# Testing Holographic Conjectures of Complexity with Born-Infeld Black Holes

## TL;DR

This paper tests holographic complexity conjectures using Born-Infeld black holes, examining the validity of the Lloyd bound near extremality and in regular spacetimes, revealing differences between the CA and CV dualities.

## Contribution

It provides a comparative analysis of the CA and CV holographic complexity conjectures using Born-Infeld black holes, highlighting conditions under which the Lloyd bound is violated or satisfied.

## Key findings

- Lloyd bound is violated near extremality in both dualities.
- In regular spacetimes, the Lloyd bound is satisfied in CV but violated in CA.
- Moving away from ground states, CA approaches saturation, while CV exceeds the Lloyd bound by a factor of π/2.

## Abstract

In this paper, we use Born-Infeld black holes to test two recent holographic conjectures of complexity, the "Complexity = Action" (CA) duality and "Complexity = Volume 2.0" (CV) duality. The complexity of a boundary state is identified with the action of the Wheeler-deWitt patch in CA duality, while this complexity is identified with the spacetime volume of the WdW patch in CV duality. In particular, we check whether the Born-Infeld black holes violate the Lloyd bound: $\mathcal{\dot{C}\leq}\frac{2}{\pi\hbar}\left[ \left( M-Q\Phi\right) -\left( M-Q\Phi\right) _{\text{gs}}\right] $, where gs stands for the ground state for a given electrostatic potential. We find that the ground states are either some extremal black hole or regular spacetime with nonvanishing charges. Near extremality, the Lloyd bound is violated in both dualities. Near the charged regular spacetime, this bound is satisfied in CV duality but violated in CA duality. When moving away from the ground state on a constant potential curve, the Lloyd bound tend to be saturated from below in CA duality, while $\mathcal{\dot{C}}$ is $\pi/2$ times as large as the Lloyd bound in CV duality.