Gauge theory of Kaluza-Klein and winding modes

We perform a Kaluza-Klein inspired rewriting of double field theory by splitting the coordinates into compact and noncompact directions. There is no truncation of the compact coordinates or their duals, and so this formulation is manifestly O(d, d) invariant, with d the number of compact directions. The action can serve as a starting point for arbitrary Kaluza-Klein ansatze. For a torus background the theory describes the full tower of Kaluza-Klein modes or, in the dual frame, of the winding modes. The Kaluza-Klein vector is a gauge field for the duality-covariantized Courant bracket algebra rather than a Lie algebra. Gauge covariance requires the inclusion of the 2-form gauge potential descending from the Kalb-Ramond field, leading to a structure resembling the tensor hierarchy of gauged supergravity.