E11, generalised space-time and IIA string theory

As advocated in hep-th/0307098 we construct the non-linear realisation of the semi-direct product of E11 and its first fundamental representation at lowest level from the IIA viewpoint. We find a theory that is SO(10,10)⊗GL(1) invariant and contains the fields of gravity, a two form and a dilaton but which depend on coordinates which belong to the vector representation of SO(10,10). The resulting Lagrangian agrees that of recent work on the so-called doubled field theory. However, the construction given in this Letter is straightforward and systematic. It also reveals the relevant underlying symmetries and opens the way to include the Ramond-Ramond, and higher level, fields together with additional coordinates of the generalised space-time.