A combinatorial formula for Wilson loop expectations on compact surfaces

Abstract

We give an almost purely combinatorial expression for Wilson loop expectations of the Yang-Mills holonomy process with values in the unitary group on a compact oriented surface, possibly with boundary and arbitrary boundary conditions. Our main result computes the non-normalized expectation of products of traces of holonomies along an arbitrary family of immersed curves with transverse self-intersections and no triple points. It is expressed as a sum over assignments of highest weights of the unitary group to the connected components of the complement of the curves. Each term is a product of a Gaussian exponential factor, dimensions of unitary representations, and local contributions at the intersection points given by the sine or cosine of an angle determined by the surrounding highest weights. As an application, we obtain a new and short proof of the Makeenko-Migdal equations on arbitrary compact surfaces.

Figures (15)

• Figure 1: This loop configuration on a surface of genus $2$ contains five loops, one of which is an isolated embedded loop. It produces a graph with 5 vertices, 11 edges, one of which is circular, and 8 faces. The face $F$ containing the left handle of the surface has a boundary with ${\sf b}_{F}=3$ connected components and Euler characteristic ${\sf e}_{F}=-3$. However, the closure of $F$ in $\Sigma$ is not homeomorphic to a torus with $3$ holes. Instead, it is homeomorphic to a torus with $3$ holes of which two distinct points of the same boundary component have been identified, to produce the vertex located on the $8$-shaped loop.
• Figure 2: On this picture, highest weights have non-negative components and are represented by Young diagrams, the lengths of the rows corresponding to the non-zero components of the highest weights. For instance, $\lambda_{{\text{\sc{e}}}_{v}}=(2,1,1,0,\ldots,0)$. The vertex $v$ is of type 1, and $\cos \theta^{\Lambda}_{v}=-\frac{1}{3}$. The vertex $w$ is of type 2, and $\theta^{\Lambda}_{w}=\theta^{\Lambda}_{v}$.
• Figure 3: This figure illustrates, in the case where $\Sigma$ is a $3$-holed sphere, the various kinds of edges that we allow ourselves to consider, and their relation to the boundary of the surface.
• Figure 4: This figure illustrates a standard identification of a $14$-gon to produce a surface of genus $2$ with $2$ boundary components. The black edges are not to be identified and give rise to the boundary components.
• Figure 5: On the left, a graphical representation of the endomorphism ${\sf I}_{e}(g)$ of ${\sf T}_{e}$. On the right, the factor $\pi^{\lambda_{\text{\sc{l}}_{e}}}\otimes {\rm id}_{T^{1}_{0}} \otimes \pi^{\lambda_{\text{\sc{r}}_{e}}}$ corresponding to the edge $e$ in ${\sf \Pi}(\Lambda)$. The arrows indicate the orientation of the edge $e$, and of the boundaries of the faces on the left and on the right of $e$. By analogy with the left part of the picture, one would expect to have, instead of the projector $\pi^{\lambda_{\text{\sc{r}}_{e}}}$, its image by the endomorphism of $\mathbb{C}[\mathsf S_{r_{e}}]$ which sends each permutation to its inverse, but this endomorphism leaves $\pi^{\lambda_{\text{\sc{r}}_{e}}}$ invariant.

Theorems & Definitions (37)

• Lemma S.1
• proof
• Definition S.2
• Theorem S.3
• Proposition 1.1
• proof
• Proposition 2.1
• Lemma 2.2
• proof
• Proposition 3.1