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# Two-dimensional Yang-Mills theory via integrable probability

Thibaut Lemoine

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Thibaut Lemoine. Two-dimensional Yang-Mills theory via integrable probability. 2025. hal-05218431

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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 $U(N)$ through probability, combinatorics and representation theory. In a first part, we define the continuous Yang-Mills measure using Markovian holonomy fields, following a construction by Lévy, then we show in a second part how to derive Migdal's 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 $U(N)$, and its $\frac{1}{N}$ expansion on a torus with an interpretation in terms of random surfaces.

# 1. INTRODUCTION

Yang–Mills theory, named after two physicists who introduced it in the ’50s, is a non-abelian gauge theory that aims to describe, in a unified way, fundamental interactions between elementary particles. It is a quantum field theory based on the Yang–Mills action, a functional on the space of connections of a principal bundle. In the framework of the *Standard model* of particle physics, it enables to describe three of the four fundamental interactions:

(1) Electromagnetism,

(2) Weak nuclear force,

(3) Strong nuclear force.

There have been many mathematical developments about Yang–Mills theory since the ’70s. Although the rigorous mathematical construction of a quantum Yang–Mills theory on a continuous four-dimensional spacetime remains an open problem (it is even one of the seven *Millenium Prize Problems* from Clay Mathematics Institute [40]), there have been significant advances in the understanding of the discrete theory in any dimension, as well as the continuous theory in two dimensions. We refer to [31, 22] for general background in quantum field theory in general and Yang–Mills theory in particular. This survey, mainly based on a probabilistic construction of Thierry Lévy [53, 56] and recent papers of the author together with Antoine

<footnote n="2020">2020 Mathematics Subject Classification. Primary 43A75, 60B15 81T13, Secondary 05A17, 81T35.</footnote>
Dahlqvist and Mylène Maïda [49, 19, 50, 52, 51], is written to target a broad audience, and aims to be an accessible introduction to the subject. We focus on an approach based on integrable probability, an evergrowing field that lies at the interface of integrable systems, discrete probabilistic models and representation theory; we refer for instance to [7, 80] for recent reviews about integrable probability in the context of the so-called KPZ universality (which is seemingly unrelated but can be tackled with similar techniques). In the course of these notes, we will play with many objects from enumerative combinatorics, which have independent interest: maps, partitions or even ramified coverings. To keep a reasonable length, we will only present them superficially, and we will detail proofs when they are short and/or important enough for the understanding. Let us briefly summarize the scope of this paper, by starting with **what is omitted**.

• We shall not develop on the lattice gauge theory on $Z^d$, which is well-defined in any dimension (especially in the physically relevant dimension 4). For recent surveys and results, we refer to [14, 15, 11] and references therein.

• We will not describe the interplay between the large $N$ asymptotics of the two-dimensional Yang-Mills measure and free probability theory, which is incarnated by the *master field* described by Singer [73]. We refer the interested reader to the papers [56, 201, 192, 67] where the master field was constructed on various surfaces, as well as the introductory notes [58].

• We will not explore the links between Yang-Mills theory on a compact surface of genus $g \ge 2$ and the integration on moduli spaces of flat connections, which was initiated by Witten [77]. We refer to [70, 71] for detailed mathematical developments.

• Finally, we will not discuss the recent results about stochastic quantization of the Yang-Mills theory in dimensions 2 and 3 [12, 18, 13], which are of independent interest but develop completely different aspects – see [16] for a review.

**1.1. Formal definition and difficulties.** Let us first recall the historical definition of the Yang-Mills measure. We will assume basic knowledge of differential geometry (Riemannian manifolds, Lie groups, principal bundles) in this subsection, but it will not be essential for the rest of the paper and any reader unfamiliar with these notions can simply skip this subsection without harm.

The two main ingredients of the theory are a spacetime and a group of symmetries that describes gauge transformations. More precisely, we take a Riemannian manifold $(M, g)$ and a compact Lie group $G$, called *structure group*. We assume that the corresponding Lie algebra $\mathfrak{g}$ is endowed with a $G$-invariant inner product $\langle \cdot, \cdot \rangle$. The spacetime and structure group can be put together into a $G$-principal bundle $\pi : P \to M$. The interactions are represented by the *Yang-Mills Lagrangian*, which is a function on the space
A of connections on P. The latter is an affine space of direction

$$ \Omega^1(M) \otimes \mathrm{ad}(P), $$

the space of ad(P)-valued 1-forms on M. The Yang-Mills action is then defined by

$$ (1.1) \qquad S_{\mathrm{YM}}(\omega) = \int_M \langle \Omega \wedge \star\Omega \rangle, $$

where $\star$ is the Hodge star operator, $\langle \cdot, \cdot \rangle$ is the structure on ad(P) induced by the invariant inner product on $\mathfrak{g}$, and $\Omega$ is the curvature of the connection $\omega \in A$.

The Yang-Mills measure on M with structure group $G$ is the measure formally defined by

$$ (1.2) \qquad d\mu_{\mathrm{YM}}(\omega) = \frac{1}{Z_{\mathrm{YM}}} e^{\frac{i}{2T} S_{\mathrm{YM}}(\omega)} \mathcal{D}\omega, $$

where $\mathcal{D}\omega$ is the Lebesgue measure on $A$, $T > 0$ is a coupling constant, and $Z_{\mathrm{YM}}$ is a normalization constant that puts the total mass to 1. A first obvious problem is that such measure, assuming that it exists, is not positive. However, by analytic continuation, it can be turned into a true probability measure by operating a *Wick rotation*:

$$ (1.3) \qquad d\mu'_{\mathrm{YM}}(\omega) = \frac{1}{Z_{\mathrm{YM}}} e^{-\frac{1}{2T} S_{\mathrm{YM}}(\omega)} \mathcal{D}\omega. $$

The resulting object is called the *Euclidean Yang-Mills measure*. Many critical issues remain: for instance, there exists no Lebesgue measure on $A$, and even if it was the case there would be no reason that the measure has finite mass. Even if we reduce the space of integration by taking the quotient by gauge symmetries, there is no well-defined corresponding measure on the quotient. In fact, such measure can be defined on an appropriate compactification [2], but nothing guarantees that the Yang-Mills measure has a density with respect to it. A partial answer was however obtained by Ashtekar et al. [3] (in the sense that they were able to compute all relevant observables within this framework, on the plane and the cylinder, with structure group SU(2)).

The first complete, rigorous constructions of path integrals for the continuous two-dimensional Yang-Mills theory (also denoted by YM₂) are due to Gross, Driver, King and Sengupta [35, 25, 69] using mainly stochastic analysis. Another construction was later obtained by Lévy [53, 56], using a combination of discrete gauge theory developed by Witten [77] and continuous Markov processes. We develop this approach in Section 2: if we replace M by a graph $\mathbb{G}$ and the principal bundle $\pi: P \to M$ by a discrete trivial bundle $\pi': \mathbb{G} \times G \to \mathbb{G}$, we can make sense of appropriate integrals with respect to (1.3). The collection of measures on appropriate graphs will be the finite-dimensional marginals of a continuous Markov process called the *Yang-Mills holonomy field*, which can then be constructed by a Kolmogorov extension theorem. The distribution of this process is, in some sense, the
“right” continuous Yang–Mills measure (see the end of §2.5 about why it is true).

1.2. **Harmonic analysis and partition function.** In two dimensions, the first interesting quantity to study is the partition function $Z_{YM}$. The partition function for a structure group $G$ on a compact connected oriented surface of genus $g$ and area $t$ only depends on $G, g$ and $t$, and can be computed explicitly as an integral on the group $G$ using the previously introduced setting; we will denote by $Z_G(g, t)$ this partition function. Section 3 will be devoted to derive a Fourier expansion of $Z_G(g, t)$, using general results of the representation theory of compact groups. We will thus recover, in a self-contained way, a formula initially found by Migdal [63], long before a mathematically rigorous construction of the measure (which is remarkable!).

1.3. **Large $N$ limit and random surfaces.** Since the seminal paper of 't Hooft [74], it is known that Yang–Mills theory for $SU(N)$ as $N$ goes to infinity gives an understanding of gauge theory for $SU(3)$ at strong coupling, which is related to the question of the confinement of quarks (it is also named “multicolor QCD” by physicists, because $SU(3)$ Yang–Mills corresponds to the quantum chromodynamics, and $N=3$ corresponds to the number of colors of quarks and gluons). The question of the asymptotic behavior of observables of the theory for matrix Lie groups with size going to infinity has been developed since then under the name “large $N$ regime”. We will consider the case of $G = U(N)$ in Section 4, and describe the asymptotics of $Z_{U(N)}(g, t)$ for all genera when $N$ tends to infinity, giving a proof of several results conjectured by physicists in the '90s [32, 24, 68, 23].

Another aspect of the large $N$ regime, already present in [74] but mainly developed in the two-dimensional setting by Gross and Taylor [32, 34, 33], is the relationship between Yang–Mills theory and string theory, namely a theory of random surfaces/graphs. They conjectured that the partition function and Wilson loop expectations on a compact surface $\Sigma_g$ of genus $g \ge 0$ with structure group $SU(N)$ could be rewritten as sums (or integrals) over ramified coverings $X \to \Sigma_g$. There have been many combinatorial developments of this interaction, nowadays known as *gauge/string duality*[^1], and it is still an active field in physics [66, 1], but everything was written in terms of formal power series. This gauge/string duality is also conjectured to hold for other classical groups, such as $U(N)$, $SO(N)$ or $Sp(N)$, and the first proof of a true asymptotic expansion related to this gauge/string duality on compact surfaces was achieved for $U(N)$ when the base space is a torus [52]. A complete solution, including the asymptotic expansion for all compact classical groups and a random surface formula, is provided in a forthcoming paper [51]. We will go through the main results and ideas in Section 5.

<footnote>[^1]: In fact, such dualities are also conjectured for other types of gauge theories, such as supersymmetric Yang–Mills theory. The two most cited papers from high energy physics ever – up to date – are related to another kind of gauge/string duality called AdS/CFT correspondence [62, 78].</footnote>
As a side note, another promising direction about the interpretation of Yang-Mills measure in terms of random surfaces has been recently achieved in the lattice $Z^d$ in any dimension [11, 6] through the means of Weingarten calculus, which is another combinatorial toolbox coming from representation theory of compact classical groups. In this case, the trivial topology of the underlying spacetime makes computations easier and the authors of the aforementioned papers have been able to prove elegant expressions in terms of surface sums for the relevant observables, which are Wilson loop expectations, both in the finite $N$ and the large $N$ regime.

## 2. TWO-DIMENSIONAL YANG-MILLS MEASURE

**2.1. Surfaces and maps.** In these notes, a *surface* will always denote a smooth real differential manifold of dimension 2. We shall start with a definition and some properties of topological maps, which will play the role of our discretized spacetime. We refer to [48, 27] for more details on maps.

**Definition 2.1.** A graph is a triple $\mathbb{G} = (V, E, I)$ where $V$ and $E$ are respectively the sets of *vertices* and *edges*, and an *incidence relation* $I \subset V \times E$, such that the cardinal of $\{v \in V : (v, e) \in I\}$ is 1 or 2 for all $e \in E$.

The incidence relationship is an abstract encoding of the intuitive idea that the vertices are at the boundaries of edges. If $\mathbb{G} = (V, E, I)$ is a graph,

(1) Two edges $e_1, e_2 \in E$ are *adjacent* if they are incident to the same vertex $v \in V$. They form a *double edge* if they are incident to the two same vertices $v_1, v_2 \in V$;

(2) An edge $e \in E$ is a *loop* if it is incident to only one vertex.

**Definition 2.2.** A topological map is a graph (possibly with loops and double edges) $\mathbb{G} = (V, E, I)$ endowed with an embedding $\theta : \mathbb{G} \to \Sigma$, where $\Sigma$ is a surface, such that:

* The images of two distinct vertices $v_1, v_2 \in V$ by $\theta$ are distinct points of $\Sigma$,

* The images of edges $e \in E$ are continuous curves $\theta_e : [0, 1] \to \Sigma$ that only meet at their endpoints $\underline{e} = \theta_e(0)$ and $\bar{e} = \theta_e(1)$,

* The complement in $\Sigma$ of the skeleton $\text{Sk}(\mathbb{G}) = \bigcup_{e \in E} \theta_e$ of $\mathbb{G}$ divides into one or several connected components $f_1, \dots, f_k$, each simply connected, called *faces*.

We denote by $F = \{f_1, \dots, f_k\}$ the set of faces of the map.

Usually, we will denote $\mathbb{G} = (V, E, F)$ the map, and the embedding will be implicit. An example of map is given in Figure 1. Edges $e \in E$ have an orientation by default, in the sense that the curve $\theta(e)$ inherits the orientation of its parametrized path, and we shall consider the action of $\{\pm 1\}$ on $E$ given by:

* $e^1$ is simply $e$,
<figure><img src="image_2.png" /><figcaption>FIGURE 1. An oriented map of genus 1, embedded in a torus. It has 4 vertices, 7 edges and 3 faces.</figcaption></figure>

• $e^{-1}$ is the edge $e$ with reverse orientation, i.e. corresponds to the parametrized path $\theta_e^{-1}: t \mapsto \theta_e(1-t)$.

A topological map is an example of two-dimensional CW-complex in algebraic topology: its vertices (resp. edges, faces) are its 0-cells (resp. 1-cells, 2-cells), and the skeleton of the map is exactly the skeleton of the complex. Besides, if the underlying surface is oriented, then the faces naturally inherit this orientation.

**Examples 2.3.** Among the most studied maps, let us mention triangulations (resp. quadrangulations): they are maps where all faces are triangles (resp. quadrilaterals). The study of random triangulations/quadrangulations is still a very active field, see for instance [38, 10].

Topological maps can be put into equivalence classes up to orientation-preserving homeomorphisms, so that they do not depend on the embedding: it leads to the notion of *combinatorial map*. These objects have attracted a growing interest in the last 30 years<fnref n="2" />. A beautiful application of the study of maps is the classification of surfaces, see [47].

**Theorem 2.4 (Classification of surfaces).** Let $\Sigma$ be a compact connected surface without boundary.

• If $\Sigma$ is orientable, then it is homeomorphic to one of the following:

(1) A sphere,

(2) The connected sum of $g \ge 1$ tori.

In this case, it is said to have genus $g$, with $g = 0$ by convention if it is homeomorphic to a sphere.

• If $\Sigma$ is not orientable, then it is homeomorphic to the connected sum of $k \ge 1$ real projective planes.

This theorem is proved by showing that any such surface can be obtained as the gluing of a polygon along its oriented sides (or two disks in the case of

<footnote n="2">²Physicists were actually interested in them much earlier, starting with the work of 't Hooft [74], but became popular in the mathematics community after Kontsevich used them to prove Witten's conjecture [45].</footnote>
the sphere). The polygon is a 4g-gon in the case of an orientable surface of genus g ≥ 1, which can be seen as a map with one face, called fundamental map (as the polygon is often called fundamental polygon). The sides of the polygon, in counterclockwise order, can be chosen to be

$$[a_1, b_1] \dots [a_g, b_g] = a_1 b_1 a_1^{-1} b_1^{-1} \dots a_g b_g a_g^{-1} b_g^{-1},$$

and the $a_i, b_i$ represent oriented edges of the map, with the convention that two edges of the same label (up to orientation) are glued together. An example of map in genus 2 is given in Figure 2 below. Going from the planar representation to the the embedded representation is an exercise of visualization: say that one wants to follow a small loop oriented counterclockwise (in black) around the vertex of the map, then record all oriented edges that are crossed from the right. The loop crosses the following edges: $b_1, a_1^{-1}, b_1^{-1}, a_1, b_2, a_2^{-1}, b_2^{-1}, a_2$. The homotopy classes of the edges of the fundamental map generate the fundamental group of the underlying surface

$$\pi_1(\Sigma) = \langle a_1, b_1, \dots, a_g, b_g \mid [a_1, b_1] \dots [a_g, b_g] = 1 \rangle,$$

where we also denoted by $a_i, b_i$ the homotopy classes of the corresponding edges.

For any map $\mathbb{G} = (V, E, F)$ embedded in a closed orientable surface $\Sigma$ of genus $g \ge 0$, the numbers of vertices, edges and faces are related through Euler's formula

$$(2.1) \qquad |V| - |E| + |F| = 2 - 2g = \chi(\Sigma),$$

where $\chi(\Sigma)$ is the Euler characteristic of the underlying surface.

In this paper, we will only consider closed orientable surfaces; a more general construction is detailed in [56]. We will henceforth denote a surface of genus g a compact, connected, oriented surface of genus g.

**Definition 2.5.** Let $\mathbb{G} = (V, E, F)$ be a topological map. A path $\gamma$ in $\mathbb{G}$ is either a single vertex $v \in V$ (it is a constant path), or a concatenation of

<figure><img src="image_3.png" /><figcaption>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.</figcaption></figure>
edges $e_1 \cdots e_n$, where $\bar{e}_i = e_{i+1}$ for all $1 \le i \le n-1$. In the second case, we write $\gamma = \underline{e_1}$ and $\bar{\gamma} = \overline{e_n}$ its endpoints, and $\ell(\gamma) = n$ its length. The path is a loop if $\gamma = \bar{\gamma}$.

Constant paths in a map are also considered as loops by convention. For any loop $\gamma$ in $\mathbb{G}$, we call *basepoint* the vertex $\gamma = \bar{\gamma}$. We denote by $P(\mathbb{G})$ (resp. $P_v(\mathbb{G})$, $L(\mathbb{G})$, $L_v(\mathbb{G})$) the set of paths (resp. paths of base $v \in V$, loops, loops of base $v \in V$) in $\mathbb{G}$.

## 2.2. Heat kernel on compact Lie groups.

**Definition 2.6.** A Lie group is a group $G$ endowed with a structure of smooth real manifold, such that the multiplication and the inversion are smooth.

In what follows, we will denote by $1 = 1_G$ its neutral element. We need two objects from differential and integral calculus on $G$: the Laplace-Beltrami operator and the Haar measure. To be more specific, the Haar measure only requires a structure of a topological group, and the Laplacian a structure of a Riemannian manifold.

**Definition 2.7.** Let $G$ be a locally compact topological group. A nonnegative Radon measure $\mu$ on $G$ is *left-invariant* if it satisfies for all $f \in \mathcal{C}_c(G)$

$$ \int_G f(gx)d\mu(x) = \int_G f(x)d\mu(x), \quad \forall g \in G. $$

It is *right-invariant* if it satisfies for all $f \in \mathcal{C}_c(G)$

$$ \int_G f(xg)d\mu(x) = \int_G f(x)d\mu(x), \quad \forall g \in G. $$

A positive left-invariant (resp. right-invariant) on $G$ is called *left Haar measure* (resp. *right Haar measure*).

A theorem due to von Neumann and Kakutani [75, 41] states that for any compact topological group $G$, there exists a unique (up to a constant factor) Haar measure, which is simultaneously a left and a right Haar measure, and which has finite mass. We denote by $dg$ the corresponding probability measure.

If $G$ is a compact topological group and $f, g: G \to \mathbb{R}$ are Borel functions, their convolution is defined by

$$ f * g(x) = \int_G f(y)g(y^{-1}x)dy. $$

**Definition 2.8.** Let $G \subset \text{GL}(N, \mathbb{C})$ be a compact matrix Lie group, whose Lie algebra $\mathfrak{g} = T_1G$ is endowed with a $G$-invariant inner product $\langle \cdot, \cdot \rangle$. Let $\{X_1, \dots, X_d\}$ be an orthonormal basis of $\mathfrak{g}$ for this inner product. The
Laplace-Beltrami operator on G is the differential operator $\Delta_G : \mathcal{C}^2(G) \to \mathbb{C}$ defined by

$$ (2.2) \qquad \Delta_G f(g) = \sum_{i=1}^{d} \frac{d^2}{dt^2} \Big|_{t=0} f(ge^{tX_i}). $$

The heat kernel on G is the convolution semigroup $(p_t)_{t \ge 0}$ solution of

$$ (2.3) \qquad \frac{d}{dt} p_t(g) = \frac{1}{2} \Delta_G f(g), \quad \forall g \in G, $$

with initial condition $\lim_{t \to 0^+} p_t = \delta_1$, where the limit has to be understood in the sense of distributions.

From a probabilistic point of view, the heat kernel is the distribution of a Brownian motion on G starting from 1. This approach has been fruitful to study the heat kernel on compact groups [55, 59]. In Section 3 we will use representation theory to obtain a spectral decomposition of the heat kernel on $G = U(N)$ associated with this inner product.

**2.3. Discrete Yang-Mills measure.** As we have seen in the introduction, the Yang-Mills measure is formally a measure on the space of connections on the underlying spacetime. In the case of a graph, this space becomes finite-dimensional, so that the rigorous definition is within reach.

**Definition 2.9.** Let $\mathbb{G} = (V, E, I)$ be a graph, and $G$ be a compact matrix Lie group.

(1) The trivial $G$-principal bundle above $\mathbb{G}$ is $P = G^V$.

(2) A section of $P$ is simply an element of $P$, i.e. a function $\sigma : V \to G$.

(3) A connection on $P$ is a function $\omega : E \to G$

$$ \omega : \left\{ \begin{array}{ccc} E & \to & G \\ e & \mapsto & \omega(e), \end{array} \right. $$

that satisfies

$$ \omega(e^{-1}) = \omega(e)^{-1}, \quad \forall e \in E. $$

(4) Given a path $\gamma = e_1 \cdots e_n$ in $\mathbb{G}$, the holonomy of the connection $\omega$ along $\gamma$ is given by

$$ \mathrm{hol}(\omega, \gamma) := h_{\gamma} = \omega(e_1) \cdots \omega(e_n). $$

(5) If moreover $\mathbb{G}$ is a map, the curvature of $\omega$ is the function

$$ \Omega^{\omega} : \left\{ \begin{array}{ccc} F & \to & G \\ f & \to & \mathrm{hol}(\omega, \partial f), \end{array} \right. $$

where $\partial f$ is the boundary of $f$ oriented counterclockwise. We say that $\omega$ is flat if $\Omega^{\omega}(f) = 1$ for all $f \in F$.
There is a caveat in the previous definition: normally, connections and their curvature are respectively defined as $\mathfrak{g}$-valued 1-forms and 2-forms, whereas they are defined in Definition 2.9 as $\mathcal{G}$-valued forms. This conversion from the Lie algebra to the Lie group turns the linear structure of connections into a multiplicative structure, and that is because we implicitly identified connections with their holonomies along edges. The curvature of a connection depends on a choice of basepoint for the boundary of each face, but all choices of basepoint give values of curvatures that are in the same conjugacy class in $G$. In the end, we will consider configurations to be discrete connections modulo gauge transformations, and in this quotient the definition of curvature is therefore uniquely defined. Besides, observe that one can interpret the curvature as a $\mathcal{G}$-valued 2-form on $\mathcal{G}$. More generally, 0-forms (resp. 1-forms, 2-forms) correspond to functions on the 0-cells (resp. 1-cells, 2-cells) of the map $\mathcal{G}$ seen as a CW complex. For more details on this viewpoint, see [20, Section 2] for maps, or [15, Section 2] for the lattice $\mathbb{Z}^d$.

**Examples 2.10.**

* If $\mathcal{G} = \mathbb{Z}^d$ and $G = \mathbb{Z}_2$, a $\mathcal{G}$-principal bundle endowed with a connection defines the model of *Ising gauge theory* [15]: the sections of the bundle define a spin configuration, and the connection defines an interaction between adjacent spins, with the condition that the connection is flat. A generalized model has been recently studied in [29].

* If we take $G = U(1)$ and an arbitrary $\mathcal{G}$, we recover the configuration space of *cycle-rooted spanning forests* (CRSF) studied by Kenyon [42].

The space of connections is clearly in bijection with $G^{|E|}$, which is a compact Lie group. The *uniform measure* on connections is taken to be the Haar probability measure on $G^{|E|}$.

**Definition 2.11.** Let $\mathcal{G} = (V, E, I)$ be a graph endowed with a $\mathcal{G}$-connection $h$. A *gauge transformation* is a function $k: V \to G$. The gauge group $G^V$ acts on $G^E$ as follows:

$$ (2.4) \qquad (j \cdot \omega)(e) = j(\bar{e})^{-1}\omega(e)j(e), $$

and the *configuration space* of the discrete gauge theory on $\mathcal{G}$ with structure group $G$ is the quotient $\mathcal{C}_{\mathcal{G}}^G = G^E/G^V$ by this action.

Intuitively, the discrete Yang-Mills measure would be a measure on $G^E$ invariant by this action, and is expected to descend to a measure on $\mathcal{C}_{\mathcal{G}}^G$. We can endow the set $\Omega^1(\mathcal{G}, G) = G^E$ with two different sigma-algebras:

* The *cylindrical* sigma-algebra $\mathcal{C}$ is the smallest sigma-algebra that makes the applications $h \mapsto h_\ell$ measurable for all loop $\ell$ in $\mathcal{G}$;

* The *invariant* sigma-algebra $\mathcal{I}$ is the smallest sigma-algebra that makes the applications $f(h_{\ell_1}, \dots, h_{\ell_n})$ measurable, for any $n \ge 1$,
any $f: G^n \to \mathbb{R}$ invariant by diagonal action of $G$ by conjugation, and any collection $(\ell_1, \dots, \ell_n)$ of loops on $\mathbb{G}$ with same basepoint.

Naturally, only the second one can be expected to descend to quotient into a sigma-algebra on $G^E/G^V$. By invariance of the Haar measure by translation, the Haar measure $d\omega$ on $\Omega^1(\mathbb{G}, G)$ also descends to a measure on $G^E/G^V$, that we shall also denote by $d\omega$.

**Remark 2.12.** This configuration space is related to the notion of spin networks [4]. If one does not assume $G$ compact, this setting remains well-defined, and seems to be suitable for a formalization of quantum gravity [5, 30]. However, in this case, there is no more uniform probability measure: there are either Haar measures with infinite mass or probability measures that are not uniform.

We are now able to define the discrete Yang-Mills measure.

**Definition 2.13.** Let $\mathbb{G} = (V, E, F)$ be a topological map endowed in a surface of genus $g \ge 0$, endowed with a smooth measure of area vol and with total area $t > 0$, and $G$ be a compact matrix Lie group. The Yang-Mills measure on $\mathbb{G}$ with structure group $G$ is the measure $\mu_{\mathbb{G},G,\Sigma,\text{vol}}$ on $\Omega^1(\mathbb{G}, G)$ defined by

$$ (2.5) \qquad \mu_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) = \frac{1}{Z_G(g,t)} \prod_{f \in F} p_{\text{vol}(f)}(\Omega^\omega(f))d\omega, $$

where

$$ (2.6) \qquad Z_{\mathbb{G},G}(g,t) = \int_{G^E} \prod_{f \in F} p_{\text{vol}(f)}(\Omega^\omega(f))d\omega $$

is the partition function.

Equation (2.5) is named *Driver-Sengupta formula*, from the two mathematicians who proved it: Driver in the plane [25] and Sengupta on compact surfaces [69]. As the heat kernel on $G$ is invariant by conjugation, the measure does not depend on the parametrization of the boundary of each face for the definition of the curvature, and the measure descends to a measure on the configuration space $\mathcal{C}_{\mathbb{G}}^G$. By construction, the measure is also invariant by area-preserving diffeomorphisms.

**Example 2.14.** Consider the map given in Figure 3, which has genus 1. It is described by $\mathbb{G} = (V, E, F)$, where $V = \{v_1, v_2\}$, $E = (a, b, c, d)$ and $F = \{f_1, f_2\}$ where $f_1$ has boundary $\partial f_1 = d$ and $f_2$ has boundary $\partial f_2 = d^{-1}c^{-1}aba^{-1}b^{-1}c$. If we set $t_i = \text{vol}(f_i)$ for $i \in \{1, 2\}$, as well as $t = t_1+t_2$ the total area of the underlying surface, the Driver-Sengupta formula becomes

$$ \mu_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) = \frac{1}{Z_G(1,t)} p_{t_1}(\omega(d)) p_{t_2}(\text{hol}(\omega, d^{-1}c^{-1}aba^{-1}b^{-1}c)) d\omega. $$
<figure><img src="image_4.png" /><figcaption>FIGURE 3. A map of genus 1. The edges with same label are glued together.</figcaption></figure>

Using the bijection $G^E \simeq G^{|E|} = G^4$, and denoting respectively by $x_1, x_2, x_3, x_4$ the variables of integration corresponding to the edges $a, b, c, d$, one gets

$$ \mu_{G,G,\Sigma,\text{vol}}(d\omega) = \frac{1}{Z_G(1,t)} p_{t_1}(x_4)p_{t_2}(x_4^{-1}x_3^{-1}x_1x_2x_1^{-1}x_2^{-1}x_3)dx_1dx_2dx_3dx_4. $$

In particular,

$$ Z_G(1,t) = \int_{G^4} p_{t_1}(x_4)p_{t_2}(x_4^{-1}x_3^{-1}x_1x_2x_1^{-1}x_2^{-1}x_3)dx_1dx_2dx_3dx_4. $$

Using the semigroup property of heat kernel yields

$$ Z_G(1,t) = \int_{G^3} p_t(x_3^{-1}x_1x_2x_1^{-1}x_2^{-1}x_3)dx_1dx_2dx_3, $$

and as $p_t(gxg^{-1}) = p_t(x)$ for all $x, g \in G$, one finally gets

$$ Z_G(1,t) = \int_{G^2} p_t([x_1, x_2])dx_1dx_2, $$

where $[x, y] = xyx^{-1}y^{-1}$ is the commutator of elements of $G$.

*Remark 2.15.* In the literature, this definition of the Yang–Mills measure is sometimes described as a discrete gauge theory for the *heat kernel action*, or *Villain action*, as opposed to the *Wilson action* defined by Wilson in [76] by

$$ (2.7) \qquad S(\omega) = \sum_{f \in F} \Re(\text{Tr}(1 - \Omega^\omega(f))). $$

In the case of the Wilson action, the Yang–Mills measure with coupling constant $\beta > 0$ is defined as the Gibbs measure of inverse temperature $\beta$ and Hamiltonian $S$, that is,

$$ (2.8) \qquad \mu(d\omega) = \frac{1}{Z} e^{-\beta S(\omega)} d\omega. $$

The advantage of the Wilson action, mainly used in lattice gauge theory [15, 11], is that it defines a simpler model in terms of statistical mechanics,
but going from the discrete to the continuous model seems more complicated in general and requires renormalization – which is not the case for the heat kernel action, as we shall see in a moment. Note that in two dimensions, recent papers establish a correspondence between Villain gauge theory and Wilson gauge theory [17, 72]. In the sequel, we shall always use the heat kernel action.

**Remark 2.16.** The Yang-Mills measure can also be defined for a surface $\Sigma$ with boundary. A way to do it is to see $\Sigma$ as a subset of a surface $\Sigma_0$ without boundary but with marked simple loops whose union gives $\partial\Sigma$, and the map embedded in $\Sigma$ is a graph $\mathcal{G}$ embedded in $\Sigma_0$ such that $\partial\Sigma$ is a concatenation of edges of $\mathcal{G}$, and whose faces that do not belong to $\Sigma$ are marked faces that do not contribute to the definition of the measure. One can then impose boundary conditions, cf. [56, 19]. In particular, the measure on $\mathbb{R}^2$ can be defined as a measure on a planar map (i.e. a map embedded in the sphere) with a marked face, the “unbounded face”.

A fundamental property of the Yang-Mills measure is its invariance by subdivision. It implies that the partition function does not depend on the graph, but only on the topology and area of the surface.

**Theorem 2.17.** Let $\mathcal{G} = (V, E, F)$ be a topological map embedded in a surface $\Sigma$ of genus $g \ge 0$, and $\mathcal{G}' = (V, E', F')$ be a topological map obtained by removing one edge of $\mathcal{G}$, endowed with the same embedding. If we denote by $\mathcal{R}$ the restriction map from $G^{E'}$ to $G^E$ induced by the one from $\mathcal{G}$ to $\mathcal{G}'$, then

$$ (2.9) \qquad (\mathcal{R})_* \mu_{\mathcal{G},G,\Sigma,\text{vol}} = \mu_{\mathcal{G},G,\Sigma,\text{vol}}. $$

In particular, the partition function $Z_{\mathcal{G},G}(g,t)$ does not depend on $\mathcal{G}$, and satisfies

* If $g \ge 1$,

$$ (2.10) \qquad Z_{\mathcal{G},G}(g,t) = Z_G(g,t) = \int_{G^{2g}} p_t([x_1, y_1] \cdots [x_g, y_g]) dx_1 dy_1 \cdots dx_g dy_g. $$

* If $g = 0$,

$$ (2.11) \qquad Z_{\mathcal{G},G}(0,t) = Z_G(0,t) = p_t(1). $$

*Proof.* The proof consists into two steps. We first prove the result for the unnormalized measures $\tilde{\mu}_{\mathcal{G},G,\Sigma,\text{vol}}$, then we show that the partition function does not depend on the graph. These two steps will imply the result for the normalized measures.

**Step 1.** Let $e \in E$ be the edge removed from $\mathcal{G}$ to get $\mathcal{G}'$, so that $E' = E \cup \{e\}$. If we require that $\mathcal{G}$ remains a map, it means that $e$ borders two faces $f_1, f_2$ of $\mathcal{G}$, whose union is a single face $f_* \in F'$, see Figure 4. Set $F_0 = F \setminus \{f_1, f_2\}$ and $F'_0 = F' \setminus \{f_*\}$. Let $\Phi: G^{E'} \to \mathbb{R}$ be a bounded measurable function. It corresponds to a function on $G^{E'} \sim G^{|E'|}$ invariant
by the action of $G^{V'}$, that we shall also denote by $\Phi$. According to the Driver-Sengupta formula,

$$ \int_{G^{E'}} \Phi(\omega) \mathcal{R}_* \tilde{\mu}_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) = \int_{G^{E}} \Phi \circ \mathcal{R}(\omega) \prod_{f \in F'} p_{\text{vol}(f)}(h_{\partial f}) d\omega. $$

Let us label arbitrarily the edges of $E'$ by $(e_1, \dots, e_k)$, where $k = |E|$, so that $E = \{e_1, \dots, e_k, e\}$.

$$ \begin{aligned} & \int_{G^E} \Phi \circ \mathcal{R}(\omega) \prod_{f \in F'} p_{\text{vol}(f)}(h_{\partial f}) d\omega \\ &= \int_{G^{k+1}} \Phi(e_1, \dots, e_k) \prod_{f \in F'} p_{\text{vol}(f)}(h_{\partial f}) de_1 \dots de_n de \\ &= \int_{G^k} \Phi(e_1, \dots, e_k) \prod_{f \in F'} p_{\text{vol}(f)}(h_{\partial f}) de_1 \dots de_n, \end{aligned} $$

where the second equality comes from integrating with respect to $e$. Using the same arguments as before,

$$ \int_{G^E} \Phi \circ \mathcal{R}(\omega) \tilde{\mu}_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) = \int_{G^{k+1}} \Phi(e_1, \dots, e_k) \prod_{f \in F} p_{\text{vol}(f)}(h_{\partial f}) de_1 \dots de_k de. $$

Set

$$ P(e_1, \dots, e_k) = \Phi(e_1, \dots, e_k) \prod_{f \in F_0} p_{\text{vol}(f)}(h_{\partial f}), $$

and $t_1 = \text{vol}(f_1)$, $t_2 = \text{vol}(f_2)$. On the one hand,

$$ \int_{G^E} \Phi \circ \mathcal{R}(\omega) \tilde{\mu}_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) = \int_{G^{k+1}} P(e_1, \dots, e_k) p_{t_2}(h_{\partial f_1}) p_{t_1}(h_{\partial f_2}) de_1 \dots de_k de, $$

<figure><img src="image_5.png" /><figcaption>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.</figcaption></figure>
and on the other hand

$$ (2.13) \qquad \int_{G^{E'}} \Phi(\omega) \mathcal{R}_* \tilde{\mu}_{G,G,\Sigma,\text{vol}}(d\omega) = \int_{G^k} P(e_1, \dots, e_k) p_{t_1+t_2}(h_{\partial f_*}) d e_1 \dots d e_k. $$

There are words $\alpha, \beta$ in the edges of $G'$ such that $\partial f_1 = e\alpha$, $\partial f_2 = \beta e^{-1}$ and $\partial f_* = \beta\alpha$. We deduce that $\partial f_* = \partial f_2 \partial f_1$.

Let us integrate the RHS of (2.12) with respect to $e$ and use the semigroup property of the heat kernel:

$$ \int_{G^E} \Phi \circ \mathcal{R}(\omega) \tilde{\mu}_{G,G,\Sigma,\text{vol}}(d\omega) = \int_{G^k} P(e_1, \dots, e_k) p_{t_1+t_2}(\partial f_2 \partial f_1) p_{t_1}(e\alpha) d e_1 \dots d e_k. $$

$$ \begin{aligned} \int_{G^E} \Phi \circ \mathcal{R}(\omega) \tilde{\mu}_{G,G,\Sigma,\text{vol}}(d\omega) &= \int_{G^k} P(e_1, \dots, e_k) p_{t_1+t_2}(\partial f_2 \partial f_1) p_{t_1}(e\alpha) d e_1 \dots d e_k \\ &= \int_{G^{E'}} \Phi(\omega) \mathcal{R}_* \tilde{\mu}_{G,G,\Sigma,\text{vol}}(d\omega), \end{aligned} $$

which shows the invariance of the unnormalized measures.

**Step 2.** The partition function $Z_{G,G}(g, t)$ is the integral of the unnormalized measure $\tilde{\mu}_{G,G,\Sigma,\text{vol}}$, which is invariant by subdivision as previously shown. If $g \ge 1$, we can obtain (2.10) by applying the Driver-Sengupta formula to a map of genus $g$ with one face. We can take as a map a $4g$-gon with boundary $[a_1, b_1] \dots [a_g, b_g]$. Finally, if $g = 0$, we apply the Driver-Sengupta formula to a planar map with two faces separated by a simple loop. $\square$

**2.4. The group of loops.** The main observables of Yang-Mills theory are Wilson loops $W_\ell^\chi = \chi(h_\ell)$, where $\ell$ is a loop traced in $G$ and $\chi: G \to \mathbb{C}$ is a central function on $G$, that is

$$ \chi(k^{-1}gk) = \chi(g), \quad \forall g, k \in G. $$

In practice, we shall only consider the following Wilson loops:

$$ W_\ell = \text{tr}(h_\ell), $$

where $\text{tr} = \frac{1}{N}\text{Tr}$ is the normalized trace on $M_N(\mathbb{C})$, because they form a dense generating family in the algebra of Wilson loops when $G$ is a compact classical group, according to a theorem of Lévy [54]. They constitute random variables, whose expectation is either called *Wilson loop expectation*, or *vacuum expectation value* (VEV). A first remark is that Wilson loop expectations vanish for open paths.

**Proposition 2.18.** Let $\gamma$ be an open path in $G$. Then $\mathbb{E}[W_\gamma] = 0$.

*Proof.* It is a straightforward consequence of gauge invariance. $\square$

<figure><img src="image_6.png" /></figure>

According to the previous proposition, it is sufficient to focus on Wilson loops for loops in the graph. We will see what structures can be put in the set of loops in the graph $G$. A first remark is that one can endow $P(G)$ with an operation of multiplication, given by concatenation of non-empty words: if $\gamma = e_1 \dots e_m$ and $\gamma_2 = e'_1 \dots e'_n$ and if $\overline{e_m} = e'_1$, then $\gamma_1 \cdot \gamma_2 =$
<figure><img src="image_7.png" /><figcaption>FIGURE 5. The loop on the left and on the right are in the same reduction class.</figcaption></figure>

$\gamma_1\gamma_2 = e_1 \dots e_m e'_1 \dots e'_n$. One can also extend the concatenation to constant words: $\gamma \cdot \bar{\gamma} = \underline{\gamma} \cdot \gamma = \gamma$ by convention. For any path $\gamma = e_1 \dots e_n \in P(\mathbb{G})$, we define the reverse path $\gamma^{-1} = e_n^{-1} \dots e_1^{-1}$. Even if we restrict ourselves to loops, there is no satisfying algebraic structure: $L(\mathbb{G})$ is not stable by multiplication, and there is no neutral element for the inversion. A simple way to overcome the first issue is to fix a common basepoint for all loops. It might seem to be restrictive, but we will see in a moment that it is not, because of the way we deal with the second issue.

In order to get a neutral element, we put an equivalence class on loops as follows: $\ell_1$ and $\ell_2 \in L(\mathbb{G})$ are in the same reduction class if one can go from one to the other by a finite number of insertion and removals of edges. More concretely,

$$e_1 \dots e_i \tilde{e} \tilde{e}^{-1} e_{i+1} \dots e_n \sim e_1 \dots e_i e_{i+1} \dots e_n.$$

Two equivalent loops in a triangulation of a surface of genus 2 are shown in Figure 5 (sides of the polygons are glued pairwise according to their labels in order to get a genus 2 surface). This equivalence relation is related to the tree-like equivalence sometimes found in the literature [31, 39]. We shall denote by $RL_v(\mathbb{G}) = L_v(\mathbb{G})/\sim$ the set of reduced loops. The following result is simple, yet quite useful.

**Proposition 2.19.** For all $v \in V$, $RL_v(\mathbb{G})$ is a group.

We will state a few important results about this group, that come from [56]. But first, let us explain why fixing a basepoint is not a big deal for our purpose. Fix a map $\mathbb{G} = (V, E, F)$ of any genus, and consider a marked vertex $v \in V$. If we want to compute the Wilson loop expectation of a loop $\ell$ that does not go through $v$, here is what one can do. Let $v_0$ the initial basepoint of $\ell$, and choose an oriented path $c$ in $\mathbb{G}$ going from $v$ to $v_0$. Then the loop $\ell_v = c\ell c^{-1}$ has basepoint $v$ and is in the same reduction class as $\ell$. See Figure 6 for an example. In the case where $\ell = \partial f$ is the boundary of a
<figure><img src="image_8.png" /><figcaption>FIGURE 6. Construction of a loop $\ell_v$ in the same reduction class as $\ell$ but based at $v$.</figcaption></figure>

face $f \in F$, then the loop $\ell_v$ is called *lasso* of base $v$ associated to the face $f$. A priori, there are many possible $\ell_v$ for a given $\ell$, but if we fix a spanning tree $T$ of $\mathbb{G}$ (i.e. a connected subgraph without cycles that goes through all vertices of $\mathbb{G}$), there is a unique way to take a path from $v$ to the boundary of $f$ through $T$. Lassos are one of the main tools to prove the following two results.

**Lemma 2.20** ([56], Lemma 1.3.33). For any map $\mathbb{G} = (V, E, F)$ of genus $g \ge 0$ and for any $v \in V$, the group $\text{RL}_v(\mathbb{G})$ is free of rank $|E| - |V| + 1 = 2g - 1 + |F|$.

**Proposition 2.21** ([56], Prop. 2.4.2). Let $\mathbb{G} = (V, E, F)$ be a map or genus $g$ and $v \in V$ be a fixed vertex. Set $k = |F|$. Then:

(1) The group $\text{RL}_v(\mathbb{G})$ has the following presentation

$$ \langle a_1, b_1, \dots, a_g, b_g, c_1, \dots, c_k \mid [a_1, b_1] \dots [a_g, b_g] = c_1 \dots c_k \rangle, $$

where the homotopy classes of $(a_1, b_1, \dots, a_g, b_g)$ generate the fundamental group of $\Sigma$, and $c_1, \dots, c_k$ are lassos enclosing the faces of $\mathbb{G}$.

(2) For any bounded measurable function $f: G^{2g+k} \to \mathbb{C}$,

$$ Z_G(g, t) \int_{G^{2g+k}} f(H_{a_1}, H_{b_1}, \dots, H_{a_g}, H_{b_g}, H_{c_1}, \dots, H_{c_k}) \mu_{\mathbb{G},G,\Sigma,\text{vol}}(d\omega) \\ = \int_{G^{2g+k-1}} f(x_1, y_1, \dots, x_g, y_g, z_1, \dots, z_k) \prod_{i=1}^{k} p_{\text{vol}(f_i)}(z_i) \prod_{i=1}^{g} dx_i dy_i \prod_{j=1}^{k-1} dz_j, $$

by setting $z_k = z_{k-1}^{-1} \dots z_1^{-1}[x_1, y_1] \dots [x_g, y_g]$, and with the convention that $H_\ell = H_\ell(\omega)$ as a random variable for any loop $\ell$.

**Proposition 2.21** yields another interpretation of the discrete Yang-Mills measure: it is a measure on the *representation space*

$$ \text{Hom}(\text{RL}_v(\mathbb{G}), G) $$
which is invariant by the simultaneous adjoint action of $G$ on all generators. It descends to a measure on

$$ \mathrm{Hom}(\mathrm{RL}_v(\mathbb{G}), G)/G, $$

which is actually the same space as the *character variety*

$$ \chi(\Sigma_F, G) = \mathrm{Hom}(\pi_1(\Sigma_F), G)/G, $$

for the surface $\Sigma_F$ obtained from $\Sigma$ by removing one point from the interior of each face $f \in F$. Indeed, the groups $\mathrm{RL}_v(\mathbb{G})$ and $\pi_1(\Sigma_F)$ have the same presentation, therefore they are isomorphic. The character variety $\chi(\Sigma_F, G)$ has a rich structure that we will not develop here; we refer to [47] for an introduction.

**2.5. Continuous extension.** The property of invariance by subdivision satisfied by the discrete Yang-Mills theory enables to define it for graphs that are finer and finer, and one could expect to extend it into a “continuous” measure by taking a projective limit of graphs. The underlying argument is the same as Kolmogorov’s extension theorem to prove the existence of the Brownian motion from its finite-dimensional marginals. The moral is that one does not lose any generality to use the discrete Yang-Mills measure, as long as we consider observables associated to fixed loops: under loose regularity assumptions, they can always be drawn in some map and all computations can be done in the discrete setting.

To define the continuous limiting object, we need to choose a correct space of paths in the surface that contains at least all paths that can be traced in maps.

**Definition 2.22.** Let $\Sigma$ be a smooth compact surface. A *parametrized path* on $\Sigma$ is a path $\gamma : [0, 1] \to \Sigma$ that is either constant, or Lipschitz-continuous, with a speed bounded from below by a positive constant. A *path* on $\Sigma$ is an equivalence class of parametrized paths on $\Sigma$ for the equivalence relation

$$ \gamma \sim \gamma' \Leftrightarrow \exists \Phi : [0, 1] \to [0, 1] \text{ bi-Lipschitz increasing homeo. s.t. } \gamma = \gamma' \circ \Phi. $$

We denote by $P(\Sigma)$ the set of paths in $\Sigma$, and $L(\Sigma)$ the set of *loops* in $\Sigma$, i.e. closed paths.

The reason why we require non-constant paths to be Lipschitz-continuous with speed bounded from below is that it enables to choose a parametrization with constant speed. The main result of this subsection, due to Lévy [56], is the following.

**Theorem 2.23.** Let $\Sigma$ be a compact surface endowed with a smooth measure of area vol, and $G$ a compact Lie group whose Lie algebra is endowed with an invariant inner product. There exists a family of $G$-valued random variables $(H_\ell)_{\ell \in L(\Sigma)}$ such that:

(1) For any map $\mathbb{G} = (V, E, F)$ embedded in $\Sigma$ and any fixed $v \in V$, the distribution of $(H_\ell)_{\ell \in \mathrm{RL}_v(\mathbb{G})}$ is given by Proposition 2.21 (ii).
(2) For any family $(\ell_n)_{n \ge 1}$ of loops that converges<fnref n="3" /> to a loop $\ell$, then the sequence of random variables $(H_{\ell_n})_{n \ge 1}$ converges in probability to $\ell$.

The continuous Yang–Mills measure on $\Sigma$ is the distribution of $(H_\ell)_{\ell \in L(\Sigma)}$.

The proof of this result relies on many technical developments in measure
theory and Riemannian geometry, that we will skip here. Instead, we will
rather make a few remarks:

* The right point of view to get a projective system of measures is to consider graphs whose edges are embedded into piecewise geodesic paths, which requires fixing a Riemannian metric on $\Sigma$.

* However, one can show that all discrete Yang–Mills measures are invariant by area-preserving diffeomorphisms, therefore they do not depend on the choice of the Riemannian metric.

* The choice of regularity of the loops comes from the fact that they can be obtained as limits of piecewise geodesic loops. It is also true for rectifiable paths, which are continuous with finite length. Note that having finite length does not depend on the choice of a metric, which justifies point (ii) in Theorem 2.23.

To conclude this section, we must clarify an important subtlety regard-
ing the continuous Yang–Mills measure. In the construction of the discrete
Yang–Mills measure, we mentioned that discrete connections were implicitly
identified to their holonomies along loops, which was quite acceptable. It
becomes much less obvious at the continuous level, and the convoluted con-
struction of the continuous Yang–Mills measure can leave a doubt whether
it is indeed related to the formal Yang–Mills measure (1.3). Lévy and Nor-
ris gave a positive answer [61], and proved a large deviation principle of the
Yang–Mills measure on a compact surface with area $t$ when $t \to 0$, whose rate
function is proportional to the Yang–Mills action $S_{YM}$. Replacing random
connections by random holonomies is here harmless because the observables
of the theory are essentially Wilson loops, which are central functions of
the holonomies. This point of view has been used with a relative success
in an attempt of defining quantum gravity, and it is called the loop repre-
sentation, see [31]. Another relationship between the Yang–Mills functional
and the Yang–Mills measure was conjectured in [12]: in this paper, Chan-
dra, Chevyrev, Hairer and Shen constructed a Markov process governed
by the stochastic Yang–Mills equation, and asked whether this process had
the Yang–Mills measure described in Theorem 2.23 as a unique invariant
measure. Chevyrev and Shen later gave a positive answer to this question
[18].

<footnote n="3">³In the sense of uniform convergence of paths with their constant speed parametrization.</footnote>
### 3. CHARACTER EXPANSION OF THE PARTITION FUNCTION

In this section, we will recall the main results of representation theory of compact groups required to compute the Fourier transform of the partition function, which will be stated in Theorem 3.13. We shall mainly refer to [9, 28].

#### 3.1. Representations of Lie groups.

**Definition 3.1.** Let $G$ be a Lie group. A real (resp. complex) *representation* of $G$ is a couple $(\rho, V)$ where $V$ is a real (resp. complex) vector space and $\rho: G \to GL(V)$ is a smooth group morphism. The *degree* of $\rho$, also called its *dimension*, is the dimension of the vector space $V$.

A representation $(\rho, V)$ of $G$ is said to be *irreducible* if the only subspaces of $V$ stable by $\rho(g)$ for any $g \in G$ are $\{0\}$ and $V$. Irreducible representations somehow constitute the building blocks of representation theory.

Note that for any finite-dimensional representation $(\rho, \mathcal{H})$ of a compact Lie group $G$ in a Hilbert space$^4 \mathcal{H}$, there is an inner product on $\mathcal{H}$ such that the representation is unitary, i.e.

$$ \|\pi(g)v\| = \|v\|, \quad \forall g \in G, \forall v \in \mathcal{H}. $$

Indeed, if we start from an arbitrary inner product $\langle \cdot, \cdot \rangle_0$, the new inner product $\langle \cdot, \cdot \rangle$ defined by

$$ \langle v, w \rangle = \int_G \langle \rho(g)v, \rho(g)w \rangle_0 dg, \quad \forall (v, w) \in \mathcal{H}^2, $$

turns $\rho$ into a unitary representation.

**Lemma 3.2 (Schur).** Let $(\rho_1, V_1)$ and $(\rho_2, V_2)$ be two irreducible complex representations of a Lie group $G$, and $A \in \text{Hom}(V_1, V_2)$ be a linear application such that $A\rho_1 = \rho_2 A$ (it is said to be $G$-equivariant). Then one of the following assertions is true:

(1) $A = 0$,

(2) $A$ is an isomorphism.

Furthermore, if $V_1 = V_2$ and $\rho_1 = \rho_2$, then there exists $\lambda \in \mathbb{C}$ such that $A = \lambda I_{V_1}$.

Schur's Lemma enables to define an equivalence relation on irreducible representations, and we denote by $\hat{G}$ the set of equivalence classes, sometimes called the *dual* of $G$. Note that if $G$ is compact, then $\hat{G}$ is countable, and if $G$ is abelian then $\hat{G}$ is also a group.

<footnote>$^4$Here, it might be overkill to speak about a Hilbert space, since it is a finite-dimensional vector space, but we rather use this denomination to underline the fact that it is endowed with an inner product. In general, there might be unitary representations in infinite-dimensional Hilbert spaces, e.g. the left and right regular representations act on $L^2(G)$.</footnote>
3.2. **Peter–Weyl theorem.** Let $G$ be a compact group and $(\rho, \mathcal{H})$ be a unitary irreducible representation of $G$. The *matrix coefficients* of $\rho$ are functions of the form

$$g \mapsto \langle \rho(g)u, v \rangle,$$

pour $u, v \in \mathcal{H}$. The space of all matrix coefficients is denoted by $\mathcal{M}_{\rho}$, and if $\{e_1, \dots, e_k\}$ is an orthonormal basis of $\mathcal{H}$, $\mathcal{M}_{\rho}$ is generated by the *elementary matrix coefficients*

$$\rho_{ij} : g \mapsto \langle \rho(g)e_i, e_j \rangle, \quad \forall 1 \le i, j \le k.$$

**Theorem 3.3** (Schur's orthogonality relations I). Let $(\rho, \mathcal{H})$ be a unitary irreducible representation of a compact Lie group $G$. For any $u, v, u', v' \in \mathcal{H}$,

$$(3.1) \qquad \int_G \langle \rho(g)u, v \rangle \overline{\langle \rho(g)u', v' \rangle} dg = \frac{1}{d_{\rho}} \langle u, u' \rangle \langle v, v' \rangle.$$

In particular, if $\{e_1, \dots, e_{d_{\rho}}\}$ is an orthonormal basis of $\mathcal{H}$, then

$$(3.2) \qquad \int_G \rho_{ij}(g) \overline{\rho_{k\ell}(g)} dg = \frac{1}{d_{\rho}} \delta_{ik} \delta_{j\ell}.$$

**Theorem 3.4** (Schur's orthogonality relations II). Let $(\rho, \mathcal{H})$ and $(\rho', \mathcal{H}')$ be two irreducible unitary representations of a compact Lie group $G$. If $\rho$ and $\rho'$ are not equivalent, then $\mathcal{M}_{\rho}$ and $\mathcal{M}_{\rho'}$ are orthogonal in $L^2(G)$.

A fundamental consequence of these orthogonality relations is the following theorem:

**Theorem 3.5** (Peter–Weyl). Let $G$ be a compact Lie group. For any $\lambda \in \hat{G}$, denote by $\mathcal{M}_{\lambda}$ the space of matrix coefficients of a representation of the class $\lambda$. We have

$$(3.3) \qquad L^2(G) = \bigoplus_{\lambda \in \hat{G}} \mathcal{M}_{\lambda}.$$

One of the main applications of this theorem is Plancherel's formula for compact groups, that we shall see in the next subsection. Beforehand, let us rewrite the Peter–Weyl theorem under a more specific form. A function $f: G \to \mathbb{C}$ on a compact Lie group is said to be *central* if for all $(g, h) \in G^2$, $f(ghg^{-1}) = f(h)$. It is in particular the case of the *characters*

$$\chi_{\rho}: g \mapsto \mathrm{Tr}(\rho(g)),$$

for any finite-dimensional representation $\rho$ of $G$. By convention, if $\rho$ is an irreducible representation of the class $\lambda \in \hat{G}$, we will denote by $\chi_{\lambda}$ the character associated to $\rho$. In particular, $\chi_{\lambda} \in \mathcal{M}_{\lambda}$.

**Theorem 3.6** (Peter–Weyl II). For any compact Lie group $G$, the set $\{\chi_{\lambda}, \lambda \in \hat{G}\}$ is a Hilbert basis of the space $Z^2(G) \subset L^2(G)$ of square integrable central functions.
### 3.3. Fourier transform.

**Definition 3.7.** The Fourier transform of $f \in L^2(G)$ is the function $\hat{f}: \hat{G} \to \text{End}(\mathcal{H}_\lambda)$ defined by

$$ \hat{f}(\lambda) = \int_G f(g) \rho_\lambda(g^{-1}) dg, $$

where $(\rho_\lambda, \mathcal{H}_\lambda)$ is an irreducible unitary representation of $G$ in the class $\lambda$.

**Theorem 3.8 (Plancherel).** The map $f \mapsto \hat{f}$ is a unitary isomorphism of Hilbert spaces between $L^2(G)$ and the space of sequences $A = (A_\lambda)_{\lambda \in \hat{G}}$ of operators $A_\lambda \in \text{End}(\mathcal{H}_\lambda)$ such that

$$ \|A\|^2 = \sum_{\lambda \in \hat{G}} d_\lambda \text{Tr}(A_\lambda^* A_\lambda) < \infty. $$

Besides, for any $f \in L^2(G)$,

$$ (3.4) \qquad f(g) = \sum_{\lambda \in \hat{G}} d_{\lambda} \text{Tr}(\hat{f}(\lambda) \rho_{\lambda}(g)). $$

One can give an interpretation of (3.4) in terms of convolution:

**Corollary 3.9.** Let $f \in L^2(G)$ for a compact Lie group $G$. The following identity holds in $L^2(G)$:

$$ (3.5) \qquad f(g) = \sum_{\lambda \in \hat{G}} d_{\lambda} (f * \chi_{\lambda})(g). $$

*Proof.* For any $\lambda \in \hat{G}$, take any representation $(\rho_\lambda, \mathcal{H}_\lambda)$. By (3.4) and the definition of Fourier coefficients,

$$ f(g) = \sum_{\lambda} d_{\lambda} \text{Tr} \left( \int_G f(h) \rho_{\lambda}(h^{-1}) dh \rho_{\lambda}(g) \right), $$

and by the linearity of the trace,

$$ f(g) = \sum_{\lambda} d_{\lambda} \int_{G} f(h) \text{Tr}(\rho_{\lambda}(h^{-1}) \rho_{\lambda}(g)) dh. $$

The result follows then from straightforward simplifications. $\square$

<figure><img src="image_9.png" /></figure>

### 3.4. Character expansion of the heat kernel.
Recall that the heat kernel on $G$ is the convolution semigroup $(p_t)_{t \ge 0}$ that satisfies the heat equation

$$ (3.6) \qquad \begin{cases} \frac{d}{dt} p_t(g) = \frac{1}{2} \Delta_G p_t(g), & \forall g \in G, \forall t > 0, \\ \lim_{t \downarrow 0} p_t = \delta_{1_G}. \end{cases} $$

It clearly depends on the choice of the inner product on $\mathfrak{g}$. As in the case of heat equation in $\mathbb{R}^d$, one can get an explicit formula for the heat kernel in terms of eigenvalues and eigenfunctions of the Laplacian by the means of Fourier transform. In our case, the eigenfunctions of $\Delta_G$ are in fact the irreducible characters.
**Proposition 3.10.** Let G be a compact Lie group. The set {$\chi_{\lambda}, \lambda \in \hat{G}$} is a Hilbert basis of eigenvectors of $\Delta_G$. Let us denote by $c_2(\lambda) \ge 0$ the eigenvalue of $-\Delta_G$ associated to $\chi_{\lambda}$.

The main result of this subsection is the following.

**Theorem 3.11.** Let G be a compact Lie group. The heat kernel $(p_t)_{t \ge 0}$ admits the following decomposition, which holds pointwise for $t > 0$ and in $L^2(G)$ for all $t \ge 0$:

$$ (3.7) \qquad p_t(g) = \sum_{\lambda \in \hat{G}} e^{-\frac{t}{2}c_2(\lambda)} d_{\lambda}\chi_{\lambda}(g), \quad \forall g \in G. $$

*Proof.* By Plancherel's formula, we have for any $t > 0$ and $g \in G$

$$ p_t(g) = \sum_{\lambda \in \hat{G}} d_{\lambda}(p_t * \chi_{\lambda})(g). $$

If we use the linearity of the integral and apply the heat equation,

$$ \frac{d}{dt}(p_t * \chi_{\lambda})(g) = \left( \left( \frac{d}{dt}p_t \right) * \chi_{\lambda} \right)(g) = \frac{1}{2}((\Delta_G p_t) * \chi_{\lambda})(g). $$

Besides, integrating by parts yields

$$ (\Delta_G p_t) * \chi_{\lambda} = (p_t * \Delta_G \chi_{\lambda}), $$

thus an application of the identity $\Delta_G \chi_{\lambda} = -c_2(\lambda)\chi_{\lambda}$ gives

$$ \frac{d}{dt}(p_t * \chi_{\lambda})(g) = -\frac{1}{2}c_2(\lambda)(p_t * \chi_{\lambda})(g), \quad \forall g \in G. $$

We can conclude by solving the differential equation for $\lambda \in \hat{G}$. $\square$

<figure><img src="image_10.png" /></figure>

**Example 3.12.** In the abelian case $G = U(1)$, one recovers a classical expression of the heat kernel on the circle, or equivalently on $[0, 2\pi]$ with periodic boundary conditions: recall that the irreducible characters of $U(1)$ are $\chi_n : e^{i\theta} \mapsto e^{in\theta}$, for any $n \in \mathbb{Z}$, and one can check that $\Delta_{\chi_n} = -n^2\chi_n$, thus Theorem 3.11 yields

$$ p_t(e^{i\theta}) = \sum_{n \in \mathbb{Z}} e^{-\frac{t}{2}n^2 + in\theta}. $$

**3.5. Exact formula for the partition function.** We have seen that the partition function $Z_G(g, t)$ can be written as an integral over $G^{2g}$ involving the heat kernel $p_t$, for $t > 0$. The goal of this subsection is to obtain the following character expansion of this matrix integral, which was first found by Migdal by other means [63].

**Theorem 3.13.** Let G be a compact Lie group, $\Sigma_g$ a surface of genus $g \ge 0$ and area $t > 0$. The Yang-Mills partition function satisfies

$$ (3.8) \qquad Z_G(g, t) = \sum_{\lambda \in \hat{G}} e^{-\frac{t}{2}c_2(\lambda)} d_{\lambda}^{2-2g}. $$
A first step towards the proof of this result is to decompose the partition function in a series involving integrals over irreducible characters.

**Lemma 3.14.** Let $G$ be a compact Lie group. For any $g \ge 1$,

$$ (3.9) \qquad Z_G(g,t) = \sum_{\lambda \in \hat{G}} e^{-\frac{t}{2}c_2(\lambda)} d_\lambda \int_{G^{2g}} \chi_\lambda([x_1, y_1] \cdots [x_g, y_g]) \prod_{i=1}^g dx_i dy_i. $$

*Proof.* It is a straightforward consequence of (2.10) and Theorem 3.11. $\square$

It remains to understand how to compute the integral of an irreducible character applied to a product of commutators of uniform random variables. To do so, we will need the following two propositions.

**Proposition 3.15.** Let $G$ be a compact group, and $\lambda \in \hat{G}$ an equivalence class of irreducible representations. For any $x, y \in G$,

$$ (3.10) \qquad \int_G \chi_\lambda(xgyg^{-1})dg = \frac{\chi_\lambda(x)\chi_\lambda(y)}{d_\lambda}. $$

*Proof.* For any $x$, the function

$$ F_{\lambda,x}(y) = \int_G \chi_\lambda(xgyg^{-1})dg $$

is central as a function of $y$. By Schur's lemma, it is therefore proportional to $\chi_\lambda$:

$$ F_{\lambda,x}(y) = \frac{1}{d_\lambda} F_{\lambda,x}(1_G)\chi_\lambda(y) = \frac{1}{d_\lambda} \chi_\lambda(x)\chi_\lambda(y). $$

<figure><img src="image_11.png" /></figure>

**Proposition 3.16.** Let $G$ be a compact group and $\lambda, \mu \in \hat{G}$ two equivalence classes of irreducible representations.

$$ (3.11) \qquad \chi_\lambda * \chi_\mu = \frac{\chi_\lambda}{d_\lambda} \delta_{\lambda,\mu}. $$

*Proof.* Let $(\rho_\lambda, \mathcal{H}_\lambda)$ be a unitary representation in the class $\lambda$. Denote by $\rho_\lambda^{ij}$ the matrix coefficients of the representations, i.e.

$$ \rho_\lambda^{ij}(x) = \langle \rho_\lambda(x)e_i, e_j \rangle, \quad \forall 1 \le i, j \le d_\lambda. $$

We have

$$ \chi_\lambda(xy) = \mathrm{Tr}(\rho_\lambda(xy)) = \mathrm{Tr}(\rho_\lambda(x)\rho_\lambda(y)) = \sum_{i,j=1}^{d_\lambda} \rho_\lambda^{ij}(x)\rho_\lambda^{ji}(y). $$

Using the same trick for a representation $\rho_\mu$ of the class $\mu$, we get

$$ \chi_\lambda(xy)\chi_\mu(y^{-1}) = \chi_\lambda(xy)\overline{\chi_\mu(y)} = \sum_{i=1}^{d_\lambda} \sum_{j=1}^{d_\lambda} \sum_{k=1}^{d_\mu} \rho_\lambda^{ij}(x)\rho_\lambda^{ji}(y)\overline{\rho_\mu^{kk}(y)}. $$
Integrating with respect to $y$ yields

$$ \sum_{i,j,k} \rho_{\lambda}^{ij}(x) \int_{G} \rho_{\lambda}^{ji}(y) \rho_{\mu}^{kk}(y^{-1}) dy = \sum_{i,j,k} \rho_{\lambda}^{ij}(x) \int_{G} \rho_{\lambda}^{ji}(y) \overline{\rho_{\mu}^{kk}(y)} dy, $$

and Schur's orthogonality relation yields

$$ \chi_{\lambda} * \chi_{\mu}(x) = \frac{1}{d_{\lambda}} \sum_{i=1}^{d_{\lambda}} \rho_{\lambda}^{ii}(x) \delta_{\lambda,\mu}, $$

which is the expected result. □

*Remark 3.17.* The previous proposition gives the structure constants of $Z^2(G)$ as a convolution algebra, if we recall that it is generated by the irreducible characters.

The main integration result that we will need is the following.

**Proposition 3.18.** Let $G$ be a compact group, and $\lambda \in \hat{G}$ be an equivalence class of irreducible representations. For any $g \ge 1$,

$$ (3.12) \qquad \int_{G^{2g}} \chi_{\lambda}([x_1, y_1] \cdots [x_g, y_g]) \prod_{i=1}^{g} dx_i dy_i = d_{\lambda}^{1-2g}. $$

*Proof.* We proceed by induction on $g$. The case $g = 1$ is a consequence of Proposition 3.15 applied to $x = x_1$ and $y = x_1^{-1}$:

$$ \int_{G^2} \chi_{\lambda}(x_1 y_1 x_1^{-1} y_1^{-1}) dx_1 dy_1 = \int_G \frac{\chi_{\lambda}(x_1) \chi_{\lambda}(x_1^{-1})}{d_{\lambda}} = 1, $$

where the last equality follows from orthogonality relations. Assume that the equality holds for a fixed $g$. Then

$$ \int_{G^{2g+2}} \chi_{\lambda}([x_1, y_1] \cdots [x_g, y_g][x, y]) \prod_{i=1}^{g} dx_i dy_i dx dy = \frac{1}{d_{\lambda}} \int_{G^{2g+1}} \chi_{\lambda}([x_1, y_1] \cdots [x_g, y_g]x) \chi_{\lambda}(x^{-1}) \prod_{i=1}^{g} dx_i dy_i dx. $$

The RHS can be rewritten

$$ \frac{1}{d_{\lambda}} \int_{G^{2g}} (\chi_{\lambda} * \chi_{\lambda})([x_1, y_1] \cdots [x_g, y_g]) \prod_{i=1}^{g} dx_i dy_i = \frac{1}{d_{\lambda}^2} \int_{G^{2g}} \chi_{\lambda}([x_1, y_1] \cdots [x_g, y_g]) \prod_{i=1}^{g} dx_i dy_i, $$

by using the formula of convolution of characters. □

*Proof of Theorem 3.13.* We simply combine Lemma 3.14 and Proposition 3.18. □
## 4. LARGE N LIMIT OF THE PARTITION FUNCTION

We can now specialize to the case of the unitary group

$$G = U(N) = \{U \in \mathrm{GL}(N, \mathbb{C}) : U^* = U^{-1}\},$$

whose Lie algebra is the space of skew-Hermitian matrices

$$\mathfrak{u}(N) = \{X \in \mathcal{M}_N(\mathbb{C}) : X^* = -X\} = i\mathcal{H}_N,$$

where $\mathcal{H}_N = \{X \in \mathcal{M}_N(\mathbb{C}) : X^* = X\}$ is the space of Hermitian matrices, on which is notably defined the Gaussian Unitary Ensemble. We will endow $\mathfrak{u}(N)$ with the following inner product:

$$\langle X, Y \rangle = N \mathrm{Tr}(XY^*) = -N \mathrm{Tr}(XY).$$

We will first recall the expressions of relevant quantities, such as the dimension and Casimir of irreducible representations, and then we will provide results about the asymptotics of the partition function.

### 4.1. Highest weight theory for $U(N)$.
In this subsection, we gather standard results of the representation theory of unitary groups in order to get explicit formulas for the character, dimension and Casimir of all irreducible representations. Good references are [9, 28] for instance. Note that we need to specify an inner product on $\mathfrak{u}(N)$ in order to give an expression of the Casimir, because it is an eigenvalue of $-\Delta_{U(N)}$, which depends on a choice of a metric on $G$.

**Theorem 4.1 (Weyl's character formula).** *The irreducible representations of $U(N)$ (resp. $SU(N)$) are in bijection with the $N$-tuples $(\lambda_1, \dots, \lambda_N) \in \mathbb{Z}^N$ such that $\lambda_1 \ge \dots \ge \lambda_N$ (resp. $\lambda_1 \ge \dots \ge \lambda_N = 0$), called highest weights. Besides, for any irreducible representation of highest weight $\lambda$ and any $U \in \mathrm{U}(N)$ or $\mathrm{SU}(N)$ with eigenvalues $(x_1, \dots, x_N)$,*

$$(4.1) \qquad \chi_{\lambda}(U) = s_{\lambda}(x_1, \dots, x_N),$$

where $s_{\lambda}$ is the Schur function associated to $\lambda$:

$$s_{\lambda}(x_1, \dots, x_N) = \frac{\det\left[x_i^{\lambda_j+N-j}\right]_{1\le i,j\le N}}{\det\left[x_i^{N-j}\right]_{1\le i,j\le N}}.$$

An example of highest weight of $U(N)$ is given in Figure 7.

**Remark 4.2.** These Schur functions are symmetric Laurent polynomials for $U(N)$ and symmetric polynomials for $SU(N)$.

**Corollary 4.3 (Weyl's dimension formula).** Let $\lambda \in \hat{U}(N)$ (or $\hat{SU}(N)$). Its dimension satisfies

$$(4.2) \qquad d_{\lambda} = \prod_{1 \le i < j \le N} \frac{\lambda_i - \lambda_j + j - i}{j - i}.$$
**Theorem 4.4.** Assume that $u(N)$ is endowed with the inner product $\langle X, Y \rangle = N\mathrm{Tr}(X^*Y)$. For any $\lambda \in \hat{U}(N)$,

$$ (4.3) \qquad c_2(\lambda) = \frac{1}{N} \left( \sum_{i=1}^{N} \lambda_i^2 + \sum_{1 \le i < j \le N} (\lambda_i - \lambda_j) \right). $$

Although we will not need it here, the Casimir s of irreducible representations of SU(N) are slightly different:

$$ c_2(\lambda) = \frac{1}{N} \left( \sum_{i=1}^{N} \lambda_i^2 + \sum_{1 \le i < j \le N} (\lambda_i - \lambda_j) \right) - \frac{1}{N^2} \left( \sum_{i=1}^{N} \lambda_i \right)^2, \quad \forall \lambda \in \hat{SU}(N). $$

**4.2. Witten zeta functions.** If G is a compact connected simply connected group, we define its Witten zeta function as the meromorphic continuation of the series

$$ \zeta_G(s) = \sum_{\lambda \in \hat{G}} d_{\lambda}^{-s}, $$

for all $s \in \mathbb{C}$. This name has been chosen by Zagier [79] because the function appears in the work of Witten [77], and because it generalizes the Riemann zeta function: indeed, for any $s \in \mathbb{C}$,

$$ \zeta_{\mathrm{SU}(2)}(s) = \sum_{n \ge 1} \frac{1}{n^s} = \zeta(s). $$

For the sake of simplicity, we will admit that for any $s > 1$,

$$ \sup_N \zeta_{\mathrm{SU}(N)}(s) < \infty, $$

(see [49, Prop. 2.2] for a proof). The main result we care about is the following:

<figure><img src="image_12.png" /><figcaption>FIGURE 7. The highest weight $\lambda = (3, 2, 2, 1, -1, -3, -3)$ of U(7).</figcaption></figure>
**Lemma 4.5.** For all $s > 1$,

$$ (4.4) \qquad \lim_{N \to \infty} |\zeta_{\mathrm{SU}(N)}(s) - 1| = 0. $$

*Proof.* Let us start by using Weyl’s dimension formula (Theorem 4.3), which directly yields that the irreducible representation of highest weight $\lambda = (0, \dots, 0)$ has dimension 1. Now assume that $\lambda \neq (0, \dots, 0)$; we perform a change of variables

$$ m_1 = \lambda_1 - \lambda_2 + 1, $$

$$ m_2 = \lambda_2 - \lambda_3 + 1, $$

$\vdots$

$$ m_{N-1} = \lambda_{N-1} - \lambda_N + 1. $$

The condition $(\lambda_1 \ge \dots \ge \lambda_N = 0)$ is equivalent to $m_1, \dots, m_{N-1} \ge 0$. We get

$$ d_{\lambda} = \prod_{1 \le i < j \le N} \frac{m_i + \dots + m_{j-1}}{j - i}. $$

This expression is clearly nondecreasing in the coefficients $m_i$, therefore the smallest possible values of $d_\lambda$ are those of $\lambda_1 = (1, 0, \dots, 0)$, $\lambda_2 = (1, 1, 0, \dots, 0)$, ..., $\lambda_{N-1} = (1, \dots, 1, 0)$. In other terms, for any $\lambda \in \tilde{\mathrm{SU}}(N) \setminus \{(0, \dots, 0)\}$ and any $1 \le i \le N-1$,

$$ d_{\lambda} \ge d_{\lambda_i} = \binom{N}{i}. $$

In particular, for any nontrivial highest weight $\lambda$, we find that $d_\lambda \ge N$. From this point the proof becomes elementary: for any $1 < s' < s$,

$$ \zeta_{\mathrm{SU}(N)}(s) - 1 = \sum_{\substack{\lambda \in \tilde{\mathrm{SU}}(N) \\ \lambda \neq (0, \dots, 0)}} d_{\lambda}^{-s} = \sum_{\substack{\lambda \in \tilde{\mathrm{SU}}(N) \\ \lambda \neq (0, \dots, 0)}} d_{\lambda}^{-s' - (s-s')} \le N^{-(s-s')} \sum_{\substack{\lambda \in \tilde{\mathrm{SU}}(N) \\ \lambda \neq (0, \dots, 0)}} d_{\lambda}^{-s'}, $$

and the RHS converges to 0 as $N$ tends to infinity. $\square$

<figure><img src="image_13.png" /></figure>

The previous lemma, which has many consequences, was proved for a large class of groups by Guralnick-Larsen-Manack [37] and rediscovered by the author later. It will imply the convergence of the partition function $Z_{U(N)}(g, t)$ for $g \ge 2$. Before giving the details, let $\theta: \mathbb{C} \to \mathbb{C}$ be the Jacobi theta function defined by the series

$$ \theta(z) = \sum_{n \in \mathbb{Z}} z^{n^2}. $$

Note that it is in particular holomorphic on $\{z \in \mathbb{C} : |z| < 1\}$. The asymptotics of partition functions on genus $g \ge 2$ are the following, as first stated in [68] and proved in [49].
**Theorem 4.6.** For any $g \ge 2$ and $t > 0$, we have the following convergence:

$$ (4.5) \qquad \lim_{N \to \infty} Z_{U(N)}(g, t) = \theta(q_t), $$

by setting $q_t = e^{-t/2}$.

*Proof.* We use the following bijection:

$$ \left\{ \begin{array}{rcl} \widehat{SU}(N) \times \mathbb{Z} & \to & \widehat{U}(N) \\ (\lambda, n) & \mapsto & \lambda + n := (\lambda_1 + n, \dots, \lambda_N + n). \end{array} \right. $$

It is easy to check that $d_{\lambda+n} = d_{\lambda}$ for all $\lambda \in \widehat{SU}(N)$ and $n \in \mathbb{Z}$. Consequently,

$$ Z_{U(N)}(g, t) = \sum_{\lambda \in \widehat{SU}(N)} \sum_{n \in \mathbb{Z}} q_t^{c_2(\lambda+n)} d_{\lambda+n}^{2-2g} = \sum_{\lambda \in \widehat{SU}(N)} \left( \sum_{n \in \mathbb{Z}} q_t^{c_2(\lambda+n)} \right) d_{\lambda}^{2-2g}. $$

We separate the term $(0, \dots, 0)$ in the sum over $\lambda$, and notice that $c_2(n, \dots, n) = n^2$:

$$ Z_{U(N)}(g, t) = \sum_{n \in \mathbb{Z}} q_t^{n^2} + \sum_{\substack{\lambda \in \widehat{SU}(N) \\ \lambda \neq (0, \dots, 0)}} \sum_{n \in \mathbb{Z}} q_t^{c_2(\lambda+n)} d_{\lambda}^{2-2g}. $$

However,

$$ c_2(\lambda + n) = \frac{1}{N} \left( \sum_{i} (\lambda_i + n)^2 + \sum_{i<j} (\lambda_i - \lambda_j) \right) \geq n^2, $$

given that $\lambda_1 \ge \dots \ge \lambda_N = 0$. Finally,

$$ 0 \le Z_{U(N)}(g, t) - \theta(q_t) \le \theta(q_t) \sum_{\substack{\lambda \in \widehat{SU}(N) \\ \lambda \neq (0, \dots, 0)}} d_{\lambda}^{2-2g}, $$

and the RHS converges to zero by Lemma 4.5, because $2g - 2 > 1$ for any $g \ge 2$. □

**Remark 4.7.** The only highest weights of $U(N)$ that contribute to the limit are the constant highest weights $(n, \dots, n)$ with $n \in \mathbb{Z}$. If we remember that $\widehat{U}(1) = \mathbb{Z}$ and we compute the dimension and Casimir of irreducible representations of $U(1)$, we find that

$$ Z_{U(1)}(g, t) = \theta(q_t), \quad \forall g \ge 0, \forall t > 0. $$

The previous theorem can thus be interpreted as follows: for all $g \ge 2$ and $t > 0$,

$$ \lim_{N \to \infty} Z_{U(N)}(g, t) = Z_{U(1)}(g, t). $$

The partition function becomes asymptotically the one of an abelian gauge theory, which is a phenomenon that one can also recover in another regime, where $N$ is fixed and we let $g \to \infty$ [50]. As we will discover in the next subsections, it is no more the case if $g \le 1$.
### 4.3. Discrete Gaussian measure in a cone and random partitions.
In the case $g = 1$, we see that

$$Z_{U(N)}(1, t) = \sum_{\lambda \in \hat{U}(N)} e^{-\frac{t}{2}c_2(\lambda)},$$

therefore all techniques from the previous subsection become useless because they were based on lower bounds of dimensions. We thus require an understanding of the asymptotics of Casimirs. A probabilistic approach, proposed in [52], is the following: consider for any real $t > 0$ and any integer $N \ge 1$ the probability measure $\mathcal{G}_{N,t}$ on $\hat{U}(N)$ defined by

$$(4.6) \qquad \mathcal{G}_{N,t}(\lambda) = \frac{1}{Z_{U(N)}(1,t)} e^{-\frac{t}{2}c_2(\lambda)}, \quad \forall \lambda \in \hat{U}(N).$$

If we recall that the Casimir is a quadratic expression of $\lambda$, the measure can be interpreted as a discrete Gaussian measure on $\hat{U}(N)$, which is a cone of $\mathbb{Z}^N$. The interpretation becomes even more obvious in the abelian case: on $\hat{U}(1) = \mathbb{Z}$ we have

$$\mathcal{G}_{1,t}(n) = \frac{1}{\theta(q_t)} e^{-\frac{t}{2}n^2},$$

where $q_t = e^{-\frac{t}{2}}$. We will denote by $\mathbb{E}_{N,t}$ the expectation with respect to $\mathcal{G}_{N,t}$. In the case $N = 1$, the moments are fairly explicit.

**Proposition 4.8.** For any $t > 0$ and $k \in \mathbb{N}$,

$$(4.7) \qquad \mathbb{E}_{1,t}[n^{2k}] = \frac{1}{\theta(q_t)} \left( q \frac{d}{dq} \right)^k \theta(q_t), \quad \mathbb{E}_{1,t}[n^{2k+1}] = 0.$$

*Proof.* For odd moments, it is a consequence of the fact that $n \mapsto n^{2k+1}e^{-\frac{t}{2}n^2}$ is an odd function on $\mathbb{Z}$. For even moments, we can obtain the formula by induction from

$$\left( q \frac{d}{dq} \right) q^{n^2} = n^2 q^{n^2}.$$

□

Surprisingly, it has been discovered that the measure $\mathcal{G}_{N,t}$ can be expressed in terms of $\mathcal{G}_{1,t}$ and a measure on the set of integer partitions. Recall that a partition of an integer $n$ is a nonincreasing family $\alpha = (\alpha_1, \dots, \alpha_r)$ of positive integers: $\alpha_1 \ge \dots \ge \alpha_r > 0$. Its length is denoted by $\ell(\alpha) = r$ and its size by $|\alpha| = n$. One can also write $\alpha \vdash n$. Partitions can be represented by Young diagrams, which are piles of cells, aligned in the left, such that the $i$-th row contains $\alpha_i$ cells (see Figure 8).

The set $\mathcal{P}_n$ of partitions of $n$ is in bijection with the dual of the symmetric group $\hat{S}_n$, and the set of all partitions is the disjoint union $\mathcal{P} = \bigsqcup_{n \ge 1} \mathcal{P}_n$. For any $q \in (0,1)$, let the $q$-uniform measure on $\mathcal{P}$ be the probability measure

$$\mathcal{U}_q(\alpha) = \phi(q) q^{|\alpha|}, \quad \forall \alpha \in \mathcal{P},$$
where $\phi : (0, 1) \to \mathbb{R}$ is defined as the convergent infinite product $\phi(q) = \prod_{m=1}^{\infty} (1 - q^m)$.

A first “naive” way to relate highest weights to partitions consists in shifting the coefficients of a highest weight so that all coefficients are nonnegative: if $\lambda = (\lambda_1 \ge \cdots \ge \lambda_N) \in \hat{U}(N)$, then we can write $\lambda = \Phi_N(\alpha, n)$, where $\alpha \in \mathcal{P}$ and $n \in \mathbb{Z}$ are defined by $n = \lambda_N$, $\ell(\alpha) = \max\{i \le N-1 : \lambda_i > n\}$, and

$$ \alpha_i = \lambda_i - \lambda_N, \quad \forall i \le \ell(\alpha). $$

We get a first bijection

$$ \hat{U}(N) \simeq \{\alpha \in \mathcal{P} : \ell(\alpha) \le N - 1\} \times \mathbb{Z}. $$

It is reminiscent of the construction used in the study of Witten zeta functions, if we note that

$$ \widehat{SU}(N) \simeq \{\alpha \in \mathcal{P} : \ell(\alpha) \le N - 1\}. $$

Let $\alpha = \alpha_\lambda$ be the partition obtained from $\lambda \in \hat{U}(N)$ from this construction. If we try to express $c_2(\lambda)$ in terms of $\alpha$ and $n$, we get

$$ \begin{aligned} c_2(\lambda) &= \frac{1}{N} \left( \sum_{i} \lambda_i^2 + \sum_{1 \le i < j \le N} (\lambda_i - \lambda_j) \right) \\ &= \frac{1}{N} \left( \sum_{i=1}^{\ell(\alpha)} (\alpha_i + n)^2 + (N - \ell(\alpha))n^2 \right. \\ & \qquad \left. + \sum_{1 \le i < j \le \ell(\alpha)} (\alpha_i - \alpha_j) + (N - \ell(\alpha)) \sum_{i=1}^{\ell(\alpha)} \alpha_i \right) \\ &= |\alpha| + n^2 + \frac{1}{N} \left( \sum_{i=1}^{\ell(\alpha)} \alpha_i^2 + (2n - \ell(\alpha))|\alpha| + \sum_{1 \le i < j \le \ell(\alpha)} (\alpha_i - \alpha_j) \right). \end{aligned} $$

It is not clear how it behaves for large $N$, given a partition $\alpha$ of fixed size: for instance, consider two extremal cases, $\alpha_1 = (1, \dots, 1, 0)$ and $\alpha_2 =$

<figure><img src="image_14.png" /><figcaption>FIGURE 8. The Young diagram of the partition $(7, 7, 6, 3, 1)$.</figcaption></figure>
$(N-1, 0, \dots, 0)$, so that $\ell(\alpha_1) = |\alpha_1| = N-1$, $\ell(\alpha_2) = 1$ and $|\alpha_2| = N-1$. Then

$$c_2(\Phi_N(\alpha_1, 0)) = 2 - \frac{2}{N},$$

whereas

$$c_2(\Phi_N(\alpha_2, 0)) = 3N - 6 + \frac{3}{N},$$

which shows that both partitions do not have comparable contributions. Intuitively, we expect that the highest weights that contribute to the limit are those which have small Casimirs, i.e. $c_2(\lambda) = O(1)$ as $N \to \infty$. Actually, such highest weights are obtained by perturbation of a constant highest weight $(n, \dots, n)$ with two small partitions $\alpha$ and $\beta$ as in Figure 9. They have been used in particular in [49, 50] under the name “almost flat highest weights”, and they are somehow related to the rational representations of $\text{GL}(N, \mathbb{C})$, see [43] for instance. Let us start with a general statement. We shall use cutoff parameters $A_N = \lfloor(N+1)/2\rfloor - 1$ and $B_N = N - \lfloor(N+1)/2\rfloor$, and remark that $A_N, B_N \sim N/2$ as $N \to \infty$.

**Proposition 4.9.** There exists a bijection

$$\hat{U}(N) \simeq \Lambda_N := \{(\alpha, \beta, n) \in \mathcal{P} \times \mathcal{P} \times \mathbb{Z} : \ell(\alpha) \le A_N, \ell(\beta) \le B_N\}.$$

It is given by

$$\lambda_N(\alpha, \beta, n) = (\alpha_1 + n, \dots, \alpha_{\ell(\alpha)} + n, n, \dots, n, n - \beta_{\ell(\beta)}, \dots, n - \beta_1).$$

*Proof.* It is enough to exhibit the reciprocal bijection:

$$(\lambda_N)^{-1}(\lambda) = (\alpha_{\lambda}, \beta_{\lambda}, n_{\lambda}),$$

where $n_{\lambda} = \lambda_{\lfloor(N+1)/2\rfloor}$, and $\alpha_{\lambda}$ and $\beta_{\lambda}$ are the partitions defined by $(\alpha_{\lambda})_i = \lambda_i - n_{\lambda}$, $(\beta_{\lambda})_i = n - \lambda_{N-i}$. $\square$

We will also need to introduce a new function on partitions, called the *content*: if we denote by $\square$ a cell of the Young diagram corresponding to

<figure><img src="image_15.png" /><figcaption>FIGURE 9. An almost flat highest weight.</figcaption></figure>
α and $R(\square)$ (resp. $C(\square)$) the index of its row (resp. column), then we set
$c(\square) = C(\square) - R(\square)$. Let

$$
K(\alpha) = \sum_{\square \in \alpha} c(\square) = \sum_{1 \le i \le l(\alpha)} \sum_{1 \le j \le \alpha_i} (j - i)
$$

be the *total content* of the partition α. The Casimir of the highest weight
λ<sub>N</sub>(α, β, n) has a very convenient expression:

**Corollary 4.10 ([49]).** For any $(\alpha, \beta, n) \in \Lambda_N$,

$$
(4.8) \quad c_2(\lambda_N(\alpha, \beta, n)) = |\alpha| + |\beta| + n^2 + \frac{2}{N}(K(\alpha) + K(\beta) + n(|\alpha| - |\beta|)).
$$

The following result, which is one of the main results of [52], shows that a
random highest weight $\lambda \in \hat{U}(N)$ distributed according to $\mathcal{G}_{N,t}$ can be seen
as a coupling of two $q_t$-uniform random partitions and an integer distributed
according to $\mathcal{G}_{1,t}$, where $q_t = e^{-\frac{t}{2}}$.

**Theorem 4.11 ([52]).** Let $N \ge 1$ be an integer and $t > 0$ be a real number.
If $\lambda \sim \mathcal{G}_{N,t}$, then for any test function $F : \hat{U}(N) \to \mathbb{R}$,

$$
(4.9) \qquad \mathbb{E}_{N,t}[F(\lambda)] = \frac{\theta(q_t)}{Z_{U(N)}(1,t)\phi(q_t)^2} \\
\qquad \times \mathbb{E}[F(\lambda_N(\alpha, \beta, n)) q_t^{\frac{2}{N}(K(\alpha)+K(\beta)+n(|\alpha|-|\beta|))} \mathbf{1}_{\Lambda_N}(\alpha, \beta, n)],
$$

where $(\alpha, \beta, n)$ are independent random variables such that $\alpha, \beta$ are $q_t$-uniform and $n \sim \mathcal{G}_{1,t}$.

The proof of this theorem is a direct consequence of the definitions and
of Corollary 4.10. It enables the following result:

**Corollary 4.12.** For any $t > 0$,

$$
(4.10) \qquad \lim_{N \to \infty} Z_{U(N)}(1, t) = \frac{\theta(q_t)}{\phi(q_t)^2}.
$$

This limit was first written in physics [23], and proved in [49] by combina-
torial techniques. It can be recovered from the previous theorem by taking
$F(\lambda) = 1$ for any $\lambda \in \hat{U}(N)$ and by using two key estimates: a global bound
on $c_2(\lambda_N(\alpha, \beta, n))$ that enables to perform a (slightly modified) dominated
convergence, and deviation inequalities for the measure $\mathcal{U}_q$ for any $q \in (0, 1)$
that ensure that $\mathbb{E}[\mathbf{1}_{\Lambda_N}(\alpha, \beta, n)] = 1 + O(e^{-c\text{st}\cdot N})$. As we will see later, this
framework has much more to offer, because it can lead to an asymptotic
expansion up to any order, as well as an interpretation in terms of random
surfaces.
4.4. **Douglas–Kazakov phase transition.** The careful reader will observe that we have seen the asymptotics of $Z_{U(N)}(g, t)$ for any $g \ge 1$, but the case of the sphere remains to be treated. Surprisingly, the corresponding regime is completely different and requires another approach. From Migdal's formula, one can prove that

$$ (4.11) \qquad Z_{U(N)}(0,t) = \frac{e^{\frac{t}{24}(N^2-1)+N(N-1)\log N}}{\prod_{1\le i<j\le N} (j-i)^2} \sum_{\lambda \in \bar{U}(N)} e^{-N^2 \mathcal{J}_t(\hat{\mu}_\lambda)}, $$

where $\hat{\mu}_\lambda$ is the empirical measure of rescaled highest weights:

$$ \hat{\mu}_{\lambda} = \frac{1}{N} \sum_{i=1}^{N} \delta_{\lambda_i + \frac{N+1}{2} - i}, $$

and $\mathcal{J}_t$ is a functional on the space of Borel probability measures on $\mathbb{R}$ given by

$$ \mathcal{J}_t(\mu) = - \iint_{(x,y) \in \mathbb{R}^2, x \neq y} \log |x-y| \mu(dx)\mu(dy) + \frac{t}{2} \int_{\mathbb{R}} x^2 \mu(dx). $$

In this case, the partition function diverges in the large $N$ limit, but the free energy $\lim_{N \to \infty} \frac{1}{N^2} \log Z_{U(N)}(0,t)$ exists and is finite. It was conjectured by Douglas and Kazakov [24] and proved by different methods by Boutet de Monvel and Shcherbina [8] and Lévy and Maïda [60].

**Theorem 4.13** (Douglas–Kazakov phase transition). For all $t \ge 0$, the limit

$$ (4.12) \qquad F(t) = \lim_{N \to \infty} \frac{1}{N^2} \log Z_{U(N)}(0,t) $$

exists. It defines a function that is $\mathcal{C}^2$ on $[0, \infty)$ and $\mathcal{C}^\infty$ on $[0, \pi^2) \cup (\pi^2, \infty)$, whose third derivative has a discontinuity at $\pi^2$.

A popular method to compute the free energy nowadays is by using large deviation techniques. For instance, the proof from [60] relies on [36, Theorem 1.2], which provides a quite general large deviation principle for random Hermitian matrices.

## 5. GAUGE/STRING DUALITY ON A TORUS

Let $\mathbb{T}$ be a compact two-dimensional torus. It is a compact surface of genus 1 as a smooth real manifold, but it is also a Riemann surface, i.e. a complex manifold of dimension 1. As such, it can be defined as the quotient $\mathbb{C}/\Lambda$ where $\Lambda = \mathbb{Z} \oplus \tau\mathbb{Z}$ is a lattice generated by $(1, \tau)$ with $\Im(\tau) > 0$.

**Definition 5.1.** Let $R = \{x_1, \dots, x_k\}$ be a finite subset of $\mathbb{T}$ and $n \ge 1$ be an integer. A ramified covering of $\mathbb{T}$ of degree $n$ with ramification locus $R$ is a Riemann surface $X$ together with a continuous surjective map $\pi : X \to \mathbb{T}$ such that:
(1) For any $x \in T \setminus R$, the preimage of an open neighborhood $U \subset T \setminus R$ of $x$ by $\pi$ is the disjoint union of open sets $V_1, \dots, V_n$ of $X$,

(2) For all $x \in R$, there is an integer $n_x \ge 2$ such that $\pi^{-1}(\{x\})$ contains $n+1-n_x$ points (instead of $n$). The number $n_x$ is the ramification order, and $x$ is called a generic ramification point if $n_x = 2$.

Fix a basepoint $x_0 \in T$. For simplicity, we shall always assume that coverings are not ramified over $x_0$ (otherwise we can change the basepoint). The monodromy of the ramified covering $X \to T$ along loops based at $x_0$, which is basically the same object as holonomy for principal $G$-bundles, yields a bijection between the set of ramified coverings of degree $n$ with ramification locus $R = \{x_1, \dots, x_k\}$ and the space

$$ \mathrm{Hom}(\pi_1(T \setminus R, x_0), S_n). $$

The group of automorphisms of such coverings is $S_n$, and the space of equivalence classes of ramified coverings of degree $n$ with $k$ ramification points of $T$ is

$$ \mathcal{R}_{n,k} = \mathrm{Hom}(\pi_1(T \setminus R, x_0), S_n)/S_n, $$

where $R = \{x_1, \dots, x_k\} \subset T$ is an arbitrary finite subset of size $k$ (indeed, the fundamental groups of $T \setminus \{x_1, \dots, x_k\}$ and $T \setminus \{x'_1, \dots, x'_k\}$ for any $\{x_1, \dots, x_k\}$ and $\{x'_1, \dots, x'_k\}$ are isomorphic). To avoid too heavy notations, we will also denote by $X$ the equivalence class of the corresponding covering $X \in \mathrm{Hom}(\pi_1(T \setminus R, x_0), S_n)$. Remark that $\mathcal{R}_{n,k}$ is analogous to character varieties introduced in the end of §2.4, but with the finite group $G = S_n$. The enumeration of ramified coverings is usually considered up to equivalence, which corresponds to the counting measure on $\mathcal{R}_{n,k}$ (and more specifically, on subsets with specific ramification profiles, see [26, §2.1]). We can form the countably infinite space

$$ \mathcal{R} = \bigsqcup_{n,k \ge 0} \mathcal{R}_{n,k} $$

of all possible ramified coverings of the torus, up to automorphisms. The counting measure on $\mathcal{R}$ is $\mu$ such that

$$ \int_{\mathcal{R}} f(X) \mu(dX) = \sum_{n,k \ge 0} \sum_{X \in \mathcal{R}_{n,k}} f(X) = \sum_{n,k \ge 0} \sum_{\substack{X \to T \\ \deg(X)=n \\ \chi(X)=-k}} \frac{f(X)}{n!}, $$

where $\chi(X)$ is the Euler characteristic of $X \to T$. Any random model built on $\mathcal{R}$ would be a model of random surfaces, suitable for a string theory (although finding the corresponding Lagrangian, or string action, might be a challenge), and we will describe one in this subsection: for any $n, k$ we consider the Hurwitz space $\mathcal{H}_1(n, k) \subset \mathcal{R}_{n,k}$ of (orientable) ramified coverings of degree $n$ with $k$ generic ramifications: the monodromy of a loop around any ramification point has is in the conjugacy class of a transposition. It is one of the simplest models of random ramified coverings, and also a very
natural one because it is related to random walks on the Cayley graph of the symmetric group generated by transpositions [64].

By the Riemann–Hurwitz formula, one can check that $\mathcal{H}_1(n, 2k + 1)$ is empty for any $k$, and that any $X \in \mathcal{H}_1(n, 2k)$ has genus $g = k + 1$. Counting such ramified coverings is a standard problem in enumerative geometry, considered for instance by Eskin and Okounkov [26], and we denote by $H_1(n, 2k) = \#\mathcal{H}_1(n, 2k)$ the size of the Hurwitz space, also known as a Hurwitz number. Let

$$ \mathcal{F}_k(q) = \sum_{n \ge 1} q^n H_1(n, 2k) $$

be the generating function of such numbers. Using a general formula due to Frobenius [48, Theorem A.1.10], we have:

$$ \sum_{\alpha \in \mathcal{P}} q^{\vert \alpha \vert} K(\alpha)^{2k} = \mathcal{F}_k(q). $$

By similar arguments,

$$ \sum_{\alpha \in \mathcal{P}} q^{\vert \alpha \vert} \vert \alpha \vert^{\ell} K(\alpha)^{2k} = \sum_{n \ge 1} q^n n^{\ell} H_1(n, 2k) = \left( q \frac{d}{dq} \right)^k \mathcal{F}_k(q). $$

Using Theorem 4.11 and expanding the exponential terms up to order $p \ge 1$, we obtain (up to technical details that are developed in [52]) the following:

**Theorem 5.2.** For any $p \ge 1$, and any $t > 0$, we have an asymptotic expansion

$$ (5.1) \qquad Z_{U(N)}(1, t) = a_0(t) + \frac{a_1(t)}{N^2} + \dots + \frac{a_p(t)}{N^{2p}} + O(N^{-2p-2}). $$

The coefficients $(a_k(t))_{k \ge 0}$ do not depend on $p$, and they are explicit functionals of $\mathcal{F}_k(q_t)$ and $\theta(q_t)$.

In a certain sense, which is developed in the forthcoming paper [51], the asymptotic expansion (5.1) can be rewritten as an integral over $\mathcal{R}^2$ with respect to a measure that randomizes the number of generic ramification points and the degree of ramified coverings (respectively using a Poisson measure conditioned to take even values, and a geometric measure), and such that the ramified coverings, conditioned to their degree $n$ and number of (generic) ramification points $2k$, are uniform in the Hurwitz space $\mathcal{H}_1(n, 2k)$. The precise measure (which has infinite mass, hence it cannot be normalized) is

$$ \rho_t(dX) = \sum_{n \ge 0} e^{-\frac{t}{2}n} \sum_{k \ge 1} \frac{t^{2k}}{(2k)!} \sum_{Y \in \mathcal{H}_1(n, 2k)} \delta_Y(dX), $$

where the Dirac mass is applied to equivalence classes in $\mathcal{R}$. We obtain two random ramified coverings $X_1, X_2 \to \mathbb{T}$ which are coupled by a random integer $n$, and this coupling reflects the coupling of random partitions obtained in Theorem 4.11.
**Theorem 5.3 ([51]).** For any real $t > 0$, and any integers $p \ge 0$ and $N \gg 1$ large enough, there is an explicit functional $\Phi_{p,t,N} : \mathbb{R}^2 \to \mathbb{R}$ such that

$$ (5.2) \quad Z_{U(N)}(1,t) = \int \Phi_{p,t,N}(X,Y) N^{\chi(X)+\chi(Y)} \rho_t(dX) \rho_t(dY) + O(N^{-2p-2}), $$

where the integral is over $(X, Y) \in \mathbb{R}^2$ such that $X$ and $Y$ only have generic ramifications, and $\chi(X) + \chi(Y) \ge -2p$.

Similar results hold for other groups, such as $SU(N)$, $SO(N)$ or $Sp(N)$, and all proofs rely on a similar description of $Z_G(1, t)$ as the partition function of a Gaussian measure on $\hat{G}$, or equivalently, the trace of the operator $e^{t \Delta_G}$ as a linear operator on $Z^2(G)$.

To conclude, another promising direction towards the gauge/string duality is the use of *Gromov-Witten invariants*. Indeed, they correspond to intersections of some cohomology classes of moduli spaces of maps $\Sigma \to M$, for a symplectic manifold $M$ and a Riemann surface $\Sigma$, and have been used in the last decades to describe correlation functions and partition functions of string theories from a more “cohomological” point of view [46, 44]. When the target $M$ is a compact Riemann surface, Hurwitz theory coincides with a part of Gromov-Witten theory, according to a result by Okounkov-Pandharipande [65], and an application to Yang-Mills theory on a torus is also described in [51], but we will not elaborate further because it would lead us to a much more geometric realm.

## ACKNOWLEDGEMENTS

These notes combine the content of a talk given in 2024 at the International Congress of Mathematical Physics and a lecture on two-dimensional Yang-Mills theory given in 2025 at École Normale Supérieure-PSL, and the author is grateful to the respective organizations for the opportunities to speak there. Many thanks to Elias Nohra for a careful feedback and useful suggestions on the manuscript.

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