Swampland and the Geometry of Marked Moduli Spaces

TL;DR

This paper introduces new conjectures about the geometry of marked moduli spaces in physical theories, suggesting they are contractible and have unique shortest paths, supported by evidence in theories with high supersymmetry.

Contribution

It proposes two novel Swampland principles regarding the geometric structure of marked moduli spaces and provides supporting evidence for theories with 8 or more supercharges.

Findings

Marked moduli spaces are conjectured to be contractible.

There is a conjecture of unique shortest paths in these spaces.

Evidence supports these conjectures in highly supersymmetric theories.

Abstract

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 shortest path 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.