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Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

Antoine Dahlqvist, Thibaut Lemoine

TL;DR

This work analyzes the large $N$ limit of the two-dimensional Yang--Mills measure on compact orientable surfaces with classical matrix groups. It establishes that Wilson loops contained in a disc converge in the large $N$ limit to the planar master field, while simple non-contractible loops vanish in probability, using a strategy based on the convergence of partition functions via harmonic analysis on compact groups. The approach avoids Makeenko--Migdal equations and instead derives the disc and non-contractible-loop results from a robust control of partition functions and their representations. The results unify and extend prior plane and sphere findings to general closed surfaces, laying groundwork for a comprehensive understanding of master fields in 2D YM at large $N$ and for future exploration of more intricate loop classes.

Abstract

We compute the Large N limit of several objects related to the two-dimensional Euclidean Yang-Mills measure on compact connected orientable surfaces of genus larger or equal to one, with a structure group taken among the classical groups of order N. Our result shows the convergence of all Wilson loops for all loops within a topological disc and all simple loops.

Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

TL;DR

This work analyzes the large limit of the two-dimensional Yang--Mills measure on compact orientable surfaces with classical matrix groups. It establishes that Wilson loops contained in a disc converge in the large limit to the planar master field, while simple non-contractible loops vanish in probability, using a strategy based on the convergence of partition functions via harmonic analysis on compact groups. The approach avoids Makeenko--Migdal equations and instead derives the disc and non-contractible-loop results from a robust control of partition functions and their representations. The results unify and extend prior plane and sphere findings to general closed surfaces, laying groundwork for a comprehensive understanding of master fields in 2D YM at large and for future exploration of more intricate loop classes.

Abstract

We compute the Large N limit of several objects related to the two-dimensional Euclidean Yang-Mills measure on compact connected orientable surfaces of genus larger or equal to one, with a structure group taken among the classical groups of order N. Our result shows the convergence of all Wilson loops for all loops within a topological disc and all simple loops.
Paper Structure (27 sections, 29 theorems, 159 equations, 5 figures)

This paper contains 27 sections, 29 theorems, 159 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setup and statement of results
  3. Heat kernel on compact Lie groups
  4. Heat kernel
  5. Classical compact Lie groups
  6. 2D-maps and multiplicative functions
  7. Topological maps on compact surfaces
  8. Topological maps on $\mathbb{R}^2$ or an open disc
  9. Multiplicative functions
  10. Wilson loops
  11. Area-weighted maps
  12. Two dimensional Yang--Mills measure
  13. Uniform measure on $\mathcal{M}(\mathrm{P}(M),G)$
  14. Discrete Yang--Mills measure
  15. Yang--Mills holonomy field
  16. ...and 12 more sections

Key Result

Proposition 2.1

Assume that $M$ is a topological map on a compact, connected, orientable surface $\Sigma$ endowed with a Riemannian metric, such that $\partial\Sigma$ has positive and finite length. Then, any connected component $C$ of $\partial\Sigma$ is the drawing of an element of $\mathrm{L}(M)$.

Figures (5)

  • Figure 1: An example of topological map on a torus (on the left), and its representation as an abstract graph (on the right) whose opposite edges of the same colour are identified.
  • Figure 2: An example of area-weighted map on a torus.
  • Figure 3: An area-weighted map on the torus. The loop $def$ is included in the disc $U$ with boundary $t$.
  • Figure 4: The lifting of the loop $\ell$ of Fig. \ref{['fig:exemple']} in the plane.
  • Figure 5: The surface on the left can be seen as the result of a binary gluing of two surfaces along a separating loop (top right), or of one surface along a nonseparating loop (bottom right).

Theorems & Definitions (62)

  • Remark
  • Proposition 2.1
  • Lemma 2.2
  • Remark : Gauge equivalence
  • Proposition 2.3
  • Lemma 2.4: Sen0Lev2
  • Lemma 2.5: Sen0Lev2
  • Proposition 2.6: Lev2Lev3
  • Remark
  • Theorem 2.7: Lev2
  • ...and 52 more