Gauging of Flat Groups in Four Dimensional Supergravity

TL;DR

We address how Scherk–Schwarz spontaneously broken $N=8$ supergravity in four dimensions can be realized from a purely four-dimensional perspective by gauging a nontrivial electric flat group of $E_{7,7}$, exploiting an $\mathfrak{so}(1,1)$ grading. The authors identify non‑isomorphic electric subgroups and show that Scherk–Schwarz gaugings lie outside previous classifications, with a nonsemisimple flat-group structure formed by translations and a Cartan generator. The work provides the explicit four‑dimensional Lagrangian with non‑Abelian field strengths, axion covariant derivatives, and a scalar potential required by the gauging, and extends the framework to $N=2$ theories with cubic prepotentials, highlighting a five‑dimensional origin and brane‑flux connections. This unifies no‑scale and Scherk–Schwarz type gaugings across different amounts of supersymmetry and has implications for string compactifications with fluxes and spontaneous supersymmetry breaking in 4D effective theories.

Abstract

We show that N=8 spontaneously broken supergravity in four dimensions obtained by Scherk-Schwarz generalized dimensional reduction can be obtained from a pure four dimensional perspective by gauging a suitable electric subgroup of E_{7,7}. Owing to the fact that there are non isomorphic choices of maximal electric subgroups of the U-duality group their gaugings give rise to inequivalent theories. This in particular shows that the Scherk-Schwarz gaugings do not fall in previous classifications of possible gauged N=8 supergravities. Gauging of flat groups appear in many examples of string compactifications in presence of brane fluxes.