# RG flow and Ricci flow

It looks like the Laplace operator in the nonlinear sigma model (say the Polyakov action) is different from the Laplace-Beltrami operator, how can one get the Ricci flow as a low order approximation to the perturbative renormalization group flow?

The Lagrangian in the Polyakov action is the trace of the pullback metric $\phi^*g$, or induced metric $\phi^*g_{\mu \nu}(x) = g_{ij}(\phi(x)) \partial_\mu \phi^i(x) \partial_\nu \phi^j(x)$ for $\phi: (\Sigma, \gamma) \rightarrow (M, g)$,

$S( g; \phi) = \frac{T}{2} \int_\Sigma dvol \,\gamma^{\mu \nu} \phi^*g_{\mu \nu} = - \frac{T}{2} \int_\Sigma dvol \, g_{ij} \phi^i \partial^\mu\partial_\mu \phi^j$

The usual notations are $\Delta = \partial^\mu \partial_\mu$ and $\Delta_g = Tr_g(- \Delta)$. But this $\Delta_g$ is obviously different from the
Laplace-Beltrami operator $\Delta_g = g^{ij}\partial_i \partial_j$ in the Ricci flow.