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

TL;DR

This work analyzes M-theory on a twisted seven-torus with fluxes in the regime where all four-dimensional antisymmetric tensors are dualized to scalars, revealing an enhanced $\mathrm{E}_{7(7)}$ symmetry. By employing the embedding tensor formalism and free differential algebra (FDA) methods, the authors identify a dual gauge algebra within $\mathrm{E}_{7(7)}$ that includes both electric and magnetic-like generators, tied to the Scherk--Schwarz twist $\tau_{IJ}{}^K$ and 4-form flux $g_{IJKL}$. They show how dualization and a suitable symplectic rotation yield a consistent, electric-frame gauging, corresponding to a generalized Scherk--Schwarz reduction and related to a flat group in particular cases. The dual and original gauge structures coincide upon quotienting by an abelian ideal, providing a coherent picture of how tensor dualization, Higgs/Stueckelberg mechanisms, and $\mathrm{E}_{7(7)}$ covariance interplay in maximal four-dimensional supergravity arising from M-theory compactifications. Overall, the paper clarifies how dual gaugings fit into the maximal supergravity framework and offers a robust route to connect FDA descriptions with conventional gauged supergravity via embedding-tensor techniques.

Abstract

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 E_{7(7)} symmetry is recovered. We find that the Scherk--Schwarz and flux gaugings define a ``dual'' gauge algebra, subalgbra of 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.