Entanglement C-functions of defects and interfaces in $\mathcal{N}=4$ supersymmetric Yang-Mills theory

TL;DR

This work analyzes planar codimension-one defects and interfaces in $ ext{N}=4$ SYM using the D3/D5 holographic setup in the probe limit. By calculating the holographic entanglement entropy for a ball centered on the defect, the authors define a defect $C$-function $C(a)=(a\partial_a-1)S_{ ext{def}}(a)$ that monotonically decreases along defect RG flows triggered by a mass deformation, and they extend the framework to interfaces with dissolved D3-branes where a four-dimensional flow arises. They show that, while a monotone defect $C$-function persists for the interface, its interpretation as a count of degrees of freedom is subtle, motivating alternative measures such as four-dimensional $A$-functions (e.g., $A_{ ext{CTT}}$, $A_{ ext{LM}}$, and $ ilde A_{ ext{LM}}$). The analysis reveals that some $A$-functions remain finite and capture UV/IR Weyl anomaly data, but their monotonicity can depend on the interface parameters (notably the dissolved D3-brane charge $n_3$). The results highlight the subtle interplay between defect versus ambient degrees of freedom and point toward backreaction-enabled studies and broader entangling-region explorations to fully map RG flow diagnostics in defect CFTs.

Abstract

We consider planar codimension-one defects and interfaces in $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory, realized by the D3/D5-brane intersection. Working in the probe limit, where the number of D5-branes is small compared to the number of D3-branes, we obtain analytic results for the holographic entanglement entropy of a ball-shaped region centered on the defect. A defect renormalization group flow is triggered by giving the defect hypermultiplets a mass, which corresponds to separating the D3- and D5-branes. Along this flow the entanglement C-function decreases monotonically. We also allow the D5-branes to carry worldvolume flux corresponding to dissolved D3-branes, in which case the setup describes an interface between two copies of $\mathcal{N}=4$ SYM theory with different gauge groups, where an RG flow is triggered by a mass term for vector multiplets. Here we again find monotonic behavior of the entanglement C-function, although its interpretation as a measure of effective degrees of freedom is problematic. We investigate possible alternative measures of degrees of freedom.

Figures (12)

• Figure 1: (a): A $d$-dimensional Euclidean CFT on a $d$-sphere $\mathrm^{d} S$^d$$, with a $p$-dimensional conformal defect wrapping a maximal $\mathrm^{p} S$^p$$. We denote by $F_\mathrm{def}$ the defect's contribution to the free energy in this configuration. (b): The CFT in Lorentzian signature, in $d$-dimensional Minkowski space $M_d$. The thick, vertical line represents a planar $p$-dimensional defect, i.e. a defect with worldvolume $M_p$. The relative entropy of a ball centered on the defect, represented by the gray disk in the figure, may be used to define a $C$-function Casini:2023kyj.
• Figure 2: The D3/D5 intersection giving rise to the codimension-one defect or interface in $\mathcal{N}=4$ SYM theory that we consider. The horizontal and vertical lines represent stacks of coincident D3 and D5-branes, respectively. Some number $n_3$ of the D3-branes may end on the D5-branes, in which case the defect forms an interface between two copies of $\mathcal{N}=4$ SYM theory with different rank gauge groups. An RG flow may be triggered by separating the D5-branes and any D3-branes that end on them from the rest of the D3-branes.
• Figure 3: Cartoon of the holographic description of the D3/probe D5 system. The box at the top represents the conformal boundary of AdS_5 AdS$_{5}$, with the bulk of AdS_5 AdS$_{5}$ extending below. The blue parallelogram represents the stack of coincident probe D5-branes, which intersect the boundary along a flat hypersurface, corresponding to the location of the defect in the dual QFT. If the D5-branes carry dissolved D3-brane charge, then the rank of the gauge group either side of the interface is different. The gray disk represents the ball-shaped region, centered on the defect, for which we will compute the entanglement entropy. Holographically, the entanglement entropy is proportional to the area of the RT surface.
• Figure 4: The shaded region shows the typical form of the integration region for $S_\mathrm{brane}$ in the $(\tau,\zeta,s)$ coordinates used in equation \ref{['eq:Sbrane_integral']}. This plot shows the exact integration region for $q=a=1$, but the shape of the integration region is qualitatively similar for other non-zero values of $q$ and $a$.
• Figure 5: Cross-sections of AdS_5 AdS$_{5}$ for D3/probe D5 system, with RT surfaces. The top panels show the projection onto the $(x,z)$ plane, where $x$ is the boundary direction orthogonal to the defect and $z$ is the bulk coordinate. The D5-branes correspond to the shaded region. The bottom panels show the projection onto the $(r,z)$ plane, where $r$ is a radial polar coordinate on the plane of the defect. The D5-branes are represented by the vertical, purple lines. (Left) When $q=0$, the D5-branes extend to $z=1/\mu$ straight into the bulk, orthogonal to the boundary in the $r$ direction and the RT-surfaces. The maximal value $\sigma$ of $\mu z$ on the intersection of the D5-branes with the RT surface is $\sigma = 1$ for all $a \geq 1$. (Right) For $q > 0$ the D5-branes bend and extend to infinity in a direction parallel to the boundary. As a result, $\sigma$ grows smoothly to its maximal value of $\sigma=1$ as $a \to \infty$.