Abstract
Motivated by its well defined higher dimensional origin, a detailed study of $D=4$ $\mathcal{N}=8$ supergravity with a dyonically gauged $\textrm{ISO}(7) = \textrm{SO}(7) \ltimes \mathbb{R}^7$ gauge group is performed. We write down the Lagrangian and describe the tensor and duality hierarchies, focusing on an interesting subsector with closed field equations and supersymmetry transformations. We then truncate the $\mathcal{N}=8$ theory to some smaller sectors with $\mathcal{N}=2$ and $\mathcal{N}=1$ supersymmetry and SU(3), $\textrm{G}_2$ and SO(4) bosonic symmetry. Canonical and superpotential formulations for these sectors are given, and their vacuum structure and spectra is analysed. Unlike the purely electric ISO(7) gauging, the dyonic gauging displays a rich structure of vacua, all of them AdS. We recover all previously known ones and find a new $\mathcal{N}=1$ vacuum with SU(3) symmetry and various non-supersymmetric vacua, all of them stable within the full $\mathcal{N}=8$ theory.
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