Nonperturbative Conditions for Local Weyl Invariance on a Curved World Sheet

TL;DR

This paper explores nonperturbative conditions for local Weyl invariance on curved world sheets, using renormalization group techniques to analyze anomalies and derive beta functions related to target space effective actions.

Contribution

It introduces a nonperturbative approach to Weyl invariance on curved backgrounds, linking beta functions to target space actions and clarifying the role of redundant operators.

Findings

Derived nonperturbative beta functions for tachyon, graviton, and dilaton.

Established a nonperturbative Curci-Paffuti relation including tachyon beta function.

Connected beta functions to target space effective actions via a kappa function.

Abstract

We investigate Weyl anomalies on a curved world sheet to second order in a weak field expansion. Using a local version of the exact renormalization group equations, we obtain nonperturbative results for the tachyon/graviton/dilaton system. We discuss the elimination of redundant operators, which play a crucial role for the emergence of target space covariance. Performing the operator product expansion on a curved world sheet allows us to obtain the nonperturbative contributions to the dilaton $\beta$ function. We find the $\beta$ functions, after suitable field redefinitions, to be related to a target space effective action through a $\kappa$ function involving derivatives. Also we can establish a nonperturbative Curci-Paffuti relation including the tachyon $\beta$ function.