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Algebraic traversable wormholes

Eyoab Bahiru

TL;DR

The paper addresses how traversable wormholes can arise in AdS/CFT from purely algebraic data by introducing a new large $N$ limit built from an algebra at infinity and half-sided modular translations. It constructs an operator $L_N$ via modular inclusions, showing that bounded functions of $L_N$ acting on the thermofield double render the eternal black hole traversable in the extreme large-$N$ limit, without relying on gravitational scattering inputs. Crucially, the algebraic computation reproduces the boundary correlator modifications associated with traversability, matching results previously obtained from bulk shock-wave analyses (as in Maldacena–Stanford–Yang). The work thus provides an operator-algebraic route to encode bulk traversability, suggesting deep links to quantum teleportation and endomorphisms in type III von Neumann algebras and offering a framework for exploring the holographic bulk from boundary algebraic data beyond perturbative gravity.

Abstract

We propose a new large $N$ limit which at the extreme limit is dual in the bulk to a back-reacted traversable wormhole, by making use if a novel definition of algebra at infinity, an algebra familiar in the literature from the study of quasi-local algebras. We also compute, from a purely algebraic perspective, the effects registered by a left universe observer due to a unitary fluctuation on the right universe of the traversable wormhole, and reproduce a result from an earlier computation by Maldacena, Stanford and Yang \cite{Maldacena:2017axo}.

Algebraic traversable wormholes

TL;DR

The paper addresses how traversable wormholes can arise in AdS/CFT from purely algebraic data by introducing a new large limit built from an algebra at infinity and half-sided modular translations. It constructs an operator via modular inclusions, showing that bounded functions of acting on the thermofield double render the eternal black hole traversable in the extreme large- limit, without relying on gravitational scattering inputs. Crucially, the algebraic computation reproduces the boundary correlator modifications associated with traversability, matching results previously obtained from bulk shock-wave analyses (as in Maldacena–Stanford–Yang). The work thus provides an operator-algebraic route to encode bulk traversability, suggesting deep links to quantum teleportation and endomorphisms in type III von Neumann algebras and offering a framework for exploring the holographic bulk from boundary algebraic data beyond perturbative gravity.

Abstract

We propose a new large limit which at the extreme limit is dual in the bulk to a back-reacted traversable wormhole, by making use if a novel definition of algebra at infinity, an algebra familiar in the literature from the study of quasi-local algebras. We also compute, from a purely algebraic perspective, the effects registered by a left universe observer due to a unitary fluctuation on the right universe of the traversable wormhole, and reproduce a result from an earlier computation by Maldacena, Stanford and Yang \cite{Maldacena:2017axo}.

Paper Structure

This paper contains 11 sections, 3 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Traversability
  3. Traversable wormholes
  4. Gao-Jafferis-Wall traversable wormholes
  5. Shock waves and traversable wormholes
  6. Algebraic traversable wormholes
  7. The action of half sided modular translations
  8. Algebraic computation
  9. Discussion and outlook
  10. Algebra at infinity
  11. Algebra at infinity and Half sided modular inclusions

Key Result

Theorem 1

For a given state $\omega$ on $\mathcal{Q}$, the algebra at infinity $\mathcal{Q}_{\infty,\omega}$ is a sub algebra of $\mathcal{Q}_{ce,\omega}$ and the following statement are equivalent;

Figures (2)

  • Figure 1: (above) is the Penrose diagram of a wormhole in the presence of a shock wave with positive energy very close to the horizon (shown is green), i.e, sent from the very past in CFT$-1$. The future horizon of the left black hole and the past horizon of the right black hole will 'miss' each other because of the time advance geodesics receive in the presence of the shock wave. However (below), if the energy of the shock wave is negative (shown in dark red), the infalling geodesics will translate backwards along the trajectory of the shock wave and can escape to the left black hole's asymptotic universe.
  • Figure 2: $\mathcal{N}$ is a subalgerba of $\mathcal{M}$ including operators with $t>t_{0}$ for some positive $t_{0}$, while $J\mathcal{N}J$ is a subalgerba of $\mathcal{M}^{'}$ including operators with $t<-t_{0}$. Under modular evolution by a positive parameter $s$, $\mathcal{N}$ is traslated into an even smaller sub algebra of $\mathcal{N}$ and $\mathcal{M}$ ans similarly for $J\mathcal{N}J$. $J\mathcal{N}J$ is translated into a smaller sub alegrba because modular evoultion by a positive parameter translates operators into the past for CFT-1.

Theorems & Definitions (7)

  • Definition A.1
  • Definition A.2
  • Theorem 1
  • Definition A.3
  • Lemma 2
  • Definition A.4
  • Theorem 3