Sigma model renormalisation group flows, singularities and some remarks on cosmology

TL;DR

This work connects renormalisation group flows of two-dimensional sigma models to Ricci flow, showing that many flows encounter strong-coupling singularities before reaching fixed points. It leverages Ricci solitons to model the geometry near type I singularities and demonstrates that such solitons can equivalently be viewed as cosmological constant-like solutions, including de Sitter space, with the potential for regionally varying constants. The authors discuss how these ideas could seed cosmological interpretations within string theory and speculate about proving large-scale isotropy via geometrisation-inspired methods, while acknowledging substantial theoretical hurdles. Overall, the paper proposes a toy-model framework in which RG flow singularities yield emergent cosmological structure, offering a bridge between geometric analysis and cosmology that warrants further investigation.

Abstract

We investigate the properties of the renormalisation group (RG) flow of two-dimensional sigma models with a generic metric coupling by utilising known results for the Ricci flow. We point out that on many occasions the RG flow develops singularities, due to strong coupling behaviour, before it reaches a UV or an IR fixed point. We illustrate our analysis with several examples. We give particular emphasis to type I singularities, where the length of the curvature of the sigma model target space grows at most as $|t-T|^{-1}$ as the flow parameter $t$ approaches the singularity at $T$. For these, the geometry near the singularity is described in terms of a shrinking Ricci soliton that exhibits a cosmological constant even though the original RG flow does not. Assuming that the spacetime satisfies an RG flow equation, we use the Ricci solitons to introduce a cosmological constant in a string theory setting. This can allow for different cosmological constants at different regions of spacetime. In particular, we point out how the de-Sitter space is a solution of the theory. We also raise the question on whether the techniques used to prove the geometrisation conjecture can be applied to prove the homogeneity and isotropy of the universe at large scales.