Dualisation of Dualities, I

TL;DR

The paper systematically maps how toroidal reductions of eleven-dimensional supergravity, together with selective dualisations, shape the global symmetry of maximal supergravities across D=3–11. It shows that the maximally dualised theories have E_{11−D} scalar symmetries realized on coset spaces K(E_{11−D})\backslash E_{11−D}, while partial or inverse dualisations produce contractions or double cosets, and sometimes break these symmetries in the full Lagrangian. A unified framework using positive-root structures ties axions to roots and dilatons to Cartan elements, with doubled Lagrangians providing a consistent even-dimensional realization of duality groups. The analysis also clarifies how higher-rank fields transform under these symmetries and how Ramond–Ramond versus NS-NS dualisations affect the global symmetry content. Overall, the work reveals a rich landscape of inequivalent, non-locally related maximal supergravities, each with distinct duality structures and abelian shift symmetries, organized by an E_{11−D} umbrella and its contractions.

Abstract

We analyse the global (rigid) symmetries that are realised on the bosonic fields of the various supergravity actions obtained from eleven-dimensional supergravity by toroidal compactification followed by the dualisation of some subset of fields. In particular, we show how the global symmetries of the action can be affected by the choice of this subset. This phenomenon occurs even with the global symmetries of the equations of motion. A striking regularity is exhibited by the series of theories obtained respectively without any dualisation, with the dualisation of only the Ramond-Ramond fields of the type IIA theory, with full dualisation to lowest degree forms, and finally for certain inverse dualisations (increasing the degrees of some forms) to give the type IIB series. These theories may be called the GL_A, D, E and GL_B series respectively. It turns out that the scalar Lagrangians of the E series are sigma models on the symmetric spaces K(E_{11-D})\backslash E_{11-D} (where K(G) is the maximal compact subgroup of G) and the other three series lead to models on homogeneous spaces K(G) \backslash G\semi \R^s. These can be understood from the E series in terms of the deletion of positive roots associated with the dualised scalars, which implies a group contraction. We also propose a constrained Lagrangian version of the even dimensional theories exhibiting the full duality symmetry and begin a systematic analysis of abelian duality subalgebras.