Complexity Equals Anything II

TL;DR

This paper introduces a broad new class of gravitational observables in asymptotically Anti-de Sitter space that generalize existing complexity proposals, exhibiting universal late-time growth and the switchback effect, and are linked to the gravitational symplectic form.

Contribution

It extends previous work by defining a new class of observables that unify various holographic complexity proposals and demonstrates their universal features and symplectic structure.

Findings

New class of gravitational observables generalizing complexity proposals

Observables grow linearly and show switchback effect in thermofield double state

Variations encoded in gravitational symplectic form and mapped to CFT

Abstract

We expand on our results in arXiv:2111.02429 to present a broad new class of gravitational observables in asymptotically Anti-de Sitter space living on general codimension-zero regions of the bulk spacetime. By taking distinct limits, these observables can reduce to well-studied holographic complexity proposals, e.g., the volume of the maximal slice and the action or spacetime volume of the Wheeler-DeWitt patch. As with the codimension-one family found in arXiv:2111.02429, these new observables display two key universal features for the thermofield double state: they grow linearly in time at late times and reproduce the switchback effect. Hence we argue that any member of this new class of observables is an equally viable candidate as a gravitational dual of complexity. Moreover, using the Peierls construction, we show that variations of the codimension-zero and codimension-one observables are encoded in the gravitational symplectic form on the semi-classical phase-space, which can then be mapped to the CFT.