An action for F-theory: $SL\left(2\right)R^{+}$ exceptional field theory

We construct the 12-dimensional exceptional field theory (EFT) associated to the group $SL\left(2\right)\times R^{+}$. Demanding the closure of the algebra of local symmetries leads to a constraint, known as the section condition, that must be imposed on all fields. This constraint has two inequivalent solutions, one giving rise to 11-dimensional supergravity and the other leading to F-theory. Thus $SL\left(2\right)\times R^{+}$ EFT contains both F-theory and M-theory in a single 12-dimensional formalism. (paper)