Complexity Equals Anything II

TL;DR

<3-5 sentence high-level summary> The paper broadens the holographic complexity program by introducing a large class of codimension-zero bulk observables in AdS that reduce to known proposals (CV, CA, CV2.0) in appropriate limits and exhibit universal late-time linear growth and switchback for thermofield double states. It develops a two-step construction: extremize a bulk functional over codimension-zero regions and then evaluate a second functional on the resulting region, with the two bulk functionals and four boundary functionals enabling a vast design space. The authors connect these observables to the gravitational symplectic form via the Peierls construction, providing a holographic interpretation of their conjugate variations and showing how variations of codimension-zero and -one observables map to boundary data. They also discuss the extension to an squared framework (Complexity = Anything$^2$), the null-limit behavior, and several explicit examples (including constant-mean-curvature slices and Weyl-squared deformations) while outlining open issues such as uniqueness, matter content, and higher-curvature corrections.

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.

Figures (11)

• Figure 1: The codimension-zero region $\mathcal{M}$ (orange shaded patch) illustrated in an eternal AdS black hole background. The future and past boundaries of $\mathcal{M}$ are denoted by $\Sigma_\pm$, respectively, and are both anchored on the same boundary time slice $\Sigma_\textrm{\tiny CFT}$.
• Figure 2: Left: Sketch of a codimension-zero region where boundaries $\Sigma_\pm$ are constant mean curvature slices, determined by the functional in eq. \ref{['eq:CMCfunc']}. Right: Taking the limit $\alpha_\pm \to 0$, the boundaries $\Sigma_\pm$ become null surfaces, and the region becomes the WDW patch.
• Figure 3: The effective potentials $\mathcal{U}(P_v^\varepsilon, r)$ for the future boundary $\Sigma_+$ (left) and the past boundary $\Sigma_-$ (right), respectively. Hence $\varepsilon=+ \,(-)$ in the left (right) panel. For this figure, we have chosen $d=2$ and $\alpha_\pm=1=\alpha_{\textrm{\tiny B}}$.
• Figure 4: Illustration of the relation between the conserved momentum $P_v^{\varepsilon}$ and the minimal radius $w_{\varepsilon,\rm min}$ in eq. (\ref{['ellipse']}), for $\alpha_{\textrm{\tiny B}}=1=\alpha_\pm$, $x=1$ and $d=2$. The solutions $P_v^{\varepsilon}(+)$ in eq. (\ref{['turn44']}) correspond to the black arcs, while $P_v^{\varepsilon}(-)$ correspond to the red arcs. The critical momenta $P_\infty^\varepsilon(\pm)$ in eq. (\ref{['limits22']}) correspond to the maximum and minimum vertical extrema of the ellipses marked by the horizontal dashed lines. The corresponding turning points $w_f(\pm)$ given in eq. (\ref{['limits33']}) are marked by the vertical dashed lines.
• Figure 5: The relation between the boundary time $\tau$ and the conserved momentum $P_v^\varepsilon$. We set $\alpha_{\textrm{\tiny B}}=1=\alpha_\pm$, $x=1$ and $d=2$ in both panels and $\varepsilon=+,-$ for the left and right plot, respectively.