Infinite Distances and Factorization

The information metric provides a unique notion of distance along any continuous family of physical theories, placing two theories further apart the more distinguishable they are. As such, the information metric is typically singular at quantum critical points, reflecting the fact they make qualitatively different scale-invariant predictions when compared to other theories in the family. However, such singularities are always at finite distance. The goal of this paper is to study infinite distance metric singularities and understand what type of physics generates them. We argue that infinite distance limits in the information metric correspond to theories in which expectation values factorize, and that unitarity restricts the asymptotic form of the metric to have a universal logarithmic singularity whose coefficient only depends on how the factorization limit is approached. We illustrate this relationship in a set of wide-ranging examples. Furthermore, we argue that it provides a particularly simple bottom-up motivation for the seemingly universal behavior of quantum gravity in asymptotic regions of moduli space described by the Swampland Distance Conjecture: gravity universally couples to stress-energy and thus abhors factorization limits. The towers of light fields that appear in these limits in so many examples serve to decouple gravity and consistently allow for factorization. We make this more precise via holography and suggest a way to consistently realize infinite distance limits without a tower of light fields.