Tameness, Strings, and the Distance Conjecture

TL;DR

This paper combines the Distance and Tameness Conjectures to establish path-independent statements about towers of states near infinite distance points in field space, using tame geometry and cosmic string solutions.

Contribution

It introduces a framework using tame geometry to address path-dependence in the Distance Conjecture and connects cosmic string solutions to the structure of effective theories near boundaries.

Findings

Finiteness of sectors near infinite distance points due to tameness.

Reconstruction of multi-field dependence from one-dimensional paths.

Cosmic string solutions serve as tools to analyze boundary regions in 4D theories.

Abstract

The Distance Conjecture states that an infinite tower of modes becomes exponentially light when approaching an infinite distance point in field space. We argue that the inherent path-dependence of this statement can be addressed when combining the Distance Conjecture with the recent Tameness Conjecture. The latter asserts that effective theories are described by tame geometry and implements strong finiteness constraints on coupling functions and field spaces. By exploiting these tameness constraints we argue that the region near the infinite distance point admits a decomposition into finitely many sectors in which path-independent statements for the associated towers of states can be established. We then introduce a more constrained class of tame functions with at most polynomial asymptotic growth and argue that they suffice to describe the known string theory effective actions. Remarkably, the multi-field dependence of such functions can be reconstructed by one-dimensional linear test paths in each sector near the boundary. In four-dimensional effective theories, these test paths are traced out as a discrete set of cosmic string solutions. This indicates that such cosmic string solutions can serve as powerful tool to study the near-boundary field space region of any four-dimensional effective field theory. To illustrate these general observations we discuss the central role of tameness and cosmic string solutions in Calabi-Yau compactifications of Type IIB string theory.