Fluid Membranes and 2d Quantum Gravity

We study the 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, bending rigidity and Gaussian rigidity. We look for non-trivial fixed points but we find no evidence for a crumpling transition at $T\mathrm{\ne }0$. 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.