Geometric flows and the Swampland

TL;DR

This paper explores the application of geometric flows, particularly Perelman's flow, to string theory and the swampland program, revealing connections between flow dynamics and the infinite tower of states predicted by the distance conjecture.

Contribution

It introduces a novel approach using geometric flows to analyze string theory models and the swampland, linking flow equations to the distance conjecture's infinite tower of states.

Findings

Flow equations derived from Perelman's entropy function.

Flow trajectories can preserve Einstein equations with additional energy-momentum terms.

Flow contributions reproduce the infinite tower of states with exponentially decreasing masses.

Abstract

After an introductory chapter on the quantum supersymmetric string, in which particular attention will be devoted to the techniques via which phenomenologically viable models can be obtained from the ultraviolet microscopic degrees of freedom, and a brief review of the swampland program, the technical tools required to deal with geometric flows will be outlined. The evolution of a broad family of scalar and metric bubble solutions under Perelman's combined flow will be then discussed, together with their asymptotic behaviour. Thereafter, the geometric flow equations associated to a generalised version of Perelman's entropy function will be derived and employed in defining the action-induced flow associated to a given theory for a scalar field and a dynamical metric. The problem of preserving Einstein field equations along the corresponding moduli space trajectories will be cured by allowing a supplementary energy-momentum tensor term to appear along the flow. In a particular example, such contribution will be shown to precisely reproduce the infinite tower of states with exponentially dropping masses postulated by the distance conjecture.