On the inconsistency of Fayet-Iliopoulos terms in supergravity theories

Motivated by recent discussions, we revisit the issue of whether globally supersymmetric theories with nonzero Fayet-Iliopoulos terms may be consistently coupled to supergravity. In particular, we examine claims that a fundamental inconsistency arises due to the conflicting requirements which are imposed on the $R$-symmetry properties of the theory by the supergravity framework. We also prove that certain kinds of Fayet-Iliopoulos contributions to the supercurrent supermultiplets of theories with nonzero Fayet-Iliopoulos terms fail to exist. A key feature of our discussion is an explicit comparison between results from the chiral (or “old minimal”) and linear (or “new minimal”) formulations of supergravity, and the effects within each of these formalisms that are induced by the presence of nonzero Fayet-Iliopoulos terms.

Figure 1

The general relation between the individual Noether currents and the supercurrent supermultiplet. While the Noether currents follow directly from their associated symmetries, they can each be modified through the addition of appropriate so-called improvement terms which preserve their divergences and charges. When suitably improved, these currents can then be mapped into the lowest components of a single supercurrent superfield through mapping relations which depend on the particular constraints that the supercurrent supermultiplet is assumed to obey.

Figure 2

Comparison between the symmetries of our original theory and those of our chirally compensated superconformal theory. In general, the existence of a local $U(1{)}_{\mathrm{FI}}$ symmetry in either theory implies the existence of a corresponding global symmetry in the other theory. In the presence of a nonvanishing FI term, the local $U(1{)}_{\mathrm{FI}}$ symmetries in the two theories are distinct as a result of the nonvanishing $\xi$ dependent $U(1{)}_{\mathrm{FI}}$ charges of the chiral-compensator fields $\Sigma$, $\overline{\Sigma }$. Thus, in the chiral formalism, the presence of a nonzero FI term requires the existence of additional global symmetries in both theories. However, when the FI term is eliminated, the local $U(1{)}_{\mathrm{FI}}$ symmetries in both theories can be identified with each other. In this case, the global $U(1{)}_{\mathrm{FI}}$ symmetry within the local $U(1{)}_{\mathrm{FI}}$ symmetry of either theory serves as the global symmetry required by the local $U(1{)}_{\mathrm{FI}}$ symmetry of the other theory, and no additional global symmetries are needed in either theory.