Divergences in the rate of complexification

TL;DR

This paper investigates the growth of complexity in holographic states under perturbation, revealing divergences in complexity growth rates that challenge existing conjectures relating complexity and energy bounds.

Contribution

It demonstrates that both the complexity equals action and complexity equals volume conjectures exhibit UV divergences in complexity growth rates under time-dependent perturbations.

Findings

Complexity growth rates have UV divergences.

Instantaneous energy remains UV finite.

Current conjectures conflict with the proposed energy bound.

Abstract

It is conjectured that the average energy provides an upper bound on the rate at which the complexity of a holographic boundary state grows. In this paper, we perturb a holographic CFT by a relevant operator with a time-dependent coupling, and study the complexity of the time-dependent state using the \textit{complexity equals action} and the \textit{complexity equals volume} conjectures. We find that the rate of complexification according to both of these conjectures has UV divergences, whereas the instantaneous energy is UV finite. This implies that neither the \textit{complexity equals action} nor \textit{complexity equals volume} conjecture is consistent with the conjectured bound on the rate of complexification.