A Remark on Conformal Anomaly and Extrinsic Geometry of Random Surfaces

TL;DR

This paper explores how conformal anomalies affect the critical behavior of low-dimensional random surfaces with extrinsic curvature, using a small D expansion to identify fixed points and speculate on phase diagrams.

Contribution

It introduces a small D expansion approach to analyze conformal anomalies in random surfaces with extrinsic curvature, revealing fixed points and phase behavior.

Findings

Existence of a non-trivial infra-red fixed point.

Phase diagram insights in the $(,D)$ plane.

Qualitative understanding of numerical simulations in 3D and 4D.

Abstract

In low dimensions, conformal anomaly has profound influence on the critical behavior of random surfaces with extrinsic curvature rigidity $1/\a$. We illustrate this by making a small $D$ expansion of rigid random surfaces, where a non-trivial infra-red fixed point is shown to exist. We speculate on the renormalization group flow diagram in the $(\a,D)$ plane. We argue that the qualitative behavior of numerical simulations in $D=3, 4$ could be understood on the basis of the phase diagram.