AdS3 axion wormholes as stable contributions to the Euclidean gravitational path integral

This paper generalizes the stability of Euclidean Giddings-Strominger axion wormholes from 4D Minkowski to AdS3 spacetime by explicitly constructing and analyzing regular classical solutions, thereby establishing them as stable contributions to the 3D gravitational path integral with potential implications for resolving wormhole paradoxes like the factorization problem.

The Gist

The Big Picture: The "Universe Soup"

Imagine the universe isn't just a single, smooth sheet of fabric, but a bubbling pot of soup where every possible shape and connection of space-time is constantly popping in and out of existence. In the world of quantum gravity, physicists try to calculate the "recipe" for this universe by adding up every possible shape (topology) that space could take.

For a long time, there was a scary thought: some of these shapes are wormholes (tunnels connecting different parts of space). If these wormholes are real and stable, they cause a major problem. They act like a leak in the universe's logic, breaking the rule that distant parts of the universe should be able to act independently (a principle called "cluster decomposition"). It's like if two people on opposite sides of the world could instantly whisper secrets to each other without any phone or signal, breaking the rules of how the world works.

To fix this, many physicists hoped that these wormhole shapes were unstable—like a house of cards that collapses the moment you try to build it. If they collapse, they don't count in the recipe, and the universe remains safe.

The New Discovery: The Wormholes are Sturdy

This paper, by Andrew Loveridge and Hao-Yu Sun, investigates a specific type of wormhole in a 3-dimensional universe (a simplified model of our reality) that has a negative curvature (like a saddle or a Pringles chip, known as AdS3).

They found that these wormholes do not collapse. They are stable.

Here is how they broke it down:

1. Building the Wormhole (The Classical Solution)

The authors built a mathematical model of these wormholes.

• The Shape: They looked at wormholes shaped like spheres, donuts (torus), and more complex hyperbolic shapes.
• The Glue: To keep the wormhole from pinching shut, they used a "magnetic" field (which, in this 3D world, acts like a particle called an axion). Think of this field as the air pressure inside a balloon that keeps it inflated.
• The Result: They proved that you can glue two halves of a wormhole together to make a smooth, complete tunnel that doesn't have any sharp edges or "cracks" (singularities). It is a perfectly valid shape for space to take.

2. Testing the Stability (The Stress Test)

Just because a shape exists doesn't mean it's stable. The authors performed a "stress test" by shaking the wormhole with tiny ripples (perturbations) to see if it would fall apart.

• The Shake: They imagined wiggling the magnetic field and the shape of space slightly.
• The Outcome: In most cases, the wormhole resisted the wiggles and returned to its original shape. It is a stable minimum.
• The Twist (The Donut): There was one tricky case: the donut-shaped (torus) wormhole. Initially, it looked like it might be unstable. However, the authors realized that the rules of the universe (boundary conditions) in this specific 3D model forbid the specific type of wiggle that would break it. Once you apply the correct rules, even the donut wormhole is stable.

3. Calculating the Cost (The Action)

In physics, every shape has a "cost" (called an action). Nature prefers low-cost shapes. The authors calculated this cost for their wormholes.

• They found the cost depends on how much "magnetic charge" (the air pressure in our balloon analogy) is inside the wormhole.
• The more charge, the higher the cost, but the shape remains a valid option for the universe to choose.

Why This Matters (According to the Paper)

The paper concludes that these wormholes are real, stable contributors to the quantum gravitational path integral.

• The Paradox: Because they are stable, they must be included in the calculation of how the universe works.
• The Problem: Including them leads to the "factorization problem" mentioned earlier. It suggests that the universe might not be able to keep its distant parts independent, which creates a conflict with our current understanding of quantum mechanics and how the universe behaves (specifically in the dual "CFT" theory that describes the edge of this universe).

The Bottom Line

The authors have shown that in this specific 3D model of gravity, the "house of cards" (the wormholes) is actually made of steel. They are stable, smooth, and mathematically sound.

This means the "easy fix" for the wormhole paradox—that they just don't exist because they are unstable—is not working for this type of universe. The paradox remains, suggesting that either our understanding of quantum gravity needs a major overhaul, or there are other, more complex effects (like those from String Theory) that we haven't fully accounted for yet. The paper does not solve the paradox; it simply proves that the wormholes are sturdy enough to be part of the problem.

Technical Summary

Technical Summary: AdS3 Axion Wormholes as Stable Contributions to the Euclidean Gravitational Path Integral

Problem Statement
The inclusion of topologically non-trivial spacetimes, specifically wormholes, in the Euclidean gravitational path integral presents a fundamental tension in quantum gravity. While such contributions are necessary for background independence and have successfully reproduced black hole thermodynamics and the Page curve, they introduce pathologies such as the violation of cluster decomposition, random couplings, and the factorization problem in dual Conformal Field Theories (CFTs). A potential resolution to these paradoxes, proposed by Hertog et al. [26], is that such wormhole solutions might be unstable saddle points and thus excluded from the path integral. However, subsequent analyses [28, 30] demonstrated that Giddings–Strominger axion wormholes are stable in 4D asymptotically flat spacetime under specific boundary conditions (fixing axion charge and boundary metric), challenging the instability hypothesis. The stability of these solutions in Anti-de Sitter (AdS) backgrounds, particularly in the context of the AdS/CFT correspondence where cluster decomposition violations are critical, remained an open question due to computational difficulties in 4D.

Methodology
This paper generalizes the stability analysis of axion wormholes to three-dimensional asymptotically AdS spacetime ($AdS_3$). In $D=3$, the axion field responsible for Giddings–Strominger wormholes is dual to a $U(1)$ gauge field (the "dual photon"), allowing for a tractable analysis using the ADM formalism.

1. Classical Solutions: The authors construct classical Euclidean solutions for Einstein-Hilbert gravity with a negative cosmological constant ($\Lambda = -1/l^2$) coupled to a $U(1)$ gauge field. They derive solutions for three distinct spatial topologies:

   • Spherical topology ($S^2$).
   • Torus topology ($T^2$).
   • Hyperbolic quotients (genus $g > 1$).
     The solutions are constructed by "gluing" two half-wormhole geometries (defined by a minimal throat radius $\tau_{min}$) to form complete, non-singular two-sided wormholes.

2. Stability Analysis: The authors perform a perturbative stability analysis around these classical solutions. They decompose fluctuations into Scalar-Vector-Tensor (SVT) modes. Given the absence of bulk propagating gravitons in 3D, the analysis focuses on the vector sector of the $U(1)$ field and its gravitational backreaction (shift vector perturbations).

   • Spatially Varying Modes: The authors demonstrate that electromagnetic and gravitational perturbations decouple for all non-zero eigenmodes of the spatial Laplacian. The quadratic action for these modes is shown to be positive definite.
   • Zero Modes: For the torus topology, which admits constant vector zero modes, the authors explicitly compute the perturbative action. Crucially, they apply boundary conditions appropriate for $D=3$: fixing the normal component of the field strength (equivalent to fixing the electric charge) rather than the potential. Under these conditions, the zero-mode perturbations are forbidden, ensuring stability.

3. Action Computation: The Euclidean action is computed for the half-wormhole solutions (which can be doubled for two-sided solutions) using the ADM action plus the Gibbons-Hawking-York boundary term and holographic counterterms.

Key Contributions and Results

• Explicit Construction: The paper provides explicit, regular classical solutions for axion wormholes in $AdS_3$ with spherical, toroidal, and hyperbolic topologies. The solutions are shown to be non-singular, with finite curvature invariants (Kretschmann scalar) and field strengths, provided the magnetic charge $n > 0$.
• Stability Proof: The primary result is the demonstration that these $AdS_3$ wormhole solutions are stable saddle points.
  • For spherical and hyperbolic topologies, stability holds for all modes, including zero modes (which do not exist for these topologies due to the Poincaré-Hopf theorem).
  • For the torus topology, stability is achieved specifically because the physically correct boundary conditions for $D=3$ (fixing field strength/charge) eliminate the potentially unstable zero modes. This contrasts with 4D flat space results where stability depended on the choice of boundary conditions.
• Action Values: The authors compute the Euclidean action for the torus and sphere cases.
  • For the torus, the action is $I_{torus} = n\pi / \sqrt{8\alpha\kappa}$, showing a linear dependence on the magnetic charge units $n$.
  • For the sphere, a more complex expression is derived, which asymptotically approaches a linear scaling with $n$ for large charges.
• Boundary Condition Nuance: The paper highlights that in $D=3$, the dual of the bulk gauge field corresponds to a conserved charge in the boundary CFT only if the field strength is fixed at the boundary. Fixing the potential (Neumann conditions for the dual scalar) would violate unitarity bounds in the CFT. This distinction is critical for the stability of the torus solution.

Significance and Claims
The authors claim that their findings suggest it is increasingly unlikely that the "wormhole paradoxes" (factorization, loss of cluster decomposition) can be resolved simply by asserting that such topologies are unstable saddle points. Since these $AdS_3$ solutions are stable and contribute to the path integral, they inherently introduce the associated pathologies in the dual CFT description.

The paper positions these results as a step toward understanding the gravitational path integral in a context where holographic applications are well-studied. The authors note that while their solutions differ from 4D results in important respects (specifically regarding boundary conditions), the persistence of stable wormholes suggests that any resolution to the paradoxes must lie elsewhere, potentially within the UV completion of the theory (e.g., String Theory). They suggest that future work should examine the inclusion of additional fields (such as dilatons) and Chern-Simons terms, or explore the tensionless limit of string theory, to see if these effects render the solutions unstable or redundant. The paper concludes that the stability of these solutions necessitates a re-evaluation of how topological contributions are treated in the context of unitarity and locality in quantum gravity.