Finite Transformations in Doubled and Exceptional Space

In the double field theory, gauge symmetries are realized as generalized diffeomorphisms in the doubled spacetime. By consistency of the theory, dependence of tensor fields on the doubled coordinates is strongly constrained. This causes finite transformation law highly complicated, both technically and conceptually. In this paper, we propose a new, physically intuitive approach to finite gauge transformations by utilizing untwisted form of vector fields. In our approach, finite gauge transformation law is expressed in terms of diffeomorphisms on a maximal null subspace on which dynamical degrees of freedom live and of a local rotation of this null subspace embedded inside the doubled space. We show that our finite transformation automatically satisfies the composition law. We also show that ours is free from the so-called Papadopoulos problem, so can describe background with nontrivial three-form flux $H_3$. Added advantage of our approach is straightforward applicability to general extended field theories. We demonstrate this by explicitly obtaining finite transformation law in the SL(5) exceptional field theory.