Real-time gravitational replicas: Low dimensional examples

TL;DR

This work demonstrates that stationary points of the real-time gravitational path integral, including replica-wormhole saddles, can be concretely realized and analyzed in low-dimensional settings. By combining JT gravity and holographic 2d CFTs, the authors compute Rényi entropies via both Euclidean constructions and Lorentzian real-time evolutions, showing exact agreement between the two formalisms. They develop and apply a concrete toolkit—covering-space versus fundamental-domain descriptions, Schottky uniformization for disjoint intervals, and monodromy/ accessory-parameter methods—to extract entropy data from explicit bulk geometries. The results illuminate how complex-regulated, localized contributions to the gravitational action encode entropic data, supporting the broader program of real-time holography and its relevance to information-theoretic questions in gravity and black hole dynamics.

Abstract

We continue the study of real-time replica wormholes initiated in arXiv:2012.00828. Previously, we had discussed the general principles and had outlined a variational principle for obtaining stationary points of the real-time gravitational path integral. In the current work we present several explicit examples in low-dimensional gravitational theories where the dynamics is amenable to analytic computation. We demonstrate the computation of Rényi entropies in the cases of JT gravity and for holographic two-dimensional CFTs (using the dual gravitational dynamics). In particular, we explain how to obtain the large central charge result for subregions comprising of disjoint intervals directly from the real-time path integral.

Figures (11)

• Figure 1: An illustration of the real-time contours for the computation of the density matrix $\rho(t)$ (left) and traces of its powers ($\Tr(\rho(t)^3)$ on 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.
• Figure 2: The Poincaré disc geometry dual to the thermofield double (or Hartle-Hawking) state of JT gravity and its $n$-fold replica depicted here for $n=3$. We have shaded the single fundamental domain obtained by taking the replica quotient and indicated the interior boundary at $r=\epsilon$ one introduces while computing the on-shell Euclidean action contribution from a single fundamental domain.
• Figure 3: The domains in the Lorentzian geometry dual to a single fundamental domain $\widehat{{\cal M}}_n$. We have indicated both the 'ket' and 'bra' components of the spacetime ${\sf M}^k$ and ${\sf M}^b$ which are each a copy of the AdS$_{2}$ geometry past of the Cauchy slice at $t=0$. The geometry $\widehat{{\cal M}}_n$ has a fixed point locus of the replica $\mathbb{Z}_n$ action at the splitting surface $\bm{\gamma}$. The ket and bra geometries are real in the Rindler wedges, regions spacelike separated from $\bm{\gamma}$, but are complex in the Milne wedge, the causal past of $\bm{\gamma}$.
• Figure 4: The geometry in the vicinity of the splitting surface $\bm{\gamma}$ in the Lorentzian geometry dual to a single fundamental domain $\widehat{{\cal M}}_n$. We have excised a neighbourhood $\mathscr{U}_\epsilon$ of $\bm{\gamma}$ with boundary $\partial \mathscr{U}_\epsilon$ to regulate the contribution from the fixed point locus. We take $\partial \mathscr{U}_\epsilon$ to be parametrized by an arbitrary curve $\tilde{x}^+ = U(\tilde{x}^-)$ in the $\tilde{x}^\pm$ plane.
• Figure 5: Causal domains on the boundary ket spacetime ${\sf B}^k$ for a two dimensional field theory with the region $\mathcal{A}$ taken to be a spacelike segment of a boosted Cauchy slice. We indicate the regions where the resulting metric is real and complex, respectively. In general the metric is not guaranteed to be real in regions that are in the causal past of the entangling surface $\partial\mathcal{A}$ which here comprises of the two points $a_1$ and $a_2$.