Weyl Anomaly Coefficients of Holographic Defect CFTs at Weak and Strong Coupling

Abstract: We determine the type-A Weyl anomaly coefficient $b$, associated with the intrinsic scalar curvature of the defect, for the class of holographically realised co-dimension two defect CFTs (dCFTs) introduced in arXiv: 2506.14505 and arXiv: 2512.14853. At strong coupling, we employ the dual D5-brane solutions in Euclidean signature, where the defect is supported on an $S^2$ submanifold of the Euclidean $AdS_3\times S^1$ boundary. At weak coupling, we use the classical solutions of the ${\cal N}=4$ SYM equations of motion, previously conjectured to describe the defects dual to the D5-brane configurations. Notably, the coefficient $b$ is found to be negative in a finite region of parameter space. To our knowledge, this constitutes the first explicit example of an {\it interacting} unitary dCFT with $b<0$. We also compute the type-B Weyl anomaly coefficients associated with the extrinsic curvature of the defects, first at strong coupling and subsequently at weak coupling. In a certain limit, we find agreement between the weak- and strong-coupling results for both the type-A and type-B anomaly coefficients.

• The paper computes both type-A (b) and type-B (d1) anomaly coefficients for holographic defect CFTs, revealing negative b in specific parameter regimes.
• It employs holographic renormalization of probe-brane setups in type IIB theory and gauge theory analyses to match weak and strong coupling results.
• The study demonstrates anomaly non-renormalization in a BMN-like double-scaling limit with implications for entanglement entropy, displacement operators, and RG flows.

Weyl Anomaly Coefficients for Holographic Defect CFTs at Weak and Strong Coupling

Introduction

This work presents a comprehensive analysis of Weyl anomaly coefficients for two-dimensional (codimension-2) defect CFTs (dCFTs) realized holographically via probe branes in type IIB string theory. The specific classes of non-supersymmetric dCFTs considered were introduced in "Holography of a novel codimension-2 defect CFT" (Georgiou et al., 17 Jun 2025) and "Holographic interpolations of defect CFTs" (Georgiou et al., 16 Dec 2025). The central focus is the computation of both type-A ($b$) and type-B ($d_1$) defect anomaly coefficients, at weak and strong 't Hooft coupling. These coefficients characterize the response of defect degrees of freedom to background curvature (intrinsic and extrinsic), yielding insight into the structure and allowed space of unitary dCFTs.

General Framework for Defect Weyl Anomalies

The trace anomalies in the presence of a conformal defect admit localized contributions, parameterized by new central charges distinct from standard bulk anomalies. For a two-dimensional defect embedded in $d\geq3$ dimensions, the relevant anomaly structure is

$$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$

where $\mathcal{R}_\text{def}$ is the defect's intrinsic curvature, $Y^\mu_{ab}$ the traceless part of the second fundamental form, and $W_{ab}^{~~ab}$ the pullback of the bulk Weyl tensor. $b$ is a type-A coefficient (Wess-Zumino), while $d_1$ and $d_2$ are type-B (Weyl-invariant). These quantities are extracted from the logarithmic divergences in the regularized defect effective action, either via a holographic renormalization procedure or by direct computation in field theory.

Holographic Realizations and Bulk-Brane Embedding

The brane configurations analyzed employ D5-branes probing $d_1$0, where the D5 worldvolume is $d_1$1, with $d_1$2 wrapped inside $d_1$3. The solutions generically depend on several continuous parameters encoding geometric deformations, orientation, and worldvolume flux. These data map onto dual field theory characteristics describing the defect's embedding, support, and symmetry.

For the family in (Georgiou et al., 17 Jun 2025), the defect sits at an $d_1$4 in the boundary of $d_1$5, preserving no supersymmetry, and the D5 carries quantized worldvolume flux $d_1$6. The more general interpolating system of (Georgiou et al., 16 Dec 2025) includes an extra parameter $d_1$7 controlling the brane's orientation in $d_1$8; at $d_1$9 it reduces to the previous solution, while for $d\geq3$0 the flux vanishes and the brane shrinks to a singular configuration.

Calculation of Type-A Anomaly Coefficient ($d\geq3$1)

Strong Coupling (Holographic)

The anomaly coefficient $d\geq3$2 is extracted from the logarithmically divergent part of the regulated Euclidean D5-brane action evaluated on-shell. The computation involves:

• Placing the defect on a curved two-dimensional submanifold (usually $d\geq3$3), and regulating the $d\geq3$4 volume via a radial cutoff.
• Identifying the coefficient of $d\geq3$5 in the on-shell action, yielding $d\geq3$6 via $d\geq3$7.
• Careful treatment of the background RR 4-form and the worldvolume flux is necessary to match known anomaly structures in protected limits.

For (Georgiou et al., 17 Jun 2025), the result is: $d\geq3$8 where $d\geq3$9 parameterizes the brane embedding and flux. Remarkably, $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$0 can become negative for $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$1, transitions through zero, and becomes positive for larger $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$2. This is the first explicit realization of an interacting unitary dCFT with negative $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$3, contrasting with the free Dirichlet scalar result previously known.

Weak Coupling (Gauge Theory)

The computation utilizes the conjectured classical solution to $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$4 SYM equations with a codimension-2 surface operator. Evaluating the SYM action (including conformal curvature couplings for the relevant background) on this solution, and focusing on the log-divergent term,

$$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$5

This is also negative across parameter space, mirroring the strong coupling analysis.

Interpolating/Generalized System

For the more general system of (Georgiou et al., 16 Dec 2025), both the brane construction and gauge theory calculation involve additional structural complexity, but exhibit the same qualitative features---the $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$6 coefficient interpolates between the value for (Georgiou et al., 17 Jun 2025) (at $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$7) and zero (at $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$8).

Weak/Strong Coupling Agreement

In a refined BMN-like double-scaling limit (large flux $$\langle T^{\mu}_{~\mu} \rangle_\text{def} = \frac{1}{24\pi} \left( b \mathcal{R}_\text{def} + d_1 Y^\mu_{ab}Y_{\mu}^{ab} - d_2 W_{ab}^{~~ab} \right)$$9, small $\mathcal{R}_\text{def}$0 with $\mathcal{R}_\text{def}$1 fixed), the weak and strong coupling results for $\mathcal{R}_\text{def}$2 coincide: $\mathcal{R}_\text{def}$3 This provides significant evidence for the non-renormalization of this type-A defect anomaly, at least within this region of parameter space.

Type-B Anomaly ($\mathcal{R}_\text{def}$4)

For the extrinsic curvature anomaly $\mathcal{R}_\text{def}$5, both regimes are analyzed by considering defects supported on curved submanifolds (e.g., a large-radius cylinder) to isolate $\mathcal{R}_\text{def}$6 contributions.

• Strong coupling: $\mathcal{R}_\text{def}$7, with explicit dependence on model parameters. For vanishing flux, $\mathcal{R}_\text{def}$8 vanishes, as required.
• Weak coupling: Each branch (set of scalar vevs) contributes additively to $\mathcal{R}_\text{def}$9, proportional to group theory data and defect parameters. The computation matches the holographic result at leading order when the same double-scaling limit is imposed.

While the matching between weak and strong coupling is exact only at leading order, subleading terms differ, most likely due to the sensitivity of the type-B anomaly to detailed quantum corrections.

Implications, Theoretical Significance, and Future Directions

Negative $Y^\mu_{ab}$0 in Interacting Unitary dCFT: The existence of a parameter region where $Y^\mu_{ab}$1---not previously seen in interacting CFTs---demonstrates the non-universality of the sign of the type-A defect anomaly. This provides a counterpoint to expectations based on bulk $Y^\mu_{ab}$2-theorems, though the monotonicity theorems constrain only flows, not the sign of $Y^\mu_{ab}$3 itself [Jensen:2015swa].

Holographic Evidence: The robust agreement between the holographic and gauge theory calculations for $Y^\mu_{ab}$4 (and leading $Y^\mu_{ab}$5) solidifies the defect-brane dualities of (Georgiou et al., 17 Jun 2025, Georgiou et al., 16 Dec 2025). The ability of these systems to interpolate between supersymmetric and non-supersymmetric defects further enriches the space of tractable dCFTs and allows for detailed studies of RG interface phenomena and anomaly matching.

Practical Applications: The explicit values of $Y^\mu_{ab}$6 and $Y^\mu_{ab}$7 control universal parts of entanglement entropies for regions intersecting defects, constrain two- and three-point functions of displacement operators, and enter monotonicity constraints for defect RG flows.

Theoretical Outlook: Future work should explore quantum corrections to these results (e.g., $Y^\mu_{ab}$8 or $Y^\mu_{ab}$9 expansions), the behavior of higher type-B anomaly coefficients, and the implications for general defect field theories, including those with less supersymmetry or non-conformal defects. There is also a pressing need to connect these explicit anomaly computations to bulk brane stability, integrability (see (Leeuw et al., 2024)), and the full structure of the moduli space of defect conformal field theories.

Conclusion

This paper establishes a detailed quantitative portrait of Weyl anomaly coefficients for a class of non-supersymmetric holographic dCFTs, for the first time demonstrating negative $W_{ab}^{~~ab}$0 in an interacting unitary theory, and confirming the non-renormalization of $W_{ab}^{~~ab}$1 between the weak and strong coupling regimes in an appropriate limit. These results greatly expand our understanding of anomaly constraints, RG flows, and allowed defect spectra in CFT, providing new benchmarks and methods for future investigations of holography, brane constructions, and defect quantum field theory.