On effective field theory of F-theory beyond leading order

We construct a proposal for effective bosonic field theory at order $\alpha'^3$ in twelve dimensions, whose compactification on a circle and on a torus respectively yields eleven-dimensional and type IIB supergravity theories at eight-derivative level. The couplings $({\partial {G_5}})^2 R^2$, $({\partial {F_4}})^2 R^2$, $({\partial {F_4}})^4$, $({\partial {G_5}})^4$ and $({\partial {G_5}})^2({\partial {F_4}})^2$ in twelve-dimensional supergravity are determined with this requirement that an ansatz of these couplings should admits a consistent truncation to the eleven-dimensional and type IIB supergravity theories. The self-duality condition of the five-form field strength in twelve dimensions is also understood by considering the RR five-form field strength of type IIB theory at linear order.