Infinite distances in the moduli space of Calabi-Yau threefolds

Abstract

In this work we analyze the Swampland Distance Conjecture for the field space of scalar fields, called the complex moduli, that arise in Calabi-Yau threefold compactifications. The Swampland Distance Conjecture states that at any infinite distance point in this moduli space an infinite tower of states will appear, rendering the effective 4D theory unphysical. This is very important when one attempts to realize inflation with a large field distance from string theory, as in essence this renders the complex moduli unable to be identified with the inflaton that causes this type of large field inflation. In this work we elucidate the above motivations for studying the Swampland Distance Conjecture, after which we connect the Swampland Distance Conjecture in physics to the mathematical Wang conjecture. For a moduli space with singularities encoded on subspaces called divisors the Wang conjecture states that the metric on the moduli space diverges if and only if the divisors have monodromies of infinite order. In a connection between infinite order monodromy matrices and infinite towers of states is postulated, bridging the gap between the Wang and Swampland Distance conjectures. Wang himself already showed that an infinite order monodromy is a necessary requirement for an infinite field distance to arise. In this work we study the mechanism behind the Swampland Distance Conjecture, by considering whether having such a monodromy is also a sufficient condition for the divergence of the field distance. We review a proof by Lee of specific cases of this sufficient condition of the Wang conjecture. The cases we revisit and clarify assume the singularity to be located on the intersection of up to two divisors. Finally we present a new result, by giving new criteria for the field distance to diverge, in the case of a singularity on one infinite and one finite divisor in two dimensions.