Swampland Constraints and Non-perturbative String Theory: Towards the boundaries of moduli space in M- and F-theory

Abstract

In this thesis we analyse general criteria which consistent theories of quantum gravity must obey according to the so-called Swampland Conjectures, and their interplay with the quantum geometry of non-perturbative string compactifications.

The Swampland Distance Conjecture predicts that infinite-distance limits in the moduli space of a gravitational EFT that can be consistently completed to a theory of quantum gravity are accompanied by an infinite tower of asymptotically massless states, with an exponentially decreasing mass scale in the geodesic distance. The Emergent String Conjecture (ESC) refines this claim by stating that said towers are furnished either by Kaluza-Klein states, signalling a decompactification along the trajectory, or by the excitations of a unique, weakly coupled, asymptotically tensionless critical string, determining the equidimensional duality frame to which we transition at the endpoint of the limit.

We commence by studying the consistency conditions imposed by the ESC on the asymptotic behaviour of quantum gravity under dimensional reduction. Consider an infinite-distance limit in which a (1+2)-dimensional membrane becomes asymptotically tensionless. If its circle reduction leads to a critical string, we show that such a membrane must parametrically decouple from the Kaluza-Klein scale in the original theory. We confirm this censorship against emergent membrane limits, i.e. trajectories in which the membrane sits at the Kaluza-Klein scale, in the hypermultiplet moduli space of Calabi-Yau threefold compactifications of M-theory. While it is possible to find putative membrane limits at the classical level, the quantum corrections to the hypermultiplet moduli space metric arising from M2-instantons obstruct such infinite-distance trajectories, turning them instead into decompactification limits to eleven dimensions.

Next, we turn our attention to the infinite-distance limits in the complex structure moduli space of elliptic Calabi-Yau threefolds. In the context of six-dimensional F-theory, these include infinite-distance trajectories in the non-perturbative open string moduli space. Geometrically, such limits are described as degenerations of elliptic threefolds whose central element exhibits non-minimal elliptic fibers, which do not admit a crepant resolution in the fiber. For this reason, F-theory models presenting codimension-one non-minimal singular elliptic fibers are usually discarded. We set out to understand the geometry and physics of the infinite-distance non-minimal singularities of Calabi-Yau threefolds. Our analysis shows how these non-crepant singularities can be removed by a systematic sequence of blow-ups of the base, leading to a union of log Calabi-Yau spaces glued together along their boundaries. We identify criteria for the blow-ups to give rise to open chains or more complicated trees of components, characterise the base geometry of the resulting log Calabi-Yau spaces, determine the line bundles defined over them and explain how to extract the gauge algebra for F-theory probing such reducible spaces. Focusing on those limits associated with the appearance of non-minimal singularities in the elliptic fiber over genus-zero curves in a Hirzebruch surface base, we determine the asymptotic physics that they lead to. As our main result, we interpret the central fiber of a subclass of these degenerations as endpoints of decompactification limits to theories with six-dimensional defects. The genus-zero single infinite-distance limit degenerations of Hirzebruch models also give rise to emergent string limits, whose endpoints are at global weak coupling. We analyse the geometrical conditions that must be met in order for such global weak coupling to be possible. Our results rely on an adiabatic limit to gain information on the asymptotically massless states from the structure of vanishing cycles. Whenever possible, we employ F-theory/heterotic duality to compare our analysis to the heterotic dual description. Our findings provide further evidence for the Emergent String Conjecture, aligning with general expectations from quantum gravity.