Topological and Time Dependence of the Action-Complexity Relation

TL;DR

This paper investigates how the action/complexity duality in holography depends on time and the topology of the bulk spacetime, revealing that topology changes affect the proportionality factor in the duality.

Contribution

It demonstrates the sensitivity of the action/complexity relation to spacetime topology and confirms the duality's validity across different topologies with a modified proportionality.

Findings

Complexity depends on boundary time in a manner consistent with bulk action rate changes.

The duality holds for both black holes and geon spacetimes, with a factor of 4 difference.

Topology influences the proportionality factor in the action/complexity relation.

Abstract

We consider the dependence of the recently proposed action/complexity duality conjecture on time and on the underlying topology of the bulk spacetime. For the former, we compute the dependence of the CFT complexity on a boundary temporal parameter and find it to be commensurate with corresponding computations carried out in terms of the rate of change of the bulk action on a Wheeler deWitt (WDW) patch. For the latter, we compare the action/complexity relation for $(d+1)$-dimensional Schwarzschild AdS black holes to those of their geon counterparts, obtained via topological identification in the bulk spacetime. The complexity/action duality holds in both cases, but with the proportionality changed by a factor of 4, indicating sensitivity to spacetime topology.