Comments on Holographic Complexity

TL;DR

The paper analyzes UV divergences in two holographic complexity proposals, CV and CA, showing that divergence coefficients are local boundary geometric invariants and extending the framework to subregions. It derives the leading and subleading divergent structures, including a universal logarithmic term in CA, and discusses normalization ambiguities of null boundaries. It then proposes covariant subregion versions for both CV and CA and examines their divergence patterns, highlighting how entangling surfaces contribute new terms and how certain cancellations or ambiguities arise in simple examples. The results illuminate the geometric nature of holographic complexity and suggest directions for refining covariant definitions and exploring broader bulk geometries. Overall, the work emphasizes the locality of divergences, the role of boundary and entangling-surface data, and the need for a deeper covariant understanding of complexity in holography.

Abstract

We study two recent conjectures for holographic complexity: the complexity=action conjecture and the complexity=volume conjecture. In particular, we examine the structure of the UV divergences appearing in these quantities, and show that the coefficients can be written as local integrals of geometric quantities in the boundary. We also consider extending these conjectures to evaluate the complexity of the mixed state produced by reducing the pure global state to a specific subregion of the boundary time slice. The UV divergences in this subregion complexity have a similar geometric structure, but there are also new divergences associated with the geometry of the surface enclosing the boundary region of interest. We discuss possible implications arising from the geometric nature of these UV divergences.

Figures (7)

• Figure 1: Showing the extremal volume construction for the CV conjecture for $AdS_3$. We regulate the volume by introducing a cutoff surface at $z=\delta$, where $\delta$ is the short-distance cutoff in the boundary theory.
• Figure 2: Wheeler-DeWitt patch with two different regularizations. In both cases, the WDW patch terminates at the regulator surface: (a) The edge of the WDW patch is the time slice on the asymptotic boundary. The action contains a GHY surface term and two joint terms from the new boundary at $z=\delta$. (b) The edge of the WDW patch is the time slice in the regulator surface. The action contains null joint term from the edge at $z=\delta$.
• Figure 3: For a ball-shaped boundary region $B$, the bulk region $\widetilde{\mathcal{W}}$ is the intersection of the entanglement wedge $\mathcal{W}_{\cal E}[B]$ and the WDW patch $\mathcal{W}_\textrm{\tiny WDW}[\Sigma]$. (a) Showing details of the null joints appearing in the boundary of $\widetilde{\mathcal{W}}$. (b) Showing a cross-section of $\widetilde{\mathcal{W}}$ at $r=0$. (See the main text for the notation.)
• Figure 4: We consider the ground state $|\psi_0\rangle$ of the boundary theory but evaluate the complexity on three different time slices, $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$. Comparing the first two time slices, the complexity sees a large reduction of $\Sigma_2$ because the proper volume of this time slice is reduced by $\Delta{\cal V}=(\sqrt{\Delta\ell^2-\Delta t^2}-\Delta\ell)\,{\cal V}_\textrm{\tiny trans}$ where ${\cal V}_\textrm{\tiny trans}$ is the volume in the transverse directions. The third time slice $\Sigma_3$ is composed of null segments and so the leading divergence in the complexity vanishes.
• Figure 5: Various joints or junctions considered by Hayward Hayward:1993my. Class I junctions are shown in (a) and (b) while Class II junctions are given in (c), (d) and (e).