The Odd story of $\alpha'$-corrections

The $\alpha'$-deformed frame-like Double Field Theory (DFT) is a T-duality and gauge invariant extension of DFT in which generalized Green-Schwarz transformations provide a gauge principle that fixes the higher-derivative corrections. It includes all the first order $\alpha'$-corrections of the bosonic and heterotic string low energy effective actions and of the Hohm-Siegel-Zwiebach $\alpha'$-geometry. Here we gauge this theory and parameterize it in terms of a frame, a two-form, a dilaton, gauge vectors and scalar fields. This leads to a unified framework that extends the previous construction by including all duality constrained interactions in generic (gauged/super)gravity effective field theories in arbitrary number of dimensions, to first order in $\alpha'$.