U-duality covariant gravity

We extend the techniques of double field theory to more general gravity theories and U-duality symmetries, having in mind applications to the complete D = 11 supergravity. In this paper we work out a (3 + 3)-dimensional 'U-duality covariantization' of D = 4 Einstein gravity, in which the Ehlers group SL(2, R) is realized geometrically, acting in the 3 representation on half of the coordinates. We include the full (2 + 1)-dimensional metric, while the 'internal vielbein' is a coset representative of SL(2, R)/SO(2) and transforms under gauge transformations via generalized Lie derivatives. In addition, we introduce a gauge connection of the 'C-bracket', and a gauge connection of SL(2, R), albeit subject to constraints. The action takes the form of (2+1)-dimensional gravity coupled to a Chern-Simons-matter theory but encodes the complete D = 4 Einstein gravity. We comment on generalizations, such as an 'E-8(8) covariantization' of M-theory.