Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories

TL;DR

The paper proposes a method to optimize Euclidean path integrals for states in CFTs by varying the background geometry (a position-dependent UV cutoff) to minimize a Liouville action. In two dimensions, this optimization uniquely yields a hyperbolic metric that corresponds to time slices of AdS, providing a continuous tensor-network perspective on AdS/CFT. The authors extend the approach to excited states, finite-temperature (thermofield double) setups, and the SYK model, and show that, for reduced density matrices, the resulting geometry naturally reproduces entanglement wedges and holographic entanglement entropy. They also discuss higher-dimensional generalizations and the potential connection to computational complexity, outlining future directions such as correlation functions and time-dependent backgrounds.

Abstract

We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti--de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.

Figures (2)

• Figure 1: A computation of ground state wave functional from Euclidean path integral and its optimization, which is described by a hyperbolic geometry. The right figure schematically shows its tensor network expression.
• Figure 2: The optimization of path integral for a reduced density matrix. The upper left picture is the definition of $\rho_A$ in terms of the path integral in flat space. The upper right one is the one after the optimization and is equivalent to a geometry which is obtained by pasting two identical entanglement wedges along the geodesic (=the half circle) as shown in the lower right picture. If we start from $\rho_A^n$, we obtain the geometry in the lower left picture with $\delta=\pi (1-n)$.