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Bootstrapping Wilson lines in Critical Bosonic QED $_3$ Part 1: Large- $N$ endpoint correlators and CFT data

This paper employs a combination of diagrammatic four-point calculations and endpoint crossing equations to derive closed-form expressions for Wilson line correlators in critical bosonic QED $_3$ at large $N$ , revealing a surprising agreement between the bulk and Wilson-line spectra in non-singlet flavor channels through order $1/N$ .

Original authors: Jun Maeda, Toshiki Onagi, Soichiro Shimamori

Published 2026-10-30

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CC BY 4.0

Original authors: Jun Maeda, Toshiki Onagi, Soichiro Shimamori

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

In the universe of the very small, matter does not always behave like solid particles or smooth waves. Instead, at the precise moment when a material undergoes a fundamental change of state—like a magnet losing its magnetism or a fluid becoming a superconductor—it settles into a strange, frozen state of perfect balance known as a critical point. At this point, the system has no preferred scale; a tiny ripple looks just like a massive wave, and the rules governing the behavior of the material are dictated by symmetry and geometry rather than by the specific details of the atoms involved. Physicists call this state a conformal field theory. It is a universal language that describes how energy and information flow through these chaotic, fluctuating systems. For decades, scientists have tried to decode this language, mapping out the "spectrum" of the theory: the list of all possible vibrations or patterns the system can make, and how strongly they interact with one another.

Usually, this decoding happens by looking at the bulk of the material, the vast interior where particles interact freely. But nature often hides its most interesting secrets at the edges. Just as a shoreline is a distinct place where the ocean meets the land, the interior of a quantum system can host special lines or surfaces where the rules change. These are called defects. A line defect is like a thread running through the fabric of space, and it can carry its own unique set of vibrations and rules. One particularly important kind of line defect is the Wilson line, a theoretical thread that traces the path of an electric charge. In many theories, these lines are just mathematical tools, but in certain exotic materials, they represent real, physical objects that can end, split, or merge. Understanding what happens at the very tip of such a line—where the thread meets the rest of the universe—is crucial for understanding how these materials behave, yet it has remained one of the hardest puzzles in theoretical physics.

A team of researchers has now cracked a significant part of this puzzle by developing a new way to listen to the tips of these lines. They focused on a specific, highly complex material model known as critical bosonic QED3, which is a three-dimensional theory of charged particles interacting with a force field. This model is a prime example of a system where particles are so strongly coupled that traditional calculation methods fail. The researchers wanted to know exactly what kind of vibrations exist at the end of a charged line in this material and how those vibrations talk to the rest of the system. To do this, they combined two powerful approaches. First, they used a method called diagrammatic calculation, which involves drawing and summing up billions of possible interaction paths between particles to see what the math says. Second, they applied a technique known as the conformal bootstrap, which uses the fundamental rules of symmetry to force the math to reveal its own secrets without needing to solve every single interaction.

The researchers treated the end of the line not as a dead end, but as a special operator, a point where the line changes its identity. They imagined placing four of these endpoints in a specific arrangement and calculating how they influence one another. By doing this, they could set up a set of equations that act like a mirror, reflecting the properties of the line's interior against the properties of the bulk material. The key insight was that by looking at the endpoints, they could see the same information from two different angles: one angle showing the bulk material, and the other showing the line itself. Because the math must be consistent from both angles, the researchers could use the known behavior of the bulk to deduce the unknown behavior of the line.

The results were surprisingly precise. The team calculated the properties of these line endpoints through a high level of approximation, finding closed-form expressions that describe exactly how the endpoints behave for any integer amount of electric charge. They determined the "dimension" of the endpoint, which is a measure of how its influence fades with distance, and they calculated a quantity called the defect g-function, which acts like a measure of the entropy or information content stored on the line. Perhaps most strikingly, they found that the spectrum of vibrations on the line is directly proportional to the spectrum of vibrations in the bulk material, at least for the lowest levels of energy. This means that the line is not a chaotic, independent world; it is a tightly constrained reflection of the universe around it.

For the simplest case, where the line carries a single unit of charge, the researchers went even further. They reconstructed the entire tower of higher-spin vibrations on the line, showing how they are built from the known vibrations of the bulk. They discovered that the line does not just copy the bulk; it mixes different types of bulk vibrations together in a very specific way, creating a weighted average of their properties. This mixing explains why the line's behavior looks the way it does. The study also showed that by changing the type of particle inserted near the endpoint, they could access new, previously hidden sectors of the theory, revealing a rich structure of new operators that appear when the line interacts with the bulk.

This work provides a rigorous, non-perturbative window into the behavior of charged lines in strongly coupled systems. It demonstrates that the bootstrap method, when applied to the endpoints of defects, can extract precise data that was previously out of reach. The findings confirm that the conformal bootstrap is not just a tool for the bulk of a theory but can be extended to its edges, offering a systematic way to map the landscape of defects in quantum field theories. By solving these equations, the researchers have provided a blueprint for understanding how charged lines behave in critical materials, a step that could eventually help in understanding the deconfined quantum critical points found in real-world quantum magnets and other exotic phases of matter. The study stands as a testament to the power of combining direct calculation with the deep constraints of symmetry to reveal the hidden architecture of the quantum world.

Technical Summary: Bootstrapping Wilson Lines in Critical Bosonic QED3 (Part 1)

Problem Statement
This work addresses the challenge of extracting conformal data for line defects in interacting three-dimensional gauge theories, specifically critical bosonic QED $_3$ with $N$ complex scalar flavors. While the conformal bootstrap has successfully determined bulk CFT data for various universality classes, applying it to defects is complicated by the lack of positivity in the standard bulk-to-defect crossing equations. For line defects that can terminate, such as Wilson lines in gauge theories with dynamical matter, the endpoints act as "defect-changing operators." The paper aims to formulate and solve an "endpoint bootstrap" for these operators, utilizing the positivity of both the bulk and defect Hilbert spaces to constrain the spectrum and Operator Product Expansion (OPE) coefficients of the Wilson line and its endpoints.

Methodology
The authors combine two complementary approaches:

Diagrammatic Large- $N$ Expansion: They perform explicit perturbative calculations of four-point correlation functions involving Wilson line endpoints and bulk Hubbard–Stratonovich fields ( $\sigma$ ) in the critical bosonic QED $_3$ model. These calculations are carried out to the first subleading order in the $1/N$ expansion, keeping the Wilson line charge $q$ fixed as an arbitrary positive integer.
Endpoint Crossing Equations: They formulate crossing equations for the four-point functions of endpoint operators. Unlike standard defect bootstrap equations, these relate the bulk channel (operators in the trivial defect sector $H_{00}$ ) to the defect channel (operators living on the charge- $q$ Wilson line, $H_{qq}$ ). Crucially, the authors incorporate the full $SU(N)$ flavor structure, deriving the appropriate Racah matrices and flavor tensors to ensure the OPE weights remain manifestly positive.

The strategy involves computing the correlators diagrammatically, projecting them onto irreducible flavor representations, and then using the crossing relations to extract conformal data (dimensions and OPE coefficients) that are not directly accessible from the diagrammatic expansion alone.

Key Contributions and Results

Formulation of Flavor-Structured Endpoint Bootstrap: The paper derives the complete crossing equations for Wilson line endpoints carrying arbitrary charge $q$ in the presence of $SU(N)$ flavor symmetry. It explicitly constructs the flavor tensors and Racah matrices required to relate the bulk and defect channels, ensuring the positivity of the spectral data.
Closed-Form CFT Data at Order $1/N$ : For arbitrary fixed charge $q$ $q$ , the authors obtain closed-form expressions for:
- The endpoint scaling dimension $\Delta_{E_q}$ and the defect $g$ -function $g_q$ .
- The dimensions and OPE coefficients of the lowest non-singlet Wilson-line primaries in every flavor channel $R_k$ (where $k=0, \dots, q$ ).
- Specifically, they find $\Delta_{E_q} = \frac{q}{2} - \frac{6q(q+2)}{\pi^2 N} + O(N^{-2})$ and $g_q = 1 + \frac{8q^2}{N} + O(N^{-2})$ .
Bulk-Defect Spectral Relation: A significant finding is the discovery of a proportionality relation between the bulk and defect channel functions through order $1/N$ . For non-singlet channels ( $k \neq 0$ ), the spectral data (dimensions and weighted OPE coefficients) in the defect channel are identical to those in the bulk channel up to a normalization factor dependent on $g_q$ and $q$ . This implies that the anomalous dimensions of the lowest operators in the bulk and on the line agree at this order.
Higher-Spin Reconstruction ( $q=1$ ): For the fundamental charge $q=1$ , the authors go beyond the lowest operators. They demonstrate that the endpoint correlator measures OPE-weighted averages of anomalous dimensions for degenerate levels. By reconstructing the bulk higher-spin spectrum restricted to the line, they independently verify that the diagrammatic results for the endpoint correlator match the weighted averages of the known bulk higher-spin anomalous dimensions.
Mixed Correlators: The study extends to mixed correlators involving two endpoints and two bulk $\sigma$ fields. This setup probes new defect-changing operators created by fusing an endpoint with a bulk scalar. The authors derive crossing equations for these mixed correlators and extract the OPE-weighted dimensions and squared OPE coefficients for the resulting tower of operators. For $q=1$ , where the lowest level is non-degenerate, they determine the individual dimension and OPE coefficient for the operator $:\sigma E^{(1)}:$ .

Significance and Claims
The paper claims to provide a systematic, analytic method for uncovering the conformal data of non-supersymmetric Wilson lines in interacting gauge theories. By combining diagrammatic calculations with the endpoint bootstrap, the authors demonstrate that one can rigorously constrain defect data using bulk positivity.

The work establishes a direct link between bulk CFT data and defect data, showing that for critical bosonic QED $_3$ , the bulk and defect spectra are tightly correlated through order $1/N$ . The authors note that this approach allows for the extraction of the defect $g$ -function and endpoint dimensions without relying solely on the large- $N$ expansion, as the crossing equations provide non-perturbative constraints on the cumulative OPE weights.

The paper positions this work as a foundational step ("Part 1") toward a broader program. It suggests that this framework can be extended to:

Investigate screening and string breaking phases of Wilson lines by analyzing the stability of the defect spectrum.
Test dualities of line defects (e.g., in fermionic QED $_3$ or Chern-Simons matter theories) by matching endpoint data across dual descriptions.
Perform "spectroscopy" of fractionalized excitations (spinons) in deconfined quantum critical points via gauge-invariant endpoint insertions.
Study critical transitions in anyon condensation by analyzing junction operators.

The authors emphasize that their results are specific to the large- $N$ limit with fixed $q$ and are derived analytically, providing a benchmark for future numerical bootstrap implementations and studies of other defect types.

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