We give a method of deriving the field-strengths of all massless and massive, i.e. gauged, maximal supergravity theories in any dimension starting from the Kac-Moody algebra E11. Considering the subalgebra of E11 that acts on the fields in the non-linear realisation as a global symmetry, we show how this is promoted to a gauge symmetry enlarging the algebra by the inclusion of additional generators. Given the enlarged algebra corresponding to the massless theory, we show that this can be deformed to obtain the algebra corresponding to the gauged theory. We show how this works in detail for the case of Scherk-Schwarz reduction of IIB to nine dimensions. We then classify all the possible deformations in any dimensions and show that these are in one to one correspondence with all the possible gauged supergravities. This provides a very efficient, simple and unified derivation of the field strengths and gauge transformations of all the bosonic fields of any maximal supergravity theory.

Ten-dimensional IIB supergravity is invariant under the symmetry group SL(2.R). We give a classification of all the forms that are present in the IIB supersymmetry algebra as representations of SL(2,R), and we show under which conditions these forms are associated to branes whose effective actions are kappa symmetric. We use these results to give a simple universal SL(2,R)-invariant expression for all IIB brane actions.

The IIB supersymmetry algebra in d=10 can be extended including 10-forms. One of these is the RR 10-form, which takes part in the tadpole cancellation mechanism of the type-I string. D9-branes are electrically charged with respect to this 10-form. IIB supergravity possesses a global SL(2,R) symmetry, which the quantum theory breaks to SL(2,Z). Even if 10-forms have no dynamics, since their field-strength vanishes identically, it is possible to study their behaviour under SL(2,Z) transformations. We determine all the possible independent 10-forms as representations of SL(2,Z), and in particular we show in which multiplet the RR 10-form lies. Finally, we perform the same analysis for the IIA supersymmetry algebra, and interpret the results in terms of T-duality.