Local E(11)

TL;DR

This paper develops a method to derive the gauge algebras of all maximal supergravity theories, both massless and massive, in any dimension from the Kac-Moody algebra E11, unifying their structure through algebraic deformations.

Contribution

It introduces the concept of local E11 algebra and its deformations to systematically obtain gauge symmetries of supergravity theories in various dimensions.

Findings

Derived gauge algebras for all maximal supergravities from E11.

Demonstrated the algebraic structure for massive IIA and gauged supergravities.

Showed how deformations of E11 encode mass and gauging parameters.

Abstract

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.