E11, Romans theory and higher level duality relations

From the underlying nonlinear realisation, we compute the complete E-11 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 E-11 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.