E7(7) symmetry and dual gauge algebra of M-theory on a twisted seven-torus

We consider M-theory compactified on a twisted 7-torus with fluxes when all the seven antisymmetric tensor fields in four dimensions have been dualized into scalars and thus the ${\mathrm{E}}_{7(7)}$ symmetry is recovered. We find that the Scherk–Schwarz and flux gaugings define a “dual” gauge algebra, subalgbra of ${\mathrm{E}}_{7(7)}$, where some of the generators are associated with vector fields which are dual to part of the original vector fields (deriving from the 3-form). In particular they are dual to those vector fields which have been “eaten” by the antisymmetric tensors in the original theory by the (anti-)Higgs mechanism. The dual gauge algebra coincides with the original gauge structure when the quotient with respect to these dual (broken) gauge generators is taken. The particular example of the S–S twist corresponding to a “flat group” is considered.