Geometric flow equations for the number of space-time dimensions

TL;DR

This paper introduces D-flow, a new geometric flow equation describing how space-time geometries evolve as the number of dimensions changes, motivated by swampland conjectures and curvature invariants.

Contribution

It formulates the D-flow equations for space-time geometries and provides explicit solutions for spheres and compactifications, linking dimension variation to geometric properties.

Findings

D-flow equations derived for D-dimensional geometries

Explicit solutions for spheres and Freund-Rubin compactifications

Dimension variation impacts geometric invariants and manifold volumes

Abstract

In this paper we consider new geometric flow equations, called D-flow, which describe the variation of space-time geometries under the change of the number of dimensions. The D-flow is originating from the non-trivial dependence of the volume of space-time manifolds on the number of space-time dimensions and it is driven by certain curvature invariants. We will work out specific examples of D-flow equations and their solutions for the case of D-dimensional spheres and Freund-Rubin Compactification. The discussion of the paper is motivated from recent swampland considerations, where the number $D$ of space-time dimensions is treated as a new swampland parameter.