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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 )\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)\G⋉IR 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.implemented only on the field strengths (necessarily in the equations of motion), then all the members of the multiplet must appear in the field equations and Bianchi identities only through their field strengths. If the result of dualisations is to make it that neither of these conditions is satisfied, then the original global symmetry prior to the dualisations will be broken.The fact that the global symmetry can depend on the choice of dualisation [12], and the fact that not all dualisations are possible, are both consequences of the occurrence of nonlinear terms in the D-dimensional Lagrangian. These terms have two origins, namely the F (4) ∧ F (4) ∧ A (3) term in the original eleven-dimensional Lagrangian, and the non-linearity of the eleven-dimensional Einstein-Hilbert action. The latter implies that modifications to the field strengths in lower dimensions will arise in the Kaluza-Klein reduction process. These are sometimes called "Chern-Simons modifications," but the term is really a misnomer since they actually come from the separation between the gauge transformations originating from diffeomorphisms along the compactified directions and the other gauge symmetries. In this paper they will be called Kaluza-Klein modifications. In order to investigate these issues in more detail it is convenient to divide the discussion into two parts, namely for the subsector comprising the scalar fields, and then the remaining sectors involving the higher-degree field strengths.In any dimension D ≥ 6, the scalar sector of the D-dimensional theory that is obtained by dimensional reduction from D = 11 is unam...

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We introduce a doubled formalism for the bosonic sector of the maximal supergravities, in which a Hodge dual potential is introduced for each bosonic field (except for the metric). The equations of motion can then be formulated as a twisted self-duality condition on the total field strength G, which takes its values in a Lie superalgebra. This doubling is invariant under dualisations; it allows a unification of the gauge symmetries of all degrees, including the usual U-dualities that have degree zero. These "superdualities" encompass the dualities for all choices of polarisation (i.e. the choices between fields and their duals). All gauge symmetries appear as subgroups of finite-dimensional supergroups, with Grassmann coefficients in the differential algebra of the spacetime manifold.