The String Landscape and the Swampland. The Weak Gravity Conjecture and Scalar Fields

Our Universe is governed by few fundamental forces: the electromagnetic and the weak interactions, the strong interaction, the Higgs-mediated interaction and the gravitational interaction. While the first four forces are described by the Standard Model (and extensions of it), gravity is described by General Relativity, which is a classical (in the sense of non-quantum) theory. Convinced that gravity is a fundamental force and so has to be quantic (for consistency with all the other fundamental interactions), the way to quantize it has to be found. Animated by the belief that String Theory (which is the only consistent model we can refer to and make calculation with for now) is predictive, it is quite natural to ask ourselves how Nature has selected our Universe among the ``jungle" of possible universes allowed by String Theory itself. Nowadays, a popular approach to face such debate is represented by the so called ``Swampland program”. Among all consistent-looking 4-dimensional effective Quantum Field Theories, only few are completable in String Theory or (more generally) in Quantum Gravity in the ultraviolet. Such few theories are said to belong to the Landscape. The abstract concept of Swampland acquires consistency only if it possible to distill it out from that of Landscape. The present thesis work places itself right here. Besides the rich variety of Swampland conjectures that one can consider, we have concentrated our attention on the ``Weak Gravity Conjecture” (WGC). In words, it is the statement according to which gravity acts as the weakest force in any circumstance. After having presented its best known version (in the presence of a U(1) gauge field) and being conscious of the fact that (likely) two scalar fields (the Higgs field and the inflaton) play a crucial role in our Universe and that many scalar fields appear in theories going beyond the Standard Model and in supergravity theories, we have deepened the study of the scalar WGC (SWGC). In particular, we have presented the SWGCs recently (2017) proposed by E. Palti and we have investigated a strong SWGC proposal (SSWGC) that E. Gonzalo and L. Ibàñez have made in April (2019). We have managed to show that, despite of being a potentially powerful tool constraining the Standard Model and many inflationary models, Gonzalo and Ibàñez's conjecture exhibits various criticisms and is in tension with the physical principle that animates the WGC itself. In order to try to get to a general statement (overcoming Gonzalo and Ibàñez's claim) we have elaborated and studied some models. By imposing that gravity is the weakest force, we have got some constraints involving the relevant parameters of such theories, that are coherent with Palti's SWGCs. This gives support to its approach in dealing with the scalar versions of the WGC and has suggested to us a way to really proceed forward (beyond known results).