Real Weyl-Anomaly Constraint

• Real Weyl-Anomaly Constraint is the set of algebraic conditions derived from BRST cohomology that identifies how quantum anomalies break local scale symmetry in curved spacetimes.
• It distinguishes Type-A anomalies, constructed from Euler densities, from Type-B anomalies based on Weyl invariants, clarifying their geometric and physical implications.
• The constraint imposes strict coefficient relations in higher-order gravitational Lagrangians, ensuring regularization-independent consistency across arbitrary even dimensions.

A real Weyl-anomaly constraint refers to structural, algebraic, or coefficient relations that must be satisfied by quantum anomalies associated with Weyl (local scale) symmetry breaking in quantum field theory and gravity. Such constraints emerge from the interplay between classical Weyl invariance, the structure of quantum anomalies as classified by cohomological methods (notably the BRST formalism), and the universal features of the curvature invariants constructed in arbitrary dimensions. The algebraic classification of Weyl anomalies, and explicit coefficient relations in higher-order gravitational Lagrangians, form the technical content of these constraints, with deep implications for the formulation, consistency, and interpretation of conformally invariant massless field theories in external gravitational backgrounds.

1. Algebraic Structure and Classification via BRST Cohomology

The quantization of classically conformally invariant, massless field systems—comprising only dimensionless parameters—generally leads to the violation of Weyl invariance through the appearance of the Weyl anomaly. Algebraically, this anomaly is analyzed by adopting the BRST formalism, which unifies Weyl and diffeomorphism (coordinate) transformations. The associated BRST differential $s$ is combined with the exterior spacetime differential $d$ to produce a single operator $\widetilde{W} = s + d$ acting on the graded algebra of local total forms.

Weyl anomalies are identified with local $(n+1)$-forms $\alpha(W)$ (on an $n$-dimensional manifold) satisfying the Wess–Zumino consistency condition: $$\widetilde{W} \, \alpha(W) = 0 \quad \text{modulo trivial redefinitions}.$$ The physically relevant anomaly is then the cohomology class of $\widetilde{W}$ at form degree $n+1$ and ghost number one. Descending to form degree $n$, the top-degree form $a_1$ must obey

$$s a_1 + d(\ldots) = 0, \quad a_1 = s \zeta + (\text{constant term}),$$

which echoes the descent equations fundamental to the cohomological classification of local anomalies (0706.0340).

2. Universal Decomposition: Type-A and Type-B Anomalies

The algebraic analysis isolates two universality classes for Weyl anomalies:

• Type-A anomaly: Characterized by a nontrivial descent and constructed from the Euler density of the underlying manifold. Explicitly, in even dimensions $n=2m$, the type-A anomaly takes the form:

$$e_1^n = (-1)^m \omega\, \sqrt{-g}\, [R^{(1)} R^{(2)} \cdots R^{(m)}]\, \epsilon,$$

where $R^{(i)}$ are curvature two-forms and $\epsilon$ is the antisymmetric Levi–Civita symbol. This term is unique and directly reflects the spacetime topology, analogous to the consistent non-Abelian chiral anomaly (0706.0340).

• Type-B anomalies: Built from the contraction of Weyl tensors (and their derivatives), i.e., conformally invariant curvature invariants. Type-B anomalies do not generate new cohomology classes beyond those of existing Weyl-invariant densities; their descent is trivial.

The complete anomaly structure is algebraically encoded as

$$\alpha(W) = 2\omega\, C \cdots C\, a_{N_1\cdots N_n}(T),$$

where the $C$'s represent generalized connections involving the Weyl ghost $\omega$, the metric $g$, and invariants formed from the Weyl tensor $W_{\mu\nu\rho\sigma}$ and its derivatives.

3. Dimensional Independence and Regularization-Independence

A central outcome of the algebraic approach is that the entire classification and universal structure of Weyl anomalies in both type-A and type-B sectors are regularization independent and valid in arbitrary (even) dimensions. The analysis does not depend on specific regulators or on detailed dimensional counting. The derived descent equations are intrinsic, as in non-Abelian chiral anomalies, and generalize the well-known two-dimensional trace anomaly to higher dimensions without recourse to specific regularization artifacts.

4. Imposed Consistency Conditions: Real Constraints on Lagrangian Coefficients

The universality of the anomaly, and the requirement that the descent equations be satisfied, leads to stringent constraints on the coefficients of terms in higher-order curvature Lagrangians—these are the "real Weyl-anomaly constraints" in the sense of necessary and sufficient algebraic relations.

For example, in the context of the third-order Lovelock Lagrangian in six dimensions, a generic linear combination of the eight linearly independent curvature-cubed invariants $K_1, \ldots, K_8$ with coefficients $b_1,\ldots, b_8$ is subject to system of consistency equations such as: \begin{align*} b_4 &= \frac{2(-b_2+4b_3)}{3}, \ b_6 &= 30b_1+2b_2-10b_3, \ b_g &= 100b_1+12b_2+12b_3, \ b_5 &= -60b_1-8b_2-4b_3, \ b_7 &= 30b_1 + \frac{14b_2}{3} + \frac{28b_3}{3}, \tag{28} \end{align*} leaving three degrees of freedom in the allowed parameter space (1107.1034). Cubic Weyl invariants automatically satisfy these constraints in six dimensions, and the constraints carry over—after accounting for geometric identities—to higher-dimensional settings.

5. Descent Equations, Geometric Invariants, and Physical Implications

The presence and form of the anomaly are inextricably tied to geometric invariants constructed from the metric and curvature tensors, with the Euler density (type-A) and Weyl invariants (type-B) as central roles. The loss of Weyl invariance after quantization appears as additional terms in the trace of the energy-momentum tensor—a quantum breaking of the "tracelessness" condition. When an external gravitational field is present, the invariance under

$\delta_\omega g_{\mu\nu} = 2\omega g_{\mu\nu}$

is spoilt, and the resulting anomaly picks out precisely those terms (e.g., the Euler term and Weyl invariants) fixed by spacetime geometry and topology.

In physical terms, this imposes powerful restrictions on the renormalization properties and spectrum of conformally invariant field theories; for example, the type-A anomaly constrains the allowed spectrum and effective couplings of quantum field theories relevant at very high energy (e.g., in early-universe cosmology and quantum gravity).

6. Independence from Regularization Schemes and Application Scope

The algebraic classification demonstrates that the universal structure of the Weyl anomaly—specifically, the splitting into nontrivial (type-A) and trivial (type-B) descent classes—is unaffected by the specifics of the regularization procedure employed. This independence implies that the anomaly is not an artifact of any given method, but a robust feature of the quantum theory defined over a curved background, dictated by the interplay of geometry and quantum field theory.

These structures generalize known lower-dimensional results and clarify the nature of anomaly cancellation and quantization in high-dimensional quantum field theories, providing unambiguous input for model building in quantum gravity and string theory.

7. Summary Table: Key Concepts and Relations

Category	Main Concept/Expression	Significance
Universal Classes	Type-A (Euler density)	Unique, nontrivial descent, topological, "rare"
	Type-B (Weyl invariants)	Trivial descent, built from $W_{\mu\nu\rho\sigma}$
Constraint Eq.	$\widetilde{W}\,\alpha(W) = 0$	Anomaly cohomology, Wess–Zumino condition
Example Rel.	$b_4 = \frac{2(-b_2+4b_3)}{3},$ etc.	Consistency relations among Lagrangian coefficients
Formulas	$\alpha(W)=2\omega C\cdots C a_{N_1\cdots N_n}(T)$	Algebraic encapsulation of anomaly structure

The "real Weyl-anomaly constraint" thus refers to both the algebraic equations that the coefficients of curvature-invariant Lagrangian terms must satisfy for Weyl invariance (prior to quantization), and the inevitable universal breaking characterized by the type-A and type-B anomalies—whose form, uniqueness, and implications are dictated by BRST cohomology and geometric descent equations, independent of the regularization scheme or spacetime dimension (0706.0340, 1107.1034).