The E11 origin of all maximal supergravities. The hierarchy of field-strenghts

Starting from E-11 and the space-time translations we construct an algebra that promotes the global E-11 symmetries to local ones, and consider all its possible massive deformations. The Jacobi identities imply that such deformations are uniquely determined by a single tensor that belongs to the same representation of the internal symmetry group as the D - 1 forms specified by E-11. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible gauged maximal supergravity theory in any dimension. All the possible deformed algebras are in one to one correspondence with all the possible massive maximal supergravity theories. The hierarchy of fields inherent in the E-11 formulation plays an important role in the derivation. The tensor that determines the deformation can be identified with the embedding tensor used previously to parameterise gauged supergravities. Thus we provide a very efficient, simple and unified derivation of all the field strengths and gauge transformations of all maximal gauged supergravities from E-11. The dynamics arises as a set of first order duality relations among these field strengths.