Supergravity and the Swampland

Within the attempt of quantizing gravity and within the spirit of the Swampland Program we are going to discuss the Weak Gravity Conjecture (WGC) both as a property of a theory in the Landscape, which is interesting to investigate per se, and as a tool to address the out-standing and cosmologically relevant problem of realizing de Sitter vacua in String Theory or in four-dimensional supergravity theories. In particular, animated by the presence of scalar fields in Nature (as the Higgs field or possibly the inflaton field), we study the intriguing generalization of the WGC to forces mediated by light scalar fields. Analyzing the BPS black hole solutions in extended supergravity theories, we describe two interesting relations involving first and second derivatives of combinations of the central charges. One relation is a new identity that solely relies on the geometric properties of the scalar manifolds of extended supergravities, and the other relation is a generalization of a scalar weak gravity conjecture recently proposed by E. Palti and uses properties of the underlying black hole solution. We also provide for the first time an explicit covariant construction of the BPS squared action for such solutions. After that, we prove that, while respecting the magnetic WGC, a charged gravitino can not have parametrically small or vanishing Lagrangian mass in de Sitter vacua of extended Supergravity. This allows to place large classes of de Sitter solutions of gauged Supergravity, interestingly including all known stable solutions of the N=2 theory, in the Swampland. Inspired by the conclusions that we just presented, we are going to deal with the intensively discussed Kachru--Kallosh--Linde--Trivedi (KKLT) model, one of the few proposed constructions of de Sitter vacua in a string theory framework. We actually challenge this scenario by showing that anti-brane uplifting procedures may suffer from a tachyonic instability towards goldstino condensation. Having in mind that the embedding of the KKLT-type uplift within Supergravity includes the coupling to a nilpotent superfield, one of the cleanest ways to make its alleged pathologies evident is to possibly bring them out in the low energy 4D N=1 supergravity description. Since (within a stringy setup) anti-branes induce spontaneous supersymmetry breaking and a goldstinio sector consequently appears on their world-volume, we focus on the Volkov--Akulov (VA) model, which is the minimal supersymmetric theory that describes the low energy dynamics of a goldstino. Confining ourselves to rigid supersymmetry (as a first step), after recasting the VA model in terms of constrained superfields, we show via the exact renormalization group (ERG) technique combined with a supersymmetric rendition of the local potential approximation the emergence of composite states of the goldstino. We also provide their effective low energy characterization by means of a Kaehler potential and a superpotential. This in turn allows us to reveal an inherent non-perturbative tachyonic instability of the pure VA theory. Willing to give firmer physical substance to the goldstino condensation phenomenon, we are finally going to discuss the standard component-form 4D Volkov--Akulov action in the presence of N non-linear supersymmetries. This is an interesting ground to explore, because, as our analysis clarifies, a large number N of non-linearly realized supersymmetries corresponds to an actual large N limit for which the vacuum structure of the tree-level dual bosonic theory is controlled by the classical behaviour. Within this framework, we find that the effective scalar potential, written in terms of two composite real scalar fields, exhibits at least two stationary points, one representing the original supersymmetry breaking configuration and the other one corresponding to goldstino condensation, where supersymmetry is restored in the deep IR. This result nicely agrees with the ERG analysis.