  The ten-dimensional supergravity theory is a geometric low-energy effective
theory and the equations of motion for its fields can be obtained from string
theory by computing $\beta$ functions. With $d$ compact dimensions, we can add
to it an $O(d, d;\mathbb{Z})$ geometric structure and construct the
supergravity theory inspired by double field theory through the use of a
suitable commutative star product. The latter implements the weak constraint of
the double field theory on its fields and gauge parameters in order to have a
closed gauge symmetry algebra. The consistency of the action here proposed is
based on the orthogonality of the momenta associated with fields in their
triple star products in the cubic terms defined for $d\ge1$. This orthogonality
holds also for an arbitrary number of star products of fields for $d=1$.
Finally, we extend our analysis to the double sigma model, non-commutative
geometry and open string theory.
