# Fayet-Iliopoulos terms

It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in superspace is $$\xi \int d^4 \theta V$$ where $V$ is a vector superfield. Now if we do a gauge transformation on $V$, i.e., $$V\longrightarrow V+ i (\Phi-\bar{\Phi})$$ the FI term remains invariant since $$\int d^4 \theta \Phi=\int d^4 \theta \bar{\Phi}=0.$$

So what is the source of the gauge non-invariance in superspace ?

I think what they mean is FI-term is not gauge invariant under the full gauge symmetry of the theory, but under this remaining gauge freedom after WZ gauge, which is $U(1)$.
• yes, but also this is well known that adding FI restricts YM theory to $G = U(1)$. – John Doe Jan 22 '16 at 6:10