The $β$-symmetry of supergravity

Continuous O$(d,d)$ global symmetries emerge in Kaluza-Klein reductions of $D$-dimensional string supergravities to $D-d$ dimensions. We show that the non-geometric elements of this group effectively act in the $D$-dimensional parent theory as a hidden bosonic symmetry that fixes its couplings: the $β$-symmetry. We give the explicit $β$-transformations to first order in $α'$ and verify the invariance of the action as well as the closure of the transformation rules.