Swampland geometry and the gauge couplings

Abstract: The purpose of this paper is two-fold. First we review in detail the geometric aspects of the swampland program for supersymmetric 4d effective theories using a new and unifying language we dub domestic geometry', the generalization of special K\"ahler geometry which does not require the underlying manifold to be K\"ahler or have a complex structure. All 4d SUGRAs are described by domestic geometry. As special K\"ahler geometries, domestic geometries carry formal brane amplitudes: when the domestic geometry describes the supersymmetric low-energy limit of a consistent quantum theory of gravity, its formal brane amplitudes have the right properties to be actual branes. The main datum of the domestic geometry of a 4d SUGRA is its gauge coupling, seen as a map from a manifold which satisfies the geometric Ooguri-Vafa conjectures to the Siegel variety; to understand the properties of the quantum-consistent gauge couplings we discuss several novel aspects of suchOoguri-Vafa' manifolds, including their Liouville properties. Our second goal is to present some novel speculation on the extension of the swampland program to no-supersymmetric effective theories of gravity. The idea is that the domestic geometric description of the quantum-consistent effective theories extends, possibly with some qualifications, also to the non-supersymmetric case.