Renormalization of Couplings in Embedded Random Surfaces

TL;DR

This paper investigates how couplings in models of embedded random surfaces are renormalized through operator dressing, analyzing two limits: large string tension dominated by Nambu-Goto action and small tension dominated by extrinsic curvature, revealing non-trivial effects.

Contribution

It applies the methods of David, Distler, and Kawai to embedded random surfaces, showing how couplings are dressed in different tension regimes and how this affects the relationship between physical scales and beta functions.

Findings

Couplings are dressed by the Liouville mode in the large tension limit.

The induced metric and worldsheet metric relationship is renormalized in the small tension limit.

Coupling dependence on physical scale differs from standard beta function predictions.

Abstract

We study the dressing of operators and flows of corresponding couplings in models of {\it embedded} random surfaces. We show that these dressings can be obtained by applying the methods of David and Distler and Kawai. We consider two extreme limits. In the first limit the string tension is large and the dynamics is dominated by the Nambu-Goto term. We analyze this theory around a classical solution in the situation where the length scale of the solution is large compared to the length scale set by the string tension. Couplings get dressed by the liouville mode (which is now a composite field) in a non-trivial fashion. However this does {\it not} imply that the excitations around a physical ``long string" have a phase space corresponding to an extra dimension. In the second limit the string tension is small and the dynamics is governed by the extrinsic curvature term. We show, perturbatively, that in this theory the relationship between the induced metric and the worldsheet metric is ``renormalized", while the extrinsic curvature term receives a non-trivial dressing as well. This has the consequence that in a generic situation the dependence of couplings on the physical scale is different from that predicted by their beta functions.