Holographic Model of Defect CFTs

• Holographic Defect CFTs are gravitational dual frameworks modeling conformal theories with localized defects using explicit Type IIB supergravity solutions.
• The models employ warped product geometries and harmonic functions to encode five-brane singularities, linking D3-brane charges with detailed quiver data.
• Quantitative tests via free energy matching and RG flow analysis validate the holographic dictionary, confirming N² log N scaling and consistency with the F-theorem.

The holographic model of defect conformal field theories (DCFTs) refers to a class of gravitational duals that realize conformal field theories in the presence of localized defects or interfaces, often preserving a nontrivial subgroup of the conformal symmetry. This framework generalizes the AdS/CFT correspondence by encoding defect data in explicit supergravity backgrounds—typically involving domain walls, interface branes, and associated singularities—that precisely match the operator content, renormalization group flows, and free energies of the defect CFTs. These constructions are central in establishing quantitative dualities between 3d $\mathcal{N}=4$ linear and circular quivers and Type IIB string backgrounds, as well as in defining domain wall solutions dual to 4d $\mathcal{N}=4$ SYM coupled to lower-dimensional defects. The holographic dictionary maps physical quantities such as linking numbers, brane charges, and partition functions in the gravitational description to field theory data, allowing for stringent tests of RG monotonicity (F-theorem), computable free energies, and explicit realization of mirror symmetry.

1. Geometric Structure of the Holographic Dual

The gravitational duals to 3d $\mathcal{N}=4$ DCFTs are constructed as explicit Type IIB supergravity solutions with warped product spaces of the form

$$ds^2 = f_4^2\,ds^2_{AdS_4} + f_1^2\,ds^2_{S^2_1} + f_2^2\,ds^2_{S^2_2} + 4\rho^2\,dz\,d\bar{z}$$

where the geometry is a warped product of $AdS_4$ with two 2-spheres, $S^2_1$ and $S^2_2$, fibered over a two-dimensional Riemann surface $\Sigma$. The warp factors $f_4, f_1, f_2, \rho$ are determined via two real harmonic functions $h_1(z, \bar{z})$ and $h_2(z, \bar{z})$ on $\Sigma$. The regularity conditions imposed on $\partial \Sigma$ dictate which $S^2$ shrinks along each boundary segment, producing localized $S^3$ topologies at the endpoints and allowing for the interpretation of logarithmic singularities in $h_1$ or $h_2$ as D5- or NS5-brane sources.

Explicitly, for given harmonic functions, the warp factors are

$$\begin{aligned} f_4^8 &= 16\,\frac{N_1 N_2}{W^2} \ f_1^8 &= 16\,\frac{h_1^8 N_2 W^2}{N_1^3} \ f_2^8 &= 16\,\frac{h_2^8 N_1 W^2}{N_2^3} \ \rho^8 &= \frac{N_1 N_2 W^2}{h_1^4 h_2^4} \end{aligned}$$

where $W$ is an auxiliary function of $h_1, h_2$. The harmonic functions themselves encode the positions and charges of five-brane singularities: $$\begin{aligned} h_1 &= -i \sum_{a=1}^p \gamma_a\,\ln\tanh\left[\frac{i\pi}{4} - \frac{z-\delta_a}{2}\right] + c.c. \ h_2 &= -\sum_{b=1}^{\hat{p}} \hat{\gamma}_b\,\ln\tanh\left[\frac{z-\hat{\delta}_b}{2}\right] + c.c. \end{aligned}$$ with parameters $\gamma_a, \hat{\gamma}_b$ and locations $\delta_a, \hat{\delta}_b$ marking the positions of D5- and NS5-brane sources.

2. Holographic Dictionary and Quiver/Brane Data

The explicit holographic dictionary is established via flux quantization near each five-brane singularity, mapping to ordered partitions $(\rho, \hat{\rho})$ encoding the numbers of D3-branes ending on corresponding five-brane stacks. For D5-brane singularities, the D3-brane charge and linking number are

$l^{(a)} = \frac{N_3^{(a)}}{N_5^{(a)}}$

Analogously for NS5-branes,

$$\hat{l}^{(b)} = -\frac{\hat{N}_3^{(b)}}{\hat{N}_5^{(b)}}$$

These partitions fully specify the IR fixed points of 3d quiver gauge theories—denoted $T^{\rho}_{\hat{\rho}}(SU(N))$—and their parameters. For circular quivers, an additional integer $L$ (the D3-brane winding number around the periodic direction on $\Sigma$) appears, leading to fixed point labels $C^\rho_{\hat{\rho}}(SU(N), L)$.

In the domain wall (defect) setup, the harmonic functions are chosen such that the asymptotic $AdS_5 \times S^5$ regions are uncapped and remain at the “ends” of the strip. The corresponding solution describes a 4d $\mathcal{N}=4$ SYM on a half-space (or with an interface), coupled to a 1/2 BPS 3d defect theory. The bulk moduli—such as asymptotic dilaton values and radii $L_+, L_-$—map to gauge couplings and gauge group ranks on the left/right of the defect.

3. Five-Brane Singularities, Consistency, and Mirror Symmetry

The five-brane singularities (logarithmic branch points in $h_1$ or $h_2$) directly reflect the brane construction of the 3d quivers. The residues at these points match the five-brane charges, and the linking numbers (the partitions $\rho$, $\hat{\rho}$) extracted from asymptotic flux quantization automatically obey the S-rule constraints necessary to avoid decoupled free sectors in the IR: the inequalities match field theory rules for consistent quiver data.

Mirror symmetry, exchanging Coulomb and Higgs branches of the 3d $\mathcal{N}=4$ quiver theory, corresponds in the gravitational solution to exchanging the roles of $h_1$ and $h_2$—or, equivalently, D5 and NS5 singularities.

4. Free Energy Computations and the F-Theorem

The duality is quantitatively checked by computing the three-sphere free energy, both from the field theory (via supersymmetric localization and matrix model reduction) and from the gravity side (as the regularized on-shell type IIB supergravity action). In the large $N$ limit for theories such as $T[SU(N)]$, the free energy exhibits the characteristic scaling: $$F_{CFT} = -\log|Z| \sim \frac{1}{2} N^2 \log N + O(N^2)$$ On the gravity side, the effective action reduces to four-dimensional Einstein gravity with internal “volume” $$\operatorname{vol}_6 = 32 (4\pi)^2 \int_\Sigma d^2x\,(-W) h_1 h_2$$, leading to the same leading $N^2 \log N$ scaling.

This matching serves as a precise check of the GKPW prescription and the correctness of the holographic dual. Furthermore, for RG flows among different $T^\rho_{\hat{\rho}}(SU(N))$ fixed points—the flows induced by subdividing partitions (i.e., “splitting” the quiver)—the free energy decreases monotonically, in agreement with the three-dimensional F-theorem.

Observable	Gravity Dual (Supergravity Side)	Field Theory (Quiver SCFT Side)
Linking numbers	D3 charge per five-brane (from quantized flux)	Node-by-node D3 brane partition in the quiver
Free energy $F$	Regularized on-shell Type IIB action	$S^3$ partition function via matrix model
Mirror symmetry	Exchange $h_1 \leftrightarrow h_2$	Exchange Coulomb/Higgs branches

5. Domain Wall Solutions and Defect Systems

Defect SCFTs are engineered by domain wall solutions in which the Riemann surface $\Sigma$ is non-compact and connects two asymptotic $AdS_5 \times S^5$ regions via an $AdS_4$ “throat.” These solutions correspond to 4d $\mathcal{N}=4$ SYM theories on the left and right, possibly with different gauge couplings and ranks, coupled to a 3d defect at the interface. The bulk data (asymptotic moduli, radii, and dilaton) are directly mapped to gauge and flavor data of the 4d/3d defect coupled system.

In this context, the holographic dictionary is refined to account for the D3-brane charges in both asymptotic regions, the precise structure of the domain wall, and the quiver data inheriting from the five-brane singularity structure.

6. Quantitative Tests and Implications

• Consistency Checks: Linking numbers and five-brane data automatically satisfy all gauge theory inequalities for IR consistency, providing direct evidence that the gravitational construction captures the field theory fixed point.
• Free Energy Matching: On both sides, the universal $N^2\log N$ scaling (for maximally symmetric quivers and their splitting patterns) demonstrates precise quantitative agreement and allows for field-theoretic tests of the F-theorem holographically in three dimensions.
• Explicit Formulas: The holographic data (warp factors, harmonic functions, partition integer labels) realize a direct, invertible map to the quiver description, proving the completeness of the gravitational encoding.

The overall construction not only expands the catalog of explicit IIB supergravity backgrounds dual to 3d $\mathcal{N}=4$ SCFTs, but also provides a robust platform for testing field-theoretic conjectures (such as the F-theorem) and for analyzing the dynamics of defect conformal field theories, including cases with domain walls, junctions, and generalizations to circular quivers and more complex interface systems (Assel, 2013).