Holographic Complexity for Defects Distinguishes Action from Volume

TL;DR

This paper analyzes how a conformal defect in a 2d DCFT affects holographic complexity by comparing the CV and CA proposals in a defect AdS$_3$ geometry with a thin AdS$_2$ brane. It shows that the CV prescription acquires a defect-induced logarithmic divergence in the full-boundary complexity, linked to the central charge $c_T$ and the Affleck--Ludwig boundary entropy, while the CA prescription remains insensitive to the defect. The work extends the analysis to subregions, finding that CA subregion complexity around a symmetric defect also omits defect contributions, whereas CV subregion complexity carries a defect-dependent log term. Free-field QFT models with defects are discussed to connect holographic results to field theory and highlight the potential importance of zero modes. Overall, the dramatic CV–CA discrepancy in the defect setting provides a sharp testbed for understanding holographic complexity and guides future explorations in higher dimensions and top-down models.

Abstract

We explore the two holographic complexity proposals for the case of a 2d boundary CFT with a conformal defect. We focus on a Randall-Sundrum type model of a thin AdS$_2$ brane embedded in AdS$_3$. We find that, using the "complexity=volume" proposal, the presence of the defect generates a logarithmic divergence in the complexity of the full boundary state with a coefficient which is related to the central charge and to the boundary entropy. For the "complexity=action" proposal we find that the complexity is not influenced by the presence of the defect. This is the first case in which the results of the two holographic proposals differ so dramatically. We consider also the complexity of the reduced density matrix for subregions enclosing the defect. We explore two bosonic field theory models which include two defects on opposite sides of a periodic domain. We point out that for a compact boson, current free field theory definitions of the complexity would have to be generalized to account for the effect of zero-modes.

Figures (13)

• Figure 1: Constant time slices of the two AdS patches on the two sides of the defect, corresponding to the metric \ref{['DCFTsol2']}, glued together at the location of the defect along $y=\pm y^*$ curves. Lines of constant $r$ are indicated in red and lines of constant $y$ are indicated in green.
• Figure 2: Extension of the cutoff surface in the region of the defect following lines of constant $r$. This generates a cutoff surface which is perpendicular to the defect and connects smoothly the two sides. We have indicated in light blue the region inside the cutoff surface.
• Figure 3: Illustration of the (future half of the) WDW patch in defect AdS$_3$. $S_3^L$ and $S_3^R$ are the two half cones, already present for the case of vacuum AdS$_3$. $S_2^L$ and $S_2^R$ are the additional boundaries of the WDW patch in the defect region, fixed by parts of the lightcones generated from the points $\theta=0,~\pm\pi$ on the boundary ($\phi=\pi/2$). Those null surfaces are smoothly connected across the defect and they terminate along a ridge at the top of the WDW patch. The yellow surfaces correspond to the defect brane, where the left and right patches are glued together.
• Figure 4: Cross section of the boundary of the WDW patch for different times $t$ denoted by the green and blue lines inside and outside the defect region respectively. In the defect region the boundary of the WDW patch is fixed by the light cone emanating from the boundary at $\theta=0$, indicated by solid green curves, and it meets the lightcone surface coming from $\theta=\pi$ along a ridge at $\theta=-\pi/2$ (and $t=\pi/2$). The rest of the boundary of the WDW patch is the conical region fixed by straight infalling light rays coming from different boundary points along lines of constant $\theta$ and its cross section for different times $t$ is indicated by the blue circular arcs. The plot corresponds to a defect parameter of $y^*=0.6$, see eq. \ref{['eq:defparam']}, and is presented using the $x$ and $y$ coordinates defined in eq. \ref{['eqapp:xycoords']}.
• Figure 5: Division of the constant time slice inside the cutoff surface to two different portions which we use in evaluating the volume integrals for the CV conjecture. $V_1$ is the volume in the defect region and $V_2$ is the volume outside the defect region.