Swampland constrains and infinite distance limits

In string compactifications it has been observed that common features appear, which are independent of the compactification itself, and that are believed to be  deeply related to the quantum gravitational nature of the theory. Along these lines, the Swampland program aims to understand from a bottom-up perspective the conditions that an effective field theory should obey in order to be consistently UV completed to quantum gravity. These constraints are currently expressed as a series of conjectures, which at a given energy divide the space of effective field theories into Landscape (namely, the ones that come from quantum gravity) and Swampland (the ones that despite being consistent as quantum field theories e.g. anomaly free, cannot be consistently coupled to quantum gravity). A typical feature of string compactifications is the presence of moduli, massless scalar fields with flat potential whose kinetic term in the supergravity action plays the role of a metric in a space parametrized by the moduli themselves, the so-called moduli space. Related to this, one of the most widely accepted conjectures, the Distance Conjecture, states that moving an infinite distance in the moduli space of a theory of quantum gravity, an infinite tower of states becomes exponentially light in the geodesic distance. In this talk, we will focus on some aspects of this conjecture in specific string theory settings. We will explain how these infinitely many states enhance the symmetry algebras in decompactification limits of Heterotic and CHL strings on tori to the affine (and possibly twisted) version of the ones in the higher dimensional theory. Moreover, we show that in the simple setting of symmetric moduli spaces (such as those of toroidal M- and string theory compactifications), there is a natural connection between the geometry of the moduli space and the string spectrum that leads to the Distance Conjecture.