Complexity and the bulk volume, a new York time story

Abstract

We study the boundary description of the volume of maximal Cauchy slices using the recently derived equivalence between bulk and boundary symplectic forms. The volume of constant mean curvature slices is known to be canonically conjugate to "York time". We use this to construct the boundary deformation that is conjugate to the volume in a handful of examples, such as empty AdS, a backreacting scalar condensate, or the thermofield double at infinite time. We propose a possible natural boundary interpretation for this deformation and use it to motivate a concrete version of the complexity=volume conjecture, where the boundary complexity is defined as the energy of geodesics in the Kähler geometry of half sided sources. We check this conjecture for Bañados geometries and a mini-superspace version of the thermofield double state. Finally, we show that the precise dual of the quantum information metric for marginal scalars is given by a particularly simple symplectic flux, instead of the volume as previously conjectured.Here, we are denoting linearized deformations by |δΨ λ ≈ |Ψ λ+δλ − |Ψ λ , the bulk field φ is dual to the operator sourced by λ, its conjugate momentum is π, and Σ is a bulk initial value surface. This is the main result of [21] and it gives a

Aspects of the first law of complexity

“…Hence we may conclude that our results will agree with those arising from the analysis of coherent states in both [125] and [126].…”

“…Similar independent discussions have appeared in [125,126] relating the volume variation to an integral of the matter stress tensor perturbation, as in (3.116), even though their derivations are different. We show below their results are equivalent to our explicit volume extremisation.…”

“…Recall that in section 3.3.3, we commented on the relation of our results for δC V with similar variations studied in [125,126]. Here, we would like to consider the possible connection of our first law of complexity (2.25) with the second law in [25].…”

“…First, in [125], 22 the volume (complexity) variation is related to the integral of the matter stress tensor…”

“…on the spacelike surface defined by setting the timelike WDW coordinate τ = 0. The description of the AdS d+1 geometry in [125] uses FRW coordinates…”