On gauged maximal d=8 supergravities

We study the gauging of maximal $d=8$ supergravity using the embedding tensor formalism. We focus on SO$(3)$ gaugings, study all the possible choices of gauge fields and construct explicitly the bosonic actions (including the complicated Chern-Simons terms) for all these choices, which are parametrized by a parameter associated to the 8-dimensional SL$(2,\mathbb{R})$ duality group that relates all the possible choices which are, ultimately, equivalent from the purely 8-dimensional point of view. Our result proves that the theory constructed by Salam and Sezgin by Scherk-Schwarz compactification of $d=11$ supergravity and the theory constructed in Ref.~\cite{AlonsoAlberca:2000gh} by dimensional reduction of the so called ``massive 11-dimensional supergravity'' proposed by Meessen and Ort\'{\i}n in Ref.~\cite{Meessen:1998qm} are indeed related by an SL$(2,\mathbb{R})$ duality even though they have two completely different 11-dimensional origins.
Comment: 28 pages