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Two-dimensional Yang-Mills theory via integrable probability

Thibaut Lemoine

Abstract

In this paper, we review the construction and large $N$ study of the continuous two-dimensional Yang--Mills theory with gauge group $\mathrm{U}(N)$ through probability, combinatorics and representation theory. In the first part, we define the continuous Yang--Mills measure using Markovian holonomy fields, following a construction by Lévy, then we show in the second part how to derive the character expansion of the partition function for any compact structure group from this setting. We continue with two developments obtained in the last few years by Dahlqvist, Lemoine, Lévy and Maïda with similar approaches with respect to the partition function: its large-$N$ asymptotics on all compact surfaces for the structure group $\mathrm{U}(N)$, and its $\frac{1}{N}$ expansion on a torus with an interpretation in terms of random surfaces.

Two-dimensional Yang-Mills theory via integrable probability

Abstract

In this paper, we review the construction and large study of the continuous two-dimensional Yang--Mills theory with gauge group through probability, combinatorics and representation theory. In the first part, we define the continuous Yang--Mills measure using Markovian holonomy fields, following a construction by Lévy, then we show in the second part how to derive the character expansion of the partition function for any compact structure group from this setting. We continue with two developments obtained in the last few years by Dahlqvist, Lemoine, Lévy and Maïda with similar approaches with respect to the partition function: its large- asymptotics on all compact surfaces for the structure group , and its expansion on a torus with an interpretation in terms of random surfaces.

Paper Structure

This paper contains 22 sections, 34 theorems, 134 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Formal definition and difficulties
  3. Harmonic analysis and partition function
  4. Large-$N$ limit and random surfaces
  5. Two-dimensional Yang--Mills measure
  6. Surfaces and maps
  7. Heat kernel on compact Lie groups
  8. Discrete Yang--Mills measure
  9. The group of loops
  10. Continuous extension
  11. Character expansion of the partition function
  12. Representations of Lie groups
  13. Peter--Weyl theorem
  14. Fourier transform
  15. Character expansion of the heat kernel
  16. ...and 7 more sections

Key Result

Theorem 2.4

Let $\Sigma$ be a compact connected surface without boundary.

Figures (9)

  • Figure 1: An oriented map of genus 1, embedded in a torus. It has 4 vertices, 7 edges and 3 faces.
  • Figure 2: A map with one face in genus 2, traced in the plane (left) with edges of the same color pairwise identified, and traced in a genus 2 surface (right). It only has one vertex, marked by a black dot.
  • Figure 3: A map of genus 1. The edges with same label are glued together.
  • Figure 4: A portion of $\mathbb{G}$ (on the left) and $\mathbb{G}'$ (on the right). The faces $f_1$ and $f_2$ of $\mathbb{G}$ are separated by the edge $e$ (in red). After removal of $e$, $f_1$ and $f_2$ are replaced by $f_*$ (on the right). All other faces and edges remain unchanged.
  • Figure 5: The loop on the left and on the right are in the same reduction class.
  • ...and 4 more figures

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.4: Classification of surfaces
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.11
  • Remark 2.12
  • ...and 59 more