Comments on double field theory and diffeomorphisms

As the theory is subject to a section condition, coordinates in double field theory do not represent physical points in an injective manner. We argue that a physical point should be rather one-to-one identified with a `gauge orbit' in the coordinate space. The diffeomorphism symmetry then implies an invariance under arbitrary reparametrizations of the gauge orbits. Within this generalized sense of diffeomorphism, we show that a recently proposed tensorial transformation rule for finite coordinate transformations is actually (i) consistent with the standard exponential map, and further (ii) compatible with the full covariance of the `semi-covariant' derivatives and curvatures after projectors are properly imposed.