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On the 2D Yang-Mills/Hurwitz Correspondence

Jonathan Novak

Abstract

In this paper, we show that in the large $N$ limit two-dimensional Yang-Mills theory with $U(N)$ gauge group becomes mixed Hurwitz theory, in the sense that the $1/N$ expansion of the chiral partition function receives contributions from both classical and monotone Hurwitz theory for all but finitely many compact orientable spacetimes.

On the 2D Yang-Mills/Hurwitz Correspondence

Abstract

In this paper, we show that in the large limit two-dimensional Yang-Mills theory with gauge group becomes mixed Hurwitz theory, in the sense that the expansion of the chiral partition function receives contributions from both classical and monotone Hurwitz theory for all but finitely many compact orientable spacetimes.
Paper Structure (48 sections, 24 theorems, 166 equations, 2 figures)

This paper contains 48 sections, 24 theorems, 166 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Graph theory
  3. Hurwitz theory
  4. Yang-Mills theory
  5. Organization
  6. Hurwitz-Cayley Graph
  7. Elementary features
  8. Jucys-Murphy correspondence: combinatorial form
  9. Jucys-Murphy correspondence: matricial form
  10. Jucys-Murphy correspondence: polynomial form
  11. Weakly monotone walks
  12. Fourier transform
  13. The operator $\Omega_q^{-1}$
  14. Summary
  15. Yang-Mills Theory
  16. ...and 33 more sections

Key Result

Theorem 2.1

There exists a unique strictly monotone walk between every pair of permutations, and this walk is a geodesic.

Figures (2)

  • Figure 1: Jucys-Murphy labeling of $\mathrm{S}^4$.
  • Figure 2: Jucys-Murphy correspondence in $\mathrm{S}^4$.

Theorems & Definitions (30)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6: Novak:BCP
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 3.1: Gross-Taylor formula
  • ...and 20 more