On Fayet-Iliopoulos terms and de Sitter vacua in supergravity: some easy pieces

Abstract

We clarify a number of issues on Fayet-Iliopoulos (FI) terms in supergravity, keeping the formalism at a minumum and making use of explicit examples. We explain why, if the U(1) vector is massive everywhere in field space, FI terms are not genuine and can always be redefined away or introduced when they are not present. We formulate a simple anomaly-free model with a genuine FI term, a classically stable de Sitter (dS) vacuum and no global symmetries. We explore the relation between $ \mathcal{N} $ = 2 and $ \mathcal{N} $ = 1 FI terms by discussing $ \mathcal{N} $ = 1 truncations of $ \mathcal{N} $ = 2 models with classically stable dS vacua.