Abstract
In this paper, we study the overlaps of wavefunctionals prepared by turning on sources in the Euclidean path integral. For nearby states, these overlaps give rise to a Kähler structure on the space of sources, which is naturally induced by the Fubini–Study metric. The Kähler form obtained this way can also be thought of as a Berry curvature and, for holographic field theories, we show that it is identical to the gravitational symplectic form in the bulk. We discuss some possible applications of this observation, in particular a boundary prescription to calculate the variation of the volume of a maximal slice.
Highlights
The Ryu–Takayanagi formula [1] opened an intriguing connection between quantum gravity and quantum information
It is highly desirable to understand better the bulk duals of boundary quantities that only depend on wave functionals, hoping that they will teach us about the structure of the Hilbert space in quantum gravity
We have studied the overlaps of nearby path integral states in holographic quantum field theories
Summary
The Ryu–Takayanagi formula [1] opened an intriguing connection between quantum gravity and quantum information. There have been many advances in understanding the holographic duals of quantum information measures in AdS/CFT, see [2] for a review Most of these quantities are tied to subregions or mixed states and there has not been much progress in understanding measures associated with pure states. We are going to give a precise mapping between the antisymmetrized overlap of states (naturally understood as a Kähler form in the space of state deformations) and the bulk symplectic form This generalizes the work of [5] and identifies the precise bulk dual. The states in consideration are prepared by turning on arbitrary (possibly complex) Euclidean sources, which correspond to classical geometries This new entry in the dictionary has similarities with the relative entropy for subregions: it compares two neighbouring states and the bulk and boundary quantities are equal [6]. Volume of the extremal time slice, which we will explore further in [7]
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