Second-Order Renormalization Group Flow of Three-Dimensional Homogeneous Geometries

TL;DR

This paper analyzes the second-order Renormalization Group flow on three-dimensional homogeneous geometries, revealing behaviors similar to Ricci flow in some cases and complex phase space dynamics in others, including singularity formation and geometric collapse.

Contribution

It provides a detailed phase plane analysis of the second-order RG flow on various 3D geometries, extending understanding beyond Ricci flow and identifying parameter-dependent behaviors.

Findings

Flow behaves like Ricci flow on $ ext{SU}(2)$.

Flow develops singularities or expands depending on curvature in $ ext{H}(3)$ and $ ext{H}(2) imes ext{R}$.

Configuration space partitioned into regions with different singularity behaviors for $ ext{Nil}$, $ ext{Sol}$, and $ ext{SL}(2, ext{R})$ geometries.

Abstract

We study the behavior of the second order Renormalization Group flow on locally homogeneous metrics on closed three-manifolds. In the cases $\mathbb R^3$ and $\text{SO}(3)\times \R$, the flow is qualitatively the same as the Ricci flow. In the cases $\text{H}(3)$ and $\text{H}(2)\times \R$, if the curvature is small, then the flow expands as in the Ricci flow case, while if the curvature is large, then the flow contracts and forms a singularity in finite time. The main focus of the paper is the flow on the $\text{SU}(2)$, $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ 3-geometries, with two of the three principal directions set equal. The configuration spaces for these geometries are two dimensional, and we can consequently apply phase plane techniques to the study. For the $\text{SU}(2)$ case, the flow is everywhere qualitatively the same as Ricci flow. For the $\text{Nil}$, $\text{Sol}$, and $\text{SL}(2,\R)$ cases, we show that the configuration space is partitioned into two regions which are delineated by a solution curve of the flow that depends on the coupling parameter: in one of the regions, the flow develops cigar or pancake singularities characteristic of the Ricci flow, while in the other both directions shrink. In the $\text{Nil}$ case we obtain a characterization of the full 3-dimensional flow.