Local E11

We give a method of deriving the field-strengths of all massless and massive maximal supergravity theories in any dimension starting from the Kac-Moody algebra E-11. Considering the subalgebra of E-11 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. We show how this works in eleven dimensions, and we call the resulting enlarged algebra E-11(local). Torus reduction to D dimensions corresponds to taking a subalgebra of E-11(local), called E-11,D(local), that encodes the full gauge algebra of the corresponding D-dimensional massless supergravity. We show that each massive maximal supergravity in D dimensions is a non-linear realisation of an algebra (E) over tilde (local)(11,D). We show how this works in detail for the case of Scherk-Schwarz reduction of IIB to nine dimensions, and in particular we show how (E) over tilde (local)(11,9) arises as a subalgebra of the algebra E-11,10B(local) associated to the ten-dimensional IIB theory. This subalgebra corresponds to taking a combination of generators which is different to the massless case. We then show that (E) over tilde (local)(11,D) appears as a deformation of the massless algebra E-11,D(local) in which the commutation relations between the E-11 and the additional generators are modified. We explicitly illustrate how the deformed algebra is constructed in the case of massive IIA and of gauged five-dimensional supergravity. These results prove the naturalness and power of the method.