The Conformal Anomaly and a new Exact RG

TL;DR

This paper introduces a generalized Exact Renormalization Group (RG) framework for scalar field theory in curved space that explicitly incorporates the conformal anomaly, providing a new approach to regularization and recovering known anomalies in two dimensions.

Contribution

It proposes a novel generalization of the Exact RG to curved space that explicitly includes the conformal anomaly and introduces a regularization method using free fields at non-critical fixed points.

Findings

Successfully recovers the integrated conformal anomaly in 2D

Derives the Polyakov action for the Gaussian theory in 2D

Provides a regularization scheme that removes divergent vacuum terms

Abstract

For scalar field theory, a new generalization of the Exact RG to curved space is proposed, in which the conformal anomaly is explicitly present. Vacuum terms require regularization beyond that present in the canonical formulation of the Exact RG, which can be accomplished by adding certain free fields, each at a non-critical fixed-point. Taking the Legendre transform, the sole effect of the regulator fields is to remove a divergent vacuum term and they do not explicitly appear in the effective average action. As an illustration, both the integrated conformal anomaly and Polyakov action are recovered for the Gaussian theory in d=2.