Holographic Complexity and Volume

TL;DR

The paper sharpens holographic complexity by refining the CV-duality: complexity is proportional to the volume of maximal bulk slices, but the appropriate divisor is the maximal fall-time $\tau_f$ from the horizon to the final slice. It introduces a volume current, a local, divergence-free object tied to a maximal foliation, and shows how its flux encodes the growth of complexity, including a second-law-like behavior at horizons and a UV-to-IR flow interpretation. The authors derive global volume inequalities that imply monotonic growth on boost-symmetric backgrounds and test CV-duality across AdS-Vaidya quenches, rotating BTZ, Kerr, and AdS-Rindler spacetimes, finding consistent scaling with $T_H S_{BH}$ when using $\tau_f$. Overall, the work strengthens the CV framework, provides a constructive tool (the volume current) and clarifies the regimes in which CV captures thermal-state complexity, while acknowledging the heuristic nature of the chosen divisor outside strict thermal equilibrium.

Abstract

The previously proposed "Complexity=Volume" or CV-duality is probed and developed in several directions. We show that the apparent lack of universality for large and small black holes is removed if the volume is measured in units of the maximal time from the horizon to the "final slice" (times Planck area). This also works for spinning black holes. We make use of the conserved "volume current", associated with a foliation of spacetime by maximal volume surfaces, whose flux measures their volume. This flux picture suggests that there is a transfer of the complexity from the UV to the IR in holographic CFTs, which is reminiscent of thermalization behavior deduced using holography. It also naturally gives a second law for the complexity when applied at a black hole horizon. We further establish a result supporting the conjecture that a boundary foliation determines a bulk maximal foliation without gaps, establish a global inequality on maximal volumes that can be used to deduce the monotonicity of the complexification rate on a boost-invariant background, and probe CV duality in the settings of multiple quenches, spinning black holes, and Rindler-AdS.

Figures (14)

• Figure 1: Shockwave geometry dual to a perturbed thermofield double state, with a maximal volume hypersurface anchored at late time on the left and early time on the right.
• Figure 2: Left: Illustration of the boundary foliation $\Sigma{(\tau)}$ with 3 slices in the foliation. Right: Illustration of the corresponding bulk foliation by maximal slices, and the volume flow.
• Figure 3: Plot of the flow lines of the BTZ volume current (in solid green) on a quarter of a Penrose diagram. Left: the flow lines are shown together with the Schwarzschild coordinate grid lines (dotted black), and the final slice (solid blue). Right: the flow lines are shown together with the maximal slices (dashed red).
• Figure 4: An SSA-like inequality is obeyed between the four maximal slices shown (solid red and dashed orange).
• Figure 5: On the left are two Cauchy slices, $\sigma_1$ (in continuous blue) and $\sigma_2$ (in dashed blue), on the boundary of the Poincaré patch of an asymptotically AdS spacetime. On the right the corresponding $\sigma_+$ and $\sigma_{-}$ are in dashed blue and continuous blue, respectively.