E11 must be a symmetry of strings and branes

Abstract

We construct the non-linear realisation of the semi-direct product of E 11 and its vector representation in five and eleven dimensions and find the dynamical equations it predicts at low levels. Restricting these results to contain only the usual fields of supergravity and the generalised space-time to be the usual space-time we find the equations of motion of the five and eleven dimensional maximal supergravity theories. Since this non-linear realisation contains effects that are beyond the supergravity approximation and are thought to be present in an underlying theory we conclude that the low energy effective action of string and branes must possess an E 11 symmetry.

A brief review of E theory

“…The approach of reference [31] focused on finding duality equations which were first order in derivatives and it found the correct equations for the forms which were uniquely determined but there were unresolved issues with the graviton sector. In references [33,34] the invariant second order equations were found, they were unique and when one retained only the low levels fields and the level zero coordinates they were precisely those of eleven dimensional supergravity. This essentially proved the E 11 conjecture.…”

“…Of course to get the full transformation one must combine the transformations of equation (6.18) The detailed construction of the equations of motion which follow from the E 11 ⊗ s l 1 non-linear realisation was given in reference [34] following earlier results in references [33] and [31]. We refer the reader to this reference and confine ourselves here to stating the result.…”

“…As a result, the non-linear realisation carried out with these limitations did not lead uniquely to eleven dimensional supergravity and one had to fix several constants whose values were not determined by the calculation. A more systematic approach was taken in references [31,48,33] and [34] where the non-linear realisation of E 11 ⊗ s l 1 at low levels was constructed for the fields up to an including the dual graviton as well as the low level coordinates of the l 1 representation. These references enforced not only the Lorentz group symmetries of I c (E 11 ) but also the much more powerful symmetries at the next levels.…”

“…The local I c (E 11 ) variations of the Cartan forms are straightforward to compute, using the E 11 algebra and they are given by [31,33,34]…”

“…As explained above if this index is made into a tangent index, that is, G A,α = (E −1 ) A Π G Π,α it transforms only under the local I c (E 11 ) transformations, the transformation just being that for the inverse vielbein of equation (6.8). One finds that the Cartan forms, when referred to the tangent space, transforms on their l 1 index as [31,33,34] δG a,…”