The boundary dual of the bulk symplectic form

TL;DR

This work identifies a precise duality between overlaps of boundary states prepared by Euclidean sources and the bulk gravitational symplectic form. By pulling back the Fubini-Study metric, it shows that the boundary space of sources carries a Kahler form that matches the bulk symplectic structure via the extrapolate dictionary, with the push-to-Lorentzian data yielding the initial-data symplectic form on a Cauchy slice. The approach yields a boundary expression for the variation of the volume of an extremal bulk slice and provides a covariant framework relating quantum information measures to gravitational dynamics. The results pave the way for broader applications to holographic complexity, extremal volumes, and holographic RG, with planned extensions to quantum corrections and more general gravity theories.

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 Kahler structure on the space of sources, which is naturally induced by the Fubini-Study metric. The Kahler 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.

Figures (1)

• Figure 1: Using conservation we can push the symplectic form from the Euclidean upper hemisphere $(\partial X)^+$ to an arbitrary slice $\Sigma$ anchored at $t_E=0$.