An Alliance in the Tripartite Conflict over Moduli Space

TL;DR

This paper compares four different proposals for measuring distances in the moduli space of metrics, analyzing their implications for supersymmetric black hole geometries and the emergence of massless particles.

Contribution

It provides a detailed comparison of four distance proposals on moduli space, highlighting their agreement and disagreement in specific geometric contexts.

Findings

Distances (1) and (3) agree on finite vs. infinite distance distinctions.

Distances (2) and (3b) differ in their predictions about massless particle towers.

Certain loci exhibit infinite towers of massless particles at finite distance under some proposals.

Abstract

We investigate three proposals of distance on the moduli space of metrics: (1) a distance derived from the symplectic form of phase space, (2) a distance obtained by moving BPS objects at small velocity, (3a) a distance proposed by DeWitt and (3b) the distance used in the context of the generalised Swampland distance conjecture. In particular, we calculate these distances on a space of geometries that have the same asymptotics as the supersymmetric black hole in five dimensions. These moduli spaces contain a locus where there exists an infinite tower of massless particles, which emerges at finite distance according to proposals (1) and (3a), and at infinite distance from proposal (2) and (3b): distances (1) and (3) agree, and they disagree with distance (3b).