Fayet-Iliopoulos Terms in Supergravity and Cosmology
  We clarify the structure of N=1 supergravity in 1+3 dimensions with constant FI terms. The FI terms induce non-vanishing R-charges for the fermions and the superpotential. Therefore the D-term inflation model in supergravity with constant FI terms has to be revisited.   We also investigate the case of the so-called anomalous U(1) when a chiral superfield is shifted under U(1). In such a case, in the context of string theory, the FI terms originate from the derivative of the Kaehler potential and they are inevitably field-dependent. This raises an issue of stabilization of the relevant field in applications to cosmology.   The recently suggested equivalence between the D-term strings and D-branes of type II theory shows that brane-anti-brane systems produce FI terms in the effective 4d theory, with the Ramond-Ramond axion shifting under the U(1) symmetry. This connection gives the possibility to interpret many unknown properties of D-\bar{D} systems in the more familiar language of 4d supergravity D-terms, and vice versa. For instance, the shift of the axion field in both cases restricts the possible forms of the moduli-stabilizing superpotential. We provide some additional consistency checks of the correspondence of D-term-strings to D-branes and show that instabilities of the two are closely related. Surviving cosmic D-strings of type II theory may be potentially observed in the form of D-term strings of 4d supergravity. 
Pierre Binetruy, Gia Dvali, Renata Kallosh and Antoine Van Proeyen