Holographic Path-Integral Optimization

TL;DR

<3-5 sentences high-level summary>This work develops a holographic realization of path-integral optimization by identifying the optimization of CFT states with the maximization of Hartle-Hawking wave functions in AdS/CFT. The authors derive a gravity-based, finite-cutoff definition of holographic path-integral complexity via a tensionful end-of-the-world surface Q and a modified Hayward term, showing that the optimizing Q is a constant-mean-curvature slice with curvature tied to the tension parameter T. They demonstrate this framework across Euclidean AdS, higher dimensions, JT gravity, and Lorentzian spacetimes including AdS and dS, recovering known 2d Liouville-type actions as UV limits and revealing a potential interpretation of time as emergent from the tensor-network optimization. The results illuminate finite-cutoff corrections to complexity, connect to AdS/BCFT constructs, and propose a Lorentzian path-integral optimization program that may illuminate real-time holography and time emergence from quantum circuits.

Abstract

In this work we elaborate on holographic description of the path-integral optimization in conformal field theories (CFT) using Hartle-Hawking wave functions in Anti-de Sitter spacetimes. We argue that the maximization of the Hartle-Hawking wave function is equivalent to the path-integral optimization procedure in CFT. In particular, we show that metrics that maximize gravity wave functions computed in particular holographic geometries, precisely match those derived in the path-integral optimization procedure for their dual CFT states. The present work is a detailed version of \cite{Boruch:2020wax} and contains many new results such as analysis of excited states in various dimensions including JT gravity, and a new way of estimating holographic path-integral complexity from Hartle-Hawking wave functions. Finally, we generalize the analysis to Lorentzian Anti-de Sitter and de Sitter geometries and use it to shed light on path-integral optimization in Lorentzian CFTs.

Figures (6)

• Figure 1: The on-shell action in the shaded region $M$ computes the Hartle-Hawking wave function, related to the path-integral optimization for holographic CFTs.
• Figure 2: Solutions \ref{['surfacep']} are half planes interpolating between the boundary at $T=-(d-1)/l$ and the $\tau=0$ slice for $T=0$ (in the text we used $l=1$). The inner angle $\theta_0$ is related to $\Theta$ and $T$ via \ref{['anglet']}.
• Figure 3: Example slices $Q$ that maximize the Hartle-Hawking wave functions in global $AdS_{d+1}$ for a fixed value of $T$. They interpolate between the maximal, $\tau=0$ slice for $T=0$ (green) and one reaching to $\tau=-\infty$ for $T=-(d-1)$.
• Figure 4: Solutions \ref{['fTFDt']} shown on the cigar geometry (left) and its projection (right) for a few sample values of $T$ interpolating between $T=0$ (red) and $T=-1$ (green). The semi-circles have the range $\tau\in [-\beta/4,\beta/4]$.
• Figure 5: Solutions of the maximization in Lorentzian $AdS_{d+1}$. Timelike half-planes \ref{['AdSLTL']}, shown in blue, interpolate between the boundary at $T=-(d-1)/l$ and null plane, dashed red, in the limit of $T\to-\infty$. The counterpart of the inner angle ("rapidity") $\eta_0$ given by \ref{['etaTLADS']} is shown in blue. Spacelike half planes \ref{['AdSLSL']}, shown in green, interpolate between the null sheet for $T\to-\infty$ and the $t=0$ slice at $T=0$. Their inner angle $\tilde{\eta}_0$, given by \ref{['etaSLADS']}, is shown in green.