Renormalization Group and String Loops

TL;DR

This paper investigates how the renormalization group can be extended from string tree level to string loop corrections by focusing on the loop-correct generating functional ${\hat{Z}}$. It argues that renormalizability requires treating local and modular (Möbius) divergences on equal footing and advocates using an extended Schottky parametrization of moduli to define loop beta-functions that match an off-shell effective action. The authors provide explicit one-loop (torus and disc) and genus-two analyses, showing how tadpole and modular divergences renormalize background fields like the metric and dilaton, and propose a topological-fixture operator approach that hints at a renormalization-group–improved resummation of the genus expansion. These results suggest a path toward a consistent, possibly nonperturbative, formulation of string theory via RG flow on the string worldsheet. Overall, the work connects worldsheet Weyl invariance, loop-correct effective actions, and modular geometry to illuminate loop-level string vacua and off-shell extensions.

Abstract

Fixed points of the 2d renormalization group flow are known to correspond to tree level string vacua. We discuss how the renormalization group (or "sigma model") approach can be extended to the string loop level. The central role of the condition of renormalizability of the generating functional for string amplitudes with respect to both "local" and "modular" infinities is emphasized. Several one-loop and two-loop examples of renormalization are considered. It is found that in order to ensure the renormalizability of the generating functional one is to use an "extended" (Schottky-type) parametrization of the moduli space. An approach to resummation of the string perturbative expansion based on operators of insertion of topological fixtures is suggested.