= Supergravity moment-map constraint on D-term breaking
{title2=$\mathcal P=i\kappa^{-2}k^iD_iW/W\quad(W\ne0)$}
Let $k^i$ generate a holomorphic <gauge transformation>, with $k^i\partial_iW=-\kappa^2rW$ and <moment map> $\mathcal P=i(k^iK_i-r)$. Substitution of the <Kähler covariant derivative of a superpotential> gives the displayed identity. Consequently, a finite point with $W\ne0$ and all $D_iW=0$ also has $\mathcal P=0$. Pure <D-term> breaking with vanishing chiral <auxiliary fields> can therefore evade this argument only where $W=0$ or outside its hypotheses. In conventional two-derivative matter-coupled <supergravity>, a constant <Fayet–Iliopoulos term> gauges an <R-symmetry> and constrains the transformation of $W$; a neutral <superpotential> and an independently assigned nonzero constant shift are not generally consistent. The gauge-covariance restriction is derived in https://arxiv.org/abs/hep-th/0410053[Van Proeyen's discussion of Fayet–Iliopoulos terms and R-symmetry].
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