On Functional Representations of the Conformal Algebra

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.

v5: Published version (50 pages)

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• More constraining conformal bootstrap
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Cited by in corpus (14)

• The nonperturbative functional renormalization group and its applications
• Precision calculation of critical exponents in the $O(N)$ universality classes with the nonperturbative renormalization group
• A Holographic form for Wilson's RG
• Trace anomaly and infrared cutoffs
• A Conformal Fixed-Point Equation for the Effective Average Action
• Conformal invariance and vector operators in the $O(N)$ model
• Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group
• A Wilsonian Energy-Momentum Tensor
• Wilsonian Ward Identities
• Conformal invariance constraints in the $O(N)$ models: a first study within the nonperturbative renormalization group
• The generating functional of correlation functions as a high momentum limit of a Wilson action
• The Conformal Anomaly and a new Exact RG
• Explicit construction of the energy-momentum tensor in the large N limit
• $SO(1,d+1)$ symmetry of the Exact RG equation