Complexity and the bulk volume, a new York time story

TL;DR

The paper develops a boundary description of the bulk volume of maximal Cauchy slices by exploiting an equality between bulk and boundary symplectic forms and introducing a boundary deformation (the New York transformation) conjugate to the volume. It proposes a boundary interpretation of this deformation and uses it to motivate a concrete complexity=volume proposal based on the kinetic energy in the space of Euclidean boundary sources, with tests in AdS vacua, scalar-condensed states, Bañados geometries, and a TFD mini-superspace. A key finding is that the dual of the quantum information metric for marginal scalars is a simple symplectic flux rather than the volume, clarifying the relationship between information geometry and holographic volumes. The results bridge holographic complexity, fidelity susceptibility, and bulk volume, offering a framework for boundary measurements of geometric bulk data and suggesting future directions for a rigorous complexity conjecture and its boundary implementation.

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.

Figures (5)

• Figure 1: Two euclidean path integral states with sources prepared on manifolds of different topology (the hemisphere and the cylinder). On the left, the state lives in a single copy of the CFT Hilbert space $\mathcal{H}$ while on the right, it lives in $\mathcal{H}\times\mathcal{H}$.
• Figure 2: Using conservation we can push the symplectic form from half of the Euclidean boundary to an arbitrary bulk slice $\Sigma$ anchored to the boundary at $t_E=0$.
• Figure 3: We show the WdW patch in Lorentzian (left) and Euclidean AdS (right).
• Figure 4: The top part of the Penrose diagram of the eternal wormhole with the location of the horizon, the maximal infinite time Cauchy slice and the singularity highlighted in the coordinates \ref{['interior']}. The blue region is the top interior, and also the WDW patch of the red slice.
• Figure 5: Plot of \ref{['eq:px2']} for $d=2,...,5$, larger $d$ hugs the infinite time line more.