Scaling and Entropy for the RG-2 Flow

Let (M,g) be a closed Riemannian manifold. The second order approximation to the perturbative renormalization group flow for the nonlinear sigma model (RG-2 flow) is given by :
∂∂tg(t)=−2Ric(t)−α2Rm2(t),
where g=Riemannian metric,Ric=Ricci curvature, Rm2ij:=RirmkRrmkj, and α≥0 is a parameter. The flow is invariant under diffeomorphisms, but not under scaling of the metric. We first develop a geometrically defined coupling constant αg 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 αg. 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