On effective field theory of F-theory beyond leading order

We construct a proposal for effective bosonic field theory at order ${\alpha }^{\mathrm{\prime }3}$ 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 $(\mathrm{\partial }{G}_5{)}^2{R}^2$, $(\mathrm{\partial }{F}_4{)}^2{R}^2$, $(\mathrm{\partial }{F}_4{)}^4$, $(\mathrm{\partial }{G}_5{)}^4$ and $(\mathrm{\partial }{G}_5{)}^2(\mathrm{\partial }{F}_4{)}^2$ 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.