Anomaly Coefficients and Central Charges

Weyl-anomaly coefficients are intrinsic fixed-point data only after their density basis and normalization are fixed. In even dimensions they divide into Euler-type, Weyl-invariant, and removable terms. Some coefficients are visible in flat-space stress-tensor correlators; others require curved backgrounds or higher-point information. The word “central charge” should therefore always be accompanied by a dimension and convention.

Required background. The Trace Ward Identity and Weyl Anomaly fixes the anomaly convention. Current and Stress-Tensor CFT Data fixes flat-space tensor normalization. Helpful background. Wess–Zumino Consistency and Descent supplies the cohomological classification principle.

Type A, type B, and trivial terms

Wess–Zumino consistency requires two Weyl transformations to commute on the generating functional. On a closed even-dimensional manifold, the resulting local anomaly densities fall into three useful classes Deser and Schwimmer 1993:

• type A: the Euler density $E_d$, whose integral is topological up to normalization;
• type B: local Weyl invariants built from the Weyl tensor and its derivatives;
• trivial terms: Weyl variations of finite local counterterms.

The classification does not fix numerical conventions. Rescaling $E_d$ or a Weyl invariant inversely rescales its quoted coefficient. Comparisons require the full equation, not just the symbol $a$, $c$, or $b$.

In two dimensions,

$$\langle T^\mu{}_{\mu}\rangle=\frac{c}{24\pi}R,$$

and the same $c$ fixes the plane stress-tensor two-point function in standard complex coordinates. There is one nontrivial bulk coefficient.

In four dimensions, use

$$\langle T^\mu{}_{\mu}\rangle =\frac{1}{(4\pi)^2} \left(cW^2-aE_4+b\nabla^2R\right).$$

Here $a$ is type A, $c$ is type B, and $b$ is removable. On conformally flat backgrounds $W^2=0$, so a round sphere isolates $a$ through its logarithmic scale dependence. Flat-space two-point data instead determine $c$.

Matching the stress-tensor two-point function

Normalize the Euclidean two-point function by

$$\langle T_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle =\frac{C_T}{x^{2d}}\, \mathcal I_{\mu\nu,\rho\sigma}(x),$$

where

$$\mathcal I_{\mu\nu,\rho\sigma} =\frac12(I_{\mu\rho}I_{\nu\sigma}+I_{\mu\sigma}I_{\nu\rho}) -\frac1d\delta_{\mu\nu}\delta_{\rho\sigma}, \qquad I_{\mu\nu}=\delta_{\mu\nu}-2\frac{x_\mu x_\nu}{x^2}.$$

In the four-dimensional anomaly convention above,

$$C_T=\frac{40}{\pi^4}c.$$

Reflection positivity gives $C_T>0$ for a nontrivial unitary CFT, hence $c>0$ in this convention. The coefficient $a$ is not determined by the two-point function; it enters particular stress-tensor three-point combinations and the Euler response Osborn and Petkou 1994.

Free fields provide a normalization checksum:

Four-dimensional field	$a$	$c$	$C_T$
Real conformal scalar	$1/360$	$1/120$	$1/(3\pi^4)$
Weyl fermion	$11/720$	$1/40$	$1/\pi^4$
Maxwell field	$31/180$	$1/10$	$4/\pi^4$

These standard free-field coefficients are tabulated in Duff 1994, §§3–4. They refer to free fields with standard stress-tensor normalization and exclude gauge zero-mode subtleties in a sphere partition function. They are checks of convention, not a basis for interpolating arbitrary interacting theories.

What each observable determines

Observable	Coefficient information	Qualification
Flat-space $\langle TT\rangle$	$C_T$, hence 4D $c$ above	Requires exact tensor normalization
Flat-space $\langle TTT\rangle$	Several structures, including combinations related to $a,c$	Contact terms and basis must be matched
Logarithmic response of $S^{2n}$	Euler-type coefficient	Zero modes and radius convention explicit
Generic curved background	Type-A, type-B, and trivial terms	Enough independent geometries are needed
Energy-flux experiment in 4D	Linear combinations of $TTT$ structures	Positivity and collider-state hypotheses apply

In six and higher even dimensions, several independent type-B invariants occur. A single symbol $c$ is then inadequate. Odd-dimensional CFTs have no local bulk Weyl anomaly on a closed manifold, but they can possess universal finite sphere data and boundary or defect anomalies.

Scheme independence is not background independence

A coefficient can be invariant under finite local counterterms yet vanish on a particular background. For example, four-dimensional $c$ is universal but invisible on a conformally flat sphere because $W_{\mu\nu\rho\sigma}=0$. Conversely, a nonzero finite term in $W[S^4]$ can be shifted by allowed local counterterms even though its logarithmic coefficient is universal.

Likewise, integrating the Euler density on a manifold with boundary requires its boundary completion. Ignoring that term can make the same bulk $a$ appear to change with the shape or regulator.

Common pitfalls

Equating every $c$ with $C_T$. The relation is dimension- and convention-dependent. In 4D it is $C_T=40c/\pi^4$ only for the displayed tensor normalization.

Extracting $c$ from a round four-sphere. A round sphere is conformally flat, so it isolates the Euler response rather than $W^2$.

Calling a total derivative a new central charge. A coefficient shifted by a finite local counterterm is scheme data unless a restricted scheme has been declared.

Exercises

Check the scalar entry in the table using $C_T=40c/\pi^4$.

Solution

For a real scalar $c=1/120$, so $C_T=40/(120\pi^4)=1/(3\pi^4)$.

Why can $a$ not be reconstructed from $\langle TT\rangle$ alone in four dimensions?

Solution

Conformal symmetry fixes the two-point tensor structure up to one number, $C_T$, which matches $c$. The independent Euler coefficient $a$ requires additional information, such as stress-tensor three-point data or a curved-background Euler response.

References

• Deser, S., and Schwimmer, A. “Geometric Classification of Conformal Anomalies in Arbitrary Dimensions.” Physics Letters B 309 (1993): 279–284. arXiv. DOI.
• Duff, M. J. “Twenty Years of the Weyl Anomaly.” Classical and Quantum Gravity 11 (1994): 1387–1404. arXiv. DOI.
• Osborn, H., and Petkou, A. C. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. arXiv. DOI.