On the Geometry of the String Landscape and the Swampland

TL;DR

This paper proposes conjectures about the geometry of string landscape moduli spaces, linking their structure to swampland criteria, and supports these ideas with string theory examples, impacting models of inflation in quantum gravity.

Contribution

It introduces new conjectures relating moduli space geometry to swampland conditions and provides string theory evidence supporting these geometric constraints.

Findings

Finite non-zero diameter moduli spaces are in the swampland.

Points at infinity in moduli space correspond to infinite towers of massless states.

Near these points, the moduli space is negatively curved.

Abstract

We make a number of conjectures about the geometry of continuous moduli parameterizing the string landscape. In particular we conjecture that such moduli are always given by expectation value of scalar fields and that moduli spaces with finite non-zero diameter belong to the swampland. We also conjecture that points at infinity in a moduli space correspond to points where an infinite tower of massless states appear, and that near these regions the moduli space is negatively curved. We also propose that there is no non-trivial 1-cycle of minimum length in the moduli space. This leads in particular to the prediction of the existence of a radially massive partner to the axion. These conjectures put strong constraints on inflaton potentials that can appear in a consistent quantum theory of gravity. Our conjectures are supported by a number of highly non-trivial examples from string theory. Moreover it is shown that these conditions can be violated if gravity is decoupled.