Path integral optimization from Hartle-Hawking wave function

Abstract

We propose a gravity dual description of the path integral optimization in conformal field theories [Caputa et al., Phys. Rev. Lett. 119, 071602 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.071602], using Hartle-Hawking wave functions in anti-de 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.