Swampland geometry and the gauge couplings

TL;DR

This paper introduces 'domestic geometry', a unifying framework for describing 4d supersymmetric effective theories and their gauge couplings, and explores potential extensions to non-supersymmetric theories within the swampland program.

Contribution

It develops the concept of domestic geometry as a generalization of special K"ahler geometry for 4d SUGRA and discusses its implications for gauge couplings and the swampland.

Findings

Domestic geometry unifies 4d SUGRA descriptions.

Gauge couplings map to Siegel varieties with Ooguri-Vafa properties.

Extension of the framework to non-supersymmetric theories is proposed.

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 such `Ooguri-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.