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Almost flat highest weights and application to Wilson loops on compact surfaces

Thibaut Lemoine

Abstract

We derive new formulas for the expectation and variance of Wilson loops for any contractible simple loop on a compact orientable surface of genus $1$ and higher, in the model of two-dimensional Yang--Mills theory with structure group $\mathrm{U}(N)$. They are written in terms of a Gaussian measure on the dual of $\mathrm{U}(N)$ introduced recently by the author and M. Maïda \cite{LM3}. From these formulas, we prove a quantitative result on the convergence of the expectation and variance as $N$ tends to infinity, refining a result of the author and A. Dahlqvist \cite{DL}. We finally derive the large $g$ limit of the Wilson loop expectation and variance, by analogy with the study of integrals on moduli spaces of compact hyperbolic surfaces. Surprisingly, the variance does not vanish in this regime, but there are no nontrivial fluctuations of any order.

Almost flat highest weights and application to Wilson loops on compact surfaces

Abstract

We derive new formulas for the expectation and variance of Wilson loops for any contractible simple loop on a compact orientable surface of genus and higher, in the model of two-dimensional Yang--Mills theory with structure group . They are written in terms of a Gaussian measure on the dual of introduced recently by the author and M. Maïda \cite{LM3}. From these formulas, we prove a quantitative result on the convergence of the expectation and variance as tends to infinity, refining a result of the author and A. Dahlqvist \cite{DL}. We finally derive the large limit of the Wilson loop expectation and variance, by analogy with the study of integrals on moduli spaces of compact hyperbolic surfaces. Surprisingly, the variance does not vanish in this regime, but there are no nontrivial fluctuations of any order.
Paper Structure (16 sections, 27 theorems, 182 equations, 3 figures)

This paper contains 16 sections, 27 theorems, 182 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Yang--Mills measure and Wilson loops
  3. Almost flat highest weights
  4. Main results
  5. Related works
  6. Yang--Mills theory
  7. Random representations of surface groups
  8. Weil--Petersson model of random hyperbolic surfaces
  9. Preliminary tools
  10. Gaussian measure and observables of highest weights
  11. Branching rules
  12. Proof of Theorem \ref{['prop:wilson_loops_exp_var']}
  13. Large $N$ asymptotics
  14. Proofs for $g\geqslant 2$
  15. Proofs for $g=1$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Sigma_{g,T}$ be an orientable compact connected surface of genus $g\geqslant 1$ and of area $T$, $\ell$ be a contractible simple loop of interior area $t$ oriented clockwise. We have the following formulas:

Figures (3)

  • Figure 1: Almost flat highest weight of ${\mathrm U}(N)$.
  • Figure 2: An contractible simple loop $\ell$ (on the left) and the oriented graph associated to it (on the right).
  • Figure 3: On the first row: we have $\lambda\nearrow\mu$, with $\lambda$ on the left and $\mu$ on the right. On the second row: we have $\lambda\sim\mu$, with $\lambda$ on the left and $\mu$ on the right.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5: Hal2
  • Theorem 1.6: MP23
  • Theorem 1.7: Mag
  • Theorem 1.8: MagII
  • Proposition 2.1: LM3
  • Proposition 2.2: LM3
  • ...and 34 more