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A 2-Replica Wormhole

Mehrdad Mirbabayi

TL;DR

The paper investigates finite-n effects of dynamical gravity on replica wormholes by constructing $1<n≤2$ replica saddles in a 2d gravity-CFT system and comparing the resulting Renyi entropies to field-theory predictions. The geometry is fixed by a conformal welding condition that ties holomorphic maps F and G to a boundary reparameterization theta, while twist-operator data and the conformal anomaly yield the matter partition function $Z_n$ in terms of the welding data. A two-step numerical strategy evolves the boundary deformation across replica number and computes $Z_n$ from the gravity action and matter partitions, revealing how dynamical boundary geometry shifts Renyi entropies relative to a fixed background. The work provides a concrete, programmable framework for studying replica wormholes in low-dimensional gravity and demonstrates finite-n corrections that arise when gravity is dynamical.

Abstract

Replica geometries are not rigid when gravity is dynamical. We numerically construct $1<n\leq 2$ replica saddles in $2d$ gravity coupled to a CFT and compare the resulting Renyi entropies with the field theory result.

A 2-Replica Wormhole

TL;DR

The paper investigates finite-n effects of dynamical gravity on replica wormholes by constructing replica saddles in a 2d gravity-CFT system and comparing the resulting Renyi entropies to field-theory predictions. The geometry is fixed by a conformal welding condition that ties holomorphic maps F and G to a boundary reparameterization theta, while twist-operator data and the conformal anomaly yield the matter partition function in terms of the welding data. A two-step numerical strategy evolves the boundary deformation across replica number and computes from the gravity action and matter partitions, revealing how dynamical boundary geometry shifts Renyi entropies relative to a fixed background. The work provides a concrete, programmable framework for studying replica wormholes in low-dimensional gravity and demonstrates finite-n corrections that arise when gravity is dynamical.

Abstract

Replica geometries are not rigid when gravity is dynamical. We numerically construct replica saddles in gravity coupled to a CFT and compare the resulting Renyi entropies with the field theory result.

Paper Structure

This paper contains 4 sections, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $2d$ setup and the results
  3. Partition function
  4. Integration

Figures (2)

  • Figure 1: Left: The Penrose diagram of AdS$_2$ black hole connected along the UV boundary (dashed lines) to the non-gravitational region. CFT fields freely propagate through this boundary and are in thermal equilibrium with the black hole. We are calculating the Renyi entropies of an interval anchored on one side in the flat region. Right: The Euclidean geometry that prepares the state. Since the boundary curve between AdS and half-cylinder fluctuates, mapping it to $\sigma =0$ generically requires a nontrivial diffeomorphism $\theta(\tau)$. $b$ and $-a$ are the entangling points.
  • Figure 2: Left: Solid line shows the Renyi entropy, multiplied by $2n/(1+n)$ to make the cutoff-dependent contribution constant, and shifted by the von Neumann entropy. This combination vanishes for a fixed interval in a CFT. The dashed line shows the result obtained by linearizing the equations in $\theta-\tau$. The dot-dashed line shows the change in the proper distance between $x = -a$ and $x=0$. Right: The Schwarzian of the AdS boundary-time at $n=2$, as a measure of how the boundary has deformed. This is $1/2$ for an undeformed boundary (the horizontal line). The dashed line shows the linear approximation. The parameters are $b=0.05$, $\kappa =1$.