String theory on a torus requires the introduction of dual coordinates
conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be
compared to generalized geometry, which doubles the tangent space but not the manifold.
For a d-torus, string theory can be formulated in terms of an infinite
tower of fields depending on both the d torus coordinates and the d dual
coordinates. This talk focuses on a finite subsector consisting of a metric
and B-field (both d x d matrices) and a dilaton all depending on the 2d
doubled torus coordinates.
The double field theory is constructed and found to have a novel symmetry
that reduces to diffeomorphisms and anti-symmetric tensor gauge
transformations in certain circumstances. It also has manifest T-duality
symmetry which provides a generalisation of the usual Buscher rules to
backgrounds without isometries. The theory has a real dependence on the full
doubled geometry: the dual dimensions are not auxiliary. It is concluded
that the doubled geometry is physical and dynamical.