Holographic Complexity and Volume

TL;DR

This paper refines the holographic complexity=volume duality by introducing a normalized volume measure, explores its implications for black hole thermodynamics and complexity transfer, and tests it in various dynamic spacetime scenarios.

Contribution

It proposes a new volume measure to restore universality in CV duality and establishes a link between boundary foliation and bulk maximal surfaces, advancing the understanding of holographic complexity.

Findings

Normalized volume measure removes universality issues.

Flux of volume suggests UV to IR complexity transfer.

Second law for complexity at black hole horizons.

Abstract

The previously proposed "Complexity=Volume" or CV-duality is probed and developed in several directions. We show that the apparent lack of universality for large and small black holes is removed if the volume is measured in units of the maximal time from the horizon to the "final slice" (times Planck area). This also works for spinning black holes. We make use of the conserved "volume current", associated with a foliation of spacetime by maximal volume surfaces, whose flux measures their volume. This flux picture suggests that there is a transfer of the complexity from the UV to the IR in holographic CFTs, which is reminiscent of thermalization behavior deduced using holography. It also naturally gives a second law for the complexity when applied at a black hole horizon. We further establish a result supporting the conjecture that a boundary foliation determines a bulk maximal foliation without gaps, establish a global inequality on maximal volumes that can be used to deduce the monotonicity of the complexification rate on a boost-invariant background, and probe CV duality in the settings of multiple quenches, spinning black holes, and Rindler-AdS.