An Action for F-theory: $\mathrm{S}\mathrm{L}(2)\times {\mathbb{R}}^{+}$ Exceptional Field Theory

We construct the 12-dimensional exceptional field theory associated to the group $\mathrm{S}\mathrm{L}(2)\times {\mathbb{R}}^{+}$ . 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 $\mathrm{S}\mathrm{L}(2)\times {\mathbb{R}}^{+}$ exceptional field theory contains both F-theory and M-theory in a single 12-dimensional formalism.