Scaling and Entropy for the RG-2 Flow

TL;DR

This paper introduces a scale-invariant version of the RG-2 flow on Riemannian manifolds, develops a related entropy functional, and analyzes its properties, including monotonicity and the flow's gradient structure.

Contribution

It constructs a geometrically defined coupling constant for the RG-2 flow, introduces a modified Perelman entropy, and explores the flow's gradient properties and invariance.

Findings

Defined a scale-invariant coupling constant g

Established a monotonic entropy functional 

Proved local existence of the variational system

Abstract

Let $(\mathcal{M},g)$ be a closed Riemannian manifold. The $\textit{ second order approximation}$ to the perturbative renormalization group flow for the nonlinear sigma model (RG-2 flow) is given by : \[ \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm{Ric}(t) \, -\, \frac{\alpha}{2} \mathrm{Rm}^2(t), \] where $ g = \mathrm{Riemannian \ metric}, \mathrm{Ric} = \mathrm{Ricci \ curvature, } \ \mathrm{Rm}^2_{ij}:=\mathrm{R}_{irmk}\mathrm{R}_j^{rmk},$ and $\alpha\ge 0$ is a parameter. The flow is invariant under diffeomorphisms, but not under scaling of the metric. We first develop a $\textit{geometrically defined}$ coupling constant $\alpha_g$ that leads to an equivalent, scale-invariant flow. We further find a modified Perelman entropy for the flow, and prove local existence of the resulting variational system. The crucial idea is to modify the flow by two diffeomorphisms, the first being the usual DeTurck diffeomorphism the second being strictly related to the geometrical characterization of the coupling constant $\alpha_g$. We minimize the entropy functional so introduced to characterize a natural extension $\Lambda[g]$ of the Perelman's $\lambda(g)$--functional, and show that $\Lambda[g]$ is monotonic under the RG-2 flow. Although the modified Perelman entropy is monotonic, the RG-2 flow is not a gradient flow with respect this functional. We discuss this issue in detail, showing how to deform the functional in order to give rise to a gradient flow for a DeTurck modified RG-2 flow.