ar X iv : 0 90 4 . 46 64 v 2 [ he p - th ] 1 3 A ug 2 00 9 April 2009 arXiv : 0904 . 4664 Imperial - TP - 2009 - CH -

The zero modes of closed strings on a torus –the torus coordinates plus dual coordinates conjugate to winding number– parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L0− L̄0 = 0 condition in closed string theory. The symmetry of this double field theory includes usual and ‘dual diffeomorphisms’, together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.