Real-time gravitational replicas: Formalism and a variational principle

TL;DR

This paper develops a formalism and variational principle for real-time gravitational replica wormholes, enabling the study of their saddle points with complex metrics without relying on analytic continuation, and highlighting unitarity features.

Contribution

It introduces a variational principle for real-time replica wormholes and describes their saddle structures, including complex metrics and real regions, advancing the understanding of Lorentzian gravitational path integrals.

Findings

Constructed a variational principle for real-time replica wormholes.

Identified saddle structures involving complex metrics and real regions.

Linked saddle properties to unitarity in the dual field theory.

Abstract

This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called `splitting surface'. This feature is an important hallmark of unitarity in a field theory dual.