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

Nguyen Viet Dang, Elias Nohra

TL;DR

The paper proves that the two-dimensional Yang–Mills measure on any compact surface emerges as a universal scaling limit of a wide class of lattice gauge theories. It builds a continuum YM measure as a random distributional 1-form via a global cylinder resolution, with a bulk white-noise part and a finite-entropy singular current supported on defect lines, and it establishes a Morse-gauge framework linking actions to Lévy’s holonomy process through Driver–Sengupta formulas. The main results show convergence of lattice holonomies, correlation functions, and partition functions across all compact surfaces in anisotropic Sobolev/Hölder spaces, thereby unifying continuum and discrete perspectives. The approach yields an intrinsic construction of the YM measure, clarifies the role of zero-area bands, and demonstrates universality across Wilson, Manton, and Villain actions, underpinning the connection between gauge actions and stochastic parallel transport with robust probabilistic control. This advances both mathematical rigor in low-dimensional gauge theory and the understanding of universality in lattice as limit objects for quantum field theories.

Abstract

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.

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

TL;DR

The paper proves that the two-dimensional Yang–Mills measure on any compact surface emerges as a universal scaling limit of a wide class of lattice gauge theories. It builds a continuum YM measure as a random distributional 1-form via a global cylinder resolution, with a bulk white-noise part and a finite-entropy singular current supported on defect lines, and it establishes a Morse-gauge framework linking actions to Lévy’s holonomy process through Driver–Sengupta formulas. The main results show convergence of lattice holonomies, correlation functions, and partition functions across all compact surfaces in anisotropic Sobolev/Hölder spaces, thereby unifying continuum and discrete perspectives. The approach yields an intrinsic construction of the YM measure, clarifies the role of zero-area bands, and demonstrates universality across Wilson, Manton, and Villain actions, underpinning the connection between gauge actions and stochastic parallel transport with robust probabilistic control. This advances both mathematical rigor in low-dimensional gauge theory and the understanding of universality in lattice as limit objects for quantum field theories.

Abstract

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.
Paper Structure (72 sections, 77 theorems, 355 equations, 31 figures)

This paper contains 72 sections, 77 theorems, 355 equations, 31 figures.

Table of Contents

  1. Introduction
  2. Contribution of the paper
  3. Separation of probabilistic and geometrical considerations.
  4. A universal scaling limit.
  5. A rigorous link between the action and Lévy's holonomy process.
  6. Convergence of correlation and partition functions on all compact surfaces.
  7. Structure of the paper
  8. Revisiting Driver's construction of $\mathrm{YM}^2$
  9. The case of the plane
  10. The case of the cylinder
  11. Defining line integrals.
  12. From line integrals to stochastic parallel transport.
  13. The case of a general surface
  14. The case of a general surface.
  15. Preliminaries on Morse theory
  16. ...and 57 more sections

Key Result

Theorem 1.1

The sequence $(A_N)_{N\geqslant 0}$ converges in law, in the Sobolev space $H_\mathrm{loc}^{-(1+\varepsilon)}(\Sigma\setminus f^{-1}(\max (f));\mathfrak{g})$ of distributional $\mathfrak{g}$-valued $1$-forms, to the Morse-gauge-fixed Yang--Mills measure on $\Sigma$. Moreover, the sequence $A_N$ conv

Figures (31)

  • Figure 1: Main steps.
  • Figure 2: The rectangle $\square(r_1,r_2,\theta_2,\theta_2)$ in the cylinder.
  • Figure 3: Morse coordinates.
  • Figure 4: A pant shaped surface.
  • Figure 5: A top view of the pant with the band of zero area.
  • ...and 26 more figures

Theorems & Definitions (160)

  • Theorem 1.1: Universal scaling limit of lattice gauge theories
  • Proposition 1.2: Bulk--singular decomposition of the free boundary Yang--Mills measure
  • Theorem 1.3: Convergence of Segal amplitudes
  • Definition 2.1: Definition-Proposition
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 150 more