Bending branes for DCFT in two dimensions

TL;DR

We develop a holographic DCFT framework in which a 2D brane with dynamical matter backreacts on a 3D AdS/BTZ bulk via Israel junction conditions, linking the brane stress $S_{ij}$ to the exterior geometry. By decomposing the junction conditions into scalar components on the 1+1 brane and applying the null, weak, and strong energy conditions, we derive qualitative constraints on brane embeddings and relate them to the Engelhardt–Wall barrier theorem for extremal surfaces and holographic entanglement entropy. We obtain exact analytical solutions for a perfect-fluid brane (including the massless-scalar case) and analyze how constant brane tension and Kondo-like matter content modify the embedding and entanglement properties, including numerical results showing SEC violations driving horizon penetration. The results illuminate how backreacted DCFTs modify bulk geometry and entanglement structure, with implications for holographic Kondo models and potential extensions to higher dimensions and Chern–Simons sectors.

Abstract

We consider a holographic dual model for defect conformal field theories (DCFT) in which we include the backreaction of the defect on the dual geometry. In particular, we consider a dual gravity system in which a two-dimensional hypersurface with matter fields, the brane, is embedded into a three-dimensional asymptotically Anti-de Sitter spacetime. Motivated by recent proposals for holographic duals of boundary conformal field theories (BCFT), we assume the geometry of the brane to be determined by Israel junction conditions. We show that these conditions are intimately related to the energy conditions for the brane matter fields, and explain how these energy conditions constrain the possible geometries. This has implications for the holographic entanglement entropy in particular. Moreover, we give exact analytical solutions for the case where the matter content of the brane is a perfect fluid, which in a particular case corresponds to a free massless scalar field. Finally, we describe how our results may be particularly useful for extending a recent proposal for a holographic Kondo model.

Figures (12)

• Figure 1: Setup for the holographic description of BCFT: Asymptotically AdS bulk spacetime $N$ with conformal boundary $M$ and additional boundary $Q$. $P$ is the intersection of $M$ and $Q$. On the field theory side, we refer to $P$ as the defect and to $M$ as the ambient space.
• Figure 2: Geometry of the setup: The manifold $N$ is split into two submanifolds $N_+$ and $N_-$. The white region is excised from the manifold. For each submanifold, the position of the brane is given by $x=x_{\pm}(t,z)$ and corresponding points of the two embeddings $x_+$ and $x_-$ are identified, as indicated by double arrows. The normal vectors to $Q$ on both sides are named $n^{\pm}$, and point from $N_-$ to $N_+$. In most of the paper, we will assume $N_{\pm}$ to be BTZ metrics \ref{['BTZmetric']} of equal temperature and the embedding to be symmetric, i.e. $x_+=-x_-$.
• Figure 3: Left: The case where the brane $Q_1$ bends back to the boundary. $\varUpsilon_0$, $\varUpsilon_1$, $\varUpsilon_2$ are $Q$-deformable spacelike extremal curves which are half-circles in a Poincaré background. The brane is crossed by $\varUpsilon_3$, but as $Q_1$ is anchored at the boundary twice, $M^+$ is a finite interval, and as the point where $\varUpsilon_3$ returns to the boundary is not inside of $M^+$, it does not violate the barrier theorem. The matter fields sustaining $Q_1$ hence may satisfy WEC and SEC everywhere, and $Q_1$ is an extremal surface barrier. Right: Possible cases where branes $Q_i$ are anchored once to the boundary $M^+$, which hence extends infinitely in one direction. Several $Q$-deformable extremal spacelike curves $\varUpsilon_a$ are depicted. $Q_3$ is a trivially embedded brane with $K^+_{ij}=0$ (i.e. sustained by $S_{ij}=0$) and is an extremal surface barrier in the sense of the theorem presented above. $Q_2$ violates SEC as it has $x'_+(z)<0$ (see table \ref{['tab::table1']}), but as it falls behind $Q_3$ everywhere it is also an extremal surface barrier. $Q_4$ is crossed by $\varUpsilon_3$, and must according to the barrier theorem hence violate the WEC somewhere.
• Figure 4: Embedding of the branes given by $y=-y_*$ in AdS space with coordinates $t,\rho,\phi$. The spacetime to the right of each of these curves corresponds to the region $N_+$ in figure \ref{['fig::geometry']}, while the part of the spacetime to their left is excised. We set $t=const.$ to consider a spacelike slice with coordinates $\rho,\phi$, and compactify by plotting $\arctan(\rho)$ as radial coordinate, such that the thick black circle represents the AdS boundary. $\phi$ is the angular coordinate. The values of $\lambda$ for the branes shown are specified by $y_*=\text{arctanh}(\frac{\kappa\lambda}{2})=\{-3,-2.75,...,3\}$. Lines involving red colour (bending to the left) stand for branes with $\lambda>0$, while lines involving blue colour (bending to the right) stand for branes with $\lambda<0$. The straight vertical line is the brane with zero tension, $\lambda=0$. The black dashed lines are geodesics perpendicular to the branes, see section \ref{['sec::normalgeodesics']}.
• Figure 5: Nontrivial matching of two AdS spacetimes along a constant tension brane for the cases $\lambda>0$ and $\lambda<0$. The two figures to the left of the equality depict the embedding of the brane with respect to $N_-$ and $N_+$, respectively. The grey shaded area is then excised, and the two resulting spacetimes are glued together along the brane, as shown to the right of the equality. By assumption of symmetry, $N_-$ is always the mirror image of $N_+$. As described in the text, the resulting spacetime will have increased volume for $\lambda>0$ and smaller volume than $\text{AdS}_{3}$ for $\lambda<0$.