Navigating string theory field space with geometric flows

TL;DR

This paper extends the Swampland Distance Conjecture to flux-supported string backgrounds using geometric flows, particularly generalized Ricci flow, to analyze infinite-distance points in scalar field spaces with fluxes.

Contribution

It introduces a refined distance measure incorporating fluxes and extends the Ricci Flow Conjecture to include fluxes, enabling geometric probing of scalar field spaces with fluxes.

Findings

Refined the distance measure to include fluxes using Perelman entropy

Extended Ricci Flow Conjecture to encompass Kalb-Ramond flux

Proposed a cost function approach for flux-supported scalar field spaces

Abstract

The Swampland Distance Conjecture postulates the emergence of an infinite tower of massless states when approaching infinite-distance points in moduli space. However, most string backgrounds are supported by fluxes, and therefore depart from the purely geometric paradigm. This fact requires an extension of the Swampland conjectures to scalar field spaces with non-trivial potentials, rather than just moduli spaces. To address this task, we utilise geometric flows, in particular generalised Ricci flow, to probe the associated scalar field spaces. Considering internal spaces supported by three-form fluxes, we first show that the distance defined in terms of the Perelman entropy functional needs to be refined in order to encompass fluxes. Doing so, we extend the Ricci Flow Conjecture to include Kalb-Ramond flux besides the metric and the dilaton field. This allows us to probe infinite-distance points within these scalar field spaces in a purely geometric way. We subsequently construct a geometric flow for internal manifolds supported by Ramond-Ramond fluxes and discuss its role in the Ricci Flow Conjecture. Our analysis suggests that in the presence of fluxes the Distance Conjecture might be better characterised in terms of a cost function on the space of metrics, rather than a genuine distance.