The Scherk-Schwarz mechanism as a flux compactification with internal torsion

TL;DR

This work reframes the Scherk–Schwarz mechanism as a flux- and torsion-based compactification, showing that a suitable internal torsion background reproduces the SS phase in lower-dimensional maximal supergravity. By employing a four-dimensional embedding tensor formalism for $\mathrm{E}_{7(7)}$ and detailed group-theoretical branching, the authors map SS parameters to components of higher-dimensional fluxes and torsion, including effects from the eleven- or ten-dimensional origin. They demonstrate that, under a constant internal torsion $M^m{}_n\in \mathfrak{sl}(n-1)$, the resulting theory yields the expected nonabelian gaugings and a scalar potential, with a vacuum favored when $M\in \mathfrak{so}(n-1)$. The analysis clarifies how T-duality exchanges NS three-form flux and internal torsion within the $912$-dimensional embedding tensor, thereby linking Type II vacua across dualities and providing a unifying framework for flux/torsion compactifications and their dualities. Overall, the paper supplies a mathematically consistent picture in which SS phases, fluxes, and torsion are facets of a single $\mathrm{E}_{7(7)}$-covariant structure, enabling new avenues for solution-generation and duality studies in gauged maximal supergravities.

Abstract

The aim of this paper is to make progress in the understanding of the Scherk-Schwarz dimensional reduction in terms of a compactification in the presence of background fluxes and torsion. From the eleven dimensional supergravity point of view, we find that a general E6(6) S-S phase may be obtained by turning on an appropriate background torsion, together with suitable fluxes, some of which can be directly identified with certain components of the four-form field-strength. Furthermore, we introduce a novel (four dimensional) approach to the study of dualities between flux/torsion compactifications of Type II/M-theory. This approach defines the action that duality should have on the background quantities, in order for the E7(7) invariance of the field equations and Bianchi identities to be restored also in the presence of fluxes/torsion. This analysis further implies the interpretation of the torsion flux as the T-dual of the NS three-form flux.