Large N limit of Wilson Loops on orientable closed surfaces in the light of Koike-Schur-Weyl duality and Spin Networks

Abstract

We prove the convergence in probability of Wilson loops under the Yang-Mills measure on any closed, orientable surface of genus larger than two, for large unitary or special unitary groups. Our approach revisits and refines recent arguments for average Wilson loops under the Atiyah-Bott-Goldman measure and for the Yang-Mills measure, using Koike-Schur-Weyl duality and Spin Networks.

Figures (11)

• Figure 1: A Brauer map and its associated surface
• Figure 2: A Brauer map $\mathfrak{m}$ with $h(\mathfrak{m})=1$ and its associated surface. The intervals $I_1,I_2,I'_2,I'_3$ displayed in the right-hand side are as defined in the first case of Lemma \ref{['Lem---CutORCompress']}'s proof.
• Figure 3: Left: a Brauer map associated to a quadrangular Brauer map $(\tau_u,\pi,\tau_b,\varphi)$ with $\pi=\pi^{0,1}_{r,ac}$ and its associated colouring, in particular, $\mathfrak{b}(s)$ is $R^{-1}, W$ and $W^{-1}$ when $s$ is in respectively $\{1,\ldots,8\}, \{9,10\}$ and $\{11,12\}.$ Right: its associated surface with marked arcs in green.
• Figure B.1: The graphical representation of the morphism $P^{\alpha,\beta}_\gamma\in \mathrm{Hom}(V_{\alpha}\otimes V_\beta, H^{\gamma}_{\alpha,\beta} \otimes V_{\gamma})$ and $P^{\gamma}_{\alpha,\beta}\in \mathrm{Hom}(H^{\gamma}_{\alpha,\beta} \otimes V_{\gamma},V_{\alpha}\otimes V_\beta).$
• Figure B.2: Graphical representation of the morphisms $P_{\alpha,\beta}^\gamma\in \mathrm{Hom}(V_{\alpha}\otimes V_\beta, V_{\gamma})$ and $P_{\gamma}^{\alpha,\beta}\in \mathrm{Hom}( V_{\gamma},V_{\alpha}\otimes V_\beta)$ when $\dim(H^{\gamma}_{\alpha,\beta})=1$ and an element of $H_{\alpha,\beta}^{\gamma}\setminus\{0\}$ has been fixed.

Theorems & Definitions (165)

• Theorem 2.1
• Theorem 2.2: LevMarkovHol
• Theorem 2.3: DLI
• Theorem 2.4: DLI
• Remark 2.1
• Remark 2.2
• Example 2.1
• Theorem 2.5
• Remark 2.3
• Theorem 2.6