AdS from Optimization of Path-Integrals in CFTs

We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in 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 cut-off. 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 AdS/CFT. We confirm our procedure for excited states, the thermofield double state, the SYK 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.