arXiv2009
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 $\tilde{E}_{11,D}^{local}$. 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 $\tilde{E}_{11,9}^{local}$ 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 $\tilde{E}_{11,D}^{local}$ 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.