Weyl Anomalies, Deformations, and Flow Constraints

Changing a background metric or perturbing a CFT probes information that flat-space symmetry alone does not reveal. This chapter connects Weyl anomalies, sphere observables, beta functions, and conformal manifolds. Its central distinction is between universal physical data and quantities that change under local counterterms or coupling redefinitions.

Enter this chapter

Begin with Weyl Covariance on Curved Backgrounds to separate a change of conformal frame from the quantum response to it.

Helpful background. Local RG and Trace Identities supplies running-source methods. What Is an Anomaly? and Wess–Zumino Consistency supply the anomaly framework. UV and IR Fixed Points fixes flow orientation.

Topics

1. Weyl Covariance on Curved Backgrounds — Tracks operator weights, improvement terms, and curvature couplings.
2. The Trace Ward Identity and Weyl Anomaly — Separates running couplings, anomalies, improvements, and contact terms in the trace.
3. Anomaly Coefficients and Central Charges — Distinguishes universal anomaly coefficients from removable contributions.
4. Boundary and Defect Weyl Anomalies — Classifies curvature response localized on boundaries and defects.
5. Sphere Partition Functions and Universal CFT Data — Extracts universal terms while controlling counterterms and zero modes.
6. Local RG and Weyl Consistency Conditions — Uses commuting local Weyl transformations to constrain renormalization-group data.
7. Conformal Perturbation Theory and Beta Functions — Derives running and mixing from integrated CFT correlators.
8. Exactly Marginal Deformations and Conformal Manifolds — Tests whether marginal directions remain conformal after higher-order obstructions.
9. Monotonicity Theorems and Flow Constraints — States the dimension and hypotheses of inequalities between RG endpoints.

How the ideas fit together

Promoting couplings to spacetime-dependent sources puts metric response and running in the same generating functional. Commuting local scale transformations then imposes consistency relations among anomaly coefficients and beta functions. Those relations depend on how redundant couplings and local terms are treated; a component of a beta function is not itself a coordinate-independent observable. Osborn 1991, §§ 2–3.

At a fixed point, a beta function can vanish while a curved-background Weyl anomaly remains. Likewise, a dimension-d scalar in d spacetime dimensions is only a candidate marginal deformation: integrated operator products can generate running. Exact marginality requires the obstructions to vanish along an actual family of theories.

Review the chapter

Does the absence of a quadratic beta-function term prove that a deformation is exactly marginal?

Solution

No. It removes one perturbative obstruction in the chosen basis. Mixing with other marginal operators and higher integrated correlators can produce later terms. One must also remove redundant directions and establish vanishing of the physical beta function throughout the proposed family.

Where to go next

Continue to Exactly Marginal Deformations and Conformal Manifolds for the geometry of conformal families, or to Monotonicity Theorems and Flow Constraints to compare distinct fixed points.

References

• Osborn, H. “Weyl Consistency Conditions and a Local Renormalization Group Equation for General Renormalisable Field Theories.” Nuclear Physics B 363 (1991): 486–526. DOI.