# Induce a Fayet-Iliopoulos term

In a supersymmetric U(1) gauge theory, if I leave off the Fayet-Iliopoulos term $\kappa [V]_D$, what keeps it from being induced in loop corrections?

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My friend gave me a nice answer: the answer is in a paper by Fischler, Niles, Polchinsky, Raby and Susskind (1981).

The Fayet-Iliopoulos term will not appear quantum mechanically if the trace of the U(1) generator $Q$ across all superfields of the theory vanishes $\text{Tr}(Q)=0$.

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