⚛️ high-energy theory

Holographic RG flows and wormholes from sinusoidal scalars

This paper investigates semiclassical holographic geometries induced by sinusoidal scalar sources on two AdS boundaries, revealing that large wormholes become the dominant saddle beyond a threshold—implying exponentially large ensemble fluctuations—and correspond to exotic, multivalued RG flows that connect to a gapped infrared phase.

Original authors: Pompey Leung

Published 2026-08-11

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CC BY 4.0

Original authors: Pompey Leung

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic hologram. In this view, the three-dimensional world we see and feel is actually a projection of information stored on a distant, two-dimensional surface, much like a 3D movie is projected from a flat film strip. This idea, known as the holographic principle, suggests that the complex laws of gravity inside our universe are secretly encoded in a simpler, quantum world living on its edge. Scientists are obsessed with understanding how these two worlds talk to each other. They use a mathematical tool called a "renormalization group flow" to trace how the rules of physics change as you zoom in or out, like adjusting the focus on a camera. Usually, if you zoom deep enough into a quantum system, it either settles into a new, stable state or hits a dead end where things get "gapped" (meaning the system stops reacting and becomes rigid). But what happens if you try to connect two separate universes with a tunnel? This is the puzzle of "wormholes." In the holographic world, these tunnels are tricky because they seem to break the rule that two separate things should stay separate. If two universes are truly independent, their combined story should just be the sum of their individual stories. But wormholes suggest they are secretly linked, creating a mathematical headache for physicists trying to keep the universe's bookkeeping balanced.

This paper, written by Pompey Leung, dives into this headache by building a specific, simplified model of these wormholes. Instead of using the usual, messy ingredients, the author uses a very neat trick: "sinusoidal scalars." Imagine the boundary of our universe not as a flat, boring sheet, but as a surface covered in a perfect, repeating wave pattern, like the ripples on a pond or the stripes on a zebra. By turning on these wave-like patterns as "sources" (inputs that tweak the system), the author creates a playground to see how gravity reacts. The goal was to see if these wave-patterned universes would form wormholes, and if so, whether those wormholes would be the dominant way the universe behaves, or just a rare, weak possibility.

The results are surprising and break the pattern of what scientists expected. In previous studies, wormholes were like shy guests at a party: they would show up, but only if you turned up the volume (the source strength) just right. Even then, they were often the "second choice," with the disconnected, separate universes being the more popular option for a while. It was thought that there would be a smooth transition, a "phase change," where the universe slowly decided to switch from being two separate rooms to one connected tunnel.

However, this paper finds that with these wave-patterned inputs, the rules are completely different. The author discovers that wormholes do exist, but they appear only after a certain threshold of wave strength is reached. Once they appear, they don't just show up; they immediately take over. There is no "shy" phase where the wormhole exists but is weak. The moment the wormhole is big enough to exist, it becomes the strongest, most dominant version of the universe. It's as if you flip a switch, and instead of a dim light slowly turning on, the room instantly floods with blinding light. The "disconnected" version of the universe, where the two sides are separate, is instantly pushed aside.

The paper also explores what these wormholes mean for the "flow" of physics. The author interprets the separate universes as a "boomerang" journey: the physics starts at a certain point, wanders through some strange, exotic changes, and then magically returns to exactly where it started. It's like a runner who starts at the finish line, runs a wild loop through a forest, and ends up back at the finish line without ever stopping. The wormholes, on the other hand, represent a journey that ends in a "gapped" phase, where the physics hits a wall and stops, effectively closing the tunnel.

Crucially, the paper rules out the idea of a smooth, gradual transition between these states. In other models, there was a middle ground where the universe was undecided. Here, the transition is sharp and sudden. The author suggests that this means if wormholes exist in this specific setup, the statistical "ensemble" (the collection of all possible universes) is wildly unstable. It's not a calm, average situation; it's a chaotic one where the average doesn't tell you much because the fluctuations are huge.

The study relies on numerical simulations and mathematical calculations of "on-shell actions" (a way of measuring the energy or "cost" of a specific shape of spacetime). The author is confident in their finding that the large wormhole is always the winner once it exists, but they acknowledge that the stability of these wormholes and the exact nature of the "hidden" contributions that might fix the factorization problem are still open questions. They don't claim to have solved the mystery of wormholes forever, but they have definitely shaken up the map, showing that in this specific, wave-patterned world, the wormhole doesn't just knock on the door; it kicks it down and takes over the house the moment it arrives.

Technical Summary: Holographic RG flows and wormholes from sinusoidal scalars

Problem Statement
The paper addresses the "factorization problem" in the AdS/CFT correspondence. In a standard quantum field theory (QFT) setup, the partition function on a disjoint union of manifolds, $Z[\Sigma_1 \sqcup \Sigma_2]$ , should factorize into $Z[\Sigma_1]Z[\Sigma_2]$ . However, the gravitational path integral includes Euclidean wormhole geometries connecting the boundaries, leading to a connected contribution $Z_{WH}$ that violates this factorization. While the "ensemble interpretation" (where gravity computes an average over an ensemble of theories) resolves this by correlating the theories, the statistical properties of this ensemble depend on the dominance of wormhole saddles. Previous bottom-up models with inhomogeneous matter (specifically those with compact boundaries like $S^3$ or $T^3$ ) exhibited a Hawking–Page-like (HP-like) phase transition: a regime where wormholes exist but are subdominant to the disconnected geometry, implying a sharply peaked (self-averaging) ensemble. This paper investigates whether such a subdominant regime exists for wormholes with non-compact, planar ( $\mathbb{R}^d$ ) boundaries sourced by sinusoidal scalar fields.

Methodology
The author studies semiclassical geometries in Einstein gravity coupled to $d$ free complex scalar fields in $(d+1)$ -dimensional asymptotically AdS space.

Ansatz: The boundary sources are turned on as identical sinusoidal functions: $J_I(\vec{x}) = J e^{ikx_I}$ . This specific ansatz induces bulk geometries that are homogeneous and isotropic in the boundary directions, allowing the metric to be written in Friedmann–Lemaître–Robertson–Walker (FLRW) form.
Solutions: The author solves the coupled Einstein and Klein–Gordon equations numerically for two types of saddles:
1. Disconnected: Two copies of a single-boundary geometry.
2. Connected: $Z_2$ -symmetric Euclidean wormholes connecting two $\mathbb{R}^d$ boundaries.
RG Interpretation: The radial coordinate is interpreted as the renormalization scale. The author computes effective holographic $\beta$ -functions and $c$ -functions to characterize the flow of the dual field theory.
Phase Structure: To determine which saddle dominates the gravitational path integral, the author calculates the renormalized on-shell action for each solution. They compare the action density of the disconnected saddle against the small and large wormhole branches across the parameter space of the dimensionless source strength $\tilde{J} \equiv J/k^{d-\Delta}$ . The specific case $d=3, \Delta=2$ is used for concreteness, though the methods generalize.

Key Results

Existence of Solutions:
- Disconnected: One-boundary solutions exist for any source strength $J$ .
- Wormholes: Wormhole solutions only exist above a critical threshold of the source strength, $\tilde{J}_{crit}$ . Above this threshold, two branches of solutions appear: a "small" wormhole and a "large" wormhole. These branches merge at a "critical wormhole" point.
Holographic RG Flows:
- Boomerang Flows (Disconnected): The disconnected geometry corresponds to a "boomerang" RG flow. The theory starts at a UV fixed point, flows away, and returns to the same CFT in the IR. This flow is characterized by a multivalued $\beta$ -function (with both positive and negative branches) and a non-monotonic $c$ -function that returns to its UV value. The flow is driven by the gradient energy of the sinusoidal scalars, which renders the deformation effectively irrelevant in the IR.
- Gapped Flows (Connected): By "folding" the wormhole at the throat, the connected geometry is interpreted as a single-sided RG flow that terminates at a finite radial depth. This corresponds to a flow to a gapped phase in the IR, where the "pinching off" of the geometry manifests as a mass gap in the dual theory.
Saddle Dominance and Phase Diagram:
- The central result is the structure of the phase diagram for the two-boundary path integral.
- Unlike previous models with compact boundaries, there is no regime where wormholes are subdominant.
- For $\tilde{J} < \tilde{J}_{crit}$ , only the disconnected saddle exists.
- For $\tilde{J} \geq \tilde{J}_{crit}$ , the large wormhole is always the dominant saddle, possessing a strictly lower action than both the small wormhole and the disconnected geometry.
- The transition from the disconnected phase to the connected phase is discontinuous (a "zeroth-order" phase transition); the action difference jumps from zero to a positive value immediately as the wormhole solutions appear. There is no smooth Hawking–Page-like transition where the action difference passes through zero.

Significance and Claims
The paper claims that the absence of a subdominant wormhole regime has profound implications for the ensemble interpretation of the gravitational path integral:

Non-Self-Averaging: In models where wormholes are subdominant for a range of parameters, the ensemble is "sharply peaked" around the mean (self-averaging). In this model, because the large wormhole dominates immediately upon existence, the putative ensemble exhibits exponentially large fluctuations relative to the mean as soon as wormholes are possible. The variance of the partition function is exponentially large compared to the square of the mean.
Factorization: The results imply that factorization of the boundary partition function is abruptly spoiled for large sources, with no intermediate regime of approximate factorization.
Exotic RG Flows: The work provides a concrete semiclassical realization of exotic RG flows with multivalued $\beta$ -functions, demonstrating that such flows can arise naturally from inhomogeneous sources without requiring multiple AdS vacua.

The author notes that while these results are derived from semiclassical gravity, the lack of a known microscopic dual for this specific setup (unlike $N=4$ SYM) leaves the precise nature of the dual ensemble open. They suggest that the absence of the HP-like transition may be due to the lack of a compact length scale (present in $S^3/T^3$ models) and speculate that hidden saddles (e.g., half-wormholes) or contour deformations might be necessary to restore factorization if the dual theory is indeed a single theory rather than an ensemble.

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