# Surface Counterterms and Regularized Holographic Complexity

## TL;DR

This paper introduces a method to regularize holographic complexity by adding boundary counterterms, removing divergences, and capturing only the dynamic information, with applications to various black hole solutions.

## Contribution

It proposes a systematic way to define finite holographic complexity through boundary counterterms, extending the approach to higher dimensions and specific symmetric cases.

## Key findings

- Derived explicit counterterms for CA and CV conjectures in dimensions 5 and below.
- Applied the regularized complexity to BTZ and Schwarzschild AdS black holes.
- Calculated the complexity of formation using the regularized approach.

## 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.