Surface Counterterms and Regularized Holographic Complexity

TL;DR

This paper tackles the UV divergences of holographic complexity by introducing a covariant regularized complexity constructed from codimension-two boundary counterterms, analogous to holographic renormalization. It develops minimal subtraction counterterms for both CA and CV conjectures in bulk dimensions up to five and applies them to BTZ and Schwarzschild-AdS black holes to obtain finite, coordinate-independent results. The work demonstrates that the regularized complexity faithfully encodes bulk dynamics and matter content, enables a well-defined complexity of formation, and remains robust under coordinate changes or choices of null-normal parametrizations. It also provides explicit results and templates for higher dimensions in symmetric cases, offering a practical framework for studying complexity growth and formation across holographic spacetimes.

Abstract

The holographic complexity is UV divergent. As a finite complexity, we propose a "regularized complexity" by employing a similar method to the holographic renormalization. We add codimension-two boundary counterterms which do not contain any boundary stress tensor information. It means that we subtract only non-dynamic background and all the dynamic information of holographic complexity is contained in the regularized part. After showing the general counterterms for both CA and CV conjectures in holographic spacetime dimension 5 and less, we give concrete examples: the BTZ black holes and the four and five dimensional Schwarzschild AdS black holes. We propose how to obtain the counterterms in higher spacetime dimensions and show explicit formulas only for some special cases with enough symmetries. We also compute the complexity of formation by using the regularized complexity.

Figures (6)

• Figure 1: Penrose diagram for Schwarzschild AdS black hole and complexity in two conjectures. At the two boundaries of the black hole, $t_L$ and $t_R$ stand for two states dual to the states in TFD. $r_h$ is the horizon radius. At the left panel, $\mathcal{B}$ is the maximum codimension-one surface connecting $t_L$ and $t_R$. At the right panel, the yellow region with its boundary is the WDW patch, which is the closure (inner region with the boundary) of all space-like codimension-one surfaces connecting $t_L$ and $t_R$.
• Figure 2: Two different approaches in computing the action at the finite cut-off boundaries. Left panel: The null boundaries of the WDW patch are changed into the null sheets coming from the finite cut-off boundary and there is a null-null joint at the cut-off. (here we only show the part near $t_R$. The part near $t_L$ is similar.) Right panel: The boundaries of the WDW patch are the same, but, the original null-null joints at the AdS boundary are sliced out by a time like boundary and two null-timelike joints are added. (here we only show the right-top part of quarter. The other parts are similar.)
• Figure 3: Penrose diagram and the regularized WDW patch for $t_R=t_L=0$ in the BTZ black holes when $J=0$. The case for $M\geq0$ is shown in the left panel, where the null sheets coming from $t_R$ and $t_L$ meet each other at the surface $r=0$. The case for $M=-1$ is shown in the right panel, where the null sheets coming from $t_R$ and $t_L$ will meet each other at $r=0$ and $t=\pm\pi/2$.
• Figure 4: The WDW patch in the Penrose diagram (left panel) and the Kruskal-type coordinate (right panel) for the rotational BTZ black hole. The null sheets coming from $t_R=t_L=0$ meet each other at the surface $r=r_0\in(r_-,r_+)$.
• Figure 5: The penrose diagram and the regularized approach of the WDW patch for $t_R=t_L=0$ in the Schwarzschild AdS black holes. The null boundaries of the WDW patch come from the finite cut-off boundary and there is a null-null joint at the cut-off $r=r_m$. In addition, in order to regularize the singularity, we need to use an additional cut-off at $r=\varepsilon\rightarrow0$, so there are also some new joints and space-like boundaries.