In this short article we discuss quantization of the Fayet-Iliopoulos parameter in supergravity theories. We argue that, in supergravity, the Fayet-Iliopoulos parameter determines a lift of the group action to a line bundle, and such lifts are quantized. Just as D-terms in rigid $\mathcal{N}=1$ supersymmetry are interpreted in terms of moment maps and symplectic reductions, we argue that in supergravity the quantization of the Fayet-Iliopoulos parameter has a natural understanding in terms of linearizations in geometric invariant theory quotients, the algebro-geometric version of symplectic quotients.

• Received 7 September 2010

DOI:https://doi.org/10.1103/PhysRevD.83.085010