The Yang--Mills measure on compact surfaces as a universal scaling limit of lattice gauge models

In this article, we study the 2 dimensional Yang-Mills measure on compact surfaces from a unified continuum and discrete perspective. We construct the Yang-Mills measure as a random distributional 1-form on surfaces of arbitrary genus equipped with an arbitrary smooth area form, using geometric tools based on zero-area bands and cylindrical resolutions. This yields a canonical bulk-singular decomposition of the measure, reflecting the topology of the surface. We prove a universality theorem stating that the continuum Yang-Mills measure arises as the scaling limit of a wide class of lattice gauge theories, including Wilson, Manton, and Villain actions, on any compact surface. We study the convergence in natural spaces of distributions with anisotropic regularity. As further consequences, we obtain a new intrinsic construction of the Yang-Mills measure, independent of the previous constructions in the literature, and prove the convergence of correlation functions and Segal amplitudes on all compact surfaces.

💡 Research Summary

The paper presents a comprehensive study of two‑dimensional Yang–Mills theory on arbitrary compact surfaces, unifying the continuum construction of the Yang–Mills measure with the scaling limits of a broad class of lattice gauge models. The authors first construct a random distributional 1‑form (a “random connection”) on any closed surface Σ of genus g equipped with a smooth area form σ. Their construction relies on new geometric tools: zero‑area bands, cylindrical resolutions, and pseudo‑coordinates derived from a Morse function on Σ. By cutting Σ along the flow lines and level sets of the Morse function, they represent the surface globally as a cylinder with a finite set of defect lines. This reduction allows all probabilistic arguments to be carried out on the cylinder, where the geometry is simple, and then transferred back to the original surface.

A central analytic innovation is the introduction of anisotropic Sobolev and Hölder–Besov spaces that reflect the directional scaling inherent in the cylindrical representation. The authors define weighted anisotropic norms that assign different regularity exponents to the longitudinal and transverse directions of the cylinder. This framework captures the “bulk–singular” decomposition of the Yang–Mills measure: the bulk part enjoys higher regularity, while singular contributions concentrate near saddle points of the Morse function and along defect lines. Precise regularity estimates are proved for the random connection in these spaces, including extensions across defect lines and control near saddle neighborhoods.

On the discrete side, the paper builds a family of increasingly fine lattices (T_N) on Σ using the Morse flow. Each lattice is equipped with a piecewise‑smooth random G‑valued 1‑form A_N whose holonomy process satisfies the Driver–Sengupta formula for a given family of probability measures (Q_t dg)_{t>0} arising from Wilson, Manton, or Villain actions. The authors call this the “Morse gauge” on the lattice. They establish a local limit theorem for the associated random walks on the Lie group and derive technical estimates that guarantee uniform bounds for all considered actions.

The main universality theorem states that, as the mesh size of the lattice tends to zero, the sequence (A_N) converges in law to the continuum Yang–Mills connection constructed earlier. Convergence is proved in the anisotropic Besov topology, and tightness is shown via uniform probabilistic bounds and compact embeddings. The proof proceeds in two stages: (i) convergence on the cylinder, where the anisotropic regularity is handled directly, and (ii) “closing” the surface, where the discrete holonomy measures are disintegrated and the conditioned measures are shown to converge to the continuous holonomy process. The authors also verify convergence of correlation functions and Segal amplitudes, providing a full functional‑analytic description of the scaling limit.

In addition to the main results, the paper supplies extensive technical appendices covering pathwise integration of rough differential forms, restriction theorems for Besov spaces, Young product estimates, spectral analysis of Morse–Smale flows, and continuous/compact embeddings in anisotropic spaces. These tools underpin the rigorous treatment of the non‑Gaussian nature of the Yang–Mills measure and the gauge‑invariance constraints.

Overall, the work delivers a novel intrinsic construction of the two‑dimensional Yang–Mills measure that is independent of earlier approaches, demonstrates its universality as the scaling limit of a wide range of lattice gauge theories, and establishes convergence in finely tuned anisotropic functional spaces. This advances the mathematical foundation of low‑dimensional gauge theory and opens pathways for applying similar techniques to higher‑dimensional settings or to stochastic partial differential equations with gauge symmetry.