Towards a Geometrization of Renormalization Group Histories in Asymptotic Safety

TL;DR

This paper proposes a geometric framework that represents the entire evolution of scale-dependent spacetimes in Quantum Einstein Gravity as a single higher-dimensional manifold, revealing new insights into the structure of RG flows in asymptotic safety.

Contribution

It introduces a universal (d+1)-dimensional metric to geometrize RG histories, showing it is Ricci flat and admits a homothetic Killing vector under certain conditions.

Findings

The higher-dimensional metric is Ricci flat and admits a homothetic Killing vector.

Maximally symmetric spacetimes lead to a flat (d+1)-dimensional manifold.

Monotonicity of the cosmological constant's running is linked to the metric's non-degeneracy.

Abstract

Considering the scale dependent effective spacetimes implied by the functional renormalization group in d-dimensional Quantum Einstein Gravity, we discuss the representation of entire evolution histories by means of a single, (d + 1)-dimensional manifold furnished with a fixed (pseudo-) Riemannian structure. This "scale-space-time" carries a natural foliation whose leaves are the ordinary spacetimes seen at a given resolution. We propose a universal form of the higher dimensional metric and discuss its properties. We show that, under precise conditions, this metric is always Ricci flat and admits a homothetic Killing vector field; if the evolving spacetimes are maximally symmetric, their (d + 1)-dimensional representative has a vanishing Riemann tensor even. The non-degeneracy of the higher dimensional metric which "geometrizes" a given RG trajectory is linked to a monotonicity requirement for the running of the cosmological constant, which we test in the case of Asymptotic Safety.