Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 48 / 2 / g / Solution

Vary $F_i$ and $F_i^{\ast }$ independently. The auxiliary fields have algebraic, rather than differential, Euler-Lagrange equations:Alternatively complete the square:Substitution therefore yields the on-shell auxiliary fields and nonnegative scalar potentialThe 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.