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Large N limit of Yang-Mills partition function and Wilson loops on compact surfaces

We compute the Large N limit of several objects related to the two-dimensional Euclidean Yang-Mills measure on compact connected orientable surfaces of genus larger or equal to one, with a structure group taken among the classical groups of order N. Our result shows the convergence of all Wilson loops for all loops within a topological disc and all simple loops.


Introduction
The purpose of this article is to study a model of random matrices originating from 2D Euclidean quantum field theory, known as the two-dimensional Yang-Mills measure. This model is defined for a classical compact Lie group and a compact surface endowed with an area measure and can be understood as a gauge theory. Each loop on the surface defines an observable called Wilson loop. We are interested here in Wilson loops when the size N of the matrices goes to infinity, while the loop and the surface are fixed.
Let us attempt to give an account of the origin and state of this problem. The question seems to have first appeared in a mathematical paper in [49]. Therein, the candidate limit of Wilson loops on an arbitrary surface Σ is called master field. This paper was inspired by the large N limits considered in quantum gauge theories [12,26,41,38,22,21], which started after the landmark work of t'Hooft [51]. Since then, the Yang-Mills measure in two dimensions has been rigorously defined 1 [23,17,47,29], and the latter question has received the attention of several mathematicians 2 [53,48,2,33,19,18,11,25]. It is now known that the limit indeed exists when the surface is the plane [2,32] or the sphere [11], giving rise to two different objects. Simultaneously to and independently from [11], conditional convergence results were obtained in [25] for loops within a topological disc embedded in a compact surface, with different set of assumptions for simple loops. Both works [11,25] relied on the so-called Makeenko-Migdal equations, which allow to tackle the problem recursively in the complexity of loops. These equations first appeared in physics in [38,26] and were proved rigorously on the plane in [32,8,19] and on surfaces in [19,18].
Our main result, Theorem 2.16, shows that for all loops included within a topological disc of Σ, the limit Wilson loop converges and its limit is given by the value of the master field on the plane evaluated at the loop obtained by embedding the same disc in the plane instead of Σ, in a way that preserves the area measure. In that sense, the behaviour of the Yang-Mills measure within a topological disc is not affected asymptotically by the topological constraint imposed by the surface. Though it was probably known by physicists, we were unable to find this phenomenon mentioned explicitly in the physics literature.
Our second result Theorem 2.17 shows that for any simple non-contractible loop the Wilson loop of any iteration vanishes asymptotically. This leaves the case of loops which are neither embedded in a topological disc nor an iteration of a simple loop. We investigate this question in the next paper in the series [10].
Our argument does not rely here on the Makeenko-Migdal equations but mainly on the convergence of another quantity: the partition function. We prove the convergence of the partition function using harmonic analysis on the classical compact groups, generalising a result of [24] and of the second author [28].
The rest of the paper is divided in three sections. The first one recalls the setup of the problem and presents the main results. The second section recalls the necessary notions from representation theory of compact groups and the result of [24,28] on partition functions which we generalise to all group series of compact matrix Lie groups and all area parameter. The last one gives the proof of our main results on Wilson loops.

Heat kernel
In this text, G will denote a compact Lie group endowed with a bi-invariant inner product. The heat kernel on G is the family of smooth functions (p t , t ∈ (0, ∞)) on G satisfying d dt p t (g) = 1 2 ∆(p t )(g) for t > 0 and g ∈ G and for any continuous function f, lim t→0 G f (g)p t (g)dg = f (1).
We denote here by dg and ∆ the Haar measure on G and the Laplace-Beltrami operator associated to its inner product, while we write 1 for the identity element of G. The functions p t are central for all t > 0 and form a semigroup for the convolution product on G, that is for t > 0, p t (xgx −1 ) = p t (g), ∀x, g ∈ G and p t * p s = p t+s , ∀s > 0.

Classical compact Lie groups
In the sequel, we shall say that G N is a classical group of size N 3 if it is equal to one of the following matrix Lie groups: We fix the bi-invariant metric as follows. Assume that G N is a subgroup of GL n (C) (so that n = N for U(N ), SU(N ), SO(N ) and n = 2N for Sp(N )). We set an integer parameter 4 β which is equal to 1 if G N = SO(N ), 2 if G N = U(N ), SU(N ), and 4 if G N = Sp(N ). We endow the Lie algebra g N of G N with the scalar product It is conjugation-invariant in the sense that gXg −1 , gY g −1 = X, Y , for all X, Y ∈ g N and g ∈ G N , and defines therefore a bi-invariant metric on G N . Here Tr denotes the trace of the above matrix, that is the non-normalised sum of diagonal coefficients.
Remark. Except in case 1., there is up to constant a unique invariant inner-product. The above choice of scaling is standard in random matrices. For instance, the gaussian vector on Hermitian matrices obtained by composition of the above scalar product with the multiplication by i is the classical Gaussian Unitary Ensemble.

2D-maps and multiplicative functions
Assume that Σ is a two dimensional compact Riemannian manifold, with Riemannian distance d Σ and genus g ≥ 0. It is homeomorphic to a 4g-gon whose sides are, counterclockwise, a 1 , b 1 , a −1 1 , b −1 1 , . . . , a g , b g , a −1 g , b −1 g , such that each two edges with same letter are glued together, while respecting the orientation. The polygon is called the fundamental domain of the surface. If Σ has a nonempty boundary ∂Σ, the latter can be described as the union of n connected components (C i ) 1≤i≤n , each homeomorphic to the unit circle S 1 . We introduce here some notations used throughout this text.
If Ψ is a diffeomorphism of Σ and ∈ L(Σ), ψ( ) ∈ L(Σ) is the loop of Σ obtained by composition a parametrisation of with Ψ.

Topological maps on compact surfaces
We follow here conventions of [27,Section 1.3.2]. A graph G is a triple (V, E, I) consisting of a set V of vertices, a set E of edges and an incidence relation I such that an edge is incident to either one vertex or two distinct vertices, called endpoints. It can be given an orientation by setting, for any e ∈ E, a source e ∈ V and a target e ∈ V. An oriented graph can be then represented by a quadruple (V, E, s, t) where V is the set of vertices, E the set of oriented edges, and s, t : E → V are the functions that map respectively an edge e to its source and target. An isomorphism between two graphs , and such that A topological map M on Σ is a finite oriented graph G = (V, E, s, t) endowed with an embedding θ : G → Σ, called drawing of the graph, such that: • the vertices are drawn as distinct points of Σ, • oriented edges are drawn as oriented continuous curves that only intersect at their endpoints, • For any e ∈ E, there is an edge e −1 ∈ E such that θ(e −1 ) = θ(e) −1 , • the set F = Σ \ e∈E {θ(e)} is given by a union of open discs called faces of the map.
In order to avoid too heavy notations, we will always identify vertices and edges with their drawing, so that a map on Σ can be represented by (V, E, F) (the applications s and t will remain implicit from now on). Σ is called the underlying surface of M . If v ∈ V is a vertex in M , we denote by P(M ) (resp. L(M ), L v (M )) the set of paths (resp. loops, loops with base v) in M obtained by concatenation of oriented edges. A loop is simple when each vertex of M is the source of at most one of its edges.
Two maps , respectively on Σ 1 and Σ 2 and with associated graphs G 1 and G 2 , are equivalent if there is an orientationpreserving homeomorphism φ : Σ 1 → Σ 2 such that the restriction of φ to M 1 induces an isomorphism between G 1 and G 2 . The genus of a topological map in Σ is the genus of Σ, and it does not depend on its equivalence class.
The following result, which is a particular case of Proposition 1.3.10 in [32], allows to identify boundary components of Σ with loops of a map, up to rerooting.

Topological maps on R 2 or an open disc
A topological map M on the plane R 2 or an open disc is a topological map on the sphere S 2 together with a marked face f ∞ . The latter is called the infinite face of M , and it contains the point at infinity when one makes the identification S 2 R 2 ∪ {∞}. From now on, for any map M , be it on the plane, a disc or a compact surface, F will denote by default the set of bounded faces, andF will denote F ∪ {f ∞ } whenever it makes sense.

Multiplicative functions
For any P ⊂ P(Σ) and any compact group G, a multiplicative function h : P → G is a function that satisfies h γ −1 = h γ −1 for any γ ∈ P such that γ −1 ∈ P , and h γ1γ2 = h γ2 h γ1 for any γ 1 , γ 2 ∈ P such that γ 1 γ 2 ∈ P . We denote by M(P, G) the space of multiplicative functions from P to G. The gauge group Γ = C ∞ (Σ, G) acts on it by gauge transformations: for any j ∈ Γ and h ∈ M(P, G), Considering topological maps, gauge-invariance of multiplicative functions is equivalent to a G-invariance, as stated by the following lemma which is a particular instance of [32, Lemma 2.1.5].
Lemma 2.2. Let M be a topological map in Σ, v ∈ V and γ 1 , . . . , γ n ∈ P(M ). There exist 1 , . . . , m ∈ L v (M ) such that for any f : for some functionf : G m → C which is invariant by the diagonal action of G.
Two sigma-fields may be put on M(P(M ), G): 5 In other terms, we ask that the boundary of Σ has positive and finite length.
• The smallest sigma-field C such that for any γ ∈ P(M ) the evaluation function 6 measurable, for all v ∈ V, n ∈ N, 1 , . . . , n ∈ L v (M ) and f : G n → C G-invariant: it is the invariant sigma-field. We will mainly work with C but some results will only hold on J , therefore we will specify which sigma-field we consider.

Wilson loops
is then called a Wilson loop. When G belongs to one of the four series given above, we will be interested in the Wilson loops where is a loop of M , while for any . For most compact Lie groups G, it can be shown that the family of Wilson loops separate points of M(P(M ), G)/Γ endowed with the quotient topology. When G belongs to one of the four series of the previous section, it can further be shown [45,30] that it is enough to consider the family {W , ∈ L(M )}.

Area-weighted maps
An area-weighted map is a topological map M together with a function a : F → R + : the marked faces are excluded because their area is either considered to be infinite (in the case of R 2 ) or to be undefined (in the case of a surface with boundary). Two maps (M, a) and (M , a ) are equivalent if M and M are equivalent as maps and the associated homomorphism φ of Σ defines a bijection F : For any T > 0 we set when M is a map of a closed surface and when (M, a) is an area-weighted map we write |a| = f ∈F a f . If the surface is endowed with a Riemannian volume vol, then it induces in particular an area function vol : F → R + .

Two dimensional Yang-Mills measure
In this section we recall a definition of the Yang-Mills measure in three steps: 1. Given a topological map M , we define a uniform measure U M,C,G on M(P(M ), G) with and without constraints.
2. We define the discrete Yang-Mills measure YM M,C,a,G on an area-weighted topological map (M, a) as an absolutely continuous measure with respect to U M,C,G . 3. We define the Yang-Mills holonomy field (H ) ∈P(Σ) on any compact Riemann surface Σ with volume form vol, whose distribution YM Σ,C,G is the continuous version of YM M,C,a,G . (G/Ad) p , so that any set C of boundary conditions can be identified with a tuple C = (t 1 , . . . , t p ) with t 1 , . . . , t p ∈ G/Ad. For any t ∈ G/Ad and n ≥ 1, the set t(n) = {(x 1 , . . . , x n ) ∈ G n : x 1 · · · x n ∈ t} is a homogeneous space for the G n -action (g 1 , . . . , g n ) · (x 1 , . . . , x n ) = (g 1 x 1 g −1 2 , . . . , g n x n g −1 1 ).
We denote by δ t(n) the extension to G n of the unique G n -invariant probability measure on t(n) ⊂ G n . It can be thought of as the conditional Haar measure on G n , under the condition x 1 · · · x n ∈ t.
Let us now define a measure on (M(P(M ), G), C). Therefor, let us choose a specific labelling of E. If we denote again by L 1 , . . . , L p the connected components of ∂Σ, we choose the labels of the oriented edges e ∈ E + such that L i = e i,1 · · · e i,ni for every i. Denote finally by e 1 , . . . , e m the remaining edges of E + . Then

Discrete Yang-Mills measure
We follow here §1.5.2 of [29]. Consider a compact orientable surface Σ with genus g ≥ 0, and an area-weighted map (M, a) with graph G embedded in Σ. For each face f ∈ F, its boundary ∂f is the loop of G given by the concatenation of edges bordering f. It can be endowed with two possible orientations. The Yang-Mills with respect to U M,C,G , where Z M,a,G (t 1 , . . . , t p ) > 0 is a positive constant. Since the heat kernel defines central functions on G, each term of the above product is a well defined Wilson loop. Using the semigroup property leads to the following elementary but remarkable lemma.
depend only on the genus g of Σ, the total area T = a∈F a f and in the first case on the boundary conditions C = (t 1 , . . . , t p ). When ∂Σ is non-empty, Z M,a,G = 1.
This constant is called the partition function. We denote it by Z (g,p),T,G (t 1 , . . . , t p ), or Z g,T,G when p = 0, and we drop the subscript G if there is no ambiguity on the group. When h ∈ G p , we also write Z (g,p), where [g] ∈ G/Ad denotes the conjugacy class of an element g ∈ G. We give explicit expressions using representation theory in section 3.1.
When p ≥ 1, the Yang-Mills measure with or without boundary conditions are related by the disintegration formula When (M , a ) is another map with genus g and volume T, We have the following lemma. In particular if ∈ L(M ), the random variable W has same law under YM M ,C,a ,G and YM M,C,a,G .
We further need to define the Yang-Mills measures with a constraint along a single loop, together with a disintegration formula. Assume that the boundary of Σ is empty, ∈ L(M ) is a simple loop and t ∈ G/Ad. Denoting by = e 0,1 . . . e 0,n the edge decomposition of in M, and labeling e 1 , . . . , e m the other edges of M, the Yang Mills on M(P(M ), G) with structure group G and constraint C →t is the probability measure YM M,C →t ,a,G on (M(P(M ), G), C) with density 10 Proposition 2.6 ( [32,29]). Let (M, a) be an area-weighted map embedded in a closed surface Σ with genus g ≥ 0 and let ∈ L(M ) be a simple loop. Then for any t ∈ G/Ad, a ∈ ∆(T ), Remark. Remarkably, the Yang-Mills measure with constraints is related to the Yang-Mills measure with boundary conditions, see for instance Theorem 4.2 below. This relation plays the role of a Markov property. The holonomy fields under the Yang-Mills measure can be furthermore understood as an example of a twodimensional Markovian holonomy fields as defined in [32].

Yang-Mills holonomy field
The compatibility relation of Lemma 2.5 suggests that the measures considered can be obtained as the image of a single measure on a larger probability space. This has indeed been achieved in [29] and was then generalised in [32], leading to the notion of (continuous) Markovian holonomy fields, allowing to consider a very large family of loops at once. Though it is not crucial to our argument, we recall here their definition, as it allows to reformulate our main results in a unified continuous model. A different rigorous continuous approach has been given earlier by [17,44,46]. It relies on stochastic analysis of the white noise on the plane and the formula (6) is obtained as a consequence of the construction and it is called the Driver-Sengupta formula. The random holonomy field can be understood as the parallel transport of a random connection with curvature given by a white noise.
Recently, yet another continuous construction, defining a random connection one form has been given in [7] recovering the latter formula. Let Σ be a compact connected orientable surface, an open disc of R 2 or R 2 itself, endowed with an area measure vol. For any ∈ L(Σ), we use the same notation as in (2), for the random variable W = tr(H ).
Furthermore, whenever h ∈ M(P(Σ), G), the restriction of h to a topological map M on Σ defines an element R M (h) of M(P(M ), G). Let us finally mention that the space M(P(Σ), G) can be endowed with two sigma-fields C and J defined as in the discrete case. The continuous extension to the discrete Yang-Mills measure is provided by the following result. 3. If (γ n ) n≥0 is a sequence of P(Σ) and γ ∈ P(Σ) with γ n = γ, γ n = γ, ∀n ≥ 0 and d(γ n , γ) → 0 as n → ∞, then, under YM Σ,G,C , (H γn ) n≥0 converges in distribution towards H γ .
Moreover, if Ψ is a diffeomorphism of Σ preserving vol, then under YM Σ,C,G and YM Σ,G , W has same law as W Ψ( ) .
Proposition 2.8 ( [32]). When Σ is closed, is a simple loop and t ∈ G/Ad, there is a measure YM Σ,C →t ,G on (M(P(Σ), G), C) such that 1. For any topological map M on Σ such that is a drawing of a loop of M , Moreover the constant Z M,a,G ( ; t) only depends on ∈ L 2 (Σ) and t. We denote it by Z Σ,G ( ; t).
2. If Σ has genus g and total volume T , The random process (H γ , γ ∈ P(Σ)) with distribution given by YM Σ,C,G is called the Yang-Mills holonomy field on Σ. We are primarily interested in the random variables W for ∈ L(Σ), with structure group a classical group G N of size N .

Master fields, conjectures and main results
Following the physics literature, I. Singer raised in [49] the following question, that we will reformulate here in a slightly 11 modified form as a conjecture. Conjecture 2.9. Let Σ be a fixed two dimensional compact Riemannian manifold or R 2 or a disc of R 2 . Assume that G N is a classical group of size N with metric given by (1). Then for any loop ∈ L(Σ), there is a constant Φ Σ ( ) such that, under If Ψ is a diffeomorphism of Σ preserving its volume form, The limit function is called a master field. This conjecture has been partly proved for the plane, working with a smaller class of loops in [53,2], for unitary groups. In [33] , it was simultaneously proved for the plane and for all above group series. Recently another argument using Makeenko-Migdal equations was also given in [25].
Remark. For any loop, ∈ L(Σ), since H is a unitary matrix, |tr(H )| ≤ 1 and the convergence in probability (9) is equivalent to as N → ∞. When we say that (9) holds uniformly in the set of area vectors a, it is equivalent to the uniformity of the convergence (*) in a. To show (*), it is sufficient to show Remark. The linear extension of a master field Φ Σ to C[L(Σ)] comes automatically with a structure of non-commutative probability space [33,11]. In the case of the plane, this non-commutative distribution can be characterised using free probability ( [33] and [5]). In that case, one can recover the distribution of a free Brownian motion from the master field ( [3,33,5]). where the right-hand side is deterministic and depends continuously on a, over a ∈ R F + in the case of the plane and a ∈ ∆ M (T ) in the case of the sphere.
The work [33] was the first to show rigorously that the master field Φ R 2 satisfies a set of differential equations named after Makeenko-Migdal equations, that appeared earlier in the physics papers [38,26]. In [11], the conjecture was proved for unitary groups, in the case of the sphere.
Example 2.11 ([53, 2, 33, 11]). Assume that ∈ L(R 2 ) is simple and encloses an area t. For n ≥ 1, Denote by S 2 T the two-dimensional Euclidean sphere with total volume T . When 12 T ≤ π 2 and ∈ L(S 2 T ) is simple, cutting the sphere into two domains of area t and T − t, is a Bessel function of the first kind.

Remark.
1. Note that when n ≥ 1 and T ≤ π 2 are fixed, µ t (n) and µ t,T (n) are different functions of t. This is also true [11] when T > π 2 , though the expression of µ t,T is different. 13 Therefore, when is a simple loop enclosing a disc of area t, the expression of the master field is not the same when the surface in which is drawn is the plane or the sphere.
2. Let us highlight nonetheless two relations between the master field on the sphere and on the plane. On the one hand, it can be shown [11] that On the other hand, it follows from dominated convergence that for all t ≥ 0 Therefore, for all 3. The sequences µ t (n) and µ t,T (n), n ≥ 1 are moment sequences of measures µ t and µ t,T on the unit circle, associated to a time marginal of the free Brownian motion [3] and of the free brownian bridge [11]. Since both µ t and µ t,T are invariant by complex conjugation, (10) and (11) imply the weak convergences where for any measure on ν on the unit circle, ν k denotes the push forward of ν by z → z k . The conjecture 2.9 remains open for general surfaces. Though, using the Makeenko-Migdal equations on surfaces proved in [18], it was realised in [11,25] that it is sometimes enough to show the convergence for a restricted family of loops. This idea was exploited for general surfaces in [25] yielding the following theorem. 14 Let us say that a loop of a map M is included in a disc if it is included in an open contractible set U of Σ.
. Assume that whenever = s n with n ≥ 0 and s is a simple loop included in a disc, of an area-weighted topological map (M, a), under YM M,a,G N , the random variable W converges in probability towards a constant as N → ∞. Then this also holds true for W for any combinatorial loop included in a disc.
B. Hall conjectured further that the above assumption can be removed. Our main result implies that this conjecture holds true for any closed surface Σ and any group series of classical groups. We simultaneously prove theorem 2.12 without using the Makeenko-Migdal equations. Instead we shall use the convergence of the partition of function; a result proved by the first author in [11] for unitary groups that we recall and generalise to other group series in section 3. Remarkably the limit in the above conjecture is given in terms of the master field on the plane as follows. Assume that is a loop of an area weighted map (M, a) included in an open disc U. Considering only vertices, edges and area-weighted faces of M included in U and replacing all faces intersecting Σ \ U by a single marked face yields an area-weighted map (M U , a U ) of R 2 . We then denote by the same symbol the loop ofM U obtained by concatenating the edges of .  This result can be generalised as follows.
Theorem 2.16. Assume that Σ is a closed compact Riemann surface with Riemannian volume vol, Ψ : Assuming furthermore that Ψ −1 : D R → U can be extented continuously to D R , with piecewise continuous derivatives, for any loop ∈ L(D R ), Note that a simple loop is included in a disc if and only if it is contractible. Our second result below is concerned with simple loops which are not contractible.
Theorem 2.17. Assume that Σ is a closed compact Riemann surface with Riemannian volume vol and genus g ≥ 1, and is a simple non-contractible loop on Σ. Then, for any k ∈ Z * and any classical group G N of size N , under YM Σ,G N , Our result covers all simple loops on orientable closed surfaces. We further believe it holds true for non-orientable surfaces. In a sequel to the current work [10], we investigate the master field question for all loops with self-intersections.

Asymptotics of partition functions
Our argument relies on the convergence of partition functions of closed orientable surfaces, for classical groups, generalising results of [24] and of the second author [28]. This result was mentioned without proof in [43] for g > 1 and in [16] for g = 1 and G = U(N ) or SU(N ). For g ≥ 2, for all group series except the unitary group series, it is a simple consequence of [24]. For g = 1 it is proved in [28] for U(N ) and SU(N ). For all classical groups series, the limit involves the Jacobi theta function 15 and the Euler phi function, both defined for q ∈ C such that |q| < 1, with Let r ≥ 1 be an integer, we denote respectively byÃ r , A r , B r , C r , D r the type of U(r), SU(r + 1), SO(2r + 1), Sp(r) or SO(2r), in reference to the type of their root system (see [4] for instance). Note that we will alternatively use r as the rank of the root system or N as the size of the classical group, and their relations will be implicit: for instance, if we consider SU(N ), it will be of type A r with r = N − 1.
Conversely, if we consider the classical group of type D r , it will be SO(2r). This change of index may be confusing at first, but it will be helpful in order to deal simultaneously with all considered root systems. Considering limits when r → ∞ and when N → ∞ is equivalent.
Theorem 3.1. For any T > 0, (r, g) ∈ (N * ) 2 and any type X r , with X ∈ {B, C, D}, let us denote by Z g,T,Xr the Yang-Mills partition function on an orientable compact surface of genus g and area T with structure group of type X r (we will also write Z g,T,r when the type of the structure group is unambiguous). Set q T = e − T 2 . 1. For all g ≥ 1 and T > 0, lim r→∞ Z g,T,Xr exists and is given by the table (12) below.
As we can see, the limits for odd and even orthogonal groups are different when g = 1, which justify our choice to make the distinction between the root systems rather than the groups themselves.
Point 2 was proved for all types exceptÃ in [24], whereas 1 was proved in [28] for group series A,Ã, proving independently point 2 for typesÃ. Theorem 3.1 will be proved in Section 3. Let us also mention that, whereas the partition function on the plane or a disc in the plane is equal to 1 for any group, the case of the sphere behaves very differently, as the partition function goes to zero exponentially fast, at a speed of order r 2 ([13]). It further displays a phase transition that was discovered by and named after Douglas and Kazakov [15] and proved (in the case of unitary groups) by Lévy and Maïda [35].
We will prove Theorem 3.1 in section 3.2. Beforehand, we recall in the next section a well known expression of the partition function using representation theory of compact groups.

Character decomposition of the partition function
We shall express the partition functions Z g,T,Xr for any X ∈ {Ã, A, B, C, D} as a sum over non-increasing sequences of integers. The result of this section are well known [46,29]. This will follow from standard representation theory of compact groups, and we will prove it succinctly for the sake of completeness. The main result we want to prove is Prop. 3.7, and the remaining of the section may be skipped by anyone familiar with representation theory. Most of the results we will present can be found in [4] and [20]. 4. The character of a representation (ϕ, V ) is the function χ ϕ : G → C defined by χ ϕ (g) = Tr(ϕ(g)).
In the following, we will only consider finite-dimensional representations, unless stated otherwise. The two main results of the representation theory that we will be using are the celebrated Schur's lemma and Plancherel's theorem.

Let
then there exists α ∈ C such that A = αI, where I denotes the identity of V.
Among the numerous consequences of this lemma, one can state that the relation (13) defines an equivalence relation between irreducible representations of G, and it enables to define the set G of equivalence classes of irreducible representations. This set, sometimes called dual space of G, is countable whenever G is compact, and it is also a group when G is abelian. Another consequence of Schur's lemma is that two irreducible representations within the same class λ ∈ G have same dimension d λ and same character χ λ . Theorem 3.4 (Plancherel's theorem). For any continuous function f ∈ C ∞ (G), the following sum absolutely converges where for any From these results one can prove the following, which is a particular instance of Thm. 4.2 in [37].
Theorem 3.5. Let (p t ) t>0 be the heat kernel on G. Then for all t > 0, p t ∈ C ∞ (G) and the following sum absolutely converges, where c λ ≥ 0 is the non-negative real number such that Before we state the Fourier decomposition of Yang-Mills partition function, let us also mention an easy but useful result about the irreducible characters of compact groups.
Proposition 3.6. Let λ ∈ G be an equivalence class of irreducible representations of a compact group G. We have and Proof. Consider an element (ϕ, V ) of the equivalence class λ and set Since the Haar measure by multiplication, A z satisfies (14) of Schur's lemma. Therefore, A z = α z I for some scalars α z ∈ C with We can now write the left-hand side of (17) as while using Plancherel theorem, the left-hand side of (18) reads Proposition 3.7. Let Σ be a connected orientable surface of genus g ≥ 1. If ∂Σ is connected, then the Yang-Mills partition function on Σ with structure group of type X r and boundary condition t ∈ G/Ad is given by If Σ has no boundary, then the Yang-Mills partition function on Σ with structure group G is given by Proof. Let us start with the simplest case, which is (20). Consider the area-weighted map (M, a) of genus g with 1 vertex, 2g edges (a 1 , b 1 , . . . , a g , b g ) and 1 face f with area T > 0 and boundary Then Z g,T,Xr = Z M,a,G = G 2g p T ([x 1 , y 1 ] · · · [x g , y g ])dx 1 dy 1 · · · dx g dy g .
We now integrate out all commutators using (18), which yields G 2g The results follows. The proof of (19) is similar, using an area-weighted map with one face whose boundary is ∂f where is the simple loop corresponding to ∂Σ, oriented positively, for any h ∈ G to Now we will specify (20) to all compact classical groups: to do this, we only need to know the dual space G, and the numbers c λ and d λ for every λ ∈ G. This can be done thanks to their root systems. Most definitions and results are borrowed from [4], and can also be recovered with much clarity from Sections 2.2 and 2.3 in [39].
Definition 3.2. Let G be a compact connected Lie group.
where T is a maximal torus of G, such that the space is a distinguished basis of the lattice and r is the rank of the weight lattice. The normaliser of T in G acts by conjugation on T yielding an action on I r . The vector space V r = RΩ = I r ⊗ Z R can be endowed with an invariant inner product ·, · . In fact, if t is the Lie algebra of the maximal torus T ⊂ G, we have RΩ t * and ·, · can be taken as the dual of the inner product (1).

2.
A root of G is a non-zero weight of the adjoint representation of G. The root system Φ can be split into Φ + Φ − with Φ − = −Φ + . The elements of Φ + are called positive roots.
3. The Weyl chamber C r is the set It is an open cone in V r .

4.
A dominant weight is a weight that belongs to the closure C r of the Weyl chamber . We denote by Λ r the set of dominant weights.
5. The element ρ = 1 2 α∈Φ+ α has the property that for any ω ∈ I r ω ∈ C r if and only if ρ + ω ∈ C r .
Below are listed the root systems corresponding to the classical groups depending on their root systems. Note that we do not treat the root systems of exceptional Lie algebras E 6 , E 7 , E 8 , F 4 and G 2 , as we focus in this paper on unitary, special unitary, orthogonal and symplectic groups.
For any classical group G associated with the type X r , with X ∈ {Ã, A, B, C, D}, there is a bijection G Λ r : this is a consequence of the highest-weight theory (see [4] for instance). From now on, λ will indistinctly represent an equivalence class of irreducible representations or an element of Λ r .
Proposition 3.8. Let G N be a classical group associated with one of the types A r , A r , B r , C r , D r and λ ∈ G N Λ r be an equivalence class of irreducible representations. We have and We list below some of the objects defined previously, specified for the corresponding classical groups. Most of the explicit computations can be found in [4].
We will prove Theorem 3.1 using an asymptotic estimation of (20).

Proof of convergence of partition functions
We first recall the result of [24] and give an alternative proof based on the result of [28] for A r series.

Witten zeta function
When g ≥ 2, for the root systems A r , B r , C r and D r , a proof relies on an asymptotic estimation of the Witten zeta function The first claim we can do is that, for any root system X r with X ∈ {A, B, C, D}, any T ∈ R + and any g ≥ 2, we have Indeed, the sum in the middle contains only nonnegative terms, therefore is bounded from below by the term corresponding to λ = (0, . . . , 0) ∈ Λ r , which is equal to 1. The upper bound comes from the fact that for any λ ∈ Λ r and g ≥ 2, the number c λ is nonnegative, and d 2g−2 λ ≥ d 2 λ . Thanks to (26) we only need to prove lim r→∞ ζ Xr (2) = 1 (27) for X ∈ {A, B, C, D} in order to prove Theorem 3.1 for g ≥ 2. Proof. The case of A r is detailed in [28], we argue here that it implies the other cases. We will compare the dimensions and the sets of dominant weights for different root systems, thus we will denote by Λ Xr the set of dominant weights of type X r and d Xr,λ the dimension of an irreducible representation of the compact connected group of type X r with highest weight λ. First, let us remark that the sets of dominant weights are the same for A r , B r and C r . Let X ∈ {B, C} and λ ∈ Λ Xr a dominant weight different from (0, . . . , 0). As we have λ i ≥ 0 for any i, it is clear that It follows that, for any s > 1: We get the right limit by letting r go to infinity.
In the case of D r , some dominant weights have a negative coefficient, therefore an extra care must be taken. Let λ ∈ Λ Dr and 1 ≤ i < j ≤ r − 1, we have λ i + λ j ≥ 0 as the sum of nonnegative integers. Now, if 1 ≤ i ≤ r − 1, we also know that λ i ≤ λ r−1 ≤ |λ r | therefore λ i − λ r ≥ 0 as well. In any case, we deduce that whereλ = (λ 1 , . . . , λ r−1 , |λ r |). Any dominant weight λ of type A r corresponds to at most two different dominant weights of type r N : the same weight, and the one obtained by changing the sign of λ r . Hence, we have The right-hand side converges to 0 when r → ∞ therefore we obtain the expected limit for D r .

Discrete Gaussian random variables in a cone
In this section we prove that the convergence of Z 1,T,Xr as r → ∞ and T > 0 is fixed for types {B, C, D}.
and the associated partition function is The denomination 'Gaussian' is justified by the following remark. Recall that for any classical group G N ⊂ GL n (C) of type X r and any λ ∈ G N , It follows that for any λ ∈ Λ r , which thanks to (22) defines, after a shift by ρ, a discrete Gaussian distribution conditioned to belong to ρ + (I r ∩ C r ) = (ρ + I r ) ∩ C r . Note that if G N is a classical group of type X r , the partition function Z r,T of the Gaussian distribution on Λ r with parameter T is equal to the Yang-Mills partition function Z 1,T,Xr on a torus of area T with structure group G N . Before we prove Theorem 3.1 for g = 1, let us give a few notations that we will use. When λ = (λ 1 , λ 2 , . . .) is a sequence of nonnegative real numbers, we denote by |λ| = i≥1 λ i its total sum. We also denote by the set of integer partitions. For any α ∈ P we write (α) for its number of non-zero parts. It is well known that the generating function of partitions is given by the inverse of Euler function: , ∀q ∈ C s.t. |q| < 1.
Proof of Thm 3.1 with g = 1. Types B and C. Let us define an increasing sequence settingρ = (r − ρ 1 , . . . , r − ρ r ) and write For the types B r and C r we have Λ r {λ ∈ P : (λ) ≤ r} therefore we have Sinceρ is increasing and λ is non-increasing, It follows that where we used for the last inequality that in B r and C r cases, . We obtain λ∈P: (λ)≤r Recalling that n is 2r + 1 or 2r, both sides converge to φ(q T ) −1 when r → ∞ by dominated convergence, and we therefore obtain the expected limit. Type D. In this case, let us introduce two notations. For λ = (λ 1 , . . . , λ r ) such that λ 1 ≥ · · · ≥ λ r and m ∈ Z, we setλ = (λ 1 , . . . , λ r−1 ) and λ + m = (λ 1 + m, . . . , λ r + m). We have We find then Using the same argument as we did for the types B and C, we have By dominated convergence both bounds converge to m≥0 (2m + 1)q m T λ∈P , hence we deduce the expected limit for Z r,T .
Remark. For any T > 0, it is also possible to deduce the case g ≥ 2 from the case g = 1 using that when G is of type X r , for any non-trivial representation λ ∈ G, d λ ≥ r. Remark. A proof for the groups U(N ) and SU(N + 1) using heighest weights can be produced similarly to the other cases yielding a different proof of the result of [28], the latter being formulated in terms of Young diagrams.

Proof of Wilson loops convergence
In this section, we will prove theorems 2.14 and 2.17. The proof relies on an absolute continuity relation and the boundedness of the Yang-Mills partition function 16 , except for a class of loops considered in 4.2.2, which requires an independent argument. In the following sections, G N will denote a classical group of size N . 16 which follows from the convergence discussed in the Section 3.

Loops within in a disc
Assume that (M, a) is an area-weighted map on Σ of genus g ≥ 1 and area |a| = T , that U is an open connected set of Σ obtained by union of faces of M , with area |a U |, such that u = T − |a U | > 0. The boundary of U is given by a loop ∂U ∈ L(M U ).
Proof. Thanks to Lemma 2.5, we can assume that M has same number of faces as M U , that is, that exactly one face of M is not included in U . Denote by k be the number of edges of ∂U. Let M be the map with k + 1 vertices, k + 2g + 1 edges u 1 , . . . , u k , e, a 1 , b 1 , . . . , a g , b g and two faces U and V g , such that ∂U = u k · · · u 1 and Let us assume that M is obtained by gluing M withM U , identifying ∂U with ∂U so that the base of ∂U is sent to an endpoint of e. For any h ∈ M(P(M ), G N ), let us write x l = h a l , y l = h b l for all 1 ≤ l ≤ g and z = h e . Denote the set of non-marked faces ofM U by F U . Then YM M,a,G N (dh) can be written as Integrating over (x i , y i ) i and z the result follows by invariance by conjugation of the Haar measure and (21).
Proof of Thm. 2.14. Let us first assume that |a U | < T = |a|. For any h ∈ G N and λ ∈ G N , since the representation is unitary, χ λ ∞ = χ λ (1) = d λ . For any s > 0, we deduce from (19) that Together with the previous lemma, R M M U * (YM M,a,G N ) has a density with respect to YMM U ,a U ,G N which is bounded from above by Z g,|a|−|a U |,G N Z g,|a|,G N .
According to Theorem 3.1, the right-hand side is uniformly bounded in N . In particular for any ε > 0 and any loop ∈ L(M U ), Together with Theorem 2.10 for the plane, we conclude that the right-hand side converges towards zero as N → ∞.
To conclude it remains to prove the uniformity of the convergence in the set A of a ∈ ∆ M (T ) with a(f ) > 0 for any face f and |a U | < T. For this, it is enough to show that for any adjacent faces f 1 , f 2 of M, there is a constant K > 0 such that for all a, a ∈ A, with a(f ) = a (f ) for all faces distinct from f 1 , f 2 and |a − a | < T 2 , When f 1 , f 2 and a, a are as above, consider an area weighted map (M r , a r ) finer than (M, a), a simple loop s ∈ L(M r ) enclosing an area |a − a | and two paths α, β of M r with = αβ, such that under YM M r ,a r , W αsβ has same law as W under YM M,a . Using this identity in law, the left-hand-side of (32) equals Consider an open disc of Σ such that the only face of M r included in V is the one enclosed by s. The right-hand-side is bounded by where we used the same argument as in the first part of the proof for the penultimate inequality. Now under YMMr V ,a r V , H s has same law as a Brownian motion at time |a − a |. It follows from [31] or [9] that for all group series considered, there is a constant c > 0 independent of N with leading to the required bound (32).
Proof of Theorem 2.16. We only sketch here the additional argument and refer to [32] for more details.
The claim then follows with the same argument as for the proof of Theorem 2.14.

Simple non-contractible loops
Consider a simple non-contractible loop on a compact surface Σ of genus g ≥ 1.
Two possibilities occur when cutting Σ along . If the surface is cut into two surfaces with exactly one connected boundary component, each with genus larger than 1 17 , then the loop is separating. Otherwise, the new surface has one connected component and two boundary components, and the loop is called non-separating. Both cases are illustrated in Fig. 2 below. We refer to [50] for details on these loops. It is now important to understand how the Yang-Mills measure interacts with surgery: we therefore begin with a few results that will help us in the next section.

Separating loops
Consider first two compact connected orientable surfaces Σ 1 and Σ 2 such that ∂Σ 1 and ∂Σ 2 are connected. Denote by L 1 and L 2 the corresponding loops, oriented positively. Let ψ : L 1 → L 2 be an orientation-reversing diffeomorphism, and Σ be the gluing of Σ 1 and Σ 2 along ψ. Let be the corresponding simple loop in the surface Σ. For i ∈ {1, 2}, denote byJ i the sigma-field generated by (H 1 , . . . , H n , n ≥ 1, i ∈ L(Σ i )). For any i, a function f : M(P(Σ), , the product f 1 f 2 is measurable with respect to J and the following equality holds true for any t ∈ G N /Ad: 17 Indeed, if one of the boundary components has genus 0, it implies that is contractible.
This theorem provides a sort of spacial Markov property, which is a consequence of the semigroup property of the heat kernel. It has the following application.
where for any x ∈ G N , Proof. To lighten the notation, we will drop in this proof the subscripts G N , as the structure group remains fixed. First of all, using (7), the left-hand-side of (34) equals Let us compute Z g,T ( ; t). Using the invariance property of Yang-Mills measure by subdivision, one can assume without loss of generality that that for i ∈ {1, 2}, g , e (i) all with v (i) as source and target, and 1 face whose boundary is given by where ε 1 = 1 and ε 2 = −1. The map M can then be defined as the quotient of M 1 ∪ M 2 by the relations v (1) ∼ v (2) and (e (1) ) −1 ∼ e (2) . We obtain dy i dz i dy j dz j =Z (g1,1),T1 (t)Z (g2,1),T2 (t −1 ).

From this result and from Theorem 4.2 we deduce
The integral over M(P(M 2 ), G N ) is simply equal to Z (g2,1),T2 (t −1 ); furthermore, under YM M1,t,a , [H ] = t, so that we have Finally, using the disintegration formula (5) leads to the expected equality.
To prove Theorem 2.17, we shall need the following similar but simpler Lemma.
thus by integrating over all variables but x we obtain the expected result.
Proof of 1. of Theorem 2.17. Using Lemma 4.4 and the bound (31), we find that under YM M,a , the density of h with respect to the Haar measure is uniformly bounded by Z g1,T1,G N Z g2,T2,G N Z g,T,G N .
Now for a Haar distributed random variable H on G N belonging to X N , according to [14], whenever n = 0, tr(H n ) converges in probability towards 0, as N → ∞. The claim now follows by absolute continuity similarly to the proof of Theorem 2.14.

Non-separating loops
Let Σ be a compact connected orientable surface such that ∂Σ has two connected components; denote the corresponding loops, oriented positively, by L 1 and L 2 . Let ψ : L 1 → L 2 be an orientation-reversing diffeomorphism, and Σ be the gluing of Σ along ψ. Let be the corresponding loop in Σ. We have the following, which is a particular case of [29,Thm. 5.4.1]. We use the same notationJ for the sigma-field on M(P(Σ), G) generated by the loops in Σ .
Theorem 4.5. Let f : M(P(Σ), G) → C be aJ -measurable function. For any t ∈ G/Ad, the following equality holds true: The consequence in terms of absolute continuity is the following.
Corollary 4.6. ? Let be a non-separating loop in an area-weighted map (M, a) of genus g ≥ 1. Under YM M,a,G , the random variable H has density given by with respect to the Haar measure, and for all g ≥ 1, T > 0, Remark. The above density can also be written as Z (g,2),|a| (h, h −1 ).
Proof. Choosing M with 1 vertex, 1 face and 2g edges, one of which is , and writing u = h , It follows that the law of h under YM M,a has density with respect to the Haar measure. Expanding in characters and using (18) yields Now (17) yields I λ (u) = d 2−2g λ χ λ (u)χ λ (u −1 ), ∀u ∈ G and the claim.
It follows that under YM M,a , the density of h with respect to the Haar measure is bounded by When g ≥ 2, it follows from 3.1 that this sequence is uniformly bounded when N → ∞ and the argument is then identical to the proof of point 1.
When g = 1, the above argument of absolute continuity argument fails. Indeed for a total area T, the maximum of the density is then given by Z 0,T ,N Z 1,T ,N . ForÃ N , thanks to Theorem 3.2, and more precisely formula (28) of [35], Since F (T ) ≥ F (π 2 ) > 0 for all T ∈ (0, π 2 ], lim N →∞ Z 0,T,N = +∞. Similarly to the strategy of [33,11,25], we shall consider the expectation and the variance of the random variable W . Remark. For any z in the center of G, under YM M,a , zh has same law as h . This can be checked changing variable in formula (37). Alternatively, Schur's lemma implies that z acts by a unitary scalar in any irreducible representation and (37) implies that for any ϕ g,T (zh) = ϕ g,T (h) for any h ∈ G. Consequently whenever D z ∈ G is a diagonal matrix of multiplication by a scalar z ∈ C, under YM M,a , tr(h n ) = tr(h n ) has same law as z n tr(h n ). For X N = {A N ,Ã N }, the center is given by {ζId N +1 : ζ N +1 = 1} and {zId N : z ∈ C, |z| = 1}. It follows that for X =Ã, E[W n ] = 0 for any n = 0, whereas for X = A, E[W n ] = 0 for any n = 0 modulo N + 1. In particular, when X ∈ {A,Ã}, lim N →∞ E[W n ] = 0. Unfortunately, the argument given in the above remark does not apply when X = {B, C, D}. Also, it does not give any information about Var(W ) = E[|W | 2 ] − |E[W ]| 2 . Instead, we shall use the expansion in characters of the heat kernel and the following lemma, which is a consequence of the expression of characters as a ratio of alternated functions.
Lemma 4.7. Let G N ⊂ GL n (C) be a classical group of size N and type X r , for X ∈ {B, C, D}. For any k = 0 and λ ∈ G N , with c µ λ,k ∈ {−1, 0, 1}, ∀µ ∈ G N and µ∈ G N |c µ λ,k | ≤ n Proof. Let us introduce a few common notations for the different cases. Set Z sym = Z if r is odd or Z + 1 2 if r is even. The mapping λ ∈ G N → λ + ρ establishes a bijection between the set of highest weights and {µ ∈ Z r sym : µ 1 > · · · > µ r }. The symmetric group S r acts in an obvious way on ∆ r = {µ ∈ Z r sym : µ 1 > · · · > µ r } ∼ = {µ ∈ Z r sym : µ i = µ j , ∀i = j}/S r , where the symmetric group S r acts on µ by permutation of its components. For µ with µ i = µ j , ∀i = j, we denote by [µ] its decreasing rearrangement and σ µ ∈ S r the unique permutation such that [µ] i = µ σ(i) for all i. We will prove each case separately using the previous notations. The explicit formulae for the characters χ λ that we will use can be found in [4] or in [39,Sect. 2.3]. When µ ∈ Z r sym with µ i = µ j for some i = j, by convention χ [µ]−ρ = 0 and ε(σ µ ) = 0. The proof for all cases will rely on the following computation: for any (z 1 , . . . , z r ) ∈ (C * ) r , any (m 1 , . . . , m r ) ∈ Z r and any k ∈ N * , we have Let us prove (41). Using the Leibniz formula for determinant and the invariance of (z k + z −k ) by the action → σ( ) of the symmetric group, we have We can then put the sum over S r back in by linearity, and for any σ and any we have Summing over leads to (41). We can now prove (38) for each type of group. Case B r : let g ∈ SO(2r + 1) be an element with eigenvalues (z ±1 1 , . . . , z ±1 r , 1). We have .

It follows that
Tr(g k )χ λ (g) = r =1 ε(σ µ + k, as expected. Case D r : let g ∈ SO(2r) be an element with eigenvalues (z ±1 1 , . . . , z ±1 r ). We have Proof of 2. of Theorem 2.17 for g = 1. Let us set T = |a|. We shall compute the expectation and the variance of W under YM M,a,G N , and show that both converge to zero as N → ∞. Assume that G N is of type X r and a subgroup of GL n (C) as 18 in section 2. where for all λ ∈ G N , using the orthogonality of characters, Thanks to (39), it follows that Similarly, Using Lemma 4.7, we conclude that From (43) and (44), we conclude that the expectation and variance of W under YM M,a,G N both vanish as N → ∞. Therefore, W k converges to 0 in probability.
Remark. For the A r type, for all k = 0 and λ ∈ Λ r , c λ λ,k = 0, and the first part of the above proof yields another argument for E[W k ] = 0 for all r ≥ 1.