Real-time gravitational replicas: Formalism and a variational principle

TL;DR

This work develops a real-time, Lorentzian path integral framework for gravitational replica wormholes to compute swap Rényi entropies in holographic theories. It constructs a well-defined variational principle for Lorentzian replica saddles that necessarily involve complex metrics, introduces a splitting surface (cosmic brane) and surface terms, and relates Euclidean conical defect data to real-time boundary conditions via an $i\varepsilon$ prescription. A key result is that replica- and CPT-symmetric saddles can have real extrinsic curvature on homology wedges, enabling cancellations between bra and ket branches and yielding real Rényi entropies from the imaginary part of ket-sector actions, thereby supporting unitarity in the dual field theory. The framework sets the stage for explicit, real-time constructions (to be demonstrated in a companion paper) and offers a path to incorporating higher-curvature corrections and quantum backreaction while clarifying the connection between Euclidean replica results and real-time holography.

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.

Figures (11)

• Figure 1: A schematic illustration of the real-time (Schwinger-Keldysh) contours for the computation of the density matrix $\rho(t)$ (left) and its powers (right). The past boundary condition is supplied by the prescribed initial state $\rho_0$ and the direction of time evolution is explicitly indicated by the arrows. Forward evolution corresponds to the ket part of the state while backward evolution corresponds to the bra part. The reduced density matrix $\rho_{_\mathcal{A}}(t)$ associated with some spatial domain at $t=0$ is obtained by sewing together the $\mathcal{U}(t;t_0)$ and $\mathcal{U}(t;t_0)^\dagger$ parts of the left panel along the complementary $t=0$ domain $\mathcal{A}^c$, while leaving open the parts along $\mathcal{A}$. It's powers thus involve similar contractions between the two parts of any given copy, while the $\mathcal{A}$ parts are again contracted with neighboring copies as shown at right.
• Figure 2: A replica wormhole spacetime that contributes in to the Euclidean path integral for the computation of spectral traces of density matrices.
• Figure 3: The timefolded Schwinger-Keldysh geometry ${\cal B}$ computing the matrix elements of $\rho_{_\mathcal{A}}$ for a quantum theory on a fixed background. The forward and backward evolutions are glued together on the partial Cauchy slice $\Sigma_{_t}$, except for a cut around $\mathcal{A}$. We have also indicated the past light-cones from the entangling surfaces which serve to demarcate the Milne and the Rindler wedges defined in the main text. In the gravitational context, the boundary geometries will be of this form.
• Figure 4: The bulk domains of interest in the Lorentzian construction for either the ket or the bra spacetime. Given a partition of $\Sigma_{_t}$ into regions $\mathcal{A}$ and $\mathcal{A}^c$, any bulk Cauchy surface $\tilde{\Sigma}_t$ with $\partial {\tilde{\Sigma}}_{_t} = \Sigma_{_t}$ admits a decomposition $\tilde{\Sigma}_t = {\cal R}_{\mathcal{A}} \cup {\cal R}_{{\mathcal{A}}^c}$. These domains are separated by a bulk codimension-2 surface $\text{\bf{\textgamma}}$, which is anchored on the entangling surface. On saddle point solutions this surface approaches the extremal surface as $n\to 1$.
• Figure 5: Bulk configurations relevant to computing the bulk dual of the boundary density matrix $\rho_{_\mathcal{A}}(t)$. The forward evolution for $\ket{\Psi}$ (left) proceeds up to ${\tilde{\Sigma}}_{_t}$, while the backwards evolution for $\bra{\Psi}$ starts there (right). We have identified the bra and ket spacetime along ${\cal R}_{{\mathcal{A}}^c}$ in accord with the prescription for the bulk dual of $\rho_{_\mathcal{A}}(t)$. Further gluing the two spacetimes together along ${\cal R}_{\mathcal{A}}$ and summing over all metrics would compute the trace of $\rho_{_\mathcal{A}}(t)$.