E11, Romans theory and higher level duality relations

Abstract

From the underlying non-linear realisation we compute the complete E11 invariant equations of motion in eleven dimensions, at the linearised level, up to and including level four in the fields. Thus we include the metric, the three and six forms, the dual graviton and three fields at level four. The fields are linked by a set of duality equations, which are first order in derivatives and transform into each other under the E11 symmetries. From these duality relations we deduce second order equations of motion, including those for the usual supergravity fields. As a result the on-shell degrees of freedom are those of the eleven dimensional supergravity. We also show that the level four fields provide an eleven dimensional origin of Romans theory and lead to a novel duality relation.Comment: 29 pages, a note has been added that contains further E11 variations of the level four field

Irreducible representations of E theory

“…Surely there are an infinite number of I c (E 9 ) invariant duality relations connecting all the higher level fields to h ij and the three form A i 1 i 2 i 3 . This is consistent with the fact that the equations of motion that follow from the E 11 ⊗ S l 1 non-linear realisation, found in references [3], [4] and [18], only contain these degrees of freedom. Putting these equations of motion in light cone notation will lead to the duality relations we are discussing here.…”

“…where at the linearised level G c 1 ...c 9 ,a = ∂ [c 1 h c 2 ...c 9 ],a and the dot above the equals sign indicates that the above relation is modulo Lorentz transformations as explained in references [8], [9], [4] and [18]. We found in section that two gravity is described the independent variablesĥ i,j which appears in equation (2.19).…”

“…We recall that in E theory the dual graviton obeys the irreducibility conditionĥ [a 1 ...a 8 ,b] = 0 and so the equation at the end of the last sentence is automatically true. At level four in the non-linear realisation we find the field A a 1 ...a 9 ,b 1 b 2 b 3 which obeys a duality relation with the three form field that is given by [18,11]…”

“…The 128 degrees of freedom in the irreducible representation must be the ones that are contained in the equations of motion that follow from the E 11 ⊗ s l 1 non-linear realisation. These equations of motion were constructed in references [3] , [4] and [18] up to level four and the degrees of freedom found up to this level were just those of eleven dimensional supergravity. The fact that the irreducible representation considered in this section has the degrees of freedom of eleven dimensional supergravity and no more strongly suggests that the full equations of motion of the non-linear realisation will contain just these degrees of freedom.…”

On supersymmetric E11 exceptional field theory

“…It has been proposed by West long ago and prior to the development of exceptional field theory that the D = 11 supergravity equations of motion should emerge from an E 11 invariant theory formulated in the framework of a non-linear realisation of E 11 in the 'vector' representation, such that the dynamics would follow from an E 11 invariant set of duality equations [35][36][37]. It has been realised recently that these first order duality equations can only hold modulo certain equivalence relations [38,39]. These ambiguities are argued to be liftable by passing to equations of motion that are eventually of arbitrarily high order in derivatives.…”