Fluid membranes and $2d$ quantum gravity

We study the renormalization-group (RG) flow of two dimensional (fluid) membranes embedded in Euclidean $D$-dimensional space using functional RG methods based on the effective average action. By considering a truncation ansatz for the effective average action with both extrinsic and intrinsic curvature terms, we derive a system of beta functions for the running surface tension ${\mu }_k$, bending rigidity ${\kappa }_k$, and Gaussian rigidity ${\overline{\kappa }}_k$. We look for nontrivial fixed points but we find no evidence for a crumpling transition at $T\ne 0$. Finally, we propose to identify the $D\to 0$ limit of the theory with two dimensional quantum gravity. In this limit, we derive new beta functions for both cosmological and Newton’s constants.

Figure 1

The RG flow in the $({\Lambda }_k,G_k)$ plane obtained by integrating the beta functions (28) for ${\kappa }_{\Lambda }=1$.