Fayet-Iliopoulos terms in supergravity and cosmology

We clarify the structure of N = 1 supergravity in 1+3 dimensions with constant Fayet-Iliopoulos (FI) terms. The FI terms gxi 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 present all corrections of order gxi/M-p(2) to the classical supergravity action required by P local supersymmetry and provide a gauge-anomaly-free version of the model. We also investigate the case of the so-called anomalous U(I) when a chiral superfield is shifted under U(I). In such a case, in the context of string theory, the FI terms originate from the derivative of the Kahler 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 the 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-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 the type II theory may be potentially observed in the form of D-term strings of 4d supergravity. We study such string solutions of supergravity with constant FI terms with one half supersymmetry unbroken and explain some of the puzzling properties of the zero modes around cosmic strings, such as the difference between the numbers of fermionic and bosonic modes.