T-duality and closed string non-commutative (doubled) geometry

We provide some evidence that closed string coordinates will become non-commutative turning on geometrical fluxes and/or H-field flux background in closed string compactifications. This is in analogy to open string non-commutativity on the world volume of D-branes with B- and F-field background. The class of 3-dimensional backgrounds we are studying are twisted tori (fibrations of a 2-torus over a circle) and the their T-dual H-field, 3-form flux backgrounds (T-folds). The spatial non-commutativity arises due to the non-trivial monodromies of the toroidal Kahler resp. complex structure moduli fields, when going around the closed string along the circle direction. In addition we study closed string non-commutativity in the context of doubled geometry, where we argue that in general a non-commutative closed string background is T-dual to a commutative closed string background and vice versa. We also discuss the corresponding spatial uncertainty relations. Finally, in analogy to open string boundary conditions, we also argue that closed string momentum and winding modes define in some sense D-branes in closed string doubled geometry.