On Holographic Defect Entropy

TL;DR

The paper develops a holographic framework to study entanglement entropy in higher-dimensional defect and boundary conformal field theories, defining defect and boundary entropies via background subtraction and extracting universal defect/boundary contributions. It derives general holographic expressions for S_defect and S_boundary, isolating universal coefficients tied to Weyl anomalies and sphere free energies, and provides evidence that these defect/boundary entropies decrease along defect/boundary RG flows, akin to a higher-dimensional g-theorem. Through explicit analyses of D3/D5 DCFTs/BCFTs, T[SU(N)] defects, SUSY and non-SUSY Janus, and M-theory Janus, the work demonstrates monotonic behavior of the defect/boundary universals (D_0 and B_0) in several cases and discusses how bulk flows can produce either increase or decrease, highlighting the nuanced connection to the F-theorem in 3d and potential higher-dimensional g-functions. The findings provide concrete, gravity-side evidence for a higher-dimensional generalization of defect entropy monotonicity and establish a set of exact holographic results linking EE to defect/boundary data across multiple well-controlled theories with known field theory duals.

Abstract

We study a number of (3+1)- and (2+1)-dimensional defect and boundary conformal field theories holographically dual to supergravity theories. In all cases the defects or boundaries are planar, and the defects are codimension-one. Using holography, we compute the entanglement entropy of a (hemi-)spherical region centered on the defect (boundary). We define defect and boundary entropies from the entanglement entropy by an appropriate background subtraction. For some (3+1)-dimensional theories we find evidence that the defect/boundary entropy changes monotonically under certain renormalization group flows triggered by operators localized at the defect or boundary. This provides evidence that the g-theorem of (1+1)-dimensional field theories generalizes to higher dimensions.

Figures (6)

• Figure 1: A cartoon of a $(2+1)$-dimensional DCFT and its holographic dual. The DCFT "lives" at the boundary of the holographic dual, depicted as the shaded plane. The two horizontal directions $x^1$ and $x^2$ are the DCFT's spatial directions, while $u$ is the holographic direction. The planar defect is extended along $x^1$, as depicted by the solid red line. We compute the EE of a spherical region centered on the defect. The entangling surface ${\mathcal{M}} = \mathbb{S}^1$ is depicted as the solid black circle. The blue hemisphere is the minimal-area surface in the bulk which ends on $\mathcal{M}$, and whose area determines the EE via eq. \ref{['rt']}.
• Figure 2: A schematic depiction of the holographic duals of the $(d+1)$-dimensional DCFTs that we study, which have metrics of the form in eq. \ref{['E:generalMetric']}. We depict the space spanned by the coordinates $u$ and $x$. Fefferman-Graham (FG) patches only exist for some range of $x$ near the asymptotically $AdS_{d+1}$ regions $x \to\pm \infty$. The rest of the geometry is a "middle region" between these FG patches.
• Figure 3: A schematic depiction of the cutoff surface that we use to regulate the divergences in the spherical EE, eq. \ref{['E:defectSEE']}. In each FG patch, our cutoff surface coincides with the FG cutoff surface $z=\varepsilon$ that we used to regulate the spherical EE when the defect is absent (i.e. in pure $AdS_{d+1}$). These cutoffs give rise to cutoffs $\chi_{\pm}$ in the $x$ integration in eq. \ref{['E:defectSEE']}. In the middle region between the FG patches, our cutoff surface, parameterized by $u^c(y^a)$, continuously connects the two $z = \varepsilon$ cutoffs, but is otherwise arbitrary. In the appendix we show that our results for the universal terms in the defect or boundary entropy are insensitive to the choice of cutoff surface in the middle region.
• Figure 4: A $T[SU(N)]$ CFT arises as the low-energy limit of the $(2+1)$-dimensional ${\cal N}=4$ SYM theory with the quiver above. The $i$-th node represents an ${\cal N}=4$ SYM theory with gauge group $SU(N_i)$. The line connecting the $i$-th node to the $i+1$-th node represents an ${\cal N}=4$ hypermultiplet in the bi-fundamental representation of $SU(N_i)\times SU(N_{i+1})$. The box represents a collection of $N$ hypermultiplets in the fundamental representation of $SU(N-1)$.
• Figure 5: The $(2+1)$-dimensional $T[SU(N)]$ CFT arises in type IIB string theory as the low-energy theory of the above D3/D5/NS5-brane intersection. The solid blue vertical lines represent NS5-branes, the solid black horizontal lines represent D3-branes, and the dashed red slanted lines represent D5-branes.