Scaling and Entropy for the RG-2 Flow

Abstract

Let (M, g) be a closed Riemannian manifold. The RG-2 flow is defined by $$\begin{aligned} \frac{\partial }{\partial t} \, g(t) \, =\, -2 \mathrm {Ric}(t) \, -\, \frac{\alpha }{2} \mathrm {Rm}^2(t), \end{aligned}$$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. This is the geometric flow associated with the second-order approximation to the perturbative renormalization group flow for the nonlinear sigma model. It is invariant under diffeomorphisms, but not under scaling of the metric, the latter of which gives rise to several delicate problems from the point of view of geometric analysis. To address the lack of scaling we introduce a 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 and 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.