Defect Weyl Anomalies

• Defect Weyl anomalies are local obstructions to conformal invariance caused by defects, encoding universal geometric data via central charges.
• They are classified using cohomological methods that yield defect central charges (b, d₁, d₂), which impact key observables like entanglement entropy and Casimir energy.
• Both free field and holographic approaches verify these anomalies, demonstrating their roles in universal defect responses and governing RG flows in CFTs.

A defect Weyl anomaly is a local obstruction to conformal (Weyl) invariance in quantum field theories induced by the presence of lower-dimensional defects, such as boundaries, interfaces, or surface operators. While the bulk Weyl anomaly is encoded in the nonvanishing trace of the stress tensor in curved space (notably the type-A and type-B structures in even dimensions (0704.2472)), a defect Weyl anomaly arises due to localized contributions on the defect worldvolume and provides key data for classifying conformal defects and their universal observables.

1. Algebraic Structure and Classification

The modern classification of Weyl anomalies, including those localized on defects, is built on the cohomological analysis of the Wess-Zumino consistency conditions (0704.2472). In the presence of a defect, these conditions constrain possible anomaly terms to be integrals of local densities constructed from intrinsic curvature invariants of the defect, extrinsic curvature, and the pullback of ambient curvature tensors to the defect worldvolume.

In particular, for co-dimension-two (e.g., four-dimensional) surface defects in higher-dimensional CFTs, the defect Weyl anomaly consists of a sum of both intrinsic and extrinsic geometric invariants: $$\langle \hat{T}^\alpha_\alpha \rangle = -\frac{1}{24\pi}\big[ b\, R^\Sigma + d_1\, \tilde{\Pi}^2 - d_2\, W_{ab}^{ab}\big] + \cdots$$ where $R^\Sigma$ is the intrinsic Ricci scalar on the defect, $\tilde{\Pi}$ is the traceless part of the second fundamental form (extrinsic curvature), and $W_{ab}^{ab}$ is the pullback of the ambient Weyl tensor (Bianchi et al., 2021). The coefficients $b$, $d_1$, and $d_2$ (often called "defect central charges") encode the universal, scheme-independent data of the anomaly.

The algebraic approach guarantees that type-A (Euler-type) and type-B (Weyl-invariant polynomial-type) anomalies can be similarly defined for defects, with the defect type-A coefficients characterizing contributions to entanglement entropy and sphere partition functions (Capuozzo et al., 2023, Apruzzi et al., 25 Jul 2024), while type-B coefficients control stress tensor one-point functions and displacement two-point functions (Chalabi et al., 2021).

2. Universal Observables: Entanglement, Rényi, and Casimir Contributions

Defect Weyl anomaly coefficients universally appear in several physical observables:

• Entanglement and Rényi Entropy: For surface defects in 6d $(2,0)$ SCFTs, the defect’s contribution to the supersymmetric Rényi entropy $S_n^{\mathrm{def}}$ is a linear function of $1/n$ (the inverse Rényi index) with coefficient proportional to $2b - d_2$:

$$S^{\mathrm{def}}_n \sim \frac{2b-d_2}{6} \log\frac{\ell}{\epsilon}$$

This linearity and explicit dependence on $b$ and $d_2$ admit a closed-form expression for all $n$ (Huang et al., 16 Jan 2025). In the $n\to1$ limit, this reduces to the defect entanglement entropy.

• Supersymmetric Casimir Energy: In squashed sphere backgrounds, the supersymmetric Casimir energy localized on the defect can be written as a universal linear combination of $b$ and $d_2$:

$$E_{\mathfrak{g}} = -\frac{1}{\omega_1} \left[ \frac{d_2-b}{6}\, \omega_2 \omega_3 + \frac{2b-d_2}{24}\, \sigma_1 \sigma_2 \right]$$

where $\omega_i$, $\sigma_i$ are chemical potentials for bulk rotations and R-charges (Huang et al., 16 Jan 2025). In the chiral algebra limit ($\omega_1 = \omega_2 = 1$, $\sigma_1\sigma_2 \propto \omega_3$), the Casimir energy simplifies to $-d_2$ up to normalization.

• Entanglement Entropy in Interacting Theories: Defect anomaly coefficients govern the logarithmically divergent coefficient in defect-induced entanglement entropy, as shown for monodromy defects and for holographic defects (Bianchi et al., 2021, Chalabi et al., 2021, Capuozzo et al., 2023).

These results reveal a direct and robust link—valid in both free and interacting/strongly-coupled settings—between the local Weyl anomaly and long-range universal observables in defect CFTs.

3. Computational Approaches and Examples

Free Field Theories

• For monodromy defects in free $d=4$ scalar and Dirac fermion theories, the anomaly coefficients $b$, $d_1$, $d_2$ can be extracted exactly from one-point functions of the stress tensor and displacement operator, and from the logarithmic piece in the entanglement entropy (Bianchi et al., 2021).
• In four-dimensional Maxwell theory, Gukov-Witten–type defects are shown to have vanishing central charges $b = d_1 = d_2 = 0$ (Bianchi et al., 2021).

Holographic SCFTs

• For 2d BPS defects in 6d $\mathcal{N}=(1,0)$ SCFTs, holographic computation using probe D4-branes in AdS$_7$ backgrounds yields the A-type anomaly $a_\Sigma$ and B-type $d_2$ directly from on-shell actions and defect entanglement entropy (Apruzzi et al., 25 Jul 2024):

$a_\Sigma = 24 (q, r)\,;\quad d_1 = d_2 = 32 (q, r)$

with $(q, r) = \sum_{i,j} q_i (C^{-1})^{ij} r_j$.

• For 4d conformal defects in 6d $(2,0)$ theories, the 11d supergravity embedding provides expressions for $a_\Sigma$ and $d_2$ as functions of geometric data such as brane charges and embedding parameters. These match independent checks from the average null energy condition and holographic entanglement entropy (Capuozzo et al., 2023).

Interacting QFT on Curved Space

Anomalous contributions in interacting theories can be formulated within locally covariant field theory, where the Weyl anomaly is a local functional $A$ encoding the failure of the renormalized time-ordered products to commute with local scale (Weyl) transformations (Fröb et al., 24 Apr 2025). The trace anomaly is obtained by functional differentiation with respect to the metric, and nontrivial anomaly terms can be classified and removed when they are cohomologically trivial. For conformally coupled $\phi^4$ theory, explicit computation confirms that only geometric terms built from the Euler density $E_4$ and the Weyl tensor square $C^2$ (plus the free stress tensor trace) survive after proper redefinitions; derivative terms such as $\Box\phi^2$ can be eliminated perturbatively.

4. Physical Implications and Universality

The appearance of $b$ and $d_2$ in defect observables is a robust, universal feature of conformal defects:

• Universality: The closed-form dependence of the defect supersymmetric Rényi entropy and Casimir energy on $2b-d_2$ and $d_2$ is found in both free and holographically dual interacting theories (Huang et al., 16 Jan 2025).
• RG Flow and the $b$-Theorem: Under defect-localized relevant deformations, RG flow monotonically decreases $b$ (the analog of the $c$-theorem); $d_2$ controls the defect stress tensor one-point function (Bianchi et al., 2021).
• Strong Coupling and Holography: Holographic computations validate that anomaly coefficients computed from on-shell brane actions or minimal area surfaces govern universal defect responses and satisfy positivity/monotonicity constraints (e.g., $a_\Sigma \geq 0$, $d_2 \leq 0$) (Capuozzo et al., 2023, Apruzzi et al., 25 Jul 2024).

This points to a unifying description of defects across a broad landscape of conformal field theories.

5. Mathematical Structure and Removal of Trivial Anomalies

Consistent definition and renormalization of the defect Weyl anomaly requires a careful cohomological treatment. The variation of the generating functional under a Weyl transformation takes the schematic form: $$\delta_\omega W = \int d^dx\, \sqrt{g}\, \omega(x)\, \mathcal{A}(x)$$ with $\mathcal{A}(x)$ the (possibly defect-localized) anomaly density. Physical anomaly terms are those not removable by finite local counterterms; these are classified by computing the cohomology of the combined BRST and exterior differentials (0704.2472, Fröb et al., 24 Apr 2025). In concrete computations (as for interacting $\phi^4$ theory), only the Euler and Weyl invariant terms survive; derivative anomalies such as $\Box\phi^2$ can in fact be removed to all orders in perturbation theory (Fröb et al., 24 Apr 2025).

6. Future Directions and Open Questions

The robust link between defect anomaly coefficients and universal observables motivates several avenues:

• Determination of allowed (and physically consistent) values for $(b, d_2)$ in both free and interacting theories, and exploration of possible “defect $C$-theorems” (Huang et al., 16 Jan 2025).
• Generalizations to supersymmetric and higher-codimension defects, including ramifications for universal entropy, defect RG flows, and operator product expansion structures (Bianchi et al., 2021, Capuozzo et al., 2023).
• Application of algebraic and holographic methods to complex settings, such as surface operators in 6d $\mathcal{N}=(2,0)$ or $(1,0)$ SCFTs (Capuozzo et al., 2023, Apruzzi et al., 25 Jul 2024).

The interplay between local, cohomologically classified (defect) Weyl anomalies and long-range entropy-type observables exemplifies the contemporary synthesis of algebraic, holographic, and physical approaches to conformal field theory and its defects.