In theoretical physics, the Fayet–Iliopoulos D-term (introduced by Pierre Fayet and John Iliopoulos) is a D-term in a supersymmetric theory obtained from a vector superfield V simply by an integral over all of superspace:
\( S_{FI} = \xi \int d^4\theta \, V \)
Because a natural trace must be a part of the expression, the action only exists for U(1) vector superfields.
In terms of the components, it is proportional simply to the last auxiliary D-term of the superfield V. It means that the corresponding D that appears in D-flatness conditions (and whose square enters the ordinary potential) is additively shifted by \( \xi \) , the coefficient.
References
Fayet, P.; Iliopoulos, J. (1974). "Spontaneously broken supergauge symmetries and goldstone spinors". Physics Letters B. Elsevier BV. 51 (5): 461–464. doi:10.1016/0370-2693(74)90310-4. ISSN 0370-2693.
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