Swampland and the Geometry of Marked Moduli Spaces

We define the notion of a marked moduli space as the parameter space of a physical theory together with all of its observables. In geometric examples, this coincides with the mathematical notion of Teichm\"uller space. We propose two new Swampland principles about the geometry of marked moduli spaces: We conjecture that a marked moduli space is always contractible, and moreover, that there is a unique geodesic connecting any pair of points in it with respect to its physical metric. We provide strong evidence for these conjectures for theories with 8 or more supercharges.