U-duality covariant gravity

TL;DR

This paper develops a U-duality covariant formulation of gravity, extending double field theory techniques to include full D=4 Einstein gravity within a (3+3)-dimensional framework, with potential applications to M-theory.

Contribution

It introduces a (3+3)-dimensional U-duality covariantization of D=4 Einstein gravity, incorporating SL(2,R) symmetry and gauge connections, extending double field theory methods.

Findings

Realizes SL(2,R) symmetry geometrically in gravity

Encodes D=4 Einstein gravity within a (2+1)-dimensional framework

Provides a basis for E8(8) covariantization of M-theory

Abstract

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.