Wormholes & Holography: An Introduction

Summary

A long (89-page) pedagogical review by Arnab Kundu (Saha Institute) on wormholes and their role in holography and the black-hole information problem. It is an introduction, not a research paper: the goal is to take a reader with basic general relativity and quantum mechanics through the modern story in which wormholes — classical solutions connecting two (or more) asymptotic regions — encode quantum entanglement and act as the geometric carriers of quantum information in semiclassical gravity.

The review is organized around the Euclidean/Lorentzian divide:

• Euclidean wormholes (§4) are saddle points of the gravitational path integral connecting separate asymptotic boundaries. The headline example is the eternal AdS black hole, whose two-sided geometry (an Einstein–Rosen bridge) is dual to the thermofield double state — a maximally entangled state of two copies of the boundary CFT. This makes “geometric connectivity ↔ entanglement” concrete and is the entry point to ER=EPR. The same section treats the resolution of the information paradox via replica wormholes in JT gravity, which reproduce the Page curve and the island formula, and the encoding of multipartite entanglement in multi-boundary wormholes.

• Lorentzian wormholes (§5) require violating the averaged null energy condition (ANEC); generic matter forbids them. The review explains how holography supplies a controlled mechanism: the Gao–Jafferis–Wall double-trace deformation $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ couples the two boundaries, injects negative null energy, and renders the eternal-black-hole wormhole momentarily traversable. It then covers the Maldacena–Qi eternal traversable wormhole (a ground state of two coupled SYK systems / nearly-AdS${}_2$), braneworld constructions, and regenesis — the reappearance of a perturbation on the far side, i.e. the gravitational dual of a quantum teleportation protocol.

Because the whole narrative is built on type III${}_1$-flavoured entanglement of the thermofield double, modular structure, and relative-entropy/generalized-entropy reasoning, the review sits directly adjacent to the group’s Bell-CHSH and relative entropy programs and to the algebraic-QFT course’s Semester II material (GJW, MSY, CPW, AAJ).

Key Results

(This is a review; “results” below are the landmark results it collects and explains, with the standard original attributions.)

1. Eternal black hole = thermofield double = two-sided wormhole (Maldacena 2003). The maximally extended AdS–Schwarzschild geometry has two asymptotic boundaries joined by a non-traversable Einstein–Rosen bridge; the dual state is the TFD $|\mathrm{TFD}⟩=Z^{-1/2}{\sum }_n{e}^{-\beta E_n/2}|n{⟩}_L|n{⟩}_R$. Entanglement between the two boundary CFTs is the bridge — the seed of ER=EPR.

2. Euclidean wormhole solutions need a source of “repulsion.” In the explicit axion–dilaton constructions (§4.1), a scalar with the right kinetic sign makes the scale factor turn around and form a throat. These constrained-instanton saddles connect otherwise-decoupled boundaries and are the origin of the factorization puzzle: a connected $⟨Z_1{Z}_2⟩$ contribution that does not factorize into $⟨Z_1⟩⟨Z_2⟩$.

3. Replica wormholes restore unitarity. In JT gravity with end-of-world branes, replica saddles dominate the gravitational replica computation of fine-grained entropy past the Page time, producing the Page curve and matching the island prescription.

4. Multi-boundary wormholes carry multipartite entanglement. Higher-genus and multi-boundary geometries realize GHZ-like and pairwise entanglement patterns in the dual multi-party CFT state.

5. ANEC violation is unavoidable for traversability (§5.1). Classical matter obeying the null energy condition keeps the throat closed (defocusing requires $\int T_{kk}d\lambda <0$). A traversable wormhole therefore demands a quantum, negative-energy source.

6. Gao–Jafferis–Wall: traversability from a double-trace coupling (GJW 2017). Turning on $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ between the two boundaries of the eternal black hole produces a one-loop negative null energy that opens the throat for a signal to pass. This is the central “UV-complete example” (§5.2.1) and is the mechanism the group’s course treats algebraically (AAJ as a Connes-cocycle perturbation).

7. Maldacena–Qi eternal traversable wormhole (§5.2.2). Two SYK dots coupled by a relevant operator have a near-AdS${}_2$ ground state that is an eternal traversable wormhole — a steady-state entangled resource rather than a one-shot opening.

8. Regenesis = gravitational teleportation (§5.2.4). A perturbation thrown into one side, together with the coupling $V$, reappears as a reconstructable signal on the other side — the bulk dual of the Hayden–Preskill / quantum-teleportation protocol through the wormhole.

Methods

• Euclidean gravitational path integral / saddle-point (instanton) methods, introduced gently via the §2 double-well quantum-mechanics analogy (instantons interpolating between vacua as a toy “wormhole”).
• AdS/CFT dictionary and the Ryu–Takayanagi / generalized-entropy ($S_{\mathrm{gen}}=\mathrm{Area}/4G_N+S_{\mathrm{bulk}}$) prescription for fine-grained entropy (§3).
• Replica trick in gravity with end-of-world branes in JT gravity (§4.2.2).
• Energy-condition / Raychaudhuri analysis of geodesic focusing to derive the traversability obstruction (§5.1).
• Double-trace boundary deformations and one-loop stress-tensor computations for the GJW mechanism (§5.2).
• Coupled-SYK / nearly-AdS${}_2$ effective (Schwarzian) description for the eternal traversable wormhole (§5.2.2).
• Appendices on AdS${}_3$ isometries and the quotient/identification construction for multi-boundary geometries.

Relevance

This paper is the natural pedagogical “front door” to the wormhole literature already threaded through the wiki, and it sharpens several contact points with the group’s own programs:

• It collects, in one place, the wormhole/traversability physics that the algebraic-QFT course covers piecewise. GJW (§5.2.1), Maldacena–Qi (§5.2.2), and the TFD-as-wormhole picture (§4.2.1) are exactly the bulk-side inputs of Semester II Blocks 4–5 (AAJ, MSY). The review supplies the geometric intuition that the algebra “does not see” — precisely the algebra/bulk demarcation the syllabus flags.
• The GJW coupling $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ is a Connes-cocycle perturbation of the TFD modular flow. This is the formal bridge to the group’s relative-entropy and crossed-product machinery — see relative-entropy-wormhole-opening.
• Bell-CHSH between the two boundaries. The TFD is the canonical state on which the group already studies Bell-CHSH (course Sem II Wk 8). Traversability adds signaling between the two sides, raising the sharp question of whether opening the wormhole degrades the Tsirelson violation — see bell-chsh-across-traversable-wormhole.
• Regenesis ↔ embezzlement. The eternal traversable wormhole is a steady entangled resource through which information passes; this resonates with entanglement embezzlement in type III₁ algebras — see wormholes-and-quantum-information-qft.

Questions Raised

1. Does the GJW double-trace coupling, treated as a cocycle perturbation, change the Araki–Uhlmann relative entropy between the TFD and the deformed state in a way computable in a free-field two-wedge analogue? → relative-entropy-wormhole-opening
2. Is there a quantitative trade-off between traversability (signaling across the throat) and Bell-CHSH violation (which presumes the two algebras commute)? → bell-chsh-across-traversable-wormhole
3. Can regenesis / wormhole teleportation be cast as an (approximate) embezzlement protocol on a type III${}_1$ resource, giving the cost a relative-entropy meaning? → holographic-dual-embezzlement-protocol
4. Does the Euclidean-wormhole factorization puzzle have anything to say about, or borrow from, large-N factorization of single-trace operators — the same factorization statement that makes the boundary algebra type III${}_1$?

Related Papers

• 2025-liu-lectures-entanglement-vna — Liu’s review; develops the algebraic ER=EPR and crossed-product side of the same story.
• Maldacena, Qi (2018), Eternal traversable wormhole (arXiv:1804.00491) — the §5.2.2 construction; already a key paper in gauge-gravity-duality and the AQFT course.
• Gao, Jafferis, Wall (2017), Traversable wormholes via a double trace deformation (arXiv:1608.05687) — the §5.2.1 mechanism; AQFT course Sem II Wk 11.
• Maldacena, Stanford, Yang (2017), Diving into traversable wormholes (arXiv:1704.05333) — AQFT course Sem II Wk 14.
• Ahmad, Jefferson (2025), Algebraic perturbation theory: traversable wormholes and generalized entropy (arXiv:2501.01487) — the algebraic/cocycle treatment of GJW; AQFT course Block 4.
• Maldacena (2003), Eternal black holes in AdS (arXiv:hep-th/0106112) — the TFD/eternal-BH foundation (held in Zotero, key UNERBER6).