Path integral optimization from Hartle-Hawking wave function

We propose a gravity dual description of the path integral optimization in conformal field theories [Caputa et al., Phys. Rev. Lett. 119, 071602 (2017)], using Hartle-Hawking wave functions in anti-dc Sitter spacetime. We show that the maximization of the Hartle-Hawking wave function is equivalent to the path integral optimization procedure. Namely, the variation of the wave function leads to a constraint, equivalent to the Neumann boundary condition on a bulk slice, whose classical solutions reproduce metrics from the path integral optimization in conformal field theories. After taking the boundary limit of the semiclassical Hartle-Hawking wave function, we reproduce the path integral complexity action in two dimensions, as well as its higher- and lower-dimensional generalizations. We also discuss an emergence of holographic time from conformal field theory path integrals.

Automated summary

This paper introduces a gravity dual description of path integral optimization in conformal field theories and discusses the equivalence between maximizing the Hartle-Hawking wave function and the path integral optimization procedure, as well as its application in various dimensions.