Gauged N=4 supergravities

TL;DR

The paper develops a universal, covariant formulation of gauged $N=4$ supergravity in $D=4$ and $D=5$ using an embedding-tensor approach with tensors $f_{eta MNP}$, $\xi_{eta M}$ in $D=4$ and $f_{MNP}$, $\xi_{MN}$, $\xi_M$ in $D=5$, constrained by quadratic relations to ensure gauge closure. It provides the complete bosonic Lagrangians, topological terms, scalar potentials, and Killing-spinor equations, and shows how various known gaugings (including those from flux compactifications and Scherk–Schwarz reductions) fit into this framework, while also revealing new classes with nonzero pairs of embedding-tensor components. The authors demonstrate how the $D=5$ gaugings embed into $D=4$, and how the $D=4$ gaugings, in turn, relate to $D=3$ via dimensional reduction, illustrating the organization of deformation parameters under larger duality groups. This unifying construction clarifies the interrelations among half-maximal supergravities across dimensions and provides a covariant basis for exploring their vacua, spectra, and higher-dimensional origins in string/M-theory contexts.

Abstract

We present the gauged N=4 (half-maximal) supergravities in four and five spacetime dimensions coupled to an arbitrary number of vector multiplets. The gaugings are parameterized by a set of appropriately constrained constant tensors, which transform covariantly under the global symmetry groups SL(2) x SO(6,n) and SO(1,1) x SO(5,n), respectively. In terms of these tensors the universal Lagrangian and the Killing Spinor equations are given. The known gaugings, in particular those originating from flux compactifications, are incorporated in the formulation, but also new classes of gaugings are found. Finally, we present the embedding chain of the five dimensional into the four dimensional into the three dimensional gaugings, thereby showing how the deformation parameters organize under the respectively larger duality groups.