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A PDE approach to the 2D Yang-Mills measure

This paper introduces the concept of rough additive functions to establish a PDE-based framework for defining controlled gauge transformations and holonomies, proving a rough version of Uhlenbeck compactness via regularity structures, and demonstrating an optimal regularity gauge-fixed representation of the 2D Yang-Mills measure on the unit square.

Original authors: Ilya Chevyrev, Tom Klose, Abdulwahab Mohamed

Published 2026-07-27

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CC BY 4.0

Original authors: Ilya Chevyrev, Tom Klose, Abdulwahab Mohamed

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 is stitched together by invisible threads of force, holding atoms together and governing how particles dance. Physicists call this the "gauge theory," and the specific rulebook for these forces is called the Yang–Mills theory. It's like a cosmic game of "connect the dots," but the dots are constantly shifting, and the lines connecting them are wild, jittery, and impossible to pin down with a ruler. For decades, mathematicians have tried to write a precise instruction manual for these forces, but the instructions keep getting lost in the noise. The problem is that at the tiniest scales, these forces behave like static on a radio or the chaotic spray of a broken hose—so messy that standard math tools simply break.

To make sense of this chaos, scientists use a concept called "gauge symmetry." Think of it like a map. If you draw a map of a city, you can choose to orient North up, North down, or even North sideways. The city doesn't change; only your description of it does. In physics, the "gauge" is just your choice of orientation. The real physics—the actual forces—should look the same no matter how you orient your map. The challenge is that when you try to zoom in on the messy, jittery forces of the Yang–Mills theory, the "map" itself starts to tear apart. The math gets so rough that you can't even define what a "line" or a "curve" is anymore. This paper tackles the question: "Can we find a way to smooth out this map so we can actually see the picture, even when the underlying forces are incredibly rough?"

The authors of this paper, Ilya Chevyrev, Tom Klose, and Abdulwahab Mohamed, have developed a new mathematical toolkit to solve this puzzle. They introduce a concept they call "rough additive functions." Imagine trying to measure the distance a hiker walks through a dense, foggy forest where the path keeps changing shape. Standard math would say, "You can't measure this; the path is too jagged." But these authors say, "Wait, let's look at the steps the hiker takes and the turns they make, even if we can't see the whole path." They treat the jagged, noisy forces not as a broken mess, but as a structured, albeit very rough, object that can be measured.

Using this new idea, they prove a "rough version" of a famous theorem called Uhlenbeck compactness. In the old days, mathematicians knew that if you had a smooth, well-behaved force field, you could always find a "perfect map" (a specific gauge) where the forces looked neat and tidy. The authors show that even when the forces are as messy as a stormy sea (mathematically, "distributional connections"), you can still find a way to smooth them out, provided you use their new "rough path" tools. They didn't just guess this; they built a rigorous mathematical engine using "regularity structures"—a fancy way of saying they created a new kind of microscope that can see patterns in the noise.

The paper focuses on a square-shaped piece of space (like a tiny tile on a floor) and shows that for the 2D Yang–Mills measure (the probability of how these forces behave), there is indeed a "gauge-fixed" version that is as smooth as mathematically possible. They didn't just say it exists; they showed exactly how to construct it and proved that if you slightly change the input (the messy noise), the output (the smooth map) changes in a predictable, stable way. This is a big deal because it means we can finally treat these wild, quantum forces with the same precision we use for smoother, classical physics.

While they focused on a simple square for this study, they believe their method is strong enough to be applied to more complex shapes, like spheres or donuts, in the future. They also hint that this approach could help solve the even harder problem of 3D space, where the messiness is even worse. However, they are careful to note that the 3D version is still an open question; they haven't solved it yet, but they've built the ladder they think will help them climb it.

In short, this paper doesn't just say "the forces are messy." It says, "We have a new way to measure the mess, and using that, we can prove the forces can be organized into a clean, understandable picture." It's a breakthrough in how we describe the fundamental fabric of reality, turning a chaotic scribble into a readable sentence.

Technical Summary: A PDE Approach to the 2D Yang–Mills Measure

Problem Statement

The paper addresses the question of the small-scale regularity of the 2D Yang–Mills (YM) measure $\mu_{YM}$ on the unit square $\Lambda = [0, 1]^2$ . While the YM measure can be constructed in a gauge-invariant manner (e.g., via Wilson loops), defining a representative connection 1-form $A$ with optimal regularity has been a challenge. Previous constructions, such as the axial gauge, yield representatives with regularity $C^{-1/2-\kappa}$ , which is suboptimal compared to the expected regularity of the Gaussian Free Field (GFF), $C^{-\kappa}$ . The central question is whether one can find a gauge-fixed representative $A_g$ of the YM measure that possesses the optimal regularity $C^{-\kappa}$ (or specifically, belongs to a space $\Omega^1_\beta$ for $\beta$ close to 1) and exhibits stability properties.

Methodology

The authors employ a combination of rough path theory, regularity structures, and geometric analysis to solve a singular elliptic partial differential equation (PDE) for the gauge transformation.

1. Rough Additive Functions (RAFs)

The authors introduce a new state space called rough additive functions ( $\Omega^1_{\alpha\text{-ax}}$ ), which extends the theory of rough paths to line integrals of distributional 1-forms.

Definition: An RAF is a pair $(A, \mathbb{A})$ where $A$ is an additive function (line integral) and $\mathbb{A}$ is an iterated line integral, satisfying Chen's identity and weak geometricity.
Role: This space encodes the necessary "enhancements" (iterated integrals) required to make sense of products of distributions in the singular PDE. Crucially, the authors show that the model needed to solve the gauge-fixing equation is determined entirely by these RAFs, which are geometrically natural objects derived from the curvature (specifically, the "lasso field" of Gross).
Advantage: This approach replaces stochastic analytic arguments with pathwise rough path arguments, providing continuity and stability of the solution map.

2. Controlled Gauge Transformations

To handle the non-linear gauge transformation $A \mapsto A_g = \text{Ad}_g A - dg g^{-1}$ in the rough setting, the authors define controlled gauge transformations ( $G^{\gamma}_A$ ). These are pairs $(g, g')$ where $g$ is a path in the Lie group $G$ controlled by the underlying RAF $A$ , analogous to Gubinelli's controlled rough paths. This structure allows for the rigorous definition of the gauge-transformed connection $A_g$ via rough differential equations (RDEs).

3. Solving the Coulomb Gauge Equation

The core analytic task is to find a gauge transformation $g$ such that the transformed connection $A_g$ satisfies the Coulomb gauge condition $d^* A_g = 0$ .

The Equation: In the axial gauge ( $A_2=0$ ), this condition becomes a singular, Lie-group valued elliptic PDE for $g$ :
$\sum_{i=1}^2 \partial_i ((\partial_i g)g^{-1}) - \partial_1 (g A_1 g^{-1}) = 0$
Regularity Structures: The authors construct a specific regularity structure tailored to this equation. Unlike standard singular SPDE approaches, they do not rely on fixed-point theorems. Instead, they use a method inspired by the Implicit Function Theorem (IFT).
Strategy:
1. Scaling: They zoom into small squares to ensure the norm of the input RAF is small.
2. ODE Formulation: They formulate the solution as an ODE in the scaling parameter $\sigma$ : $\partial_\sigma g_\sigma = - (D^2 \Phi)^{-1} \partial_\sigma \Phi$ .
3. Invertibility: They prove that the linearized operator (the second derivative of the PDE map) is invertible on the space of modelled distributions.
4. Reconstruction: The solution $g$ is reconstructed from modelled distributions, yielding a connection $A_g$ with improved regularity.

4. Patching

Since the solution theory works on small squares, the authors employ a patching argument (inspired by Uhlenbeck's classical compactness) to glue local solutions into a global gauge transformation on the unit square. This involves constructing compatible gauge transformations on overlaps and using a partition of unity adapted to the rough setting.

Key Results

1. Rough Uhlenbeck Compactness (Theorem 1.1)

The paper establishes a deterministic "rough version" of Uhlenbeck's compactness theorem.

Statement: For any rough additive function $A \in \Omega^1_{\alpha\text{-ax}}$ (with $\alpha \in (4/9, 1/2)$ ), there exists a controlled gauge transformation $g$ such that the transformed connection $A_g$ belongs to $\Omega^1_\beta$ (with $\beta \in (0, 6\alpha-2)$ ).
Bounds: The regularity of $A_g$ is bounded by the norm of the input RAF: $|A_g|_\beta \lesssim (1 + \|A\|_{\alpha\text{-ax}})^{2+\beta-\alpha} \|A\|_{\alpha\text{-ax}}$ .
Continuity: The map $A \mapsto (g, A_g)$ is locally Lipschitz continuous on bounded sets. This is a significant improvement over previous works, providing explicit continuity statements for the gauge-fixing map.

2. Gauge-Fixed YM Measure (Theorem 1.2)

Applying the deterministic result to the probabilistic setting:

Construction: The YM measure in the axial gauge is lifted to a random rough additive function $A$ using a Kolmogorov continuity theorem for RAFs.
Result: There exists a representative $B$ of the YM measure in the Coulomb gauge such that $B \in \Omega^1_\beta$ almost surely.
Moment Bounds: The paper derives sharp moment bounds for the norm of the representative: $E[|B|_\beta^p]^{1/p} \lesssim p^{(2+\beta)/2 + \delta}$ . This significantly improves upon previous bounds (e.g., from [CS26a]) which were exponential in $p$ .
Stability (Wong–Zakai): The paper proves that smooth approximations of the YM curvature in the axial gauge yield smooth approximations of the gauge-fixed representative $B$ that converge in $\Omega^1_\beta$ .

Significance and Claims

The authors position this work as a significant advancement in the regularity theory of the 2D YM measure:

Optimal Regularity: The constructed representative achieves essentially optimal regularity (close to the GFF), improving upon the suboptimal regularity of axial gauge representatives.
Geometric Naturalness: The use of rough additive functions provides a geometrically natural framework. The "model" required for the regularity structure is derived directly from the RAF (iterated line/area integrals of the curvature), rather than being an ad-hoc construction.
Continuity and Stability: Unlike previous lattice-based or inductive approaches (e.g., [Che19]), this method works directly in the continuum and provides Lipschitz continuity of the gauge-fixing map. This is crucial for studying large- $N$ limits and master fields.
Generalizability: While the probabilistic construction is currently restricted to the unit square, the authors argue that the core analytic ingredient (Rough Uhlenbeck Compactness) is general. They anticipate that the results can be extended to general surfaces once a local construction of RAFs from the YM measure is established on those surfaces.
Methodological Innovation: The approach of solving geometric singular PDEs via an ODE formulation in the context of regularity structures (inspired by the IFT) is presented as a novel and potentially more transparent alternative to standard fixed-point arguments in singular SPDE theory.

The paper concludes that while the 2D case is now well-understood in terms of regularity, the extension to 3D remains an open problem due to the lack of well-defined line integrals and the need for new gauge-covariant regularisation techniques.