Mauro Carfora

The perturbative approach to nonlinear Sigma models and the associated renormalization group flow are discussed within the framework of Euclidean algebraic quantum field theory and of the principle of general local covariance. In particular we show in an Euclidean setting how to define Wick ordered powers of the underlying quantum fields and we classify the freedom in such procedure by extending to this setting a recent construction of Khavkine, Melati and Moretti for vector valued free fields. As a by-product of such classification, we prove that, at first order in perturbation theory, the renormalization group flow of the nonlinear Sigma model is the Ricci flow.

Let $(\mathcal{M},g)$ be a closed Riemannian manifold. The $\text{ second order approximation}$ 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 $g=\mathrm{Riemannian}\mathrm{metric},\mathrm{Ric}=\mathrm{Ricci}\mathrm{curvature},{\mathrm{Rm}}_{ij}^2:={\mathrm{R}}_{irmk}{\mathrm{R}}_j^{rmk},$ and $\alpha \ge 0$ is a parameter. The flow is invariant under diffeomorphisms, but not under scaling of the metric. We first develop a $\text{geometrically defined}$ 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.