On subregion action complexity in AdS$_3$ and in the BTZ black hole

TL;DR

This paper analytically computes subsystem action complexity in BTZ black hole backgrounds, revealing a simple structure relating it to entanglement entropy, and explores more complex geometries with additional finite contributions.

Contribution

It provides the first analytical calculation of subsystem action complexity in BTZ black holes and extends understanding to more complex geometries with multiple segments.

Findings

Complexity in BTZ is a sum of divergent and entanglement-related terms.

More complex geometries introduce additional finite contributions to complexity.

Analytic results for mutual action complexity of two segments are presented.

Abstract

We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS$_3$, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion.