Quantization of Fayet-Iliopoulos Parameters in Supergravity

TL;DR

This paper reframes the quantization of Fayet-Iliopoulos parameters in ${\cal N}=1$ supergravity as a consequence of requiring a honest lift of the gauged group action to a holomorphic line bundle ${\cal L}$ over the moduli space. The FI parameter thus corresponds to a linearization in geometric invariant theory (GIT), and the resulting quotients have integral Kähler classes akin to GIT quotients rather than arbitrary symplectic quotients. The authors show that lifts are quantized and that differences between lifts live in ${\rm Hom}(G,U(1))$, yielding an even-integer quantization for the ${\rm U}(1)$ case. They discuss implications for supersymmetry breaking and higher supersymmetry, including the absence of FI ambiguity in ${\cal N}=2$ theories, and connect the discussion to projective embeddings and the GIT framework. Overall, the work provides a unifying algebro-geometric perspective on FI terms, linking supergravity gaugings to linearizations and moment-map data in a projective setting, with broader implications for string compactifications and gerbe-related constructions.

Abstract

In this short note 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 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 (GIT) quotients, the algebro-geometric version of symplectic quotients.