On Functional Representations of the Conformal Algebra

TL;DR

This paper develops functional representations of the conformal algebra using Wilsonian effective actions, linking conformal invariance to the Exact Renormalization Group and providing insights into the energy-momentum tensor in conformal field theories.

Contribution

It introduces a novel functional representation involving auxiliary functionals satisfying consistency equations, connecting conformal invariance with the Wilsonian approach and energy-momentum tensor properties.

Findings

Representation relates conformal invariance to the ERG fixed-point form.

Identifies the trace of the energy-momentum tensor as a redundant, exactly marginal field.

Provides a new perspective on conformal Ward identities and their role in defining the energy-momentum tensor.

Abstract

Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the Exact Renormalization Group equation specialized to a fixed-point whereas the latter is new. This provides a concrete understanding of how conformal invariance is realized as a property of the Wilsonian effective action and the relationship to action-free formulations of conformal field theory. Subsequently, it is argued that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. Consistency of this construction implies Polchinski's conditions for improving the energy-momentum tensor of a conformal field theory such that it is traceless. In the Wilsonian approach, the exactly marginal, redundant field which generates lines of physically equivalent fixed-points is identified as the trace of the energy-momentum tensor.