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

## TL;DR

This paper proposes an optimization method for Euclidean path integrals in conformal field theories, revealing a connection to Anti-de Sitter space and holographic entanglement, with applications to various states and models.

## Contribution

It introduces a novel optimization procedure for path integrals that links CFT wave functionals to AdS geometry, extending to excited states and higher dimensions.

## Key findings

- Optimized metrics for 2D CFT vacua are hyperbolic spaces.
- Reproduces entanglement wedges and holographic entanglement entropy.
- Applicable to excited states, thermofield double, and SYK model.

## 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.