nonlinear component action.
Back to article page
