Holographic Defect CFTs Overview

• Holographic Defect CFTs are field theories characterized by spatially localized deformations and explicit symmetry breaking, modeled via AdS gravitational duals.
• They utilize constructions like Janus slicing and probe brane methods to compute observables such as entanglement entropy, defect spectra, and interface entropy.
• Recent developments merge analytic, numerical, and tensor network techniques to uncover universal properties, defect C-theorems, and RG flow dynamics in inhomogeneous settings.

Holographic Defect Conformal Field Theories (CFTs) are a class of field theories in which defects, interfaces, or boundaries are treated via their AdS gravitational duals. These systems provide a window into strongly coupled physics with intrinsic spatial inhomogeneities, and enable the computation of observables—as well as the exploration of universal structures—in settings with reduced symmetry. Such holographic constructions are realized through fully backreacted or probe brane solutions with explicitly broken translational invariance, and are central to the paper of renormalization group (RG) flows, anomaly inflow, boundary and interface entropy, defect operator spectra, and entanglement entropy in the presence of spatially localized deformations.

1. Geometric Realization: Janus Slicings and Spatially Dependent Couplings

Holographic duals of defect CFTs are engineered by deforming AdS₍d₊₁₎ geometries to accommodate relevant operator insertions whose sources are position-dependent transverse to the defect. A canonical ansatz is the Janus-like AdS_d slicing (1207.7325): $$ds^2 = f(\mu)\left[d\mu^2 + \frac{dy^2 - dt^2 + \sum_{i=2}^{d-1} dx_i^2}{y^2}\right]$$ where the scalar field depends only on the slicing coordinate, $\varphi = \varphi(\mu)$, and the transverse coordinate $x_\perp$ is mapped via $x_\perp = y \cos\mu$, $z = y \sin\mu$. The AdS boundary lies at $\mu\to0$, and for pure AdS, $f(\mu)=1/\sin^2\mu$.

The holographic dictionary yields for the scalar near the boundary: $$\varphi(z, x_\perp) \sim \left(\frac{\alpha}{x_\perp^\Delta}\right) z^\Delta + \left(\frac{\beta}{x_\perp^{d-\Delta}}\right) z^{d-\Delta}$$ and thus the position-dependent source for the dual operator is $\beta(x_\perp) \sim \beta/x_\perp^{d-\Delta}$. This realizes CFT deformations by relevant operators with explicitly spatially varying couplings: $$\delta S_{\mathrm{CFT}} = \int d^dx \, \frac{\beta}{x_\perp^{d-\Delta}} \mathcal{O}_\Delta(x)$$ which is nontrivial compared to translationally invariant flows, leading to physical interfaces (ICFTs) or, via singular limits, boundaries (BCFTs).

2. Holographic RG Flows, Bulk Singularities, and Defect/Boundary Interpolations

Solving the full backreacted scalar-gravity system with the above ansatz numerically reveals two distinct types of solutions (1207.7325):

• ICFT regime: For small source $\beta$, the geometry features two asymptotic AdS regions (at $\mu = 0$ and $\mu = \pi$), corresponding to two half-spaces of the CFT joined at the interface ($x_\perp=0$). Couplings $\beta_{\pm}$ can differ on each side, and the solution is genuinely an interface CFT.
• BCFT regime: For larger $\beta$, a bulk singularity develops at an interior $\mu = \mu_*$, eliminating one AdS boundary. The surviving boundary corresponds to a BCFT, with the singularity interpreted as flow to a massive phase on one side. The singular "end-of-the-world brane" plays an analogous role to the cut-off branes in Takayanagi or Fujita's models.

This interpolation offers a clear holographic signal for when interface deformations gap out a region, and enables the computation of observables in both ICFT and BCFT regimes, including nontrivial transitions between them.

3. Entanglement Entropy, Fefferman–Graham Coordinates, and Universal Quantities

Universal observables such as entanglement entropy are accessible, even in bulks with singularities. The Ryu–Takayanagi prescription computes the entanglement entropy of a region $A$ by the area of the bulk minimal surface $\gamma_A$ (1207.7325, Jensen et al., 2013). For strips or half-spaces in $d=2$, the time-slice metric is

$ds^2 = f(\mu)[d\mu^2 + (dy^2)/y^2]$

and the entanglement entropy is

$$S = \frac{1}{4G_N^{(3)}} \int_{\mu_\epsilon}^{\mu_*} d\mu \sqrt{f(\mu)}$$

For small $\mu$,

$f(\mu) \sim \frac{1}{\mu^2}$

and the leading divergence reproduces the expected CFT result: $S = \frac{c}{6} \ln\frac{\ell}{\epsilon} + \ln g$ where $\ln g$ is the boundary entropy (the $g$-factor). The universal part is extracted as

$$\ln g = \lim_{\epsilon\to0}\left\{\frac{1}{4G_N^{(3)}}\left(\int_{\mu_\epsilon}^{\mu_*}d\mu\sqrt{f(\mu)} - \ln\frac{\ell}{\epsilon}\right)\right\}$$

In higher $d$, the area diverges with powers and logarithms, but similar subtraction schemes yield finite, universal boundary/defect entropies.

The Fefferman–Graham (FG) expansion offers a systematic, diffeomorphism-invariant method for introducing a UV cutoff in these computations, even for complicated defect geometries. The cutoff $z=\epsilon$ in FG coordinates regulates the minimal area integral, and the required coordinate transformations tie the transverse and defect directions (Jensen et al., 2013).

4. Analytic and First-Order Formalisms for Defect Solutions

The construction of defect and interface solutions can be systematized using a first-order formalism (Korovin, 2013), where the AdS$_d$-sliced domain wall metric and scalar potential are encoded entirely in a generating function $s(\varphi_0)$, with all bulk equations collapsing to linear ODEs. This machinery yields analytic solutions for large classes of ICFTs/BCFTs.

Explicitly, the defect metric can be written as

$ds_{d+1}^2 = \eta dz^2 + e^{2\beta\varphi} ds_d^2$

with the warp factor and potential functions determined by

$$(1/4W)q'(\varphi)s'(\varphi) + q(\varphi)s(\varphi) = 1$$

Exploiting this, one can engineer families of analytic solutions corresponding to interfaces ($q(\varphi) = a/\sin^n(b\varphi)$) or boundaries ($q(\varphi) = a/\sinh^n(b\varphi)$), and precisely track the BOPE data and entanglement structure.

Additionally, ambient operators admit a boundary operator expansion (BOPE) near the defect,

$$\mathcal{O}_d(y, r_\perp) = \sum_n \frac{B^{\mathcal{O}_d}_{\mathcal{O}^n}}{r_\perp^{\Delta_d-\Delta_n}}\mathcal{O}^n_{d-1}(y)$$

and the holographic analysis connects defect spectra to probe field fluctuation masses.

5. Tensor Networks, Causal Cones, and Entanglement Structure

Tensor network constructions such as minimal-update MERA provide a powerful language for realizing the entanglement structure of defect or interface CFTs (Czech et al., 2016). In the minimally updated MERA, only the tensors in the causal cone of the defect are modified, corresponding holographically to the non-normalizable modes sourced in the bulk. Away from the defect, the usual OPE structure of the parent CFT persists.

For generalized deformations, "rayed MERA" networks are introduced, in which tensors along rays emanating from the defect center can vary independently (tied to the SO(2,1) invariance). This network structure reflects the bulk's spatially modulated fields and captures corrections to the dual geometry's kinematic space as in Janus solutions. The interplay between tensor network updates and bulk geometry offers a constructive mapping between quantum entanglement and gravitational data.

6. Universal Properties, Defect C-Theorems, and RG Flows

Recent advances have extended universal CFT results, such as the negativity of vacuum energy and bounds on operator spectra, to defect settings, both perturbatively and nonperturbatively (Fischetti et al., 2017). The universal structure of two- and three-point functions, inherited from conformal invariance, dictates the response of defect observables (such as the defect free energy and entanglement entropy) to geometric deformations.

A robust $C$-theorem for defect CFTs has been proposed and tested: the universal contribution that a defect makes to the sphere free energy (the "defect free energy") decreases monotonically along defect-localized RG flows (Kobayashi et al., 2018). In holographic settings, this is realized via the monotonic decrease in the on-shell brane action under flows between fixed points, and for codimension-one defects, this matches the defect entropy extracted from entanglement computations. Field theory and holographic tests—with both supersymmetric and non-supersymmetric defects—confirm that the defect free energy serves as the correct measure of degrees of freedom, even if the defect entropy need not decrease monotonically.

7. Impact, Methodological Innovations, and Future Directions

The holographic framework for defect CFTs provides a rich structure for investigating strongly coupled systems with explicit spatial inhomogeneity and boundaries. The interplay between fully backreacted (non-probe) solutions and interface/boundary limits allows one to paper nontrivial RG flows, the emergence and fusion of defects, singularity resolution, and the computation of universal quantities in field theories that model real material interfaces and impurities.

Methodologically, the merging of analytic, numerical, and tensor network techniques has enabled significant progress in both computing observables and elucidating structural features. Open problems include constructing more general analytic solutions with multiple and intersecting defects, understanding the role of non-normalizable vs normalizable modes in information-theoretic quantities, and extending defect $C$-theorem proofs to higher codimensions and strongly coupled flows.

Holographic defect CFTs thus remain a core laboratory for the paper of localization, RG flows, symmetry breaking, and entanglement, with applications to condensed matter, integrable systems, boundaries in quantum field theory, and quantum information.