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

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.

29 pages, 7 figures, V2: minor changes, refs added