Weyl Anomalies, Deformations, and Flow Constraints

A CFT can be probed by changing the background metric or by deforming its action. The first operation reveals Weyl anomalies and universal curvature response; the second reveals beta functions, operator mixing, conformal manifolds, and RG endpoints. This chapter develops both from one generating-functional convention and repeatedly separates universal coefficients from local counterterms, contact terms, and coordinate choices. The type-A, type-B, and removable anomaly classes follow Deser and Schwimmer 1993, printed preprint pp. 2–3, PDF; the local-source formulation and its consistency conditions are developed in Osborn 1991, §§2–3.

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.

Enter this chapter

Your question	Start here	Output
How does an operator move between conformal frames?	Weyl Covariance on Curved Backgrounds	primary weights, improvement, and curvature completion
Why can the trace be nonzero?	Trace Ward Identity and Weyl Anomaly	beta, anomaly, improvement, and contact contributions separated
Which anomaly coefficient is being measured?	Anomaly Coefficients and Central Charges	type-A, type-B, trivial, and $C_T$ conventions
What changes on a boundary or defect?	Boundary and Defect Weyl Anomalies	intrinsic, extrinsic, ambient, and displacement data
Which part of $-\log Z[S^d]$ is universal?	Sphere Partition Functions	odd-dimensional finite or even-dimensional logarithmic data
What follows from commuting local scale transformations?	Local RG and Weyl Consistency	scheme-covariant gradient-type identities
How do CFT correlators generate running?	Conformal Perturbation Theory	beta functions, anomalous dimensions, and mixing
When does a marginal operator stay marginal?	Conformal Manifolds	quotient, metric, connection, curvature, and global identifications
Which endpoint quantity must decrease?	Monotonicity Theorems	dimension-specific theorem with full hypotheses

One convention for the whole chain

In Euclidean signature set $W=-\log Z$ and define

$$\delta W=\frac12\int d^dx\sqrt g\, \langle T^{\mu\nu}\rangle\delta g_{\mu\nu} +\int d^dx\sqrt g\, \langle\mathcal O_I\rangle\delta g^I.$$

A local Weyl transformation acts as

$$\delta_\sigma g_{\mu\nu}=2\sigma g_{\mu\nu}, \qquad \Delta_\sigma W =\int\sqrt g\,[\sigma\mathcal A+\partial_\mu\sigma\mathcal Z^\mu+\cdots].$$

For constant dimensionless scalar sources, with relevant sources suppressed, the trace identity is schematically

$$T^\mu{}_{\mu} =-B^I\mathcal O_I+\nabla_\mu V^\mu+\mathcal A+\text{contacts}.$$

Here $B^I$ is the ordinary mass-scale flow $\beta_\mu^I=\mu\,dg^I/d\mu|_0$ after quotienting redundant flavor rotations; without those rotations, $B^I=\beta_\mu^I$. The minus sign follows from the positive covariant-metric stress above, as derived in the trace Ward identity. At a fixed point, $B^I=0$; the nontrivial anomaly remains on curved backgrounds. Along a deformation, integrated OPE singularities determine the flow only after their local subtractions are specified.

The universal-versus-local test

For every quantity in this chapter, apply four questions:

1. Can a finite local counterterm change it?
2. Can a coupling or operator-basis redefinition change its components?
3. Is it a separated-point observable or a contact term?
4. Does the stated theorem require positivity, a particular dimension, or complete endpoints?

The answers sort the main objects:

Object	Classification	Safe comparison
Fixed-point type-A/type-B coefficient	universal after density normalization	same dimension and tensor convention
Total-derivative anomaly	scheme dependent	only within a fixed counterterm prescription
$C_T$	separated-point CFT datum	same stress-tensor normalization
Sphere finite part in odd $d$	universal real datum under standard assumptions	zero modes and parity phases separated
Sphere finite part in even $d$	scheme dependent	compare logarithmic coefficient instead
Beta-function components	coordinate dependent	zeros and critical exponents after quotient
Zamolodchikov metric components	geometric but coordinate dependent	distances, curvature, or stated coordinates
Flow inequality	theorem under listed hypotheses	complete UV and IR endpoint data

In four dimensions the endpoint ordering of the Euler coefficient is obtained from a dilaton dispersion relation under its unitarity, locality, and asymptotic hypotheses Komargodski and Schwimmer 2011, §§2–3. It should not be read as a scheme-independent pointwise function along every flow.

Two exact checks

Round four-sphere

For the parity-even purely gravitational anomaly in the same metric-stress convention,

$$\langle T^\mu{}_{\mu}\rangle =\frac{aE_4-cC^2+b_m\nabla^2R}{(4\pi)^2},$$

the round $S^4$ of radius $r$ gives, with $F=-\log Z$, fixed renormalization scale and local power-law terms subtracted,

$$\frac{dF_{S^4}}{d\log r}=4a.$$

Here $C^2$ is Weyl-tensor squared, $R$ is scalar curvature and $E_4$ is the Euler density defined in the fixed-point trace calculation. The check uses $C^2=0$, constant curvature and $\int_{S^4}\sqrt g\,E_4=64\pi^2$. The same calculation gives $+1/90$ for a real conformal scalar with $a=1/360$, whose conformal Laplacian has no zero mode. The measure and zero-mode qualifications for other theories are explained in Sphere Partition Functions.

Compact-boson conformal manifold

For the $c=1$ compact boson in the stated radius convention,

$$\Delta_{n,w}(R) =\frac12\left(\frac{n^2}{R^2}+w^2R^2\right)+N+\bar N$$

is invariant under $R\leftrightarrow1/R$ with $n\leftrightarrow w$. The $J\bar J$ deformation changes the spectrum while preserving $c$ and a positive Zamolodchikov metric. This checks the difference between protected data, coordinate-dependent data, and a global duality quotient.

Review the chapter

Separate four trace contributions

A flat-space trace contains $-\beta_\mu^I\mathcal O_I+\nabla^2L$ and contact terms, while the curved-space expectation value contains an Euler density. Classify them.

Solution

$-\beta_\mu^I\mathcal O_I$ is running in the displayed stress convention and a chosen coupling basis; $\nabla^2L$ is an improvement or virial contribution whose removability depends on $L$ and boundaries; contact terms are local distributional Ward data; the Euler term is a nontrivial fixed-point Weyl anomaly after its density is normalized.

Diagnose a sphere comparison

Two four-dimensional calculations report different finite values of $F_{S^4}$ but the same coefficient of $\log r$. Is there a contradiction?

Solution

Not necessarily. A finite local curvature counterterm can shift the finite part, while the logarithmic coefficient tied to $a$ is universal. Compare regulator, zero modes, and counterterms before interpreting the finite difference.

Test a marginal direction

A dimension-$d$ scalar has $C_{\mathcal O\mathcal O}{}^{\mathcal O}=0$. Is it exactly marginal?

Solution

The vanishing removes one quadratic obstruction in the stated basis. Other marginal operators, redundant directions, and higher integrated correlators can still generate beta functions. Exact marginality requires the full covariant beta function to vanish to all orders along the direction.

Where to continue

• Use Boundary, Defect, and Interface CFT for displacement, folding, and defect-flow observables.
• Use Modular and Thermal Bootstrap for torus modular response and thermal KMS crossing, which are distinct consistency systems.
• Use Superconformal Bootstrap Interfaces only after protected deformation data arrive with their normalization and mixing conventions.
• When comparing calculations, specify the regulator, normalization and tolerances needed to check the scale response.

References

• Deser, S., and Schwimmer, A. “Geometric Classification of Conformal Anomalies in Arbitrary Dimensions.” Physics Letters B 309 (1993): 279–284. DOI. Open PDF; the locators above use the printed preprint pages.
• Komargodski, Z., and Schwimmer, A. “On Renormalization Group Flows in Four Dimensions.” Journal of High Energy Physics 2011, 099 (2011). arXiv. DOI.
• Osborn, H. “Weyl Consistency Conditions and a Local Renormalization Group Equation for General Renormalisable Field Theories.” Nuclear Physics B 363 (1991): 486–526. DOI.