Double Metric, Generalized Metric and $\alpha'$-Geometry

Understand

We relate the unconstrained `double metric' of the `$\alpha'$-geometry' formulation of double field theory to the constrained generalized metric encoding the spacetime metric and b-field.

• This is achieved by integrating out auxiliary field components of the double metric in an iterative procedure that induces an infinite number of higher-derivative corrections.
• As an application we prove that, to first order in $\alpha'$ and to all orders in fields, the deformed gauge transformations are Green-Schwarz-deformed diffeomorphisms.
• We also prove that to first order in $\alpha'$ the spacetime action encodes precisely the Green-Schwarz deformation with Chern-Simons forms based on the torsionless gravitational connection.