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# The Yang-Mills measure on compact surfaces as a universal scaling limit of lattice gauge models

Nguyen Viet Dang, Elias Nohra

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Nguyen Viet Dang, Elias Nohra. The Yang-Mills measure on compact surfaces as a universal scaling limit of lattice gauge models. 2026. hal-05500947

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Preprint submitted on 9 Feb 2026

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# The Yang–Mills measure on compact surfaces as a
universal scaling limit of lattice gauge models

Nguyen Viet DANG\* and Elias NOHRA†

February 9, 2026

## Abstract

In this article, we study the 2-dimensional Yang–Mills measure on compact surfaces from a unified continuum and discrete perspective. We construct the Yang–Mills measure as a random distributional 1-form on surfaces of arbitrary genus equipped with an arbitrary smooth area form, using geometric tools based on zero-area bands and cylindrical resolutions. This yields a canonical bulk–singular decomposition of the measure, reflecting the topology of the surface.

We prove a universality theorem stating that the continuum Yang–Mills measure arises as the scaling limit of a wide class of lattice gauge theories, including Wilson, Manton, and Villain actions, on any compact surface. We study the convergence in natural spaces of distributions with anisotropic regularity. As further consequences, we obtain a new intrinsic construction of the Yang–Mills measure, independent of the previous constructions in the literature, and prove the convergence of correlation functions and Segal amplitudes on all compact surfaces.

**Keywords:** Anisotropic Sobolev and Hölder spaces, holonomy process, lattice gauge theory, Morse theory, Yang–Mills measure, random distributional 1–forms, second micro-localization, scaling limit, tightness, universality.

## Contents

<table><tr><td><b>1 Introduction</b></td><td><b>3</b></td></tr><tr><td>1.1 Contribution of the paper</td><td>5</td></tr><tr><td>1.2 Structure of the paper</td><td>9</td></tr><tr><td><b>2 Revisiting Driver's construction of YM<sup>2</sup></b></td><td><b>10</b></td></tr><tr><td>2.1 The case of the plane</td><td>10</td></tr><tr><td>2.2 The case of the cylinder</td><td>11</td></tr><tr><td>2.3 The case of a general surface</td><td>17</td></tr><tr><td><b>3 Preliminaries on Morse theory</b></td><td><b>19</b></td></tr></table>

<footnote>\*IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France AND Institut Universitaire de France, Paris, France. Email: nvdang@unistra.fr</footnote>

<footnote>†Sorbonne Université, Université Paris Cité, CNRS, Laboratoire de Probabilités, Statistique et Modélisation, LPSM, F-75005, France. Email: elias.nohra@sorbonne-universite.fr</footnote>
<table>
  <tbody>
    <tr>
      <td>4</td>
      <td>A global resolution of a surface by a cylinder</td>
      <td>22</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>A global angular form and period coordinates</td>
      <td>22</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Pseudo coordinates</td>
      <td>27</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Resolution of the surface S by a cylinder with defect lines</td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Regularity of area functions.</td>
      <td>28</td>
    </tr>
    <tr>
      <td>5</td>
      <td>The Yang-Mills connection in the continuum</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>A formula for the Yang-Mills connection</td>
      <td>30</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>The motivation to deal with singularities</td>
      <td>32</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Weighted anisotropic Sobolev and Hölder norms</td>
      <td>32</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>Non weighted anisotropic semi-norms</td>
      <td>33</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>The weighted anisotropic spaces by scalings.</td>
      <td>34</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Regularity estimates in anisotropic spaces</td>
      <td>35</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Extending near defect lines and reducing to neighborhoods of saddle points.</td>
      <td>37</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Morse lattice from flowlines and level sets</td>
      <td>45</td>
    </tr>
    <tr>
      <td>7</td>
      <td>A local limit theorem</td>
      <td>47</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Technical estimates on Lie group valued random walks</td>
      <td>51</td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Assumption on the actions.</td>
      <td>51</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Technical estimates</td>
      <td>53</td>
    </tr>
    <tr>
      <td>8.3</td>
      <td>Classical lattice gauge actions</td>
      <td>56</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Scaling limit of lattice gauge theories on surfaces</td>
      <td>57</td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>A discrete Morse gauge</td>
      <td>58</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Fixed N analysis</td>
      <td>60</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Limit in law</td>
      <td>61</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Convergence in functional spaces</td>
      <td>65</td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Discretizing the spaces of distributional connections</td>
      <td>65</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>Probabilistic bounds</td>
      <td>69</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>Tightness result</td>
      <td>72</td>
    </tr>
    <tr>
      <td>11</td>
      <td>Proof of the main theorem</td>
      <td>73</td>
    </tr>
    <tr>
      <td>11.1</td>
      <td>From the cylinder back to the surface</td>
      <td>73</td>
    </tr>
    <tr>
      <td>11.2</td>
      <td>Closing the Surface</td>
      <td>73</td>
    </tr>
    <tr>
      <td>11.2.1</td>
      <td>Disintegrating the lattice measure</td>
      <td>74</td>
    </tr>
    <tr>
      <td>11.2.2</td>
      <td>Convergence of the conditioned measures</td>
      <td>77</td>
    </tr>
    <tr>
      <td>11.2.3</td>
      <td>Control on Hölder-Besov norms</td>
      <td>79</td>
    </tr>
    <tr>
      <td>A</td>
      <td>Classical results in harmonic analysis</td>
      <td>80</td>
    </tr>
    <tr>
      <td>A.1</td>
      <td>Path-wise integral of some rough differential forms</td>
      <td>80</td>
    </tr>
    <tr>
      <td>A.2</td>
      <td>Restriction in Besov spaces</td>
      <td>80</td>
    </tr>
    <tr>
      <td>A.3</td>
      <td>Young products of distributions and duality</td>
      <td>81</td>
    </tr>
    <tr>
      <td>B</td>
      <td>Recollection on the spectral analysis of Morse-Smale flows</td>
      <td>81</td>
    </tr>
    <tr>
      <td>C</td>
      <td>Continuous and compact injections in anisotropic spaces</td>
      <td>83</td>
    </tr>
    <tr>
      <td>D</td>
      <td>Symbolic index</td>
      <td>86</td>
    </tr>
  </tbody>
</table>
# 1 Introduction

Yang-Mills theory is a quantum field theory that aims to describe interactions between fundamental particles. In Feynman's path integral formulation, the study of an Euclidean quantum field theory proceeds as follows. Given an Euclidean space-time $M$— for instance $\mathbb{R}^d$ or a more general manifold— and a space of fields—for instance functions, or generalized functions— over $M$, we associate two quantities of interest:

* observables, which describe the physically relevant aspects of the fields;

* an action functional $S$, which represents a cost associated with a field's configuration.

Roughly speaking, at least for our concern, describing a quantum field theory amounts to computing the expected values of observables, where the field $\Phi$ is chosen at random according to the probability measure

$$ \mu(d\Phi) := \frac{1}{Z} \exp(-S(\Phi)) d\Phi \quad (1.1) $$

defined on the space of all possible fields. In (1.1), the constant $Z$ denotes the normalization factor ensuring that $\mu$ is a probability measure; and the measure $d\Phi$ is the formal Lebesgue measure on the space of all fields.

The main mathematical difficulty lies in the rigorous construction of this probability measure, since the space of all possible fields is typically infinite-dimensional. The first issue is that the Lebesgue measure $d\Phi$ is, in general, ill-defined. The second issue is that sometimes, the normalization constant $Z$ appears to be infinite, even at a formal level. The third issue is that the fields $\Phi$ sampled by the measure $\mu$ are expected to be too irregular for the quantity $S(\Phi)$ to be well-defined.

A typical example arises when one attempts to construct quantum mechanics as a quantum field theory. In the simplest setting, the spacetime is $M = [0, 1]$, the fields are functions $\Phi : [0, 1] \to \mathbb{R}$, the observables are the joint values of $\Phi$ at $0 \le t_1 < \dots < t_n \le 1$, and the action functional is given by

$$ S(\Phi) = \frac{1}{2} \int_{[0,1]} |\Phi'(t)|^2 dt. $$

There is little doubt that the rigorous construction of the measure $\mu$ in this case is the Wiener measure, so that a field sampled at random according to this measure is a standard Brownian motion $(B_t)_{t \in [0,1]}$. However, we emphasize that $S((B_t)_{t \in [0,1]})$ is ill-defined, since $B$ is almost surely nowhere differentiable.

A natural question is then how one can determine in which sense this measure is indeed the correct realization of $\mu$. Several results support this identification.

* The Cameron-Martin theorem, which states that the only translations of $B$ that preserve the measure up to absolute continuity are given by functions with finite action.

* Schilder's theorem, which establishes a large deviation principle and relates the action functional $S$ to the Wiener measure.

* The well-known fact that Brownian motion can be obtained as the scaling limit of discrete (finite dimensional) approximations of $\mu$.

In the case of Yang-Mills theory, the situation is somewhat analogous. The spacetime is a $d$-dimensional Riemannian manifold $M$ equipped with a volume measure that we denote by vol. The fields are differential 1-forms valued in the Lie algebra $\mathfrak{g}$ of a compact Lie group $G$, that we will assume without loss of generality matricial. The choice of the Lie group $G$ depends on the type of interactions one wishes to describe<fnref n="1" />.

<footnote n="1">¹For example, $G = U(1)$ for quantum electrodynamics, or $G = SU(3)$ for quantum chromodynamics.</footnote>
The action of a field $A \in \Omega^1(M, \mathfrak{g})$ is defined as half the $L^2$ norm of its curvature:

$$S_{YM}(A) := \frac{1}{2} \int_M \|dA + A \wedge A\|^2 \mathrm{dvol.}$$

Let us describe the observable of Yang–Mills theory, called the Wilson loops. Given a field $A \in \Omega^1(M, \mathfrak{g})$, and a closed loop $c : [0, 1] \to M$, the holonomy of $A$ along $c$, that we denote by $\mathrm{Hol}(A, c)$, is the solution to the ordinary differential equation

$$\begin{cases} \dot{y}_t = -y_t A_{c_t}(\dot{c}_t) \\ y_0 = 1_G \end{cases}$$

taken at time one. The Wilson loop observable associated with $c$ is then $\mathrm{Tr}(\mathrm{Hol}(A, c))$.<fnref n="2" />

A fundamental physical feature of Yang–Mills theory is its gauge invariance. This invariance asserts that many seemingly different 1-forms $A \in \Omega^1(M, \mathfrak{g})$ actually represent the same physical field. Indeed, any 1-form of the form

$$g \cdot A := g^{-1}Ag - g^{-1}dg,$$

where $g \in \mathcal{G} := C^\infty(M, G)$, represents the same physical field as $A$. In particular, they yield the same Wilson loop observables for all loops in $M$. Two such 1-forms are said to be gauge equivalent. Moreover, they satisfy $S_{YM}(A) = S_{YM}(g \cdot A)$. This suggests that the Yang–Mills measure can be formally factorized as

$$\mu_{YM}(dA) = \frac{1}{Z} (\exp(-S_{YM}([A])) d[A]) \otimes \mathcal{D}g, \qquad (1.2)$$

where the first factor is interpreted as a measure on $A \mod \mathcal{G}$, and $\mathcal{D}g$ denotes the (formal) Haar measure on the infinite-dimensional group $\mathcal{G}$.

We can now state the main mathematical difficulties involved in the construction of this measure, two of which were already present in the case of Brownian motion:

* there is no Lebesgue measure on the space of 1-forms;

* one expects typical fields $A$ to be so irregular that neither $dA$ is defined pointwise, nor the product $A \wedge A$ is well-defined.

Of course, in the present setting these issues are more severe than in the case of Brownian motion. Indeed, since the Brownian action is quadratic, it falls within the framework of Gaussian measures, for which a general and well-developed theory exists. In contrast, the Yang–Mills action contains the quartic term $A \wedge A$, and the resulting measure is therefore no longer Gaussian. Another difficulty, specific to gauge theories, is that the factorization suggested by (1.2) formally leads to $Z = \infty$, introducing an additional divergence that must be addressed.

Due to these mathematical complications, a rigorous construction of the Yang–Mills measure in three or more dimensions is currently unknown. However, over the last 50 years or so, the two-dimensional Yang–Mills measure has been successfully constructed by several authors, and its properties have been extensively studied.

In the physics literature, the study of two-dimensional Yang–Mills theory dates back to the work of Migdal [61] in 1975 on $\mathbb{T}^2$, whose approach was discrete, in the spirit of lattice gauge theories. This construction was later extended to surfaces of arbitrary genus in the groundbreaking work of Witten [73, 74], where deep connections with many other areas of mathematics were revealed.

From a mathematical perspective, what makes the two-dimensional case far more tractable is the fact that the theory is locally free, meaning that for any 1-form $A$, one can find a gauge-equivalent 1-form $B$

<footnote n="2">²In the case of electrodynamics, this quantity represents the change in the phase of the wave function of a particle as it travels along $c$.</footnote>
such that locally, $B \wedge B = 0$. This observation was central to the work of Driver [37], whose construction for $M = \mathbb{R}^2$ is sufficiently important for the present work that it is recalled in Section 2.1 and adapted to the cylinder in Section 2.2. Sengupta subsequently generalized this approach to the sphere [65], and then to all compact surfaces [66].

These constructions led to the so-called Driver–Sengupta formula, which describes the joint law of the holonomies along a finite collection of loops. This, in turn, gave rise to the holonomy process, a G-valued stochastic process indexed by loops on the surface, first introduced in all generality by Lévy in his PhD thesis [52].

In [52, 53], the construction yields a holonomy process that can be interpreted as a random geometric connection, in the sense that it creates one random object that associates a parallel transport to each curve on the surface. However, no analytic connection was constructed: in particular, no random element $A \in \Omega^1(M, \mathfrak{g})$ whose parallel transport generates the holonomy process was obtained.

In 2018 [24], in a remarkable advance, Chevyrev constructed for the first time a genuine Yang–Mills measure directly on distributional connections on the flat torus $\mathbb{T}^2$ as a scaling limit of discrete random connections whose holonomies satisfy the Driver–Sengupta formula. This result provided a rigorous realization of the Yang–Mills measure at the level of random connections, rather than merely at the level of holonomies. A key feature of Chevyrev’s approach is that it produces a random 1-form with isotropic regularity $0^{-}$. More precisely, the resulting object is a random distributional 1-form on the torus,

$$A = A_1 dx + A_2 dy,$$

where both components $A_1$ and $A_2$ have regularity $0^{-}$.

By contrast, Driver’s construction leads to an object with the same total regularity $(0^{-} = \frac{1}{2} - \frac{1}{2})$, but distributed heterogeneously. In fact, there is no longer a $dy$ component and $A_1$ has regularity $\frac{1}{2}$ in the $y$ direction and regularity $-\frac{1}{2}$ in the $x$ direction. This asymmetry prevents the direct application of the solution theory based on regularity structures. Chevyrev’s isotropic construction avoids this issue as both components have the same regularity $0^{-}$, and the solution theory can be applied separately and consistently to each of them.

This breakthrough stimulated a series of important developments within the stochastic partial differential equations community, aimed at understanding the stochastic quantization of gauge theories. Notable contributions in this direction include the works of Chandra–Chevyrev–Hairer–Shen [26, 27], Cao–Chatterjee [21, 22], Bringmann–Cao [19, 20], Chevyrev–Shen [25], Shen–Smith–Zhu [67], and Shen–Zhu–Zhu [68].

We also mention that it would be interesting to investigate potential connections between our approach and some other recent works on 2-dimensional gauge theories, like for instance the works of Park–Pfeffer–Sheffield–Yu [10], Sauzedde [63], or Dahlqvist–Lemoine [29, 30].

## 1.1 Contribution of the paper

Let us start by outlining that the first author, together with his coauthors in [16], introduces on a general surface a random distributional 1-form whose holonomies generate Lévy’s random holonomy process. They investigate the problem of constructing the Yang–Mills measure as a random distributional 1-form from a complementary perspective to ours, most notably through the lens of dynamical systems, leading to results of independent interest. They construct the measure directly in the continuum and study the associated holonomy process using a novel gauge : the Morse gauge. The latter, viewed as an analytic object in its own right, is studied in depth in [17], where it is developed within the frame of classical gauge theory.

**Separation of probabilistic and geometrical considerations.** The first contribution of the present work is to present a new version of the construction from [16] made possible by the introduction of zero-area bands (see Section 2.2) and pseudo-coordinates (see Section 4). Thanks to these tools, an arbitrary
surface can be represented by a cylinder (see Section 5), which allows all probabilistic arguments to be carried out on the cylinder and only afterwards pulled back to the original surface. This yields a complete separation between the analytical and geometrical aspects of the problem, on the one hand, and its probabilistic aspects, on the other.

From an analytical point of view, the focus lies on the problem of pushing back currents on the cylinder to the original surface. From a probabilistic point of view, all steps are performed directly on the cylinder.

**A universal scaling limit.** Taken together with [16], the present paper proposes, for the first time, a construction of a candidate Yang-Mills measure on the space of distributional 1-forms on a surface of arbitrary genus and smooth area form. We show that the 2-dimensional Yang-Mills measure arises as the scaling limit of a wide range of lattice gauge theories (see Sections 9 and 10). These lattice gauge theories can be interpreted as discretizations of the continuum measure on a sequence of well chosen lattices. The approach relies on a discrete analogue of the Morse gauge, and the convergence is established towards the same Morse gauge fixed Yang-Mills measure constructed in [16]. We also analyze some aspects of this convergence in Besov spaces. Beyond its foundational role in the present work, the Morse gauge [16, 17] provides a powerful and conceptually elegant framework to state and prove universality.

We further observe that this universal scaling limit provides a new, intrinsic construction of the measure, independent of any of the previous constructions in the literature. Moreover, it is possible to define the parallel transport associated with the limiting rough object using the Wong-Zakai theorem, which gives back the holonomy process. We will now state our first main result, after describing the setting of the present work.

Let $\Sigma$ be a closed compact surface of genus $g$, endowed with an area measure $\sigma$. Let $f$ be a Morse function on $\Sigma$. Let $G$ be a compact Lie group, with Lie algebra $\mathfrak{g}$ always assumed to be equipped with an invariant scalar product. Let $(Q_t dg)_{t>0}$ be a family of probability measures on $G$, arising from the Manton, Wilson, or Villain action.

Using the flow lines and level sets of a Morse function on $\Sigma$, we construct a sequence $(\mathcal{T}_N := (\mathbb{V}_N, \mathbb{E}_N, \mathbb{F}_N))_{N \ge 0}$ of increasingly fine lattice approximations of $\Sigma$ (see Section 6 for the construction of the lattice, and Section 3 for a recollection on Morse functions).

For each $N \ge 0$, we define a piecewise smooth random $\mathfrak{g}$-valued 1-form $A_N$ on $\Sigma$ such that the corresponding holonomy process $(\mathrm{Hol}(A_N, c))_{c \in \mathrm{Loops}(\mathcal{T}_N)}$ verifies the Driver-Sengupta formula associated to $(Q_t dg)_{t>0}$, meaning that $(\mathrm{Hol}(A_N, c))_{c \in \mathrm{Loops}(\mathcal{T}_N)}$ is equal to the holonomy process associated with the lattice measure $\mu_N$ defined by

$$d\mu_N((g_e)_{e \in \mathbb{E}_N}) = \prod_{F \in \mathbb{F}_N} Q_{\sigma(F)}(\mathrm{Hol}(g, \partial F)) \otimes_{e \in \mathbb{E}_N} dg_e.$$

Then, the main theorem is as follows.

**Theorem 1.1 (Universal scaling limit of lattice gauge theories).** The sequence $(A_N)_{N \ge 0}$ converges in law, in the Sobolev space $H_{\mathrm{loc}}^{-(1+\epsilon)}(\Sigma \setminus f^{-1}(\max(f)); \mathfrak{g})$ of distributional $\mathfrak{g}$-valued 1-forms, to the Morse-gauge-fixed Yang-Mills measure on $\Sigma$. Moreover, the sequence $A_N$ converges to $A$ in local anisotropic Hölder spaces of regularity $(-\frac{1}{2}, \frac{1}{2})$.

Let us outline the main steps leading to this theorem. The first step is analytical. It consists in blowing up the surface at the maximum of the Morse function $f$, thereby obtaining a new surface $S$ with one outgoing boundary component. Next, we encode this surface using pseudo-coordinates $(r, \theta)$ that behave like polar coordinates: the Morse function serves as the radial (or height) coordinate, while the base angle provides the angular one. This construction (see Section 4) yields an almost-diffeomorphism between the surface and a cylinder, together with a well-defined continuous pullback of currents from the cylinder to the surface.

The second step is probabilistic. We transfer the lattice structure onto the cylinder and define, as before, a sequence of random $\mathfrak{g}$-valued 1-forms $A_N$ realizing this time the free-boundary lattice Yang-Mills
<figure><img src="image_3.png" /><figcaption>Figure 1: Main steps.</figcaption></figure>

measure, meaning that there are no constraints on the bonds located along the boundary of the cylinder. For topological reasons discussed in Subsection 2.3, each $A_N$ decomposes into the sum of two terms,

$$A_N = A_N^{\text{bulk}} + A_N^{\text{sing}},$$

where $A_N^{\text{bulk}}$ denotes the bulk component supported on $S$, and $A_N^{\text{sing}}$ denotes the singular component supported on small tubular regions constructed by thickening some 1-dimensional sub-manifolds of $S$. We then establish the convergence of the free lattice measure in suitable functional spaces on the cylinder. Finally, the continuity of the pullback map leads the result on $S$, as stated in the following proposition.

**Proposition 1.2** (Bulk–singular decomposition of the free boundary Yang–Mills measure). The sequences $A_N^{\text{bulk}}$ and $A_N^{\text{sing}}$ are independent. Moreover, the following convergence statements hold.

* The bulk component $A_N^{\text{bulk}}$ converges in law, in anisotropic Hölder spaces of regularity $(-\frac{1}{2}, \frac{1}{2})$, to a de Rham primitive of a $\mathfrak{g}$-valued white noise on $\Sigma$.

* The singular component $A_N^{\text{sing}}$ converges point-wise to the random 1-current $\sum_{a=1}^{2g} [U_a] \log g_a$, where $([U_a])_{0\le a \le 2g}$ are currents of degree 1 supported on smooth curves in $S$, and $(g_a)_{1\le a \le 2g}$ are independent Haar-distributed random variables on $G$.

To recover the measure on the closed surface, we condition the holonomy on the boundary component to be $1_G$ (see Section 11.2). We summarize this procedure in Figure 1. Let us make three remarks.

* We have the convergence for a much wider family of actions which are recalled in Section 8.3; however, for the sake of clarity, we leave the details to the core of the paper.

* Our result is in fact stronger. It allows one to work on surfaces with boundary, to impose a prescribed constraint on the conjugacy class of the boundary holonomy, and to construct a sequence of discrete 1-forms satisfying this constraint, whose continuum limit can be identified. The proof of Theorem 1.1 actually relies on a disintegration of the free 1-forms of Proposition 1.2 along the boundary holonomy, followed by an analysis of the resulting sequence of conditioned measures. This approach naturally leads to the study of the associated partition function, which yields results of independent interest, discussed later in this introduction.

* A first universality result was established by Sengupta in [65]. He showed that, for any reasonable action (for instance the Manton or Wilson action), on the sphere, the law of the Wilson loop associated with a fixed base lattice converges to its law under the Villain action, provided the
lattice is successively refined into finer and finer lattices whose faces all have the same area. Our result, by contrast, proves the convergence of the random 1-form as a whole single object. In this sense, Sengupta's result can be viewed as a "convergence of finite-dimensional marginals", whereas ours plays the role of the "tightness" part in a classical Donsker-type invariance principle. Moreover, our result removes the equal-area assumption on faces and extends the universality statement to arbitrary compact surfaces.

* A second universality result for the Yang-Mills measure was proven in a groundbreaking paper by Chevyrev and Shen [25] when $\Sigma$ is the flat two-dimensional torus, in the context of SPDEs. The present result provides an alternative proof of this theorem, notably without SPDEs, and generalizes it to surfaces of arbitrary genus with arbitrary smooth area form and also controls the convergence of the partition function and Segal amplitudes.

**A rigorous link between the action and Lévy's holonomy process.** In Lévy's constructions [52, 53], the Yang-Mills action – which is a fundamental quantity in the theory – appears only at a formal level. The main ingredient in this construction is, however, the heat kernel. Through the Driver-Sengupta formula, it is used as a way of expressing the Yang-Mills action of a discrete connection. Although Migdal [60] argued that the heat kernel is the most physically relevant choice, its use can also be considered somewhat arbitrary. In fact, many other actions appear in the literature, and this will be discussed in Section 8.1.

This raises the natural question of whether the measure constructed by Lévy is indeed the 2D Yang-Mills measure. An argument supporting this identification is the large deviation principle for the Yang-Mills measure established by Lévy and Norris [56], which relates the measure to the Yang-Mills action. Other results in the same directions include [24, 28].

Our universality result (Sections 9 and 10), can be interpreted as further evidence for this link. Specifically, it shows that under any reasonable discretizations of the Yang-Mills action, the measure coincides, in the scaling limit, with the Yang-Mills measure in the sense of [52, 53]. This supports from a mathematical point of view the arguments of Migdal.

**Convergence of correlation and partition functions on all compact surfaces.** As a corollary of the key result of Theorem 7.2, we obtain the convergence of correlation functions and Segal amplitudes of lattice gauge theories on all compact surfaces. This generalizes the results of Balaban [12] from the planar case to surfaces of arbitrary genus. Let us give a precise statement.

Consider a compact Lie group $G$. We use conventional notations of representation theory of compact Lie groups, that we recall in Appendix D. Consider a family of probability measures $(\mu_t = \rho_t(\cdot)dg)_{t>0}$ on $G$ satisfying the conditions of Definition 7.1. Let $\Sigma$ be a compact surface of genus $g$ with $k$ boundary components $\partial\Sigma_1, \dots, \partial\Sigma_k$, endowed with an area measure $\sigma$. For a graph $\mathcal{T} = (\mathbb{V}, \mathbb{E}, \mathbb{F})$ on $\Sigma$, the partition function of the lattice gauge measure on $\mathcal{T}$ associated with $(\mu_t)_{t>0}$, and with prescribed boundary conditions

$$ \mathrm{Hol}(\partial\Sigma_1) = [g_1], \dots, \mathrm{Hol}(\partial\Sigma_k) = [g_k], \quad g_1, \dots, g_k \in G, $$

is given by

$$ Z_{\mathcal{T}}(g_1, \dots, g_k) := \int_{G^E} \prod_{i=1}^k \delta_{[g_i]}(\mathrm{Hol}(\partial\Sigma_i)) \prod_{F \in \mathbb{F}} \rho_{\sigma(F)}(\mathrm{Hol}(\partial F)) \prod_{e \in \mathbb{E}} dg_e. $$

We can give two formulas to compute $Z_{\mathcal{T}}$. The first one is the integral formula

$$ Z_{\mathcal{T}}(g_1, \dots, g_k) = \int_{G^{2g}} \left( \underset{F \in \mathbb{F}}{\star} \rho_{\sigma(F)} \right) ([a_1, b_1] \cdots [a_g, b_g] c_1 g_1 c_1^{-1} \cdots c_k g_k c_k^{-1}) \mathrm{dadbdc}. $$
The second one is the character expansion of the latter, and it is given by

$$Z_{\mathcal{T}}(g_1, \dots, g_k) = \sum_{\lambda \in \hat{G}} \left( \prod_{F \in \mathcal{F}} \frac{\hat{\mu}_{\sigma(F)}(\lambda)}{d_{\lambda}} \right) \frac{\chi_{\lambda}(g_1) \cdots \chi_{\lambda}(g_k)}{d_{\lambda}^{2g-2+k}}.$$

In the special case where $\rho$ is the heat kernel $p$ on $G$, thanks to the semi-group property, the quantity $Z_{\mathcal{T}}$ becomes independent of the graph $\mathcal{T}$. It is therefore an intrinsic property of the surface denoted by $Z_{\Sigma}$, and it reads

$$Z_{\Sigma} = \int_{G^{2g}} p_{\sigma(\Sigma)}([a_1, b_1] \cdots [a_g, b_g] c_1 g_1 c_1^{-1} \cdots c_k g_k c_k^{-1}) \otimes_{i=1}^{g} \mathrm{dadbdc} = \sum_{\lambda \in \hat{G}} e^{-\sigma(\Sigma)c_2(\lambda)} \frac{\chi_{\lambda}(g_1) \cdots \chi_{\lambda}(g_k)}{d_{\lambda}^{2g-2+k}}.$$

We are ready to state the second main theorem.

**Theorem 1.3 (Convergence of Segal amplitudes).** Let $(\mathcal{T}_n := (\mathbb{V}_n, \mathbb{E}_n, \mathbb{F}_n))_{n \ge 0}$ be a sequence of triangulations of $\Sigma$ such that there exist constants $a, A > 0$ for which

$$\liminf_{n \to \infty} \frac{\left| \left\{ f \in \mathbb{F}_n : \frac{a^2}{|\mathbb{F}_n|} \le \sigma(f) \le \frac{A^2}{|\mathbb{F}_n|} \right\} \right|}{|\mathbb{F}_n|} > 0.$$

Then, for all $(g_1, \dots, g_k) \in G^k$, $Z_{\mathcal{T}_n} \xrightarrow{n \to \infty} Z_{\Sigma}$ in the $C^\infty(G^k)$ topology.

## 1.2 Structure of the paper

First, in Section 2 we revisit Driver's construction on the plane and adapt it to the cylinder. We then explain how this construction can be intuitively generalized to arbitrary surfaces.

In Section 3, we recall the preliminaries from Morse theory that we need in Section 4 to construct a system of polar pseudo-coordinates on a surface with one outgoing boundary component. This covers every such a surface by a cylinder, and lets us transfer the construction from the cylinder to the surface. In particular, we recover in Section 5 the main formula for the Yang-Mills random 1-form as it appears in [16].

In Section 6 we use Morse theory to construct a sequence of well-chosen lattice approximations of surfaces. Then, in Section 7, we show a local limit theorem, crucial for the proof of the convergence of the conditioned measures. As a by product, we show the corollary about the convergence of the partition functions, which is a result of independent interest.

In Section 8, we establish some technical estimates for Lie groups-valued random walks that we will need in identifying the scaling limit of lattice gauge theories in Section 9.

In Section 10, we investigate the convergence of lattice models in more refined functional spaces.

Finally, in Section 11, we close the surface by means of conditioning, and conclude the proof of the main theorem.

## Acknowledgments

E.N. wishes to express his deep gratitude to his PhD advisor, Thierry Lévy, for asking him the main question of this paper, for countless long and productive discussions, and for carefully reading parts of this work. He also thanks David García-Zelada for many suggestions and close help, and Luis Zegarra for introducing him to Morse theory. Furthermore, he thanks Vladimir Bošković, Nicolas Fournier, François Jacopin, Thomas Jaffard, Thomas Le Guerch, Thibaut Lemoine, Abdulwahab Mohamed, Sami Mustapha, Breki Pálsson, Maxence Petit, Damien Simon, and Lorenzo Zambotti for many interesting discussions.
Finally, he acknowledges funding from the “Fondation CFM pour la Recherche” and thanks the foundation for providing excellent working conditions during his PhD.

N.V.D would like to express his deep gratitude to Rongchan and Xiangchan Zhu for explaining to him the results of Driver, this allowed him to start working on the subject and is the main source of inspiration. He also warmly thanks his collaborators Yannick Guedes-Bonthonneau, Reda Chaibbi, Tô Tât Dat, Gabriel Rivière for many discussions around gauge theories, probability and Yang-Mills, and also Ismael Bailleul, Laurent Charles, Ilya Chevyrev, Benoit Estienne, Léonard Ferdinand, Vladimir Fock, Colin Guillarmou, Frédéric Hélein, Thibault Lefeuvre, Thibaut Lemoine, Jiasheng Lin, Julien Marché, Antoine Mouzard, Phan Thanh Nam, Tristan Robert, Scott Smith, Nikolay Tzvetkov, Martin Vogel for interesting questions, remarks when we gave talks or just discussed the present work, which motivated us to pursue. He also thanks Benjamin Melinand for giving wonderful references on fractional Sobolev spaces. N.V.D acknowledges the support of the Institut Universitaire de France.

# 2 Revisiting Driver's construction of YM²

In this section we revisit first Driver's argument [37] for the construction of the Yang-Mills measure on the plane. We then generalize it to the cylinder with one outgoing free boundary and discuss possible generalization to all surfaces.

We consider in this section a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, equipped with a bi-invariant scalar product.

## 2.1 The case of the plane

The Yang-Mills measure on $\mathbb{R}^2$ is the probability measure on the set of 1-forms modulo gauge transformations $\Omega^1(\mathbb{R}^2, \mathfrak{g})/\mathcal{G}$ with formal expression $d\mu = \frac{1}{Z} \exp(-S_{YM}(A))dA$, where

1. the constant $Z$, also called partition function, is a normalizing constant,

2. the quantity $S_{YM}(A)$ is the Yang-Mills action of $A$, i.e. half of the squared $L^2$ norm of the curvature $F$ of $A := A_1dx + A_2dy$ defined as

$$F(A) := dA + A \wedge A = \left( \frac{\partial A_2}{\partial x} - \frac{\partial A_1}{\partial y} + [A_1, A_2]_{\mathfrak{g}} \right) dx \wedge dy,$$

3. the measure $dA$ is the formal Lebesgue measure on $\Omega^1(\mathbb{R}^2, \mathfrak{g})/\mathcal{G}$.

This measure is ill-defined for several reasons. One way of giving a rigorous meaning to this measure was achieved in the seminal work of [37]. In this article, it is discussed that every connection on $\mathbb{R}^2$ is gauge equivalent to a connection of the form

$$Adx \text{ such that } A(\cdot, 0) = 0,$$

and that the map

$$ax : \Omega^1(\mathbb{R}^2, \mathfrak{g})/\mathcal{G} \to \mathcal{A}_{ax} := \{A \,dx; A(\cdot, 0) = 0\}$$

has a constant (formal) Jacobian (or Fadeev-Popov determinant). For a connection $[A] \in \Omega^1(\mathbb{R}^2, \mathfrak{g})/\mathcal{G}$, the differential form $ax([A])$ is said to be an axial gauge representation of $[A]$. The core idea is that the Yang-Mills action becomes a quadratic functional when restricted to connections in axial gauge. Indeed

$$S_{YM}(Adx) = \left\| \frac{\partial A}{\partial y} \right\|_{L^2(\mathbb{R}^2)}^2.$$
The Yang-Mills measure induced on the slice $A_{ax}$ has therefore the formal expression

$$d\mu_{ax} := \frac{1}{Z_{ax}} \exp \left( - \left\| \frac{\partial A}{\partial y} \right\|_{L^2(\mathbb{R}^2)}^2 \right).$$

This means that in this representation, $\frac{\partial A}{\partial y}$ is a two dimensional white noise. Formally, A can be constructed by taking a white noise $\xi$ in $\mathbb{R}^2$, and defining

$$A(x, y) = \int_{0}^{y} \xi(x, u)du,$$

which is not straightforward to define mathematically. In fact, the white noise belongs to $C^{(-1)^-}(\mathbb{R}^2)$, and cannot be integrated along lines without further justification.

One way to do it goes as follows. Write the planar white noise $\xi$ as a random series $\xi = \sum_{n \ge 0} \xi_n^g(y)e_n(x)$, where $(\xi_n^g)_{n \ge 0}$ are i.i.d g-valued white noises and $(e_n)_{n \ge 0}$ is some orthonormal basis of $L^2(\mathbb{R})$ (like for instance the Hermite polynomials, the Haar basis or any orthonormal wavelet basis). The random series converges in $S'(\mathbb{R}^2)$ and the integral along the y axis can be defined as

$$\sum_{n \ge 0} \left( \int_0^y \xi_n^g(u) du \right) e_n(x).$$

It is natural to recognize in the above expression that $(\int_0^y \xi_n^g(u) du)$ is a g-valued Brownian motion $B_n^g(y)$ started at 0 and therefore to propose the random series

$$A := \sum_{n \ge 0} B_n^g(y) e_n(x)dx$$

as a natural candidate for the random YM connection on the plane $\mathbb{R}^2$. The reader will recognize that our candidate is nothing but the cylindrical Brownian motion on $L_x^2(\mathbb{R})$ where $y$ plays the role of time for the cylindrical Brownian motion.

Another way to derive a sensible object in this special gauge where the curvature is exactly $dA$, is to use Stokes theorem to formally define A as a Schwartz distribution as follows. Let $A_{ax} := Tdx$ where

$$\forall \phi \in S(\mathbb{R}^2), \langle T, \phi \rangle = \left\langle \xi, (x, y) \mapsto \int_{\mathbb{R}} \phi(x, z) [1_{0 \le y \le z} - 1_{z \le y \le 0}] dz \right\rangle.$$

Then, Driver defines a parallel transport for the random connection A thus constructed using stochastic calculus, and computes the joint law of the parallel transport for a finite set of curves. This is known as the Driver-Sengupta formula on the plane. We will revisit the main ideas but in the slightly different case of the cylinder.

## 2.2 The case of the cylinder

Let us see how we can mimic Driver's construction on a closed disk, seen as a manifold with boundary, that can also represent a spherical cap, or a cylinder with one outgoing boundary component. Let us use polar coordinates to describe our disc $D$. Concretely, it means that we have a smooth submersion $[0, 1]_r \times \mathbb{S}^1 \mapsto D$, where $\mathbb{S}^1$ is the circle parametrized by the angle $[0, 2\pi[$. Let $\sigma$ be a smooth area measure on $D$. The cylinder is obtained by blowing up the disc at the origin $(0, 0)$, the thing is that the induced area form on the cylinder, still denoted by $\sigma$, vanishes at $r = 0$. On the blow-up space, the one form $dr$ is defined globally, as well as the corresponding vector field $\partial_r$.

Distributions and currents on the cylinder are defined as the topological dual of the space of smooth forms that vanish near 1 and are smooth up to $r = 0$. Conceptually, it means our distributions are extendible near $r = 0$ but not near $r = 1$.
Based on the previous discussion, let us consider a white noise $\xi$ on $(D, \sigma)$ corresponding to the area form $\sigma$, that we will see as a random distributional 2-form. We will try to define a primitive $A$ of $\xi$ such that $A|_{r=0} = 0$, and such that $i_{\partial r} A = 0$. This means that in polar coordinates, $A$ does not have a component along $dr$. For some $\theta \in [0, 2\pi)$, let us denote by $0 \xrightarrow{\theta} r$ the curve whose polar parametrization is given by $c(t) = (t, \theta)$, $0 \le t \le r$.

**Definition 2.1 (Definition-Proposition).** Let $A$ be defined as follows by duality.

$$ \forall \alpha \in \Omega^1(D, \mathfrak{g}), \langle A, \alpha \rangle := \left\langle \xi, (r, \theta) \mapsto -\int_{r \xrightarrow{\theta} 1} \alpha \right\rangle. $$

The well-defined random de Rham current $A$ is called the axial-gauge Yang-Mills distributional 1-form. We can also define $A$ by some explicit formula:

$$ A := \sum_{n \ge 0} \left( \int_0^r \sqrt{\sigma(s, \theta)} \xi_n^g(s) ds \right) e_n(\theta) d\theta, $$

where $(\xi_n^g)_{n \ge 0}$ are i.i.d $\mathfrak{g}$-valued white noise.

*Proof.* For $a \in \Omega^1(D, \mathfrak{g})$, the function

$$ (r, \theta) \mapsto -\int_{r \xrightarrow{\theta} 1} \alpha $$

is smooth and vanishes at all orders at the boundary circle $\{r = 1\}$ (since $\alpha$ does). Moreover, the map

$$ \alpha \in \Omega^1(D, \mathfrak{g}) \mapsto \left\{ (r, \theta) \mapsto -\int_{r \xrightarrow{\theta} 1} \alpha \in C^{\infty}(D, \mathfrak{g}) \right\} $$

is continuous for the topology of test functions that vanish up to all order at $\{r = 1\}$.

To get another formula, we shall use another representation of the 2D white noise in polar coordinates. Rewrite $\xi$ as a random series using Fourier decomposition in $\theta$:

$$ \xi_{\sigma} = \sum_{n \ge 0} (\sigma(r, \theta))^{\frac{1}{2}} \xi_{n}^{g}(r) e_{n}(\theta) dr \wedge d\theta $$

where $\xi_n^g$ are i.i.d $\mathfrak{g}$-valued white noise. The factor $(\sigma(r, \theta))^{\frac{1}{2}}$ ensures we get the correct white noise normalized w.r.t. the area form $\sigma$. Then contract with the radial vector field $\partial_r$ and take the primitive in the radial direction. This yields a de Rham primitive of $\xi_\sigma$ which is the correct random distribution $A$. $\square$

By the correct we mean that it is possible to define parallel transport of this object, and that these parallel transports verify the Driver-Sengupta formula for a wide class of graphs. This is what we will do in the remaining of this section.

**Defining line integrals.** We will now define a natural parallel transport for $A$ around curves of the form $\theta_1 \xrightarrow{r} \theta_2$ for $r \in (0, 1]$ and $0 \le \theta_1 < \theta_2 \le 2\pi$. Consider our random 1-form and the curve $\theta_1 \xrightarrow{r} \theta_2$ defined as the curve parametrized in the polar coordinates as $c(t) = (r, t)$; $\theta_1 \le t \le t_2$. The first thing to note is that $dA = \xi$.

**Proposition 2.2.** The equation $dA = \xi$ holds in the sense of currents.
<figure><img src="image_4.png" /><figcaption>Figure 2: The rectangle $\square(r_1, r_2, \theta_1, \theta_2)$ in the cylinder.</figcaption></figure>

*Proof.* For $\phi \in C^\infty(D, \mathfrak{g})$, we have

$$ \langle dA, \phi \rangle = \langle A, d\phi \rangle = \left\langle \xi, (r, \theta) \mapsto \int_{[r,1]\times\{\theta\}} d\phi \right\rangle = \langle \xi, (r, \theta) \mapsto -\phi(1, \theta) + \phi(r, \theta) \rangle = \langle \xi, \phi \rangle $$

where we used the fact that our test function $\phi$ vanishes near $r = 0$. In the proof of the Lemma, the duality pairing $\langle \psi_1, \psi_2 \rangle_{\mathfrak{g}}$ means $\int_D \langle \psi_1 \wedge \psi_2 \rangle_{\mathfrak{g}}$ where we used the exterior product and the natural inner product in the Lie algebra $\mathfrak{g}$. $\square$

Then, Stokes theorem let us define the line integral of $A$ along $\theta_1 \xrightarrow{r} \theta_2$ as follows. Since $i_{\partial r}A = 0$, in whatever way we define this integral, it is natural for

$$ \int_{0 \xrightarrow{r} \theta} A = \int_{\partial \square(0,r,0,\theta)} A $$

to hold where $\square(r_1, r_2, \theta_1, \theta_2)$ is the unique rectangle in the cylinder with vertices $(r_1, \theta_1)$, $(r_2, \theta_2)$, $(r_1, \theta_2)$, $(r_2, \theta_1)$, see Figure 2. Then, we use Stokes theorem to define

$$ \int_{\partial \square(0,r,0,\theta)} A := \int_{\square(0,r,0,\theta)} dA = \xi(1_{\square(0,r,0,\theta)}). $$

We get therefore the following definition which describes completely the probability distribution of line integrals.

**Definition 2.3.** Let $r \in (0, 1]$ and $0 \le \theta_1 < \theta_2 \le 2\pi$. The line integral of $A$ along $\theta_1 \xrightarrow{r} \theta_2$ is defined as

$$ \int_{\theta_1 \xrightarrow{r} \theta_2} A = \xi(1_{\square(0,r,\theta_1,\theta_2)}). $$

The pairing $\xi(1_{\square(0,r,\theta_1,\theta_2)})$ is well-defined since the indicator function $1_{\square(0,r,\theta_1,\theta_2)} \in L^2$ hence it belongs to the Hilbert space indexing the white noise.

**From line integrals to stochastic parallel transport.** To define the parallel transport of $A$ along horizontal curves, let us recall that if $\alpha$ is a connection, the holonomy of $\alpha$ along the curve $c: [0, 1] \to M$ is given by the solution at time 1 of an ODE that can be written as

$$ \left\{ \begin{array}{l} dX_t = X_t d\int_0^t \alpha_s(c'(s))ds \\ X_0 = 1 \end{array} \right. . \qquad (2.1) $$
The idea is that the holonomy equation for A can be interpreted as a Stratonovich SDE thanks to the following proposition.

**Proposition 2.4.** Let $r \in (0, 1]$. The stochastic process

$$ (r, \theta) \mapsto \int_{0 \to \theta}^{r} A $$

is a time changed g-valued Brownian sheet with time changing function $(r, \theta) \mapsto \sigma(\Box(0, r, 0, \theta))$.

*Proof.* The process $(r, \theta) \mapsto \xi(1_{\Box(0,r,0,\theta)})$ is gaussian, and for $\theta, \theta' \in [0, 2\pi)$,

$$ \mathbb{E}[\xi(1_{\Box(0,r,0,\theta)})\xi(1_{\Box(0,r,0,\theta')})] = \sigma(\Box(0, r, 0, \theta)) \wedge \sigma(\Box(0, r, 0, \theta)). $$

**Definition 2.5.** Let $r \in (0, 1]$ and $0 \le \theta_1 \le \theta_2 \le 2\pi$. The parallel transport of A along $\theta_1 \xrightarrow{r} \theta_2$ is defined as the strong solution of the Stratonovich SDE

$$ \begin{cases} dX_t = X_t \circ d \int_{\theta_1 \xrightarrow{r} \theta_1+t} A \\ X_0 = 1 \end{cases} $$

at time $\theta_2 - \theta_1$. It is denoted by

$$ \mathrm{Hol}(A, \theta_1 \xrightarrow{r} \theta_2). $$

The main properties of this parallel transport are regrouped in the next proposition.

**Proposition 2.6.** The following hold.

1. For fixed $r \in (0, 1]$, $\mathrm{Hol}(A, \theta \xrightarrow{r} \theta + \cdot)$ is a G-valued time-changed Brownian motion. The time changing function is $t \mapsto \sigma(\Box(0, r, \theta, \theta + t))$.

2. For fixed $0 \le \theta_1 \le \theta_2 \le 2\pi$, $r \mapsto \mathrm{Hol}(A, (\theta_1 \xrightarrow{r} \theta_2)$ is G-valued time-changed Brownian motion. The time changing function is $t \mapsto \sigma(\Box(0, t, \theta_1, \theta_2 - \theta_1))$.

3. For any $0 = \theta_1 < \dots < \theta_n = 2\pi$, the family of processes $(r \mapsto \mathrm{Hol}(A, (\theta_i \xrightarrow{r} \theta_{i+1}))_{1 \le i \le n}$ are independent.

*Proof.* For the first point, the parallel transport $\mathrm{Hol}(A, \cdot)$ is the solution of

$$ \begin{cases} dX_t = X_t \circ d \int_{\theta_1 \xrightarrow{r} \theta_1+t} A \\ X_0 = 1 \end{cases}. $$

However, from Proposition 2.4, the process $\int_{\theta_1 \xrightarrow{r} \theta_1+\cdot} A$ is a time changed Brownian motion, so that the solution has the same law as the solution of $dZ_t = Z_t \sqrt{F'(t)} \circ dB_t$, where B is a Brownian motion and $F(t) = \sigma(\Box(0, r, \theta_1, \theta_1 + t))$. The density of the law of the solution at time t is hence $p_{\sigma(\Box(0,t,\theta_1,\theta_2))}(g)dg$.

However, the solution to 2.1 is strong and it is adapted to $(\int_{\theta_1 \xrightarrow{\rho} \theta_1+t} A)_{\substack{0 \le \rho \le r \\ 0 \le t \le \theta_2 - \theta_1}}$.

For the second point, fix $0 \le \theta_1 \le \theta_2 \le 2\pi$, and $0 \le r_1 < \dots < r_n \le 1$. Let us calculate the joint law of

$$ (X^1, \dots, X^n) := (\mathrm{Hol}_A\{\theta_1 \xrightarrow{r_1} \theta_2\}, \dots, \mathrm{Hol}_A\{\theta_1 \xrightarrow{r_n} \theta_2\}). $$

The vector $(X^1, \dots, X^n)$ is the strong solution to

$$ \left\{ \begin{aligned} dX_t^1 &= X_t^1 \circ d \int_{\theta_1 \xrightarrow{r_1} \theta_1+t} A ; X_0^1 = 1 \\ \vdots & \\ dX_t^n &= X_t^n \circ d \int_{\theta_1 \xrightarrow{r_n} \theta_1+t} A ; X_0^n = 1 \end{aligned} \right. . $$
However, we have for $i = 2, \dots, n$,

$$ \int_{\theta_1 \xrightarrow{r_i} \theta_1 + t} A = \xi(\mathbb{1}_{\square(0, r_i, \theta_1, \theta_1 + t)}) = \xi(\mathbb{1}_{\square(r_{i-1}, r_i, \theta_1, \theta_1 + t)}) + \xi(\mathbb{1}_{\square(0, r_{i-1}, \theta_1, \theta_1 + t)}), $$

and we have the following equality in law

$$ \left( d \int_{\theta_1 \xrightarrow{r_i} \theta_1 + t} A \right)_{i=1,\dots,n} = (F_1(t) \circ dB_t^1 + \dots + F_i(t) \circ dB_t^i)_{i=1,\dots,n}, $$

where $(B^i)_{i=1,\dots,n}$ are independent Brownian motions and

$$ F_i(t) = \frac{d}{dt} \sigma(\square(r_{i-1}, r_i, \theta_1, \theta_1 + t)). $$

The vector $(X^1, \dots, X^n)$ has thus the same law as the solution of

$$ \begin{cases} dX_t^i = X_t^i (F_1(t) \circ dB_t^1 + \dots + F_i(t) \circ dB_t^i) \\ X_0^i = 1 \\ 1 \le i \le n \end{cases}, $$

Let's calculate $d\{X^i(X_t^{i-1})^{-1}\}$. To simplify notations, we will call $Y := X^i$ and $X := X^{i-1}$. We have

$$ d(Y_t X_t^{-1}) = dY_t X_t^{-1} + Y_t dX_t^{-1} + dY_t dX_t^{-1}. $$

To compute $dX_t^{-1}$ we write

$$ 0 = d(X_t X_t^{-1}) = dX_t X_t^{-1} + X_t dX_t^{-1} + dX_t dX_t^{-1}, $$

which gives

$$ dX_t^{-1} = -X_t^{-1} dX_t X_t^{-1} - X_t^{-1} dX_t dX_t^{-1}. $$

This equation tells us that the local-martingale part of $dX_t^{-1}$ is contained in $-X_t^{-1}dX_t X_t^{-1}$, the second term containing only processes of finite variations. Therefore,

$$ dX_t^{-1} = -X_t^{-1} dX_t X_t^{-1} + X_t^{-1} dX_t X_t^{-1} dX_t X_t^{-1}. $$

Now, we have

$$ \begin{aligned} d(Y_t X_t^{-1}) &= dY_t X_t^{-1} + Y_t dX_t^{-1} + dY_t dX_t^{-1} \\ &= Y_t (Y_t^{-1} dY_t) X_t^{-1} - Y_t (X_t^{-1} dX_t) X_t^{-1} + Y_t (X_t^{-1} dX_t) (X_t^{-1} dX_t) X_t^{-1} - Y_t (Y_t^{-1} dY_t) (X_t^{-1} dX_t) X_t^{-1} \\ &= Y_t (F_1(t) \circ dB_t^1 + \dots + F_i(t) \circ dB_t^i) X_t^{-1} - Y_t (F_1(t) \circ dB_t^1 + \dots + F_{i-1}(t) \circ dB_t^{i-1}) X_t^{-1} \\ &\quad - Y_t (F_1(t) \circ dB_t^1 + \dots + F_i(t) \circ dB_t^i) (F_1(t) \circ dB_t^1 + \dots + F_{i-1}(t) \circ dB_t^{i-1}) X_t^{-1} \\ &\quad + Y_t (F_1(t) \circ dB_t^1 + \dots + F_{i-1}(t) \circ dB_t^{i-1}) (F_1(t) \circ dB_t^1 + \dots + F_{i-1}(t) \circ dB_t^{i-1}) X_t^{-1} \\ &= Y_t (F_i(t) \circ dB_t^i) X_t^{-1} \\ &= (Y_t X_t^{-1}) F_i(t) X_t (\circ dB_t^i) X_t^{-1}. \end{aligned} $$

where we have used that $\circ dB_t^i \circ dB_t^{i-1} = 0$ since $B^i$ and $B^{i-1}$ are independent. Therefore, the vector $(X_t^1, X_t^2(X_t^1)^{-1}, \dots, X_t^n(X_t^{n-1})^{-1})$ has the same solution as the vector $(Z^1, \dots, Z^n)$, solution of

$$ \left\{ \begin{aligned} dZ_t^1 &= Z_t^1 F_1(t) \circ dB_t^1 \\ dZ_t^i &= Z^i F_i(t) \operatorname{Ad}_{\{Z_t^i Z_t^{i-1} \dots Z_t^1\}} (\circ dB_t^i) ; i = 2, \dots, n \\ Z_0^i &= 1 ; i = 1, \dots, n \end{aligned} \right. . $$
<figure><img src="image_5.png" /><figcaption>Figure 3: Morse coordinates.</figcaption></figure>

Moreover,

$$ \{ \circ dB_t^1, \mathrm{Ad}_{\{Z_t^1\}} (\circ dB_t^i), \dots, \mathrm{Ad}_{\{Z_t^i Z_t^{i-1} \dots Z_t^1\}} (\circ dB_t^i) \} $$

is a family of independent white noises driving independent Brownian motions. The vector

$$ (X_t^1, X_t^2(X_t^1)^{-1}, \dots, X_t^n(X_t^{n-1})^{-1}) $$

has finally the same law as the solution to

$$ \left\{ \begin{array}{l} dZ_t^i = Z_t^i F_i(t) \circ dW_t^i \\ Z_0^i = 1 \\ i = 1, \dots, n \end{array} \right. , $$

where $(W^i)_i$ are independent Brownian motions. Therefore, $\{X_t^1, X_t^2(X_t^1)^{-1}, \dots, X_t^n(X_t^{n-1})^{-1}\}$ is an independent family and

* the law of $X_{\theta_2-\theta_1}^1$ is

$$ \mathbb{P}(X_{\theta_2-\theta_1}^1 \in dg) = p_{\sigma(\square(0,r_1,\theta_1,\theta_2))}(g)dg, $$

* the law, for some $i$, of $X_{\theta_2-\theta_1}^i (X_{\theta_2-\theta_1}^{i-1})^{-1}$ is

$$ \mathbb{P} \left( X_{\theta_2 - \theta_1}^{i} (X_{\theta_2 - \theta_1}^{i-1})^{-1} \in dg \right) = p_{\sigma(\square(r_{i-1}, r_i, \theta_1, \theta_2))}(g)dg. $$

which proves the second point.

For the third point, and since all the solutions are strong, it is enough to note from the proof of the first point that the process $\mathrm{Hol}(A, (\theta_i \to \theta_{i+1}))$ is measurable with respect to

$$ \{ \xi(f); f \in L^2(D), \mathrm{supp} f \subset (\theta_i, \theta_{i+1}) \times [0, 1] \}. \quad \square $$

As a direct consequence of the previous proposition, we get the following corollary.

**Corollary 2.7.** 1. (Segal Amplitudes) the stochastic process $(\mathrm{Hol}_A\{0 \xrightarrow{r} 2\pi\})_{r \in [0,1]}$ has the same law as the restriction of Lévy's holonomy process to the family of curves $\{0 \xrightarrow{r} 2\pi\}, r \in [0, 1]$.

2. (Driver-Sengupta formula) For any graph $\Lambda$ whose edges are horizontal or vertical lines, the process $(\mathrm{Hol}(A, c))_{c \in \mathrm{Loops}(\Lambda)}$ verifies the Driver-Sengupta formula.
<figure><img src="image_6.png" /><figcaption>Figure 4: A pant shaped surface.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5: A top view of the pant with the band of zero area.</figcaption></figure>

## 2.3 The case of a general surface

Let us start with the surface of a pant $\Sigma$ with two outgoing boundary component as shown in Figure 3, and let us try to adapt the previous reasoning to the current case. The analogue of the $r$ coordinate is now given by a Morse function $f: \Sigma \to [0, 1]$. The analogue of $\theta$ is seen when we parametrize $\Sigma$ as follows. Parametrize the bottom circle by $[0, 2\pi)$. For $r \in [0, 1]$ and $\theta \in [0, 2\pi)$, the point $(r, \theta) \in \Sigma$ is defined as the unique intersection of the level set $f^{-1}(r)$ with the gradient flow of $f$ starting at $(0, \theta)$ on the bottom circle $f^{-1}(0)$.

We would like to define a 1-form $A$ on $\Sigma$ such that $\iota_{\nabla f} A = 0$, $A|_{f^{-1}(0)} \equiv 0$, and $dA = \xi$. The obstruction is topological and comes from the fact that the function

$$ (r, \theta) \mapsto - \int_{r \to 1} \alpha $$

is not well-defined anymore. In fact, it is ambiguous on the blue lines of Figure 4, as there are two ways there of defining flow lines of $f$ that go to the top. One way of solving this issue is to cut along the green lines (of Figure 4), and inserting a small band, transforming thus the red point into two different points. This will be called resolution of the surface. A top view of this situation is shown in Figure 5.

Now, we can construct, using the same procedure as for the cylinder, a 1-form that lives outside the band that we just added. The remaining part is to see what to do inside the bands. First, note that the band should be treated as a part of the surface with 0 area, as it is merely the thickening of a curve which of course has zero area. This leads to the natural condition of imposing that the holonomy around any closed curve inside the band is always equal to $1_G$.

At first glance, we might be tempted to consider the following stronger condition

$$ A \text{ restricted to the band} \equiv 0, $$

which will turn out insufficient, as the measure we obtain with this condition is not quite the Yang-Mills measure. In fact, we can compute the joint law of the holonomies of the two outgoing component in this case. It is the solution at time $\pi$ to the following system of SDE

$$ \begin{cases} dX_t = X_t \circ d\xi(1_{\square(0,t,0,1)}) \\ dY_t = Y_t \circ d\xi(1_{\square(\pi,\pi+t,0,1)}) \end{cases}, $$
where we recall

$$ \Box(\theta_1, \theta_2, r_1, r_2) = \{(r, \theta) \in M \mid \theta_1 \le \theta \le \theta_2; r_1 \le r \le r_2\}. $$

However, the stochastic vector-valued process

$$ (\xi(1_{\Box(0,t,0,1)}); \xi(1_{\Box(\pi,\pi+t,0,1)}))_{0 \le t \le \pi} $$

has the same law as $(B_{F_g(t)}^1; B_{F_d(t)}^2)$ where $B^1$ and $B^2$ are independent Brownian motions and

$$ F_1(t) = \sigma(\Box(0, t, 0, 1)) \text{ and } F_2(t) = \sigma(\Box(\pi, \pi + t, 0, 1)). $$

Therefore,

$$ \mathbb{P}(X \in dx, Y \in dy) = p_{F_1(0.5)}(x)p_{F_2(0.5)}(y)dxdy, $$

where $(X, Y)$ is the random vector corresponding to the joint holonomies of the two outgoing boundary components of $\Sigma$. Note that this is not what we would have obtained had we computed these holonomies using the Driver–Sengupta formula which yields

$$ \mathbb{P}(X \in dx, Y \in dy) = p_{\sigma(M)}(xy)dxdy. $$

Rather, the condition to consider is that $A$ within the band should be gauge equivalent to the 0 connection. Since we have already fixed the gauge, and we would like to conserve the fact that $\iota_{\nabla_f} A \equiv 0$, the only freedom we have is to define $A$ inside the band as $cd\theta$. The most natural choice of $c$ due to gauge invariance is the log of a uniform random variable in $G$. Let us now calculate the new joint law of the holonomies of the outgoing boundary component.

Denote by $H_1$ and $H_1$ the holonomies of the two outgoing boundary components of $\Sigma$. For a bounded measurable function $f: G \times G \to \mathbb{R}$,

$$ \mathbb{E}[f(H_1, H_2)] = \mathbb{E}[f(XU, YU^{-1})], $$

where $X, Y$ are the solution to the same system as before, and $U$ is an independent uniform random variable in $G$, which represents the holonomy of each red curve in 5. We get

$$ \begin{aligned} \mathbb{E}[f(H_1, H_2)] &= \mathbb{E}[f(XU, YU^{-1})] = \int_{G^3} f(xu, yu^{-1})p_{\sigma_1}(x)p_{\sigma_1}(y)dxdydu \\ &= \int_{G^3} f(x, y)p_{\sigma_1}(xu^{-1})p_{\sigma_1}(yu)dxdydu = \int_{G^2} f(x, y)p_{\sigma(M)}(xy)dxdy, \end{aligned} $$

which now gives the correct density and gives positive arguments that the new construction yields the Yang–Mills measure.

**The case of a general surface.** The case of the pant shaped surface is a good guide to the construction of the Yang–Mills measure in Morse gauge on a general surface. We already now that we need to have a resolution of the surface in such a way that given any point on the surface, there exists exactly one path that connects this point to the boundary of the surface. This will be made clear by the introduction of pseudo coordinates in Section 4. These pseudo–coordinates will allow us to transfer the construction for the cylinder to all surfaces. The general formula is given in (5.1). Let us here only calculate the law of the holonomy of the outgoing boundary component using this representation.

Let $\Sigma$ be a surface with genus $2g$, one outgoing boundary component, and a smooth area form $\sigma$. In order to resolve the surface, we need to insert $2g$ bands of zero area. A top view can be seen in Figure 6. The holonomy of the boundary component is given by the product

$$ \mathrm{Hol}(\partial\Sigma) = X_1U_1 \cdots X_{2g}U_{2g}X_{2g+1}U_1^{-1} \cdots X_{2g}U_{2g}^{-1}, $$
<figure><img src="image_8.png" /><figcaption>Figure 6: Top view of a general surface with zero area bands.</figcaption></figure>

where $(U_i)_{1 \le i \le 2g}$ are independent and Haar distributed $G$-valued random variables, and the $X_i$ are as before, the holonomy of the De Rham primitive of the white noise in the sector within $U_i, U_{i+1}$. Let $f$ be a measurable function. We have

$$ \mathbb{E}[f(\mathrm{Hol}(\partial\Sigma))] = \int_{G^{6g}} f(x_1 u_1 \cdots x_{2g} u_{2g} x_{2g+1} u_1^{-1} \cdots x_{4g} u_{2g}^{-1}) p_{\sigma_1}(x_1) \cdots p_{\sigma_{4g}}(x_{4g}) dxdu, $$

where $\sigma_i$ is the area of the corresponding sector.

$$ \begin{aligned} \mathbb{E}[f(\mathrm{Hol}(\partial\Sigma))] &= \int_{G^{6g}} f(\alpha) p_{\sigma_1}(\alpha u_{2g} x_{4g}^{-1} \cdots u_1 x_{2g+1}^{-1} u_{2g}^{-1} x_{2g}^{-1} \cdots x_2^{-1} u_1^{-1}) p_{\sigma_2}(x_2) \cdots p_{\sigma_{4g}}(x_{4g}) dxdu \\ &= \int_G f(\alpha) d\alpha \int_{G^{6g-1}} p_{\sigma(\Sigma)}(\alpha[a_1, b_1] \cdots [a_{2g}, b_{2g}]) dadb. \end{aligned} $$

Therefore,

$$ \mathbb{P}(\mathrm{Hol}(\partial\Sigma) \in dg) = dg \int_{G^{6g-1}} p_{\sigma(\Sigma)}(g[a_1, b_1] \cdots [a_{2g}, b_{2g}]) dadb, $$

which is the same as what we would have obtained from the Driver–Sengupta formula.

# 3 Preliminaries on Morse theory

We gather the main definitions and results that we will use, namely on how to decompose surfaces using Morse theory.

Let $\Sigma$ be a smooth, compact, oriented and closed surface. We recall that $f: \Sigma \to \mathbb{R}$ is a *Morse function* if all its critical points, the set of which we denote by Crit($f$), are non-degenerate. It can be shown that the set of Morse functions is open and dense in the $C^\infty$ topology. A Morse function is said to be perfect if it has exactly $2g + 2$ critical points, where $g$ is the genus of the surface $\Sigma$, and if all its critical values are distinct. Note that the set of perfect Morse functions is also open (and nonempty) in the $C^\infty$ topology.

A fundamental property of Morse functions is the following so-called Morse Lemma.

**Lemma 3.1 (Morse Lemma).** Let $f: \Sigma \to \mathbb{R}$ be a Morse function. Then, for any $a \in \mathrm{Crit}(f)$, there exists a smooth chart $x \in U \subset M \mapsto (x_1(x), x_2(x)) \in \mathbb{R}^2$ centered at $a$, called a *Morse Chart* near $a$, in which the function $f$ reads

$$ f(x) = f(a) + \frac{1}{2} (\varepsilon_1 x_1^2 + \varepsilon_2 x_2^2), $$
<figure><img src="image_9.png" /><figcaption>Figure 7: A perfect Morse function.</figcaption></figure>

with $\epsilon_1, \epsilon_2 \in \{-1, 1\}$. Moreover,

* If $\epsilon_1 = \epsilon_2 = 1$, we say that a has index 0, or is a local minimum.

* If $\epsilon_1 = 1, \epsilon_2 = -1$, we say that a has index 1, or is a saddle point.

* If $\epsilon_1 = \epsilon_2 = -1$, we say that a has index 2, or is a local maximum.

With this Lemma at hand, we introduce the notion of adapted metrics [45, §2].

**Definition 3.2** (Adapted Metric). Let $f: \Sigma \to \mathbb{R}$ be a Morse function. We fix a Morse chart $x_a$ near every point in $a \in \text{Crit}(f)$. We say that a metric $g$ on $\Sigma$ is adapted to $f$ if it reads

$$g = dx_{a,1}^2 + dx_{a,2}^2.$$

in every Morse chart $x_a$.

Such a metric is flat near the critical locus of $f$. By a partition of unity argument, it is not hard to verify that such metrics exist. We now fix $f: \Sigma \to \mathbb{R}$ a perfect Morse function once and for all. Given a metric $g$, one can define the corresponding gradient vector field $\nabla_g f$ by duality

$$\forall x \in \Sigma, \quad d_x f = g_x(\nabla_g f(x), \cdot).$$
<figure><img src="image_10.png" /><figcaption>Figure 8: Stable and Unstable curves.</figcaption></figure>

This induces a complete and smooth flow on $\Sigma$ that we denote by $\varphi_f^t: \Sigma \to \Sigma$. One can verify that

$$ \forall t_1, t_2 \in \mathbb{R}, \forall x \in \Sigma, \quad f \circ \varphi_f^{t_2}(x) - f \circ \varphi_f^{t_1}(x) = \int_{t_1}^{t_2} \|d_{\varphi_f^t(x)}f\|_{g^*(x)}^2 dt, \quad (3.1) $$

where $g^*$ is the metric induced by $g$ on $T^*\Sigma$. In particular, $f$ is non-decreasing along the flow lines of $\varphi_f^t$. A key property of gradient flows is that, for any $x \in M$, there exists $x_-$ and $x_+$ in $\text{Crit}(f)$ such that

$$ \lim_{t \to \pm\infty} \varphi_f^t(x) = x_{\pm}. \quad (3.2) $$

We say that the pair $(f,g)$ has the Morse-Smale property if there is no gradient line connecting two distinct saddle points of $f$. It is proved in [45, Th. 14.4][50] that there exists Morse-Smale pair with $g$ being an adapted metric and $f$ a perfect Morse function. This condition is essential for defining the celebrated Morse-Witten complex. A typical example of a perfect Morse function is shown in 7.

From this point on, we will always assume that the pair $(f,g)$ is Morse-Smale, the function $f$ is perfect and the metric $g$ is adapted. Given a critical point $a$ of $f$, we define the unstable and stable manifolds of $a$ as

$$ W^u(a) := \left\{ x \in \Sigma : \lim_{t \to -\infty} \varphi_f^t(x) = a \right\} \quad \text{and} \quad W^s(a) := \left\{ x \in \Sigma : \lim_{t \to +\infty} \varphi_f^t(x) = a \right\}. $$

These are smooth embedded curves of $\Sigma$ which are respectively diffeomorphic to $\mathbb{R}^{2-\text{ind}(a)}$ and to $\mathbb{R}^{\text{ind}(a)}$. In particular when $a$ is a saddle point of $f$, these are gradient flow lines connecting the saddle point to either the minimum or maximum of $f$; check 8.

As we are working with compact surfaces, one can verify that these sub-manifolds induce de Rham currents, i.e. the unstable currents $U_a$ defined as :

$$ \forall \psi \in \Omega^{2-\text{ind}(a)}(\Sigma), \quad \langle U_a, \psi \rangle := \int_{W^u(a)} \psi, $$

and the stable currents $S_a$ defined as

$$ \forall \psi \in \Omega^{\text{ind}(a)}(\Sigma), \quad \langle S_a, \psi \rangle := \int_{W^s(a)} \psi. $$

Thanks to the local expression of the vector field, one has
<figure><img src="image_11.png" /><figcaption>Figure 9: Blown up surface.</figcaption></figure>

1. near the maximum $U_a = \delta_0(x_1, x_2)dx_1 \wedge dx_2$;

2. near the saddle points $U_a = \delta_0(x_2)dx_2$;

3. near the minimum $U_a = 1$. In fact, as $f$ has a single minimum, one has globally $U_a = 1$ in the case where $\mathrm{ind}(a) = 0$.

In the sequel, we will denote by $N^*W^u(a) := \{(x; \xi); x \in W^u(a), \xi \in (T_xW^u(a))^{\perp}\}$ (resp $N^*W^s(a)$) the conormal bundle of the unstable (resp stable) manifold $W^u(a)$ (resp $W^s(a)$).

# 4 A global resolution of a surface by a cylinder

Let us state informally the goal of this section. Resolving the surface $\Sigma$ allows us to identify it with some cylinder. This lets us transfer our computations in 2.2 from the cylinder to general surfaces. Furthermore, our resolution aims to provide a transparent discussion of the unstable currents that appear in the expression of the Yang-Mills measure, as in 2.3 and 2.3.

Consider a triple $(\Sigma, f, g)$ of a surface $\Sigma$ equipped with a Morse function $f$ and an adapted metric $g$ as above. Blow up the surface $\Sigma$ at both $\min(f)$ and $\max(f)$. This produces a surface $S$ with two boundary components, as shown in 9. We will call the one obtained by blowing up the minimum ingoing : $\partial S_{\mathrm{in}}$; and the other one, outgoing : $\partial S_{\mathrm{out}}$. We still denote by $W^{u/s}(a)$ the lift of the unstable/stable curves to the blow-up surface $S$.

## 4.1 A global angular form and period coordinates

Concretely, blowing up the surface at $\min(f)$ consists in working in polar coordinates. For $(x_1, x_2)$ a Morse chart at $\min(f)$,

$$ \nabla f = x_1 \partial_{x_1} + x_2 \partial_{x_2}. $$

Set

$$ r = \sqrt{x_1^2 + x_2^2}, \quad x_1 = r \cos(\theta), \quad \text{and} \quad x_2 = r \sin(\theta); $$
and the map

$$ (r, \theta) \in [0, \varepsilon) \times \mathbb{S}_{\theta}^{1} \mapsto x_{1} = r \cos(\theta), x_{2} = r \sin(\theta) $$

realizes the blow-up. The preimage of $min(f) = (0,0)$ is the circle $\{0\} \times \mathbb{S}^1$ which is also given in polar coordinates by the equation $r = 0$. The vector field $\nabla f$ lifts automatically to some $b$-vector field still denoted by $V$ (the subscript $b$ stands for boundary) on the resolved surface $S$, the corresponding flow is still denoted by $(\varphi_f^t)_{t \in \mathbb{R}}$. The $b$-vector fields are in particular always tangent to the boundary. We refer to [43], [70], [47, 2.1 p. 10] for more discussions on $b$-vector fields. In polar coordinates near $r = 0$, the vector field $V$ reads $V = r\partial_r$. Informally, the goal of what comes next is to extend these angular variables $\theta$, which is only defined near $min(f)$, to the whole surface $S$ in such a way that the level sets of $\theta$ are flowlines of $V$.

Let us try to give some integral formula. We work in polar coordinates $(r, \theta)$ near the blown-up minimum and where the vector field $V$ writes $V = r\partial_r$. Start from any function $\chi$ which equals 1 near the minimum and vanishes outside $r \le 3$ say. Then observe that

$$ \lim_{T \to +\infty} \chi \circ \varphi_f^{-T} = 1 $$

where the convergence holds almost everywhere, since every point

$$ x \in S \setminus \bigcup_{a \in \text{Crit}(f)_1} W^u(a) $$

is attracted by $min(f)$ under the backward flow. Therefore, using the fundamental Theorem of calculus and the definition of the following definition of the Lie derivative

$$ \frac{d}{dt} F \circ \varphi_f^t = (\mathcal{L}_V F) \circ \varphi_f^t, $$

we may write

$$ \chi \circ \varphi_f^{-T} = \chi + \chi \circ \varphi_f^{-T} - \chi = \chi + \int_0^T (-\mathcal{L}_V \chi) \circ \varphi_f^{-s} ds. $$

Now we let $T \to +\infty$ and set $\psi = -\mathcal{L}_V \chi$ which is smooth on $\Sigma$ since it vanishes near $min(f)$. This yields a continuous partition of unity adapted to the gradient flow which writes

$$ 1 = \chi + \int_{0}^{\infty} \psi \circ \varphi_{f}^{-s} ds. $$

The above suggests to define, at least at the formal level, our global angular form $\bar{\alpha}$ on $S$ by the integral formula

$$ \bar{\alpha} = \chi(r)d\theta + \int_{0}^{\infty} (\psi d\theta) \circ \varphi_{f}^{-t} dt $$

where the element $(\psi d\theta)$ is well-defined. Indeed, $d\theta$ is well-defined on the support of $\psi$ which happens to be contained in the disc $\{r \le 3\}$.

**Lemma 4.1.** The above 1-form $\bar{\alpha}$ is well-defined, smooth on $S \setminus \bigcup_{a \in \text{Crit}(f)_1} W^u(a)$, and satisfies the following.

1. For any $x$ in the domain of definition of the chart $(r, \theta)$, we have $\bar{\alpha} = d\theta$. This means that $\bar{\alpha}$ is an angular form,

2. The form $\bar{\alpha}$ is invariant by the flow $\mathcal{L}_V \bar{\alpha} = 0$, and is closed in the sense of currents $d\bar{\alpha} = 0$, with periods in $2\pi\mathbb{Z}$.
Proof. Step 1. The first step is to prove that our formula gives a 1-form that coincides with the angular form $d\theta$ near $\min(f)$. This follows from the partition of unity formula satisfied by the pair $(\chi, \psi)$ and the fact that $d\theta$ is locally invariant by the flow.

Step 2. The second step consists in showing that

$$ \int_0^\infty (\psi(r)d\theta) \circ \varphi_f^{-t}dt $$

exists, meaning that it converges in appropriate spaces. The 1-form $(\psi(r)d\theta)$ is smooth. By a result of the first author and Rivière [34], there exists an anisotropic Sobolev space $H^m(\Sigma)$ of currents adapted to the dynamics, with $C^\infty(\Sigma) \subset H^m(\Sigma) \subset D'(\Sigma)$, and where $m \in S^0(T^*M)$ is a symbol of degree 0 acting as a variable order for the Sobolev space, such that the pull-back $(\psi(r)d\theta) \circ \varphi_f^{-t}$ converges exponentially fast to the equilibrium in $H^m(\Sigma)$. Quantitatively,

$$ (\psi(r)d\theta) \circ \varphi_f^{-t} = \sum_{a \in \text{Crit}(f)} \left( \int_{W^s(a)} \psi(r)d\theta \right) U_a + \mathcal{O}(H^m(e^{-t})) = \mathcal{O}(H^m(e^{-t})) $$

where we used the fact that the 1-form $\psi(r)d\theta$ belongs to $\ker(\iota_V)$. Therefore, its integral over all stable curves $W^s(a)$ vanishes : $\int_{W^s(a)} \psi(r)d\theta = 0$. The remainder converges to 0 exponentially fast in the appropriate anisotropic Sobolev space $H^m$, namely

$$ \| (\psi(r)d\theta) \circ \varphi_f^{-t} \|_{H^m(\Sigma)} \le C e^{-Kt}. $$

The partial conclusion is that $\int_0^\infty (\psi(r)d\theta) \circ \varphi_f^{-t}dt$ converges in the anisotropic Sobolev space $H^m$ and therefore $\bar{\alpha}$ is well-defined in $H^m$. By the results of [35], the wave front set of $\bar{\alpha}$ is contained in

$$ \overline{\cup_{a \in \text{Crit}(f)_1} N^*W^u(a)}, $$

and therefore $\bar{\alpha}$ is smooth outside

$$ \overline{\cup_{a \in \text{Crit}(f)_1} W^u(a)}. $$

Step 3. We calculate $\mathcal{L}_V \bar{\alpha}$. Observe that, by construction, we have the identity $\mathcal{L}_V \chi + \psi = 0$. We must first choose some order function $m \in S^0(T^*M)$ in such a way that in the space $H^{m-1}$, we have

$$ \| (\psi(r)d\theta) \circ \varphi_f^{-t} \|_{H^{m-1}} = \mathcal{O}(e^{-Ct}) $$

for some $C > 0$. We have

$$ \begin{aligned} \mathcal{L}_V \bar{\alpha} &= \mathcal{L}_V \left( \chi d\theta + \int_0^T (\psi d\theta) \circ \varphi_f^{-t} dt \right) = -\psi d\theta - \int_0^T \frac{d}{dt} (\psi d\theta) \circ \varphi_f^{-t} dt \\ &= -\psi d\theta - ((\psi d\theta) \circ \varphi_f^{-t} - \psi d\theta) = - (\psi d\theta) \circ \varphi_f^{-T}. \end{aligned} $$

We use that $\| (\psi d\theta) \circ \varphi_f^{-T} \|_{H^{m-1}} \lesssim e^{-CT}$, which goes to 0 when $T \to +\infty$. We deduce from the above that

$$ \mathcal{L}_V \bar{\alpha} = \lim_{T \to +\infty} - (\psi d\theta) \circ \varphi_f^{-T} = 0 \in H^{m-1}. $$

Step 4. We calculate $d\bar{\alpha}$. Near $\{r=0\}$, since $\bar{\alpha} = d\theta$ we have immediately that $d\bar{\alpha} = d^2\theta = 0$ so $\bar{\alpha}$ is locally closed near $\{r=0\}$. Now we use the fact that $\bar{\alpha} \in \ker(\mathcal{L}_V)$ to propagate the closedness property to

$$ \Sigma \setminus \overline{\cup_{a \in \text{Crit}(f)_1} W^u(a)}. $$

Since $\bar{\alpha} \in \ker(\mathcal{L}_V)$, then $\Rightarrow \varphi_f^{-T^*\bar{\alpha}} = \bar{\alpha}$ for all $T > 0$. Therefore for all $T > 0$,

$$ d\bar{\alpha} = d\varphi_f^{-T^*\bar{\alpha}} = \varphi_f^{-T^*}d\bar{\alpha}. $$
<figure><img src="image_12.png" /><figcaption>Figure 10: Thee curve $\gamma_0$.</figcaption></figure>

Outside $\overline{\cup_{a \in \text{Crit}(f)} W^u(a)}$, choosing $T$ large enough allows to conclude that $d\bar{\alpha} = 0$. In fact $d\bar{\alpha}$ is closed when tested against smooth forms which vanish at $r = 0$. We need to decompose

$$ 
\begin{aligned}
d\bar{\alpha} &= d(\chi(r)d\theta) + d\int_0^\infty (\psi(r)d\theta) \circ \varphi_f^{-T}dt \\
&= d(\chi(r)d\theta) + \int_0^\infty (d(\psi(r)d\theta)) \circ \varphi_f^{-T}dt \\
&= (d\chi \wedge d\theta) + \int_0^\infty ((d\psi \wedge d\theta)) \circ \varphi_f^{-T}dt = 0
\end{aligned}
$$

where we could invert the integral and $d$ since we have absolute convergence of the integral in the appropriate anisotropic Sobolev space. Then using the identity

$$ 0 = d\chi + \int_0^\infty (d\psi) \circ \varphi_f^{-s}ds $$

and the property of the supports of $(d\psi) \circ \varphi_f^{-s}$.

The key point is that $\bar{\alpha}$ is closed but non exact but with periods in $2\pi\mathbb{Z}$ by construction. For any closed cycle $\gamma$ supported on $\Sigma \setminus \min(f)$ and such that the support is contained in the polar coordinate chart, then the claim is obvious, we have

$$ \int_{\gamma} \bar{\alpha} = \int_{\gamma} d\theta = 2i\pi k $$

where $k = \deg_{\gamma}(\min(f))$ is the degree of the closed curve $\gamma$ around $\min(f)$. Consider any closed oriented cycle $\gamma$ which is transverse to the unstable curves and whose support does not meet $\max(f), \min(f)$. We denote by $[\gamma]$ the corresponding current. Choose some oriented flowline $\gamma_0$ going from $\min(f)$ to $\max(f)$ and consider its lift, still denoted by $\gamma_0$, to the blow-up surface $S$; see Figure 10.

By Lemma B.2 that we prove in appendix, for any smooth closed curve $\gamma$ whose support does not meet $\partial S$ and is everywhere transverse to $V$, we get that

$$ \varphi_f^{T*}[\gamma] \to \sum_{a \in \text{Crit}(f)} \left( \int_{\Sigma} U_a \wedge [\gamma] \right) S_a + \left( \int_{\Sigma} [\gamma_0] \wedge [\gamma] \right) \delta_{\{0\}}^{\mathbb{R}}(r)dr $$
<figure><img src="image_13.png" /><figcaption>Figure 11: Singularities along unstable curves.</figcaption></figure>

where the current $\delta_{\{0\}}^{\mathbb{R}}(r)dr$ is just the integration current $[\partial S_{in}]$ on the incoming boundary of $S$. Then by flow invariance

$$ \int_{\gamma} \bar{\alpha} = \int_{S} \varphi_{f}^{T*}[\gamma] \wedge \bar{\alpha} \rightarrow \left( \int_{\Sigma} [\gamma_{0}] \wedge [\gamma] \right) \int_{\partial S_{in}} d\theta \in 2\pi\mathbb{Z} $$

where we used the fact that $d\theta$ integrates as zero on the preimages $W^s(a)$ of the stable curves. □

So the 1-form $\bar{\alpha}$ is our global angular form.

**Definition 4.2.** Denote by $\theta: S \mapsto \mathbb{R}/2\pi\mathbb{Z}$ the circle valued function which is the unique de Rham primitive of $\bar{\alpha}$ which belongs to $\ker(\mathcal{L}_V)$ and coincides with the actual polar coordinates $\theta$ in the chart near $\min(f)$.

The reader should be aware that the circle valued function $\theta$ is only piecewise smooth with singularities along unstable curves. This is illustrated in 11.

**A further modification of the angular form.** In our investigation, we need a further modification of the angular form so that it has a very specific form near the saddle point. This specific form is very important in our application.

**General surfaces with boundary.** In general, given any bordered surface $\Sigma$ assumed to have both, in and out boundary components, with smooth area form $\sigma$, we can choose a vector field $V$ which points inward along $\partial\Sigma_{in}$ and points outward along $\partial\Sigma_{out}$. Then, up to changing the vector field $V$ by a conformal factor ($V_b = \rho V$ where $\rho \ge 0$ is some function which vanishes at order 1 near $\partial\Sigma$ and is $> 0$ in the interior), one can easily establish that the above formula

$$ \bar{\alpha} := \chi(r)d\theta + \int_{0}^{\infty} (\psi(r)d\theta) \circ \varphi_{f}^{-t}dt $$
<figure><img src="image_14.png" /><figcaption>Figure 12: Dynamics of in and out faces.</figcaption></figure>

also defines a closed angular form in the interior of $\Sigma$ with periods in $2\pi\mathbb{Z}$. Then, the de Rham primitives $\theta$ of the above closed form $\bar{\alpha}$ yields global angular coordinates on $\Sigma$.

## 4.2 Pseudo coordinates

Once we have constructed the global angular variable $\theta$, the pair $(f, \theta)$ yields global functions on $S$ which act as coordinate functions outside $\cup_{a \in \text{Crit}(f)_1} W^u(a)$. In fact $f$ is smooth everywhere and is nondegenerate except at critical values of $f$, and $\theta$ is piecewise smooth with discontinuities along $\overline{\cup_{a \in \text{Crit}(f)_1} W^u(a)}$.

**Proposition 4.3 (Regularity of the pseudo-coordinates).** The pseudo-coordinate $\theta: S \to \mathbb{S}^1$ is piecewise $C^\infty$ on $S$ with jumps along the unstable manifolds. Both $(f, \theta)$ are regular coordinate functions on

$$S \setminus \overline{\cup_{a \in \text{Crit}(f)_1} W^u(a)}.$$

The 1-form $d\theta$ vanishes at order 1 at $\text{Crit}(f)_1$ and df vanishes at order 1 at $\text{Crit}(f)_1 \cup \partial S$.

*Proof.* By propagation of singularities, it is enough to study the regularity of $\theta$ near a saddle point, then use the dynamics to propagate the singularities. Use coordinates $(x, y) \in [-1, 1] \times [-1, 1]$ near a saddle point $a := (0, 0)$ where the flow reads $(e^{-t}x, e^t y)$ and the unstable (resp. stable) manifold $W^u(a)$ (resp. $W^s(a)$) reads locally $\{x = 0\}$ (resp $\{y = 0\}$). In the hyperbolic box $[-1, 1]_x \times [-1, 1]_y$, the in face for the dynamics reads $\{\pm 1\}_x \times [-1, 1]_y$ any flowline that enters the box must intersect the in face once; the out face reads $[-1, 1]_x \times \{\pm 1\}_y$, every flow-line entering the box must escape via the out face by intersecting it exactly once; see 12. We know $\theta$ has smooth Cauchy data from the in face $\theta(\pm 1, y) = F_\pm(y)$ where $F_\pm$ is smooth since we are coming from the minimum. Then by flow invariance

$$\theta(x, y) = 1_{\mathbb{R}_{\ge 0}}(x)F_+(xy) + 1_{\mathbb{R}_{\ge 0}}(-x)F_-(xy)$$

is a piecewise smooth function of the pair $(x, y)$ with a discontinuity exactly along the $y$-axis which is the piece of unstable manifold $\{x = 0\}$.

Calculate $d\theta$ in a quadrant, say $x > 0$ reads

$$d\theta = F'_+(xy)(ydx + xdy)$$

and let $x \to 0^+$ yields $d\theta = F'_+(0)ydx$ which vanishes at order 1 at $(x, y) = (0, 0)$. $\square$

We next introduce a crucial concept which will be useful for us in the sequel.
## 4.3 Resolution of the surface $S$ by a cylinder with defect lines

The pair $(f, \theta)$ defines a global map

$$ \Psi : (f, \theta) : S \to [\min(f), \max(f)] \times \mathbb{S}_{\theta}^{1}. \qquad (4.1) $$

Let us describe a bit the geometry of both unstable and stable curves $W^{u/s}(a)$, $a \in \text{Crit}(f)_1$ when lifted to $S$. The $2g$ unstable (resp stable) curves intersect transversally the blow up circle at $\max(f)$ (resp $\min(f)$) at exactly $4g$ points. In the angular variables, each $W^u(a) \cup W^s(a)$ corresponds to some distinguished level $\theta(a)$ so that the saddle point $a$ is given in pseudo-coordinates by $(f(a), \theta(a))$. Remove the $2g$ unstable curves and the $2g$ stable curves, this yields a union of $4g$ hexagons with 6 edges which are described as follows:

1. there is one edge coming from the blow-up $\max(f)$,

2. there is one edge coming from the blow-up $\min(f)$,

3. there are two unstable edges,

4. there are two stable edges,

5. there are two vertices which are identified with saddle points.

**Definition 4.4 (Resolution of $S$ abstract version).** These $4g$ hexagons are interiors of $4g$ surfaces with smooth corners $H_1, \dots, H_{4g}$ which are smoothly embedded inside the surface $S$, obtained by gluing smoothly these $4g$ hexagons along well chosen edges. The pair $(f, \theta)$ induces smooth maps

$$ (f, \theta) : H_i \subset S \to [\min(f), \max(f)] \times I_i, \qquad (4.2) $$

where the collection $I_1, \dots, I_{4g}$ are $4g$-intervals on the unit circle with disjoint interiors and only endpoints can coincide. So the cylinder $[\min(f), \max(f)] \times \mathbb{S}_{\theta}^{1}$ writes as a union of $4g$ rectangles $([\min(f), \max(f)] \times I_i)_{i=1}^{4g}$ which are glued along lines $([\min(f), \max(f)] \times \partial I_i)_{i=1}^{4g}$. These lines are the defect lines and are broken flowlines made of a stable curve glued with an unstable curve.

The bands of zero area are just diffeomorphic to tubular neighborhoods of the above lines and are obtained by thickening these lines as shown in 13. The restriction of the pair $(f, \theta)$ to every hexagon $H_i$, $i \in \{1, \dots, 4g\}$ is smooth.

The consequence of Proposition 4.3 is that for every hexagon $H_i$, the map $\Psi : H_i \setminus \text{Crit}(f)_1 \to \Psi(H_i \setminus \text{Crit}(f)_1)$ is a diffeomorphism up to the edges of the hexagon, the only singularities occurs at the corners corresponding to saddle points.

In practice and for the next part of the article, we will work on each of these rectangles with the pushforward of the area form $\sigma$ via the map $(f, \theta)$. All estimates and random currents are constructed on the rectangles $([\min(f), \max(f)] \times I_i)_{i=1}^{4g}$ first, then using the pair $(f, \theta)$ we will pull-back these random objects on the initial blow-up surface $S$.

## 4.4 Regularity of area functions.

The pushforward area $\Psi_*\sigma$ form reads $\sigma(f, \theta)d f \wedge d\theta$, where the density $\sigma$ is expected to blow up near critical points. This is going to affect area estimates near critical points. In a hyperbolic box in the local Morse chart, up to shifting coordinates by constants and up to some smooth change of variables, we may assume that $f = x^2 - y^2$, and $\theta = xy$. To justify this choice, we just mimick the proof of Proposition 4.3.

**Proposition 4.5.** There exists a global angular form $d\theta$ on $S$ such that for every saddle point $a \in \text{Crit}(f)_1$, there is a neighborhood $U_a$ of a contained in the Morse chart, in which $d\theta|_{U_a} = d(xy)$ where $(x, y)$ is the Morse chart corresponding to $a$.
<figure><img src="image_15.png" /><figcaption>Figure 13: Zero area bands.</figcaption></figure>

*Proof.* Assume some angular form $\theta$ is already constructed.

Let us number the saddle points as $a_1, \dots, a_{2g}$ where $g$ is the genus of the initial surface $\Sigma$. Near each saddle point $a_i$, $i \in \{1, \dots, 2g\}$, there is a local Morse coordinate system $(x_i, y_i) \in [-1, 1]^2$ where $a_i = (0, 0)$ and the dynamics is totally linear of the form $(e^{-t}x_i, e^t y_i)$. Recall that $\{\pm 1\}_{x_i} \times [-1, 1]_{y_i}$ is the in face of the hyperbolic box around $a_i$.

For every $\varepsilon \in (0, 1)$ and every $i \in \{1, \dots, 2g\}$, we define the ribbon of thickness $\varepsilon > 0$ the following set :

$$R_i(\varepsilon) := \overline{\{\varphi_f^{-t}(\{\pm 1\}_{x_i} \times [-\varepsilon, \varepsilon]_{y_i}), t \ge 0\}}$$

which contains all the flowlines starting from the blown-up minimum $\partial S_{in}$ and hitting the small intervals $\{\pm 1\}_{x_i} \times [-\varepsilon, \varepsilon]_{y_i}$ contained in the in face of the hyperbolic box around $a_i$. Geometrically, this is the union of two very thin rectangles; see Figure 14.

We make the following claim, there exists $\varepsilon > 0$ small enough such that all the ribbons of thickness $\varepsilon R_i(\varepsilon)$, $i \in \{1, \dots, 2g\}$ are two by two disjoint; see Figure 15. To prove the claim, let us argue by contradiction. Up to extraction we can always assume there is a given pair $i < j \in \{1, \dots, 2g\}^2$ and a sequence $y_i(n) \to 0^+$ when $n \to +\infty$ such that the flowline $\varphi_f^{[-\infty, 0]}((1, y_i(n)))$ ending at the in face of $a_i$, gets at distance $\le \frac{1}{n}$ from the critical point $a_j$. Then letting $n \to +\infty$, the sequence of curves $\varphi_f^{[-\infty, 0]}((1, y_i(n)))_n$, $\ge 0$ should converge in $C^0$ topology to a broken flowline using the compactness of the space of broken gradient flowlines [3, section 3.2 p. 57]. We deduce that there exists a broken flowline connecting the saddle points $a_i$ and $a_j$ which contradicts Smale's transversality.

Therefore we have $4g$ two by two disjoint rectangles that connect in faces of every saddle points and the circle $\partial S_{in}$. This decomposes the circle $\partial S_{in}$ as a union of $4g$ two by two disjoint intervals $I_1, \dots, I_{4g}$ which are in bijection with $4g$ pieces of in faces that we will denote $J_1, \dots, J_{4g}$ in such a way that $I_i = \varphi_f^{[-\infty, 0]}(J_i) \cap \partial S_{in}$. We denote by $P_i: J_i \to I_i$ the natural smooth diffeomorphism mapping $J_i$ to $I_i$
<figure><img src="image_16.png" /><figcaption>Figure 14: Ribbon object 1.</figcaption></figure>

and which is induced by the backward flow: $\mathcal{P}(u) = v$ if and only if $v = \lim_{t \to +\infty} \varphi_f^{-t}(u)$; see Figure 16. Now we modify the angular variable on $\partial S_{in}$ in such a way that on each interval $I_j = \partial S_{in} \cap R_i(\varepsilon)$, the 1-form

$$d\theta = \mathcal{P}_*(d(x_i y_i)) \qquad (4.3)$$

which intuitively represents the 1-form $d(x_i y_i)$ defined on the in face of $a_i$ and then transported back by the flow. Once we modified the angular function in a new angular function $\tilde{\theta}$ which satisfies the above constraint equation 4.3 on each of these intervals, we repeat the construction of the angular variable by applying to $\tilde{\theta}$ the procedure described in the proof of Lemma 4.1. $\square$

Thus, $d f(x, y) \wedge d\theta(x, y) = 2(x^2 + y^2)dx \wedge dy$. Therefore, we have the explicit inverse relation:

$$\Psi_*\sigma = \tilde{\sigma}(f, \theta)dx(f, \theta) \wedge dy(f, \theta) = \tilde{\sigma}(f, \theta)(f^2 + 4\theta^2)^{-\frac{1}{2}}df \wedge d\theta \qquad (4.4)$$

where $\tilde{\sigma}$ is bounded and the above yields the asymptotics of $\Psi_*\sigma$ near singular points in $\Psi(Crit(f)_1)$. Therefore for every domain $\square \subset [\min(f), \max(f)] \times \mathbb{S}^1$ of finite area with respect to $drd\theta$, $\int_{\square} \Psi_*\sigma$ is integrable.

# 5 The Yang-Mills connection in the continuum

## 5.1 A formula for the Yang-Mills connection

We will use our pseudo-coordinates to write a new global formula for the random connection $A$. For every $(r, \theta) \in [\min(f), \max(f)] \times [0, 2\pi)$, we define $\square(r, \theta)$ as the unique rectangle obtained by connecting $(0, \theta)$, $(r, \theta)$, $(0, 0)$, $(r, 0)$ by straight lines. Inspired by our discussion of Driver's result, we can write two explicit formulas for the Yang-Mills connection. Recall we use the letter $\xi$ for the white noise on the cylinder adapted to the pushforward area form $\Psi_*\sigma$, it is a random current of degree 2. Then we may write formally the random connection $A$ as:

$$A = \Psi^* \left( \partial_{\theta} \langle \xi, 1_{\square(r, \theta)} \rangle d\theta \right) + \sum_{a \in \text{Crit}(f)_1} \log(g_a) U_a, \qquad (5.1)$$
<figure><img src="image_17.png" /><figcaption>Figure 15: Ribbon object 3.</figcaption></figure>

where $U_a$ are the currents of integration on unstable curve $W^u(a)$ and where the second sum runs over critical points of index 1 (saddle points). Let us comment on the formula of equation (5.1). The function $1_{\square(r,\theta)}$ is in $L^2(\Psi_*\sigma)$ therefore it is immediate that the Brownian sheet $W := \langle \xi, 1_{\square(r,\theta)} \rangle$, reparametrized by the pushforward area $\Psi_*\sigma$, is well-defined globally as process on the cylinder, and the pull-back $\Psi^*W$ is also globally well-defined on $S$. We can also define $(\partial_\theta W) d\theta$ globally as a current of degree 1 on the cylinder, however it is not obvious whether we can pull-back this random current of degree 1 as $\Psi^* ((\partial_\theta W) d\theta)$ globally on $S$, it is only immediate outside saddle points. But the global extension is still subtle and still require some non trivial work. This is the topic of the next paragraphs.

We immediately give a second *formal* formula which involves some random series. This second formula will also suggest the correct regularity estimates for $A$.

$$A = \Psi^* \left( \sum_{n \ge 0} \left( \int_0^r \sqrt{\sigma(s, \theta)} \xi_n^g(s) ds \right) e_n(\theta) d\theta \right) + \sum_{a \in \text{Crit}(f)_1} \log(g_a) U_a \quad (5.2)$$

where $\xi_n^g$ are i.i.d $g$ valued white noise, $(e_n)_n$ is a ONB of functions on the circle $S^1$. The goal of the whole section is to give a precise mathematical meaning to the above two formulas. The proof of the correct regularity will be done when we give a rigorous meaning to the above formula and also follows directly from the control of norms in the scaling limit from discrete models. The random series $A$ will be defined on $S$ by the singular pull-back and the detailed study of the regularity of derivatives in $\theta$ of the area functional. But it already suggests that outside $\text{Crit}(f)$, the random connection $A$ is expected to have regularity $\frac{1}{2}$ – along the flow.

We start by summarizing what we know and what are the main difficulties.
<figure><img src="image_18.png" /><figcaption>Figure 16: Ribbon object 4.</figcaption></figure>

## 5.2 The motivation to deal with singularities

Let us explain what is going on and why we need the results of the present subsection. We will define some random distribution $\tilde{A}$ on the cylinder $[\min(f), \max(f)] \times \mathbb{S}^1$ in a way very similar to Driver's construction using the push-forward area $\Psi_*\sigma \in \Omega^2([\min(f), \max(f)] \times \mathbb{S}^1)$ induced from the surface area $\sigma \in \Omega^2(\mathcal{S})$. Since $\Psi$ fails to be a diffeomorphism along unstable curves, the pushforward area form $\Psi_*\sigma$ is singular at images of saddle points $\Psi(\mathrm{Crit}(f)_1)$ by $\Psi$.

**Definition 5.1** (The random connection $\tilde{A}$ upstairs). The random distribution $\tilde{A}$ defined upstairs on the cylinder is given by the formula

$$ \tilde{A} := \sum_{n \ge 0} \left( \int_0^r \sqrt{\sigma(s, \theta)} \xi_n^g(s) ds \right) e_n(\theta) d\theta \quad (5.3) $$

in the coordinates of the cylinder.

Then we integrate $\tilde{A}(r, \theta)d\theta$ along the $\theta$ direction which defines an element denoted by $W(r, \theta)$ that we control in certain anisotropic Hölder spaces outside $\Psi(\mathrm{Crit}(f)_1)$. However, the singularities of $\Psi_*\sigma$ imply that we can control $W(r, \theta)$ only in some weighted anisotropic spaces where the weight controls singularities when we approach the points in $\Psi(\mathrm{Crit}(f)_1)$. These weighted anisotropic spaces are defined by scalings directly on the cylinder. The second problem we need to handle is that $\Psi: S \to [\min(f), \max(f)] \times \mathbb{S}^1$ fails to be a diffeomorphism at critical points so it is not obvious that the pull-back of any current by $\Psi$ should be well-defined. What we will prove is that if we scale near a saddle point and this allows us to define the pull-back $\Psi^*(\tilde{A}d\theta)$ globally on $S$. We next define precisely the functional spaces.

## 5.3 Weighted anisotropic Sobolev and Hölder norms

The formula from equation 5.2 suggests that the random connection $A$ has regularity $\frac{1}{2}$- along the flow direction and $-\frac{1}{2}$- along the levels sets of the Morse function $f$. So we expect the regularity of $A$ to be anisotropic which motivates the introduction of anisotropic Banach spaces of distributions of low regularity to describe precisely our random current $A$. Our Banach spaces are somewhat reminiscent of the anisotropic Banach spaces appearing in the functional analysis of hyperbolic dynamical systems, specially the spaces from the works of Baladi-Tsujii [4, 5]. We refer the reader to the book [13] and the extensive survey [14] and the references therein.
### 5.3.1 Non weighted anisotropic semi-norms

We start by some semi-norm on the cylinder.

**Definition 5.2** (Anisotropic semi-norms on cylinder). Set $\alpha \in (0,1)$. Outside $\Psi(\text{Crit}(f)_1)$, on some product of intervals $I \times J \subset [\min(f), \max(f)]_r \times S_\theta^1$ which avoids $\Psi(\text{Crit}(f)_1)$, we define these local semi-norms as

$$ \|\tilde{A}\|_{W_{r,\theta}^{\alpha,\alpha-1;p}(I \times J)} := \|W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I \times J)} = \left( \int_{I^2} \frac{\|W(r_1, \cdot) - W(r_2, \cdot)\|_{W_{\theta,J}^{\alpha,p}}^p}{|r_1 - r_2|^{1+\alpha p}} dr_1 dr_2 \right)^{\frac{1}{p}} + \|W\|_{L^p(I \times J)} \quad (5.4) $$

for $\tilde{A}(r, u) := \partial_\theta W(r, \theta)$ and $W(0, \cdot) = 0$ $^3$ where the derivative is taken in the sense of distributions. The Hölder version reads

$$ \|\tilde{A}\|_{C_{r,\theta}^{\alpha,\alpha-1}(I \times J)} := \|W\|_{C_{r,\theta}^{\alpha,\alpha}(I \times J)} = \sup_{r_1 \neq r_2 \in I^2} \frac{\|W(r_1, \cdot) - W(r_2, \cdot)\|_{C_{\theta,J}^{\alpha}}^p}{|r_1 - r_2|^\alpha} + \sup_{(I \times J)} |W|. \quad (5.5) $$

We consider the completion of $C^\infty$ for the above seminorms where $I \times J$ runs over all product intervals avoiding $\Psi(\text{Crit}(f)_1)$. This defines local anisotropic Sobolev spaces (resp Hölder spaces).

These anisotropic semi-norms of fractional Sobolev type are defined in the spirit of the work of Gagliardo and Slobodeckij [51, Def 1.12 p. 13], [62, equation (2.2) p. 524] and involve integral formulas in position space as opposed to the Fourier based definitions. This will be very convenient for our probabilistic applications: estimating the regularities of our random connections and proving tightness of sequences of measures on piecewise affine connections.

We can also define a similar semi-norm but this time on $S$:

**Definition 5.3** (Anisotropic semi-norms on $S$). Set $\alpha \in (0,1)$. On every flowbox $\square$ avoiding the unstable curves with local coordinates $(r, \theta)$ where $V = \partial_r$, we define these local semi-norms as:

$$ \|A\|_{W_{r,\theta}^{\alpha,\alpha-1;p}(\square)} := \left( \int_{I^2} \frac{\|W(r_1, \cdot) - W(r_2, \cdot)\|_{W_{\theta,J}^{\alpha,p}}^p}{|r_1 - r_2|^{1+\alpha p}} dr_1 dr_2 \right)^{\frac{1}{p}} + \|W\|_{L^p(\square)} \quad (5.6) $$

for $\tilde{A}(r, u) := \partial_\theta W(r, \theta)$ where the derivative is taken in the sense of distributions.

After completion w.r.t the above semi-norms, this defines a local anisotropic Sobolev space outside the saddle points. A similar completion process w.r.t. the anisotropic Hölder semi-norms allows to define local anisotropic Hölder norms.

We state an easy, yet essential lemma. It plays an essential role in the sequel.

**Lemma 5.4.** For $\alpha \in (0,1)$ in the setting of definition 5.2, any distribution $T \in C^{\alpha, \alpha-1}(\square)$ can be viewed as a Hölder continuous function of $r$ valued in distributions in the $\theta$-variable of regularity $\alpha - 1$.

For $\alpha \in (0, \frac{1}{2})$, we have continuous injections

$$ W^{\alpha, \alpha-1, p} \hookrightarrow C^{\alpha-\frac{1}{p}, \alpha-1-\frac{1}{p}} \text{ for } \alpha > \frac{1}{p} $$

and

$$ C^{\alpha, \alpha-1} \hookrightarrow C^{\alpha-1} $$

whose proof is done in Proposition C.1 of the appendix. In the appendix, we also establish the necessary compact injections $C^{\alpha+\varepsilon, \alpha+\varepsilon-1} \hookrightarrow C^{\alpha, \alpha-1}$, $\forall \varepsilon > 0$ which are needed to show tightness of the sequence of measures on these anisotropic spaces.

<footnote>$^3$Since $W$ is defined globally and this acts like some half-Dirichlet condition on $W$.</footnote>
**Remark 1.** An important remark is that for every $\alpha > -1$ and every interval $[a, b] \subset \mathbb{R}$, it makes perfect sense to think about $C^\alpha([a, b])$ as the space of Hölder distribution which are supported on $[a, b]$. Moreover, we prove in Lemma A.2 in the appendix that any distribution $T$ in $C^\alpha(\mathbb{R})$ can be multiplied with the indicator function $1_{[a,b]}$ to define $T1_{[a,b]}$ which is the canonical extension of $T \in D'((a, b))$.

As a corollary of the above remark, for every $\alpha \in (0, 1)$, we can always integrate anisotropic distributions $T \in C^{\alpha, \alpha-1}([0, 1]_r \times [0, 1]_\theta)$ in the $\theta$ direction and define $W := \int_0^\theta T(r, u) du$.

Because of the regularity of $\Psi: \square \subset S \to I \times J \subset [\min(f), \max(f)] \times \mathbb{S}^1$, the pull-back $\Psi^*$: $W^{\alpha, \alpha-1, p}(\square) \to W_I^{\alpha, \alpha-1, p}(I \times J)$ is linear continuous for the topology defined by the respective anisotropic semi-norms.

### 5.3.2 The weighted anisotropic spaces by scalings.

Recall that the map $\Psi$ has singularities at the saddle points which also induces singularities of the pushforward area $\Psi_*\sigma$ at $\Psi(Crit(f)_1)$. Therefore we can only define $\tilde{A}$ outside $\Psi(Crit(f)_1)$ and we will need some functional spaces that will control how $\tilde{A}$ behaves as we zoom in closer and closer to points in $\Psi(Crit(f)_1)$. This is the main motivation to introduce some new weighted norms near the critical points and their image by $\Psi$. So there are two types of spaces we will introduce: on the cylinder which resolves $S$ and on the surface $S$ itself. Our ideas are very close to those appearing in the work of Bony on second micro-localization [7, b) page 7], [8, Eq 2.12 Def 2.5 p. 18] and also Meyer's work revisiting certain aspects of second micro-localization [59]. As usual, we rely on scalings and dyadic decompositions to define our spaces.

We next use the previous semi-norms together with local scalings to define weighted spaces of distributions.

**Definition 5.5** (Scalings and weighted semi-norms). Set $\alpha \in (0, 1)$. On the cylinder $[\min(f), \max(f)]_r \times \mathbb{S}_\theta^1$ near any element $(r_0, \theta_0)$, we define local scaling centered at $(r_0, \theta_0)$ as:

$$S_{r_0, \theta_0}^{\lambda}(r, \theta) := (\lambda(r - r_0) + r_0, \lambda(\theta - \theta_0) + \theta_0), \forall \lambda \in \mathbb{R}_{>0}.$$

We define weighted semi-norms a singular point $(r_0, \theta_0) \in \Psi(Crit(f)_1) \subset [\min(f), \max(f)]_r \times \mathbb{S}_\theta^1$ in the cylinder. We require that for a certain scaling exponent $s < 2\alpha - 1$,

$$\| \tilde{A} \|_{W_{r, \theta}^{\alpha, \alpha-1; p; s}(\text{Cyl})} := \left( \sum_{(r_0, \theta_0) \in \Psi(Crit(f)_1)} \sum_{I \times J} \sum_{n=0}^{\infty} \left( 2^{n(s-\frac{2}{p})} \| S_{(r_0, \theta_0)}^{2-n*} \tilde{A} \|_{W_{r, \theta}^{\alpha, \alpha-1; p}(I \times J)} \right)^p + \sum_{I_2 \times J_2} \| \tilde{A} \|_{W_{r, \theta}^{\alpha, \alpha-1; p}(I_2 \times J_2)}^p \right)^{\frac{1}{p}} \quad (5.7)$$

where we take the first sum over some finite cover of the form $I_r \times J_\theta$ of a certain corona of the form $\{m \in \Sigma; \text{dist}(m, \Psi(a)) \in [1, 2]\}$ centered near a singular point $\Psi(a) = (r_0, \theta_0)$ and the second sum $\sum_{I_2 \times J_2} \| \tilde{A} \|_{W_{I_2 \times J_2}^{\alpha, \alpha-1, p}}^p$ runs over a finite cover of $\{m, \text{dist}(m, \Psi(Crit(f)_1)) \ge 1\}$.

The index $\alpha$ indicates the regularity in the $r$ variable, this means regularity along the flow, the index $\alpha - 1$ indicates the regularity in the $\theta$ variable, this means transversal to the flow, the index $p \ge 2$ tells us we used $L^p$ and $\ell^p$ norms to define our Sobolev spaces and finally the exponent $s \in \mathbb{R}$ is a scaling exponent which indicates how our connection $\tilde{A}$ blows up as measured in Sobolev norms when we approach the critical point $a$; check 17.

We also define such weighted norms for anisotropic Hölder norms, for $s < 2\alpha - 1$

$$\| \tilde{A} \|_{C^{\alpha, \alpha-1; s}(\text{Cyl})} := \sup_{(r_0, \theta_0) \in \Psi(Crit(f)_1)} \sup_{I \times J} \sup_{n \ge 0} 2^{ns} \| S^{2-n*} A \|_{C^{\alpha, \alpha-1}(I \times J)}. \quad (5.8)$$
<figure><img src="image_19.png" /><figcaption>Figure 17: A corona around a critical point.</figcaption></figure>

## 5.4 Regularity estimates in anisotropic spaces

We estimate the regularity of $\tilde{A} \in D'([\min(f), \max(f)] \times S_\theta^1)$ defined by equation (5.3) in global coordinates, start from the integrated object $W(r, \theta) = \int_0^\theta \tilde{A}(r, u)du$. Recall we would like to control the anisotropic norm of $W$ on some product of intervals $I_1 \times I_2$ as follows:

$$ \|W\|_{W_{r,\theta}^{\alpha,\alpha;2p}(I_1 \times J_1)}^{2p} := \int_{I_1^2 \times I_2^2} \frac{|W(r_1, \theta_1) - W(r_2, \theta_1) - W(r_1, \theta_2) + W(r_2, \theta_2)|_{\mathfrak{g}}^{2p}}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 $$

Then we get an estimate of the form:

$$ \begin{aligned} \mathbb{E} \left[ \|W\|_{W_{r,\theta}^{\alpha,\alpha;p}(I_1 \times I_2)}^p \right] &= \int_{I_1^2 \times I_2^2} \frac{\mathbb{E} \left[ |W(r_1, \theta_1) - W(r_2, \theta_1) - W(r_1, \theta_2) + W(r_2, \theta_2)|_{\mathfrak{g}}^{2p} \right]}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\ &\le C_p \int_{I_1^2 \times I_2^2} \frac{\mathbb{E} \left( |W(r_1, \theta_1) - W(r_2, \theta_1) - W(r_1, \theta_2) + W(r_2, \theta_2)|_{\mathfrak{g}}^{2p} \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\ &\le C_p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{\square(r_1,r_2,\theta_1,\theta_2)} (\Psi_* \sigma)(r, \theta) dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \end{aligned} $$

where we used Fubini, then hypercontractivity of $W$ since it is Gaussian where $\square(r_1, r_2, \theta_1, \theta_2)$ is the rectangle bordered by $(r_1, \theta_1)$, $(r_2, \theta_1)$, $(r_1, \theta_2)$, $(r_2, \theta_2)$ then the above is bounded for $p$ large enough as soon as $\alpha < \frac{1}{2}$. If we specialize our estimate to rectangular regions $I_1 \times I_2$ in $(r, \theta)$ that avoid $\Psi(\text{Crit}(f)_1)$, then we have upper bounds of the form $\int_{\square(r_1,r_2,\theta_1,\theta_2)} (\Psi_* \sigma)(r, \theta) dr d\theta = \mathcal{O}(|r_1 - r_2| |\theta_1 - \theta_2|)$.

Now we deal with neighborhoods of singular points in $\Psi(\text{Crit}(f)_1)$. We assume without loss of generality that the singular point reads $(0, 0)$ in coordinates of the cylinder. We just need to scale the
previous estimate taking into account that the area form blows up at the singular point (0,0):

$$
\begin{align*}
\mathbb{E} \left[ \| S_{r,\theta}^{\lambda *} W \|_{W_{r,\theta}^{\alpha,\alpha;2p}}^{2p} \right] &\lesssim C_p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{\square(\lambda r_1, \lambda r_2, \lambda \theta_1, \lambda \theta_2)} (\Psi_* \sigma)(r, \theta) dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\
&\lesssim C_p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{\square(\lambda r_1, \lambda r_2, \lambda \theta_1, \lambda \theta_2)} (\Psi_* \sigma)(r, \theta) dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\
&= C_p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{\lambda r_1}^{\lambda r_2} \int_{\lambda \theta_1}^{\lambda \theta_2} (\Psi_* \sigma)(r, \theta) dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\
&= C_p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} (\Psi_* \sigma)(\lambda r, \lambda \theta) \lambda^2 dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \\
&\lesssim C_p \lambda^p \int_{I_1^2 \times I_2^2} \frac{\left( \int_{\square(r_1, r_2, \theta_1, \theta_2)} dr d\theta \right)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2.
\end{align*}
$$

where we use the bound $\Psi_*\sigma(r, \theta) = \mathcal{O}((r^2 + 4\theta^2)^{-\frac{1}{2}})$ which implies that

$$
\Psi_* \sigma(\lambda r, \lambda \theta) |_{(r, \theta) \in \square(r_1, r_2, \theta_1, \theta_2)} = \mathcal{O}(\lambda^{-1})
$$

hence $(\int_{r_1}^{r_2} \int_{\theta_1}^{\theta_2} (\Psi_* \sigma)(\lambda r, \lambda \theta) \lambda^2 dr d\theta)^p = \mathcal{O}(\lambda^p (|r_1 - r_2| |\theta_1 - \theta_2|)^p)$. From the above bound, for all $\alpha \in (0, \frac{1}{2})$, for all $\beta < \frac{1}{2}$, there exists $p$ large enough s.t. :

$$
\begin{align*}
\sum_{n \ge 1} 2^{n(\beta - \frac{2}{2p})2p} \mathbb{E} \left( \| S_{r,\theta}^{2-n*} W \|_{W_{r,\theta}^{\alpha,\alpha;2p}}^{2p} \right) &\lesssim \mathbb{E} \left( \sum_{n \ge 1} 2^{n(\beta - \frac{2}{2p})2p} \| S_{r,\theta}^{2-n*} W \|_{W_{r,\theta}^{\alpha,\alpha;2p}}^{2p} \right) \\
&\lesssim C_p \int_{I_1^2 \times I_2^2} \frac{(|r_1 - r_2| |\theta_1 - \theta_2|)^p}{|r_1 - r_2|^{1+2\alpha p} |\theta_1 - \theta_2|^{1+2\alpha p}} dr_1 dr_2 d\theta_1 d\theta_2 \sum_{n \ge 1} 2^{n(\beta - \frac{2}{2p} - \frac{1}{2})2p} < +\infty
\end{align*}
$$

where the conclusion holds true because $I_1 \times I_2$ never meets $(0,0)$. By the commutation relation $\partial_\theta S_{r_0, \theta_0}^{2-n*} = 2^{-n} \partial_\theta S_{r_0, \theta_0}^{2-n*}$,

$$
\| S^{2-n*} \tilde{A} \|_{W^{\alpha, \alpha-1, p}} = \| S^{2-n*} \partial_{\theta} W \|_{W^{\alpha, \alpha-1, p}} = 2^n \| \partial_{\theta} S^{2-n*} W \|_{W^{\alpha, \alpha-1, p}} = 2^n \| S^{2-n*} W \|_{W^{\alpha, \alpha, p}}
$$

combining with the above estimate on $\sum_{n \ge 1} 2^{n(\beta - \frac{2}{2p})2p} \mathbb{E} \left( \| S_{r, \theta}^{2-n*} W \|_{W_{r, \theta}^{\alpha, \alpha; 2p}}^{2p} \right)$, we get

$$
\mathbb{E} \left( \sum 2^{n(s-\frac{2}{2p})2p} \| S^{2-n*} \tilde{A} \|_{W^{\alpha, \alpha-1, 2p}}^{2p} \right) = \mathbb{E} \left( \sum 2^{n(s-\frac{2}{2p})2p} 2^{2np} \| S^{2-n*} W \|_{W^{\alpha, \alpha, 2p}}^{2p} \right) \lesssim \sum 2^{n(s-\frac{2}{2p})2p} 2^{np}
$$

since $\mathbb{E} (\|S^{2-n*} W\|_{W^{\alpha, \alpha, 2p}}^{2p}) = \mathcal{O}(2^{-np})$ and the series converges as soon as $s < -\frac{1}{2}$ for large enough $p$.
We deduce the following Lemma:

**Lemma 5.6** (Regularity of $\tilde{A}$). For any $s < -\frac{1}{2}$ and for the random connection $\tilde{A}$ given by equation (5.3) :

$$
\mathbb{E} \left( \| \tilde{A} \|_{W^{\alpha, \alpha-1, p, s}}^{p} \right) < +\infty. \qquad (5.9)
$$
## 5.5 Extending near defect lines and reducing to neighborhoods of saddle points.

**Clarifying various notions of distributions on some domain with smooth corners.** We state very precisely what we mean by some distribution, or current, defined on some manifold with corner $H$ whose role will be played by the Hexagons $(H_i)_{i=1}^{4g}$. Let $H$ be a surface with smooth corners which is smoothly embedded in an closed compact surface $S: H \hookrightarrow S$. Then there are mostly four classes of distributions one can define on such surface with smooth corners, each definition corresponds to some choice for the class of test functions:

1. If the test functions are in $C_c^\infty(\text{int}(H))$ means their support does not meet the boundary, then the dual space reads $\mathcal{D}'(\text{int}(H))$, this is the largest possible space of distributions defined on the open set $\text{int}(U)$.

2. If the test functions belong to the ideal $S(H)$ (by analogy with Schwartz class functions) of $C^\infty(H)$ that vanish at infinite order at the boundary $\partial H$, then the dual $S'(H)$ consists of all distributions in $\mathcal{D}'(\text{int}(H))$ which are extendible on $S$. One can also think of them as distributions obtained by restricting elements in $\mathcal{D}'(S)$ to $H$, this is some kind of extrinsic viewpoint, or as distributions in $\mathcal{D}'(\text{int}(H))$ with moderate growth when one approaches the boundary $\partial H$ a point of view pioneered by Kashiwara in [49, section 3 p. 332] and also described in detail in [31, section 2 to 5]. The terminology Schwartz is inspired from [1] and [23]. Given an extendible distribution $T \in \mathcal{D}'(\text{int}(H))$, any extension $\bar{T} \in \mathcal{D}'(S)$ is not necessarily unique.

3. If the test functions are the smooth functions in $C^\infty(H)$ which are smooth up to the boundary. In this case, the topological dual can be realized as distributions in $\mathcal{D}'(S)$ which are supported in $H$ (the rigorous proof uses Whitney's extension Theorem for smooth functions). This space is denoted by $\mathcal{D}'_H(S)$.

4. If the test functions are $C^\alpha(H)$ functions for $\alpha \in (0,1)$ and given $\bar{T} \in C^\alpha(H)'$, we identify its restriction to $\text{int}(H)$ as an element $T \in \mathcal{D}'(\text{int}(H))$, and one can recover $\bar{T}$ as an element in $\mathcal{D}'_K(S)$ from $T$ by a limiting procedure as follows:

$$ \bar{T} := \lim_{\epsilon \to 0^+} \chi_\epsilon T $$

for $(\chi_\epsilon)_\epsilon$ a family of smooth functions in $C_c^\infty(\text{int}(H))$ s.t. $\chi_\epsilon = 1$ outside some $\epsilon$-neighborhood of $\partial H$. This is the most restricted class of distributions of the four classes of distributions we just described and this is the class of distributions that we will mostly encounter in this work.

For a pedagogical presentation of the three first class of distributions with more references on these topics, we recommend to look at [31, section 2 to 5] and the references therein, we also recommend to look at the papers [1], [23, p. 157-165] for the relation with real algebraic and analytic geometry. The discussion for currents is almost verbatim, the reader just has to replace the word test functions with test forms.

**The three steps extensions.** We would like to prove the following.

**Proposition 5.7.** Let $\alpha \in (0, \frac{1}{2})$ and $s \in (-2, 0)$. Let $\tilde{A}d\theta$ be any current in the weighted anisotropic space $C^{\alpha, \alpha-1, s}(\text{Cyl} \setminus \Psi(\text{Crit}(f)_1))$ of the cylinder $\text{Cyl} := [\text{min}(f), \text{max}(f)]_r \times \mathbb{S}_\theta^1$. Then for $\epsilon$ small enough, the pull-back $\Psi^*(\tilde{A}d\theta)$ by the map $\Psi$ from equation 4.1 which is well-defined on

$$ \text{int}(S) \setminus \bigcup_{a \in \text{Crit}(f)_1} W^u(a) $$

extends uniquely to some distribution in $C^\beta(S)'$, the topological dual of $C^\beta$, for all $\beta$ such that $\beta+\alpha-1 > 0$ and the pull-back operator followed by the extension map

$$ \Psi^*: C^{\alpha, \alpha-1, s}([\text{min}(f), \text{max}(f)] \times \mathbb{S}^1 \setminus \Psi(\text{Crit}(f)_1)) \to C^\beta(S)' $$
<figure><img src="image_20.png" /><figcaption>Figure 18: Three-step extension.</figcaption></figure>

is linear continuous.

The proof of this extension is done in three steps, and is summarized in Figure 18

1. First, since $\Psi$ is discontinuous along unstable curves argue that the pull-back $\Psi^*(\tilde{A}d\theta)$ extends across unstable curves and defines a distribution in $S \setminus \text{Crit}(f)_1$. We deal with this part in Example 5.8 and Lemma 5.9.

2. Second, our distribution $\tilde{A}d\theta$ is a current in the weighted anisotropic space $C^{\alpha, \alpha-1, s}(\text{Cyl} \setminus \Psi(\text{Crit}(f)_1))$ hence it is only well-defined on the pointed space $\text{Cyl} \setminus \Psi(\text{Crit}(f)_1)$ where we deleted images under $\Psi$ of all saddle points. So we need to prove that the scaling of $\tilde{A}d\theta$ near the singular points in $\Psi(\text{Crit}(f)_1)$ which is controlled by the exponent $s$ from the weighted norm, allows us to extend $\tilde{A}d\theta$ canonically as a current in the topological dual $C^\beta(\text{Cyl})'$ for all $\beta$ such that $\beta + \alpha - 1 > 0$. This is done in Lemma 5.10.

3. Third and finally, once $\tilde{A}d\theta$ is well-defined globally on $\text{Cyl}$ and also that we are allowed to define the singular pull-back $\Psi^*(\tilde{A}d\theta)$ on the pointed surface $S \setminus \text{Crit}(f)_1$, we use all methods from the previous two steps to construct the extension of $\Psi^*(\tilde{A}d\theta)$ near all saddle points in $\text{Crit}(f)_1$. This uses both ideas from gluing and also relies on the scaling behaviour of $\Psi^*(\tilde{A}d\theta)$ as defined in the first step near each saddle point. This is done in Lemma 5.11.

**First extensions across edges of the hexagons $H_i$.** Recall each $H_i$ is our hexagon in the decomposition of $S$. Here the main technical problem consists in gluing distributions defined on each hexagon $H_i$ into some global object defined on the surface $S$ or conversely given some global distribution $T$ defined on the surface $S$, how can we localize on each piece $H_i$ by multiplying with indicators $1_{H_i}$ of $H_i$.

We begin by illustrating the whole difficulty with an example which contains all the crux of the problem.
<figure><img src="image_21.png" /><figcaption>Figure 19: Discontinuity lines.</figcaption></figure>

**Example 5.8** (Pull-back by discontinuous map followed by gluing). Let $\alpha \in (0, \frac{1}{2})$ and $\mathbb{R}^2$ with co-ordinates $(r, \theta)$. Consider the two squares $\square_1 := [-1, 1]_x \times [-1, 0]_y$ and $\square_2 := [-1, 1]_x \times [0, 1]_y$ and a map

$$ \Psi : [-1, 1]_x \times [-1, 1]_y \mapsto M $$

which is smooth on $\square_1$ and $\square_2$, in the sense of a smooth map from a differentiable manifold with corners with value into a smooth manifold $M$. The map $\Psi$ is eventually discontinuous on the wall $[-1, 1]_x \times \{0\}$ in the sense that for all $x$, $\lim_{y \to 0^+} \Psi(x, y) \in \mathbb{R}^2$ and $\lim_{y \to 0^-} \Psi(x, y) \in \mathbb{R}^2$ might differ. We furthermore assume that $\Psi$ maps the horizontal foliation $\cup_y[-1, 1]_x \times \{y\}$ to the foliation $\cup_{\theta \in \mathbb{R}} \mathbb{R}_r \times \{\theta\}$. The images $\Psi(\square_1)$ and $\Psi(\square_2)$ are disjoint smooth squares in the target space $\mathbb{R}^2$; see Figure 19.

Assume that on the target space $\mathbb{R}^2$, we have a global distribution $T \in \mathcal{D}(\mathbb{R}^2)$ which happens to be a derivative $T = \partial_{\theta}W$ in the sense of distributions, where $W \in C_{r, \theta}^{\alpha, \alpha}(\mathbb{R}^2)$. Our goal is to give a rigorous meaning to the pull-back $T_1 = \Psi^*(T1_{\Psi(\square_1)})$ and $T_2 := \Psi^*(T1_{\Psi(\square_2)})$ and justify that we glue the two pieces $T_1$ and $T_2$ as the sum $T_1 + T_2$.

First observe that $T \in C^{\alpha, \alpha-1}$ implies that the restrictions $(T1_{\Psi(\square_1)})$ and $(T1_{\Psi(\square_2)})$ are both well-defined by Lemma A.2 from the appendix.

Second, a crucial observation is that since $\Psi$ is smooth on $\square_i$ up to the boundary, using a classical result of Seeley [64] or the Whitney extension Theorem [57, Thm 4.1 p. 10], we may extend $\Psi$ smoothly to some neighborhood $\tilde{\square}_i$ of each $\square_i$, each extension is denoted by $\tilde{\Psi}_i$ and satisfies the crucial identity $\tilde{\Psi}_i|_{\square_i} = \Psi|_{\square_i}$, in such a way that the image of extensions $\Psi(\tilde{\square}_1)$ and $\Psi(\tilde{\square}_2)$ are disjoint. Since $d\Psi|_{\partial\square_i}$ is never vanishing, we know that up to taking a smaller $\tilde{\square}_i$ we may assume that the extended map $\tilde{\Psi}_i : \tilde{\square}_i \mapsto \tilde{\Psi}(\tilde{\square}_i)$ is a diffeomorphism on its image; see Figure 20. Third, each pull-back $\tilde{\Psi}_i^*T$ belongs to the space $C_{x,y}^{\alpha, \alpha-1}$. Using the continuous injection $C_{x,y}^{\alpha, \alpha-1} \hookrightarrow C_{x,y}^{\alpha-1}$ established in Lemma C.2 from the appendix, this implies $\tilde{\Psi}_i^*T$ also belongs to $C_{x,y}^{\alpha-1}(\text{int}(\tilde{\square}_i))$. Hence by Lemmas A.1 and A.2, this implies that $\tilde{\Psi}_i^*T$ can be restricted to $\square_i$ by multiplication with the indicator function $1_{\square_i}$, hence $T_i := (\tilde{\Psi}_i^*T)1_{\square_i}$ is well-defined in $\mathcal{D}'(\text{int}(\tilde{\square}_i))$ with support contained in $\square_i$. Another approach to this third step involves leafwise extensions. Since $\tilde{\Psi}_i^*T$ is $\alpha$-Hölder in $x$ valued in distributions of the $y$ variable, we may consider the leafwise restriction
<figure><img src="image_22.png" /><figcaption>Figure 20: Extending $\Psi$.</figcaption></figure>

$\tilde{\Psi}_i^*T(x, \cdot) \in C_y^{\alpha-1}$ for every $x \in [-1, 1]$; see Figure 21.

The leaf is just the interval $\{x\} \times [-1, 1]_y$. Since $\alpha - 1 > 0$, Lemma A.2 allows us to multiply $\tilde{\Psi}_1^*T(x, \cdot)$ (resp $\tilde{\Psi}_2^*T(x, \cdot)$) with $1_{[-1,0]}(y)$ (resp $1_{[0,1]}(y)$) which yields distributions $\tilde{\Psi}_1^*T(x, \cdot)1_{[-1,0]}(\cdot)$ (resp $\tilde{\Psi}_2^*T(x, \cdot)1_{[0,1]}(\cdot)$) supported on $[-1, 0]$ (resp $[0, 1]$) for all $x \in [-1, 1]$. By continuity of the product with indicators, we still work with continuous functions in $x$ valued in distributions of $y$.

Finally, the sum $T_1 + T_2$ answers our question.

Once we discussed the above key example, we can move on to our extension procedure along unstable curves.

**Lemma 5.9 (Extension along unstable curves).** Let $\alpha = \frac{1}{2} - \varepsilon$. Let $\tilde{A}\theta$ be any current in the anisotropic space $C_{I \times J}^{\alpha, \alpha-1}$ for any $I \times J \subset [\min(f), \max(f)] \times \mathbb{S}^1$ avoiding $\Psi(\text{Crit}(f)_1)$.

Then for $\varepsilon$ small enough, the pull-back $\Psi^*(\tilde{A}\theta)$ extends uniquely on $S \setminus \text{Crit}(f)_1$.

*Proof.* In fact we can almost reduce the above proof to our example 5.8. The reader who just wants to get a flavour of the proof can skip hat we wrote below and refer to our example 5.8 and look at the pictures. First observation, the map

$$ \Psi : \text{int}(H_i) \to [\min(f), \max(f)] \times \mathbb{S}^1 $$

is such that its differential $d\Psi$ is nondegenerate along $\partial H_i \setminus \text{Crit}(f)_1$ and smooth up to the boundary minus the saddle points $\partial H_i \setminus \text{Crit}(f)_1$. This follows from Proposition 4.3.

Because of the definition of $\Psi$ and the fact that $\tilde{A}\theta$ is $\alpha = \frac{1}{2} - \varepsilon$ in the $r$-direction, its pull-back $\Psi^*(\tilde{A}\theta)$ is continuous in the flow direction, valued into distributions of regularity $\alpha - 1$ in the $\theta$ variable. The continuity in the flow direction being precisely captured by the anisotropic spaces.

Therefore, we use Lemma A.2 which shows that the leafwise restriction $\Psi^*(\tilde{A}\theta)|_{\{f=\text{constant}\} \cap \text{int}(H_i)}$ has regularity $-\frac{1}{2} - \varepsilon$ along level sets $\{f = \text{constant}\} \cap \text{int}(H_i)$ which shows there exists a unique extension to $\{f = \text{constant}\}$ where we extend by 0 outside $\{f = \text{constant}\} \cap H_i$. $\square$
<figure><img src="image_23.png" /><figcaption>Figure 21: Gluing pull backs and leaf wise restrictions.</figcaption></figure>

**Various notions of scalings.** To describe extensions of distributions defined on pointed spaces, we measure their singular behaviour with scalings. This is why we need to recall the various notions of scalings that appear in our problem. Fix an open set $U \subset S$ near a saddle point and a system of Morse coordinates $(x, y)$ on $U$, the scaling is a local diffeomorphism which writes :

$$S_{x_0, y_0}^{\lambda} : (x, y) \in U \mapsto (\lambda(x - x_0) + x_0, \lambda(y - y_0) + y_0) \in U, \lambda \in (0, 1].$$

The subtle point in our discussion is that there are two scalings, the one

$$S_{x_0, y_0}^{\lambda} : (x, y) \in U \mapsto (\lambda(x - x_0) + x_0, \lambda(y - y_0) + y_0) \in U, \lambda \in (0, 1]$$

in the nice Morse coordinates near the saddle point $a$, and the scaling

$$S_{r_0, \theta_0}^{\lambda} : (r, \theta) \mapsto (\lambda(r - r_0) + r_0, \lambda(\theta - \theta_0) + \theta_0), \lambda \in (0, 1]$$

which is defined in coordinates $(r, \theta)$ in the cylinder **Cyl**. These scalings do not match because $\Psi$ fails exactly to be a diffeomorphism at $\text{Crit}(f)_1$ and a point where we need to be careful is that the weighted norms were defined using scaling on the cylinder.

**Extension of $\tilde{A}d\theta$ near $\Psi(\text{Crit}(f)_1)$ on the cylinder $\text{Cyl}$**

**Lemma 5.10.** Let $\alpha = \in (0, \frac{1}{2})$ and $s \in (-2, 0)$. Let $\tilde{A}d\theta$ be any current in the weighted anisotropic space $C^{\alpha, \alpha-1, s}(\text{Cyl} \setminus \Psi(\text{Crit}(f)_1))$ of the cylinder $[\min(f), \max(f)]_r \times S_{\theta}^1$. Then $\tilde{A}d\theta$ extends uniquely as distribution in $C^{\beta}(\text{Cyl})'$ for all $\beta$ such that $\beta + \alpha - 1 > 0$.

*Proof.* To extend the current $\tilde{A}d\theta$ near a singular point $a \in \Psi(\text{Crit}(f)_1)$, we will use the scaling of $(\tilde{A}d\theta)$ near $a \in \Psi(\text{Crit}(f)_1)$. Very similar ideas to what we are doing can be found in [59, Chapter 2 p. 43–53] and also in [32, Thm 4.4 p. 832].

Without loss of generality, $a$ is given by $(0, 0)$ in coordinates $(r, \theta)$ of the cylinder. Start from any function $\psi_0$ which equals 1 near $a = (0, 0)$, $\psi_0(r, \theta) = 1$ when $|(r, \theta)| \le 1$ and $\psi_0(r, \theta) = 0$ when $|(r, \theta)| \ge 2$.
<figure><img src="image_24.png" /><figcaption>Figure 22: Littlewood-Paley decomposition in space, annular support of $\psi_1$.</figcaption></figure>

Observe that the function $1 - \psi_0(2^n.)$ equals 1 on the complement of a ball of radius $2^{-n+1}$ near $(0,0)$ and vanishes on a ball of radius $2^{-n}$ so that the product $(1 - \psi_0(2^n.))\tilde{A}d\theta$ is well-defined globally in $D'(\text{Cyl})$ and coincides with $\tilde{A}d\theta$ on the complement of a ball of radius $2^{-n+1}$. Therefore for every test 1-form $\varphi \in C_c^\infty(\text{Cyl} \setminus \Psi(\text{Crit}(f)_1))$, $\langle \tilde{A}d\theta, \varphi \rangle = \langle (1 - \psi_0(2^n.))\tilde{A}d\theta, \varphi \rangle$ for all $n \ge N$ for $N$ large enough, since $\text{supp}(\psi_0(2^n.)) \cap \text{supp}(\varphi) = \emptyset$ for all $n \ge N$. Then we would like to understand the convergence of $\lim_{n \to +\infty} (1 - \psi_0(2^n.))\tilde{A}d\theta$ in $D'(\text{Cyl})$. For that purpose, we are going to decompose $1 - \psi_0(2^n.)$ as a telescopic series of functions each of them is supported on some annular domain avoiding $(0,0)$ of smaller and smaller dyadic size. Denote by $K$ the annular domain centered around a singular point $a \in \Psi(\text{Crit}(f)_1)$, $K = \{(r, \theta); \|(r, \theta)\|_2 \in [\frac{1}{2}, 4]\}$. Following [59, p. 48-49], the main idea is to use a spatial version of the Littlewood-Paley partition of unity, we write $1 = (1 - \psi_0) + \sum_{n=1}^\infty \psi_n$, where for all $n \ge 1$, $\psi_n := \psi_0(2^{n+1}.) - \psi_0(2^n.)$, $1 - \psi_0$ vanishes near $(0,0)$ hence we do not need to control $(1 - \psi_0)\tilde{A}d\theta$, $\psi_1$ is supported in $K$ and where each $\psi_n$ is supported in a concentric corona centered at $(0,0)$ of radius $\in [2^{-n-1}, 2^{-n+2}]$.

What is important is that $a \notin \text{supp}(\psi_n)$ for all $n \ge 1$. Also note that for every test 1-form $\varphi \in C_c^\infty(\text{Cyl} \setminus \Psi(\text{Crit}(f)_1))$, the sum $\sum_{n=1}^\infty \psi_n\varphi$ reduces in fact to some finite sum $\sum_{n=1}^N \psi_n\varphi$ since for $N$ large enough, $\text{supp}(\psi_n) \cap \text{supp}(\varphi) = \emptyset$ for all $n \ge N$. So we are left with the task to control the convergence of the series $\sum_{n=1}^\infty \psi_n \tilde{A}d\theta$ in $D'(\text{Cyl})$. For any test 1-form $\varphi \in \Omega^1(\text{Cyl})$, we get

$$ \left\langle \sum_{n=0}^{\infty} \psi_n \tilde{A} d\theta, \varphi \right\rangle = \sum_{n=0}^{\infty} \left\langle S_{(r_0, \theta_0)}^{2^{-n}*} (\psi_n \tilde{A} d\theta), S_{(0,0)}^{2^{-n}*} \varphi \right\rangle \quad (5.10) $$

where the right hand side follows from the change of variables formula. We drop the $(0,0)$ subscript in the scaling to simplify the notations.

To study the convergence of the series in spaces of distributions, let us study an individual term $\langle S^{2^{-n}*}(\psi_n \tilde{A}d\theta), S^{2^{-n}*}\varphi \rangle$. By the upper bound :

$$ | \langle S^{2^{-n}*}(\tilde{A}d\theta), \psi_1 S^{2^{-n}*}\varphi \rangle | \le 2^{-2n} \|S^{2^{-n}*}\tilde{A}\|_{C^{\alpha, \alpha-1}} \|\psi_1 S^{2^{-n}*}\varphi\|_{C^\beta} \lesssim 2^{-2n} 2^{-ns} \|\tilde{A}\|_{C^{\alpha, \alpha-1, s}} \|\varphi\|_{C^\beta(B)}, $$
using the continuous injections $C^{\alpha, \alpha-1}(K) \hookrightarrow C^{\alpha-1}(K)$ proved in Proposition C.1, $C^{\beta} \hookrightarrow (C^{\alpha-1})'$ for all $\beta + \alpha - 1 > 0$ which follows from the criterion on Young products of Hölder-Besov distributions, see A.3 in the appendix, the definition of the weighted anisotropic norms, the family $\|\psi S^{2^{-n}*}\varphi\|_{C^{\beta}(K)}$, $n \ge 1$ is bounded by the $C^{\beta}$ norm of $\varphi$ on a certain ball $B$ centered at $(0,0)$ and the $2^{-2n}$ in factor comes from $S^{2^{-n}*}d\theta = 2^{-n}d\theta$, $S^{2^{-n}*}dr = 2^{-n}dr$. So the series $\sum_{n=0}^{\infty} \psi_n \tilde{A}d\theta$ converges in $C^{\beta}(\text{Cyl})'$ for all $\beta + \alpha - 1 > 0$ and $s > -2$. $\square$

**Extension of $\Psi^*(\tilde{A}d\theta)$ to the whole surface $S$.**

**Lemma 5.11.** Under the assumptions of Lemma 5.9, the pull-back $\Psi^*(\tilde{A}d\theta)$ defined on the pointed surface $S \setminus \text{Crit}(f)_1$ extends uniquely as an element in $C^{\beta}(\text{Cyl})'$ for all $\beta$ such that $\beta + \alpha - 1 > 0$.

*Proof.* Step 1, now $\Psi^*(\tilde{A}d\theta)$ is well-defined on $S \setminus \text{Crit}(f)_1$ thanks to Lemma 5.9.

Step 2, to extend near a critical point, we will use the scaling of $\Psi^*(\tilde{A}d\theta)$ at a critical point.

Step 3, to define the pull-back $\Psi^*(\tilde{A}d\theta)$ even at the critical points, the key is to control the pull-back of the convergent series appearing in the proof of Lemma 5.10. This suggests to study $\sum_{n=1}^{\infty} \Psi^*(\psi_n \tilde{A}d\theta)$. But there is a problem since $\Psi$ is only piecewise smooth. This is why we need to use indicator functions to localize in regions where $\Psi$ is a well-defined smooth diffeomorphism exactly in the spirit of example 5.8 and Lemma 5.9. We consider the indicator function $1_{\Psi(H_i)}$ of image under $\Psi$ of one of the hexagon $H_i \subset S$ where $\Psi^{-1}: \Psi(H_i) \mapsto H_i$ is well-defined and smooth. We make the crucial observation that the domain $\Psi(H_i) \subset \text{Cyl}$ is stable by the scaling flow: $S^{2^{-n}}(\Psi(H_i)) \subset \Psi(H_i)$ for all $n \ge 0$, since in local coordinates each domain $\Psi(H_i)$ is defined in terms of linear inequalities involving $r, \theta$. In fact, in the local Morse chart in a hyperbolic box, up to shifting coordinates by constants and up to some smooth change of variables, we can always assume that $f = x^2 - y^2$, $\theta = xy$ and we work on the quadrant $H_i = \{x \ge 0, y \ge 0\}$. Exactly for the same reason as in example 5.8 and Lemma 5.9 $^4$, we need another extra ingredient, namely the existence of certain smooth extensions of our piecewise smooth diffeomorphism $\Psi$ in some neighborhood of every hexagon $H_i$, $i = 1, \dots, 4g$. As in example 5.8, we need to slightly extend the diffeomorphism $\Psi: H_i \mapsto \Psi(H_i)$ to some slightly larger neighborhood $E_i$ of $H_i$ in such a way that the extended map denoted by $\tilde{\Psi}_i: E_i \mapsto \tilde{\Psi}_i(E_i)$ is still a diffeomorphism. We used the fundamental fact that both $\Psi^*$ and $\tilde{\Psi}_i^*$ both coincide on distributions supported by $\Psi(H_i)$.

So we will study instead the series $\sum_{i=1}^{4g} \sum_n \tilde{\Psi}_i^* (1_{\Psi(H_i)} \psi_n \tilde{A}d\theta)$ where we localized, decomposed in dyadic annular domains and then resum on $i = 1, \dots, 4g$.

We rewrite the general term of the series using the following series of identities, for any test 1-form $\varphi \in \Omega^1(S)$:

$$ \begin{aligned} \langle \tilde{\Psi}_i^* (1_{\Psi(H_i)} \psi_n \tilde{A}d\theta), \varphi \rangle &= \langle \tilde{\Psi}_i^* S_{r_0, \theta_0}^{2^n} (\psi S_{r_0, \theta_0}^{2^{-n}} (1_{\Psi(H_i)} \tilde{A}d\theta)), \varphi \rangle \\ &= \langle \tilde{\Psi}_i^* S_{r_0, \theta_0}^{2^n} (\psi 1_{\Psi(H_i)} S_{r_0, \theta_0}^{2^{-n}} (\tilde{A}d\theta)), \varphi \rangle = \langle (\psi S_{r_0, \theta_0}^{2^{-n}} (1_{\Psi(H_i)} \tilde{A}d\theta)), S_{r_0, \theta_0}^{2^{-n}} (\tilde{\Psi}_i^{-1*} \varphi) \rangle \end{aligned} $$

where in the second line we use the scaling flow stability of $\Psi(H_i)$.

The singular point reads $(0,0)$. Then we have the relation

$$ x = \sqrt{\frac{r + (r^2 + 4\theta^2)^{\frac{1}{2}}}{2}}, \quad y = \sqrt{\frac{(r^2 + 4\theta^2)^{\frac{1}{2}} - r}{2}}. $$

<footnote>$^4$We cannot pull-back by a partially smooth map some functional object supported exactly on the domain of the pull-back, we need some slight enlargement</footnote>
This implies that $\Psi^{-1}$ is locally given in coordinates by

$$ \Psi^{-1} : (r, \theta) \mapsto \left( \sqrt{\frac{r + (r^2 + 4\theta^2)^{\frac{1}{2}}}{2}}, \sqrt{\frac{(r^2 + 4\theta^2)^{\frac{1}{2}} - r}{2}} \right). $$

Observe that the above formulas extend to arbitrary values of $(r, \theta) \in [-1, 1]^2$ where we recognize immediately the only non smooth point at $(r, \theta) = (0, 0)$. We need to analyze the growth of the 1-form $S_{0,0}^{2-n*}(\Psi^{-1*}(\varphi))$ in $C^\beta(K)$ (recall $K$ is the annular domain $K = \{(r, \theta); \|(r, \theta)\|_2 \in [\frac{1}{2}, 4]\}$) for large $n$ for $\beta + \alpha - 1 > 0$ and we proceed exactly as above :

$$ S_{0,0}^{2-n*}(\Psi^{-1*}(\varphi)) = \varphi_1 \left( 2^{-n} \sqrt{\frac{r + (r^2 + 4\theta^2)^{\frac{1}{2}}}{2}}, 2^{-n} \sqrt{\frac{(r^2 + 4\theta^2)^{\frac{1}{2}} - r}{2}} \right) 2^{-n} dx \\ + \varphi_2 \left( 2^{-n} \sqrt{\frac{r + (r^2 + 4\theta^2)^{\frac{1}{2}}}{2}}, 2^{-n} \sqrt{\frac{(r^2 + 4\theta^2)^{\frac{1}{2}} - r}{2}} \right) 2^{-n} dy $$

where $x, y$ are viewed as implicit functions of $(r, \theta)$. The key idea is that $(r, \theta)$ belong to the annular domain $K$ and therefore all derivatives of $\sqrt{\frac{r+(r^2+4\theta^2)^{\frac{1}{2}}}{2}}$, $\sqrt{\frac{(r^2+4\theta^2)^{\frac{1}{2}}-r}{2}}$ in $(r, \theta)$ are bounded uniformly on $K$.

So we have the decay estimate $\|S_{0,0}^{2-n*}(\Psi^{-1*}(\varphi))\|_{C^\beta(K)} \le C2^{-n}\|\varphi\|_{C^\beta(B)}$ where as above $\beta$ is chosen in such a way that $\beta + \alpha - 1 > 0$. Therefore

$$ \left| \sum_{i=1}^{4g} \left\langle \tilde{\Psi}_i^* \left( \sum_{n=1}^{\infty} \psi_n 1_{\Psi(H_i)} \tilde{A} d\theta \right), \varphi \right\rangle \right| \le \sum_{n=1}^{\infty} C 2^{-2n} 2^{-ns} \|\tilde{A}\|_{C^{\alpha, \alpha-1, s}} \|\varphi\|_{C^\beta(B)} $$

where we used again the fact that $\tilde{A} \in C^{\alpha, \alpha-1, s}$. So for all $s \in (-2, 0)$, the pull-back $\Psi^*(\tilde{A}d\theta)$ has a unique extension in $C^\beta(S)'$ for all $\beta + \alpha - 1 > 0$. □

Now, the formal definition given in 5.1 gives perfect sense as a random distributional 1-form on $S$. The next theorem shows that the parallel transport generated by this 1-form verifies the Driver-Sengupta formula.

**Theorem 5.12 (Driver-Sengupta Formula).** The random 1-form $A$ defined in 5.1 induces via stochastic differential equations, on any graph on $\Sigma$ whose edges are either flow lines or level sets, holonomies verifying the Driver-Sengupta formula.

*Proof.* The idea is to repeat the proof in 2.6 4g times, in each of the portions of $\Sigma$ within two consecutive unstable curves. We get the global formula by independence of the white noise throughout these portions, followed by the insertion of the uniform random variables associated with the zero-area bands, as in 2.3.

We would like to relate our random connection with the one appearing in [16]. For any smooth oriented curve $\gamma$ everywhere transverse to the flow, we need to show that

$$ W_{\gamma} := \int_{\gamma} \Psi^{*} (\tilde{A} d\theta) \qquad (5.11) $$

is well-defined as a $g$-valued normal random variable with variance the area of a certain rectangle defined in terms of the curve $\gamma$. A key consequence from its definition is that $\Psi$ preserves the orientation. Beware that the curve $\gamma$ might intersect some unstable curve transversally hence its image $\Psi(\gamma) \subset \text{Cyl}$ under $\Psi$ is only piecewise smooth : $\Psi(\gamma) = \cup_{i=1}^k \gamma_i$ as a disjoint union of smooth curves. By transversality with the
flow and since this transversality is preserved under mapping by $\Psi$, $\gamma_i \subset \text{Cyl}$ is transverse with $\partial_r$ and we can describe each curve $\gamma_i$ as some graph: $\theta \in I_i \subset [0, 2\pi] \mapsto (\theta, r_i(\theta))$ where $r_i$ is a smooth function. A key consequence from its definition is that $\Psi$ preserves the orientation this is why we get the same orientation for all curves $\gamma_i$, they run counterclockwise if the initial curve $\gamma$ is oriented counterclockwise. Another consequence of transversality is that the open intervals $\text{int}(I_i)_{i=1,...,k}$ are two by two disjoint. We decompose $\Psi_*[\gamma] = \sum_{i=1}^k [\gamma_i]$ where the r.h.s. is a finite sum of currents of integration of degree 1 on the curves $\gamma_i$, $i = 1,..., k$. The above integral in equation 5.11 can only defined probabilistically as follows :

$$ \int_{\gamma} \Psi^* (\tilde{A} d\theta) = \langle \Psi_*[\gamma], \tilde{A} d\theta \rangle_{\text{Cyl}} = \sum_{i=1}^{k} \xi(1_{\square_i}) $$

where each domain $\square_i$ is defined as follows

$$ \square_i := \{(r, \theta); 0 \le r \le r_i(\theta), \theta \in I_i\} \subset \text{Cyl} $$

and each term $\xi(1_{\square_i})$ is well-defined since each $1_{\square_i} \in L^2(\text{Cyl}, \Psi_*\sigma)$ belongs to the Cameron Martin space of $\xi$. Then from the above identities

$$ \mathbb{E} \left( \int_{\gamma} \Psi^* (\tilde{A} d\theta) \otimes \int_{\gamma} \Psi^* (\tilde{A} d\theta) \right) = \mathbb{E} \left( \left( \sum_{i=1}^{k} \xi(1_{\square_i}) \right) \otimes \left( \sum_{i=1}^{k} \xi(1_{\square_i}) \right) \right) = \sum_{i=1}^{k} \sigma(\square_i) \text{Id}^g \otimes \text{Id}^g $$

where we used the crucial fact that all mutual intersections $\square_i \cap \square_j, i \ne j$ have measure 0. Now observe that $\sum_{i=1}^k \sigma(\square_i)$ is the area on $S$ of the set $\{x \prec \gamma, x \in S\} := \overline{\{\varphi_f^{-t}(x); x \in \gamma\}}$ which implies we recover the result that $W_\gamma$ has normal law with variance equals to the area $\sigma(\{x \prec \gamma\})$ which is one of the main result of [16] on the Gaussian part of the free boundary Yang-Mills measure. □

We use the notion of strong solutions for Stratonovich SDE, when we reparametrize our driving Brownian motion by some $C^{1-}$ function, this notion of strong solution still exists and the area function is only $C^{1-}$ so there is no problem here. We refer the reader to a separate note for the detailed proof [33] or the appendix of [16].

# 6 Morse lattice from flowlines and level sets

We introduce a dyadic decomposition of our surface into rectangles whose sides are either flowlines of $\nabla f$ or level sets of $f$. The construction is simplified because of the existence of the global pseudo-coordinates $(r, \theta)$ on $\Sigma$.

There are $2g + 2$ critical levels $c_1 < \dots < c_{2g+2}$ of the Morse function $f$, we decompose each piece $[c_i, c_{i+1}]$ into $2^n$ levels of the form

$$ c_i + \frac{k}{2^n} (c_{i+1} - c_i), \quad k \in \{0, \dots, 2^n\}. $$

Therefore we have $(2g + 1)2^n$ level sets of the Morse function of the form

$$ f^{-1}(c_i + \frac{k}{2^n} (c_{i+1} - c_i)), \quad i \in \{1, \dots, 2g+1\}, k \in \{0, \dots, 2^n\}. $$

These levels are also mapped to a certain level decomposition of the cylinder

$$ [\min(f), \max(f)] \times \mathbb{S}^1 $$

under the map $\Psi$. Recall we blow-up the surface $\Sigma$ at the minimum $a_1$, at the minimum $a_1$ the blown-up space becomes a circle parametrized by some angle $\theta \in [0, 2\pi]$. We consider $0 < \theta_1 < \dots < \theta_{4g} < 2\pi$ the
<figure><img src="image_25.png" /><figcaption>Figure 23: Morse lattice.</figcaption></figure>

4g intersections of the stable curves of the flow $(\Phi^t)_{t \in \mathbb{R}}$ with the blow-up circle. These corresponds to the intervals $(I_i)_{i=1}^{4g}$ as

$$I_1 = [\theta_1, \theta_2], I_2 = [\theta_2, \theta_3], \dots, I_{4g} = [\theta_{4g}, \theta_1].$$

Now we choose $2^n$ directions

$$\theta_i + \frac{k}{2^n}(\theta_{i+1} - \theta_i), \quad i \in \{1, \dots, 4g-1\}, \quad k \in \{0, \dots, 2^n-1\},$$

this decomposes each interval $I_i$, $i = 1, \dots, 4g$ into $2^n$ dyadic subintervals. Now we consider $2^n(4g)$ level sets of the global angular variable $\theta$, this yields a union of $2^n 4g$ integral curves of $\Phi^t$, $t \in \mathbb{R}$ which have these directions and intersect these $2^n 4g$ curves with our $(2g+1)2^n$ levels of the Morse function $f$. For every $n$, the above picture describes a sequence of graphs $(\Lambda^n)_n$ on our surface compatible with the Morse function and of dyadic resolution $2^{-n}$. Under the map $\Psi$, this is also canonically identified with a grid decomposition of

$$\Psi(H_i) = [\min(f), \max(f)] \times I_i.$$

The grid is made of either flowlines of $\Phi^t$ or smooth arcs orthogonal to the flow and contained in levels of $f$. Our lattice gauge model will be defined on this Morse grid. An illustrative example of a Morse grid is shown in 23

The important fact is that for any compact that does not meet the saddle points, we have that the area of a dyadic face $\simeq 2^{-2N}$ (of course the constant becomes worst when we approach the critical points).

**Lemma 6.1** (Areas of small squares and dyadic corona decomposition). We denote by $F^N$ the faces of $\Lambda^N$. Fix an arbitrary $\varepsilon > 0$, then there exists $c_\varepsilon < C_\varepsilon$ such that for every $N$, for any face $\square$ of the grid $\Lambda^N$ that does meet $\cup_{a \in \text{Crit}(f)_1} B(a, \varepsilon)$, we have

$$c_\varepsilon 2^{-2N} \le \int_{\square} \sigma \le C_\varepsilon 2^{-2N}. \qquad (6.1)$$
Moreover, for all $N$, for all squares that are at approximate distance $2^{-j}$ (measured in the coordinates $(r, \theta)$) of some given singular point in $\Psi(\text{Crit}(f)_1)$ $j = 1, \dots, N$,

$$c2^j 2^{-2N} \le \int_{\Box} \sigma \le C2^j 2^{-2N}. \qquad (6.2)$$

where the constants $c, C$ do not depend on $j, N$.

Beware that it is important in the previous Lemma that $\varepsilon$ does not depend on $N$.

*Proof.* This is an immediate consequence of the fact that $\Psi_*\sigma$ fails to be smooth at saddle points. Let us estimate the asymptotic of the worst dyadic faces of the decomposition which are close to the saddle points :

$$\int_{\Box} \sigma \sim \int_{k2^{-N}}^{(k+1)2^{-N}} \int_{\ell2^{-N}}^{(\ell+1)2^{-N}} (f^2 + 4\theta^2)^{-\frac{1}{2}} d\int f d\theta = \int_{k}^{(k+1)} \int_{\ell}^{(\ell+1)} 2^N (f^2 + 4\theta^2)^{-\frac{1}{2}} 2^{-2N} d\int f d\theta = \mathcal{O}(2^{-N}).$$

<figure><img src="image_26.png" /></figure>

We introduce a discrete corona decomposition that will be crucial to define discrete versions of the weighted norms. We should prove a scaled estimate of the following form. We would like to control the area of those dyadic squares that approach very close to the saddle points where $\Psi_*\sigma$ is singular. Consider a small dyadic square $\square(r, r+2^{-N}, \theta, \theta+2^{-N})$ at a fixed distance from the saddle points. Assume w.l.o.g. that $(0,0)$ is our saddle point. For instance all dyadic squares at distance between $\frac{1}{2}$ and $1$ from the saddle point write $\square(k2^{-N}, (k+1)2^{-N}, \ell2^{-N}, (\ell+1)2^{-N})$ where $2^{N-1} \le k, \ell \le 2^N$, so $r = k2^{-N}, \theta = \ell2^{-N}$.

So at every scale $j = 1, \dots, N-1$, $\square(k2^{-N}, (k+1)2^{-N}, \ell2^{-N}, (\ell+1)2^{-N})$ where $2^{N-j-1} \le k, \ell \le 2^{N-j}$. So all dyadic squares can be decomposed as a union

$$\bigcup_{j=1,...,N-1} \left( \bigcup_{2^{N-j-1} \le k, \ell \le 2^{N-j}} \square(k2^{-N}, (k+1)2^{-N}, \ell2^{-N}, (\ell+1)2^{-N}) \right)$$

where for each scale $j$, a square $\square(k2^{-N}, (k+1)2^{-N}, \ell2^{-N}, (\ell+1)2^{-N})$ will be contained in a region (a shell) at distance $\approx 2^{-j}$, there are approximately $(2^{2(N-j)})$ such squares.

For such a square, we have an estimate of the form

$$\int_{\square(k2^{-N}, (k+1)2^{-N}, \ell2^{-N}, (\ell+1)2^{-N})} \sigma \approx 2^{-2N} 2^j = 2^{j-2N}.$$

# 7 A local limit theorem

Let $G$ be a compact unitary Lie group with Lie algebra $\mathfrak{g}$, that we equip with a bi invariant dot product $\langle \cdot, \cdot \rangle$. In this section, we would like to show two results of independent interest, that we will use in the sequel. The first one aims to study the convergence of the densities in the central limit theorem, and the second one can be seen as some type of a non-commutative Donsker theorem in which the random variables are not independent. Before we proceed, let us mention that Theorem 1.3 is a direct corollary of Theorem 7.2.

Let us give a brief description of the result of this section. Let $(\mu_t)_{t>0}$ be a family of probability measures such that for small $t$, $\mu_t$ is concentrated around the identity, in a way that will be made precise later. Suppose that the density $\rho_t$ of $\mu_t$ exists and enjoys some properties that we will discuss later. Then, we would like roughly to study the following convergence in $C^\infty(G)$

$$\underbrace{\rho_{\frac{1}{n}} * \cdots * \rho_{\frac{1}{n}}}_{n \text{ times}} \xrightarrow{n \to \infty} p_1,$$
where $p_1$ is the heat kernel at time 1. The general idea is that we already know the convergence in $D'(G)$, and therefore the only additional argument we need is the boundedness in $C^\infty(G)$. Let us now be more precise on the assumptions.

**Definition 7.1.** Let $s > 0$. A family of probability measures $(\mu_t)_{t>0}$ is said to have the property $C(s)$ if

1. for each $t > 0$, $\mu_t$ is ad-invariant and invariant by inversion,

2. for all $v, w \in \mathfrak{g}$, $\int_G \langle \log(g), v \rangle_{\mathfrak{g}} \langle \log(g), w \rangle_{\mathfrak{g}} \mu_t(dg) = t \langle v, w \rangle_{\mathfrak{g}} + o(t)$,

3. $\int_G |\log g|^3 \mu_t(dg) = O(t^{\frac{3}{2}})$,

4. For all $t > 0$, $\mu_t$ has the following Fourier decay,

$$ \exists C > 0; \forall \lambda \in \hat{G}, \forall t > 0, |\hat{\mu}_t(\lambda)| \le \dim(V_\rho)C \left(1 + \sqrt{tc_2(\rho)}\right)^{-s} \quad (7.1) $$

where $c_2(\lambda)$ is the second Casimir number of the representation $\lambda$.

We would like to show the following theorem.

**Theorem 7.2.** Let for each $n \in \mathbb{N}$, $s_1^n, \dots, s_n^n \in \mathbb{R}_+$ such that $s_1^n + \dots + s_n^n = 1$, and such that there exists $a > 0$, and $A > 1$ for which

$$ \liminf_{n \to \infty} \frac{|T_n(a, A)|}{n} > 0, \text{ with } T_n(a, A) := \left\{ i = 1, \dots, n; \frac{a^2}{n} \le s_i^n \le \frac{A^2}{n} \right\}. $$

Then, if $(\mu_t := \rho_t dg)_{t>0}$ is a family of probability measures verifying the condition $C(s)$ for some $s > 0$,

$$ \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \xrightarrow{n \to \infty} p_1 \text{ in } C^\infty(G), $$

where $p_1$ is the heat kernel.

*Proof.* The central limit theorem tells us that $\rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \xrightarrow{n \to \infty} p_1$ in $D'(G)$. The Proposition 7.3, whose proof is the subject of the remaining of the section, shows that for all $k \in \mathbb{N}$, the sequence $(\rho_{s_1^n} \star \cdots \star \rho_{s_n^n})_{n \ge 1}$ is bounded in $C^k$. The result follows. $\square$

**Proposition 7.3.** Using the same notations as in the previous theorem, for all $k \in \mathbb{N}$, the sequence $(\rho_{s_1^n} \star \cdots \star \rho_{s_n^n})_{n \ge 1}$ is bounded in $C^k$.

The proof of the above proposition needs many intermediate steps. Let us start with some preparatory work.

**Lemma 7.4.** Using the same notations, we have for all $k$

$$ \| \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \|_{C^k} \le \sum_{\lambda \in \hat{G}} (1 + |c_2(\lambda)|)^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right|. $$

*Proof.* Using Fourier analysis on $G$, we can write

$$ \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} = \sum_{\lambda \in \hat{G}} \left( \prod_{i=1}^{n} \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right) \chi_\lambda, \text{ where } \hat{\rho}_t(\lambda) = \int_G \chi_\lambda(g) \mu_t(dg). $$

Note that

$$ \| \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \|_{C^k} := \|(1 + \Delta)^{\frac{k}{2}} \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \|_{\infty}, $$
and that

$$ (1 + \Delta)^{\frac{k}{2}} \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} = \sum_{\lambda \in \hat{G}} (1 + c_2(\lambda))^{\frac{k}{2}} \left( \prod_{i=1}^{n} \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_{\lambda}} \right) \chi_{\lambda}, $$

which gives

$$ \| \rho_{s_1^n} \star \cdots \star \rho_{s_n^n} \|_{C^k} \le \sum_{\lambda \in \hat{G}} (1 + |c_2(\lambda)|)^{\frac{k}{2}} \left| \prod_{i=1}^{n} \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_{\lambda}} \right| \| \chi_{\lambda} \|_{\infty}. $$

□

Next, we will use the previous lemma to bound the sum using ideas from micro local analysis. There will be three separate regimes in each of which we will bound the terms differently.

**The small $\sqrt{c_2(\lambda)/n}$ regime.**

**Lemma 7.5.** There exists $M, \delta_M > 0$ such that for all pairs $(t, \lambda) \in \mathbb{R}_+ \times \hat{G}$ such that $\sqrt{tc_2(\lambda)} \le M$, we have $|\hat{\mu}_t(\lambda)| \le d_{\lambda}e^{-\delta_M c_2(\lambda)t}$.

*Proof.* First Taylor expand the character $\chi_{\lambda}$ near the identity using the pseudo-coordinates defined by $g = \exp \sum_i x^i(g)\xi_i$, where $(\xi_i)_i$ is a basis of $\mathfrak{g}$. This reads :

$$ \chi_{\lambda}(x) = \chi_{\lambda}(0) + x^i \partial_{x^i} \chi_{\lambda}(0) + \frac{x^i x^j}{2} \partial_{x^i x^j}^2 \chi_{\lambda}(0) + \mathcal{O}(|x|^3 \|\chi_{\lambda}\|_{C^3}). $$

Now we replace this Taylor expansion in the definition of $\hat{\mu}_t(\lambda)$, this reads

$$ \begin{aligned} \hat{\mu}_t(\lambda) &= \int_G (\chi_\lambda(0) + x^i \partial_{x^i} \chi_\lambda(0) + \frac{x^i x^j}{2} \partial_{x^i x^j}^2 \chi_\lambda(0) + \mathcal{O}(|x|^3 \|\chi_\lambda\|_{C^3})) \mu_t \\ &= \int_G (\chi_\lambda(0) + x^i \partial_{x^i} \chi_\lambda(0) - |x|^2 c_2(\lambda) (\chi_\lambda)(0) + \mathcal{O}(|x|^3 \|\chi_\lambda\|_{C^3})) \mu_t \\ &= d_\lambda (1 - c_2(\rho)t + \mathcal{O}(c_2(\lambda)^{\frac{3}{2}} t^{\frac{3}{2}})), \end{aligned} $$

where we observe that all the crossed terms $x^i(g)x^j(g)$ for $i \ne j$ integrated against the measure $\mu_t$ contribute $o(t)$, then we use the fact that our coordinates are related to the Casimir and the fact that characters are eigenfunctions of the Casimir–Laplace operator $\Delta_G$ on $G$ with eigenvalue $-c_2(\lambda)$ and we have used classical bounds on the Hölder–Besov norms of the characters. This implies the desired bound by since we can find a $\delta_M$ such that

$$ d_{\lambda}(1 - c_{2}(\lambda)t + \mathcal{O}(c_{2}(\lambda)^{\frac{3}{2}}t^{\frac{3}{2}})) \le d_{\lambda}e^{-\delta_{M}c_{2}(\lambda)t}. $$

□

**Lemma 7.6.** We have

$$ \sup_{n \ge 0} \sum_{\lambda \in \hat{G}} \frac{|c_2(\lambda)|^{\frac{k}{2}}}{\sqrt{c_2(\lambda)/n}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_{\lambda}} \right| < \infty. $$

*Proof.* We have

$$ \begin{aligned} \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_{\lambda}} \right| &\le \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i \in T_n(a.A)} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_{\lambda}} \right| \\ &\le \sum_{\lambda \in \hat{G}} |c_2(\lambda)|^{\frac{k}{2}} \exp \left( - \frac{|T_n(a, A)|}{n} A^2 \delta_M c_2(\lambda) \right) < \infty. \end{aligned} $$

□
The big $\sqrt{c_2(\lambda)/n}$ regime.

**Lemma 7.7.** There exists $M_2$ such that for all pairs $(t, \lambda) \in \mathbb{R}_+ \times \hat{G}$ such that $\sqrt{tc_2(\lambda)} \ge M_2$, we have
$$|\hat{\mu}_t(\lambda)| \le d_\lambda (1 + \sqrt{tc_2(\lambda)})^{-\frac{s}{2}}.$$

*Proof.* By our Fourier's decay assumption, There exists $C > 0$ and such that for all $\lambda \in \hat{G}$,

$$|\hat{\mu}_t(\lambda)| \le d_\lambda C (1 + \sqrt{tc_2(\lambda)})^{-s} \le d_\lambda C (1 + M_2)^{-\frac{s}{2}} (1 + \sqrt{tc_2(\lambda)})^{-\frac{s}{2}}.$$

The proof follows by taking $M_2$ as large as needed. $\square$

**Lemma 7.8.** We have

$$\sup_{n \ge 0} \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| < \infty.$$

*Proof.* We have in this case,

$$
\begin{align*}
\sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| &\le \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i \in T_n(a,A)} \left( 1 + \sqrt{s_i^n c_2(\lambda)} \right)^{-\frac{s}{2}} \\
&\le \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \left( 1 + A \sqrt{\frac{c_2(\lambda)}{n}} \right)^{-\frac{s|T_n(a,A)|}{2}} \\
&\le \sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} \frac{|c_2(\lambda)|^{\frac{k}{2}}}{\left( 1 + \sqrt{\frac{c_2(\lambda)}{n}} \right)^{\frac{s|T_n(a,A)|}{2}}} \\
&= \sum_{j \ge \log[2]} \frac{|c_2(\lambda)|^{\frac{k}{2}}}{\left( 1 + \sqrt{\frac{c_2(\lambda)}{n}} \right)^{\frac{s|T_n(a,A)|}{2}}} \\
&\le \sum_{j \ge \log[2]} \frac{2^{k(j+1)}}{\left( 1 + \sqrt{\frac{2j}{n}} \right)^{\frac{s|T_n(a,A)|}{2}}} \\
&= \sum_{j \ge \log[2]} \frac{2^{k(j+1)}}{\left( 1 + \sqrt{\frac{2j}{n}} \right)^{\frac{s|T_n(a,A)|}{2}}} |\{\lambda \in \hat{G}; 2^{2j} \le c_2(\lambda) < 2^{j+2}\}|.
\end{align*}
$$

By Weyl's law, there exists $\kappa > 0$ such that, for all $j \ge 0$, $|\{\lambda \in \hat{G}; 2^{2j} \le c_2(\lambda) < 2^{j+2}\}| \le \kappa 2^{\dim(G)}j$.
Assume now, without loss of generality that $2^\gamma := \sqrt{n}$ and $2^\alpha := \frac{M_2}{a}$, this gives

$$\sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| \le \sum_{j \ge \gamma + \alpha} 2^{kj + k + \dim(G)} j - \frac{s|T_n(a,A)|(j-\gamma)}{2}.$$

Choose $n_0$ big enough so that for all $n \ge n_0$, $k + \dim(G) - \frac{s|T_n(a,A)|}{2} \le -1 - k$, this gives

$$\sum_{\substack{\lambda \in \hat{G} \\ \sqrt{c_2(\lambda)/n} \ge \frac{M_2}{a}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| \le 2^{(k+\dim(G))(\gamma+\alpha)+k-\frac{s|T_n(a,A)|\alpha}{2}},$$

which is bounded uniformly in $n$ for $n$ large enough since $\liminf_{n \to \infty} \frac{|T_n(a,A)|}{n} > 0$. $\square$
## The intermediate regime.

**Lemma 7.9.** For all non trivial representation $\lambda$, we have the strict inequality $|\hat{\mu}_t(\lambda)| < d_\lambda$.

*Proof.* We use the inversion invariance of our measure $\mu_t$ and the unitarity of the representation $\lambda$:

$$ \int_G \chi_\rho \mu_t = \frac{1}{2} \int_G \chi_\rho(g) + \chi_\rho(g^{-1}) \mu_t = \int_G \mathrm{Re}\{\chi_\lambda(g)\} \mathrm{d}\mu_t(g). $$

Then the lemma follows from the fact that for $g$ outside a set of zero measure, we have $-d_\lambda < \mathrm{Re}\{\chi_\lambda(g)\} < d_\lambda$. $\square$

**Lemma 7.10.** We have

$$ \sup_{\substack{n \ge 0 \\ \frac{M_1}{a} \le \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} \sum_{\substack{\lambda \in \hat{G} \\ \frac{M_1}{a} \le \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| < \infty. $$

*Proof.* Set

$$ \delta := \sup_{\substack{\lambda \in \hat{G} \\ M \le \sqrt{c_2(\lambda)/n} \le M_2}} \frac{|\hat{\mu}_t(\lambda)|}{d_\lambda}. $$

By the previous lemma, $0 < \delta < 1$. Then, since $\log \delta < 0$,

$$ \begin{aligned} \sum_{\substack{\lambda \in \hat{G} \\ \frac{M_1}{a} \le \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \prod_{i=1}^{n} \left| \frac{\hat{\rho}_{s_i^n}(\lambda)}{d_\lambda} \right| &\le \sum_{\substack{\lambda \in \hat{G} \\ \frac{M_1}{a} \le \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \delta^n \le \sum_{\substack{\lambda \in \hat{G} \\ \frac{M_1}{a} \le \sqrt{c_2(\lambda)/n} \le \frac{M}{A}}} |c_2(\lambda)|^{\frac{k}{2}} \delta^{\frac{c_2(\lambda)a}{M_1}} \\ &\le \sum_{\lambda \in \hat{G}} |c_2(\lambda)|^{\frac{k}{2}} \exp \left( \frac{c_2(\lambda)a}{M_1} \log \delta \right) < \infty. \end{aligned} $$

*Proof of Proposition 7.3.* It is the direct consequence of the combination of Lemmas 7.4, 7.6, 7.8, and 7.10. $\square$

# 8 Technical estimates on Lie group valued random walks

In the next section, we want to study the scaling limit of some lattice gauge theories. Roughly speaking, a lattice gauge theory associated to a graph is a collection of random matrices, one for each face of the graph, and such that the law of each matrix depends on the area of the face it is associated to.

Therefore, to be able to define lattice gauge theories, we need a family of probability measures $(\mu_t)_{t>0}$, where $t$ is meant to be replaced later on by the area of the face.

The idea in the next section, is to show that a large class of lattice gauge theories, in the scaling limit, converge in some sense to the Yang-Mills measure. To do this, we will need some technical estimates on the family $(\mu_t)_{t>0}$ that we will do in the current section.

## 8.1 Assumption on the actions

First, we will discuss the assumptions on the family $(\mu_t)_{t>0}$. The set of assumptions is not optimal; however, we will stick to them to simplify the presentation. Consider a family $(\mu_t)_{t>0}$ of probability measures on $G$. We assume that

1. For all $t > 0$, $\mu_t$ is symmetric and ad-invariant.
2. For all $t > 0$, $\mu_t$ has a density $\rho_t$ with respect to the Haar measure on $G$.

3. For all $v, w \in \mathfrak{g}$,

$$ \int_G \langle v, \log a \rangle \langle w, \log a \rangle d\mu_t(a) = \langle v, w \rangle t + o_{v,w}(t). $$

4. For all integers $N$, there exists a constant $C_N > 0$ such that

$$ \sup_{g \in G} \left\{ \left( 1 + \frac{\operatorname{dist}(g, 1_G)}{\sqrt{t}} \right)^N \rho_t(g) \right\} \le C_N t^{-\frac{\operatorname{dim}(G)}{2}}. $$

5. For all $\beta > 0$, there exists $C_\beta > 0$ such that

$$ \forall t \le 1, \int_G |\log x|^{2\beta} \rho_t(x) dx \le C_\beta t^\beta. $$

Such a one parameter family is said to have the property (H). The letter H stands for heat like. Let us briefly comment on each of the assumptions.

* The symmetry in the first item means that in the lattice gauge theory associated to $(\mu_t)_{t>0}$, the law of the holonomy around a face does not depend on the cyclical order of the face. The Ad-invariance ensures the gauge invariance of the model.

* The second item allows easier handling of the problem without a big loss.

* The third item is essential because it characterizes the leading term of the central limit theorem.

* The fourth item means we have super-polynomial concentration of the measure near $1_G$. It important because, as we will see in the next section, we need a concentration property for products of matrices picked under the family $(\mu_t)_{t>0}$.

* The fifth item is not essential but facilitates many results and is verified by a wide family of actions. It says that the moments of $(\mu_t)_{t>0}$ behave like those of a Gaussian.

The following lemma is particularly useful because it tells us that computing integrals in a small (but large enough) ball near $1_G$ is almost the same as computing the integral over all of $G$.

**Lemma 8.1.** For all $\eta < \frac{1}{2}$, for all $p > 1$, there exists $C_{\eta,p}$ such that

$$ \mu_t(B(0, t^\eta)^c) \le C_{\eta,p} t^p. $$

*Proof.* Let $\delta > 0$, we have, for all integers $N$,

$$ \mu_t(B(0, \delta)^c) \le C_N t^{-\frac{\dim(G)}{2}} \int_{B(0, \delta)^c} \frac{1}{\left(1 + \frac{\operatorname{dist}(g, 1_G)}{\sqrt{t}}\right)^N} dg \le \frac{C_N t^{-\frac{\dim(G)}{2}}}{\left(1 + \frac{\delta}{\sqrt{t}}\right)^N} \le C_N t^{\frac{N-\dim(G)}{2}} \delta^{-N}. $$

Now take $\delta = t^{\frac{1-\epsilon}{2}}$, we get

$$ \mu_t(B(0, \delta)^c) \le C_N t^{\frac{N\epsilon - \dim(G)}{2}}. $$

It is enough to take $N$ big enough so that $N\epsilon - \dim(G) > 2p$. $\square$

When $\rho_t = p_t$ is the heat kernel, the measure enjoys the semi-group property. This tells us that for two random variables distributed according to $p_{t_1}dg$ and $p_{t_2}dg$ respectively, their product follows $p_{t_1+t_2}dg$. This gives an automatic bound on the concentration of the product near $1_G$. This property is very important in studying lattice gauge theories, and appears when merging faces. It fails when $\rho$ is no longer the heat kernel. The next lemma, which relies essentially on the BDG inequality, aims to compensate for this loss in the case of a general measure.
**Lemma 8.2.** Under the (H)-property, for all $\beta > 0$, there exists $C_\beta > 0$ such that

$$ \forall k \ge 1, \forall s_1, \dots, s_k \le 1, \int_G |\log x|^{2\beta} \rho_{s_1} * \dots * \rho_{s_k}(x) dx \le C_\beta (s_1 + \dots + s_k)^\beta. $$

*Proof.* The proof is essentially the same as the proof of lemma 5.4 in [24]. □

## 8.2 Technical estimates

In Section 9, we will show that lattice gauge theories converge to a universal limit. We will do this using characteristic functions. Here we describe the exact technical expression we need to bound, without further motivation. The motivation, however, will be clear from the next section.

Consider, for each $N \ge 0$, a sequence of positive real numbers $(s_i^{(N)})_{0 \le i \le 2^{N-1}}$. Let also $(X_i^{(N)})_{0 \le i \le 2^{N-1}}$ be a sequence of independent $G$-valued random variables such that

$$ \forall 0 \le i \le 2^N - 1, \mathbb{P}(X_i^{(N)} \in dg) = \mu_{s_i^{(N)}}(dg). $$

Let finally $(M_k^{(N)})_{0 \le k \le 2^{N-1}}$ be a family of elements in $\mathfrak{g}$ such that

$$ \max_{1 \le k \le 2^{N-1}} \|M_k^{(N)}\| \le 1. $$

We would like to understand the limiting behavior of

$$ E_{2^{N}-1}^{(N)} := \mathbb{E} \left[ \prod_{k=0}^{2^{N}-1} \exp i \left\langle \log \left( \prod_{i=0}^{k+1} X_i^N \right) - \log \left( \prod_{i=0}^{k} X_i^N \right), M_k^{(N)} \right\rangle \right], $$

under the assumptions on the family of measure, as it will be important in the identification of the limit of lattice gauge theories. We will proceed by induction.

**Lemma 8.3.** We have

$$ E_{2^{N}-1}^{(N)} := \mathbb{E} \left[ F_{s_{2^{N}-1}^{(N)}} \left( \prod_{i=0}^{2^{N}-2} X_i^N, M_{2^{N}-1}^{(N)} \right) \prod_{k=0}^{2^{N}-2} \exp i \left\langle \log \left( \prod_{i=0}^{k+1} X_i^N \right) - \log \left( \prod_{i=0}^{k} X_i^N \right), M_k^{(N)} \right\rangle \right], $$

where

$$ F_t(g, v) := \int_G \exp \left( i \langle v, \log(ag) - \log(g) \rangle \right) d\mu_t(a). $$

*Proof.* By usual properties of the conditional expectation. □

The strategy will be to decompose $F_t(g, v)$ in the sum of two terms, a leading term, and an error term that we will bound tightly. This will induce the decomposition of

$$ F_{s_{2^{N}-1}^{(N)}} \left( \prod_{i=0}^{2^{N}-2} X_i^N, M_{2^{N}-1}^{(N)} \right) $$

in two terms as well :

* a leading deterministic term that will happen to be

$$ A_{2^{N}-1}^{(N)} := 1 - \frac{\|M_{2^{N}-1}^{(N)}\|^2}{2} s_{2^{N}-1}^{(N)}. $$
• an error random term $R^{(N)}$. The random part of the error term is a polynomial function (with possible fractional powers) of $\prod_{i=0}^{2^N-2} X_i^{N}$, which follows $\mu_{s_0^{(N)}} \star \cdots \star \mu_{s_{2^N-2}^{(N)}}$. Therefore, Lemma 8.2 lets us easily bound it.

This will tell us that

$$E_{2^N-1}^{(N)} = \left( 1 - \frac{\|M_{2^N-1}^{(N)}\|^2}{2} s_{2^N-1}^{(N)} \right) E_{2^N-2}^{(N)} + \mathbb{E}[|R^{(N)}|],$$

and will let us start with the induction.

The aim of the two following lemmas is to decompose $F_t(g, v)$.

**Lemma 8.4.** Let $\delta > 0$ be such that $\exp : B(1_G, \delta) \to \mathfrak{g}$ is a diffeomorphism on its image. Let $p$ be an integer greater than 2. There exists $C_p > 0$ such that if $|X|, |Y| \le \delta$,

1. there exists $\Lambda_Y \in \text{End}(\mathfrak{g})$, such that $|\log(\exp X \exp Y) - X - Y - \Lambda_Y(X)| \le C_p (|X|^2|Y| + |X||Y|^p)$;

2. we have $|\log(\exp X \exp Y) - X - Y| \le C_p |X||Y|$.

*Proof.* It is a direct consequence of the BCH formula taken up to order $p$. $\square$

**Remark 2.** Note that the presence of the linear function $\Lambda_Y$ can absorb as many higher-order commutator terms containing exactly one $X$, enabling us to increase the power $p$ in $Y$ on the right side as needed. This will be very useful because, thanks to symmetry, $\int_G \Lambda_Y(\log a) \mu_t(da) = 0$, and we can therefore add this term whenever we need it, gaining powers in $Y$.

**Lemma 8.5.** For each $p \ge 2$, there exists $C_p > 0$ such that, for all $t \in [0, 1]$, $g \in G$ and $v \in \mathfrak{g}$,

$$F_t(g, v) = 1 - \frac{\|v\|^2}{2} t + R_t(g, v),$$

where,

$$|R_t(g, v)| \le C_p((1 + \|v\|^2) |\log g|t + |\log g|^p \sqrt{t} + (\|v\| + \|v\|^2)t^p + \|v\|^3 t^{\frac{3}{2}} + \|v\|^2 |\log g|^2 t).$$

**Remark 3.** In the subsequent of this section, the notation $O(r)$ means a quantity whose absolute value is smaller than an absolute constant times $r$. An absolute constant is a constant that only depends on the fixed settings (such as the diameter of $G$).

*Proof.* We have

$$\begin{aligned} F_t(g, v) &= \int_G \exp i \langle v, \log(ag) - \log(g) \rangle_{\mathfrak{g}} d\mu_t(a) \\ &= \int_{B(0, t^\eta)} \exp i \langle v, \log(ag) - \log(g) \rangle_{\mathfrak{g}} d\mu_t(a) + \int_{B(0, t^\eta)^c} \exp i \langle v, \log(ag) - \log(g) \rangle_{\mathfrak{g}} d\mu_t(a). \end{aligned}$$

Fix $\eta < \frac{1}{2}$. From Lemma 8.1, the second member of the second side verifies, for all $p$,

$$\int_{B(0, t^\eta)^c} \exp i \langle v, \log(ag) - \log(g) \rangle_{\mathfrak{g}} d\mu_t(a) \le \mu_t(B(0, t^\eta)^c) \le C_p t^p.$$

To treat the first member, let us develop the exponential. We have for all $g \in B(0, t^\eta)$,

$$\exp i \langle v, \log(ag) - \log(g) \rangle = 1 + i \langle v, \log(ag) - \log(g) \rangle - \frac{\langle v, \log(ag) - \log(g) \rangle^2}{2} + O(\|v\|^3 |\log a|^3).$$
Now,

$$ \int_{B(0,t^n)} 1 d\mu_t(a) = \int_G 1 d\mu_t(a) - \int_{B(0,t^n)^c} 1 d\mu_t(a) = 1 + O(C_p t^p). $$

Moreover,

$$ \int_G \langle v, \log(ag) - \log(g) \rangle d\mu_t(a) = \int_G \langle v, \log(ag) - \log(g) \rangle d\mu_t(a) + O(2 \operatorname{diam}(G) \|v\| C_p t^p). $$

Now, using the fact that for $\Gamma \in \mathfrak{g}^*$, $\int_G \Gamma(\log x)\mu_t(dx) = 0$, we have, from the first item of the Lemma 8.4,

$$ \begin{aligned} \int_G \langle v, \log(ag) - \log(g) \rangle d\mu_t(a) &= \int_G \langle v, \log(ag) - \log(g) - \log(a) - \Lambda_{\log(g)}(\log(a)) \rangle d\mu_t(a) \\ &= O \left( C_p \int_G \left( |\log g| |\log a|^2 + |\log a| |\log g|^p \right) d\mu_t(a) \right), \end{aligned} $$

and from the fifth assumption on the family of measures,

$$ \int_G \langle v, \log(ag) - \log(g) \rangle d\mu_t(a) = O(C_p |\log g| t + C_p \sqrt{t} |\log g|^p). $$

Similarly,

$$ \int_{B(0,t^n)} \langle v, \log(ag) - \log(g) \rangle^2 d\mu_t(a) = \int_G \langle v, \log(ag) - \log(g) \rangle^2 d\mu_t(a) + O(\|v\|^2 C_p t^p). $$

and

$$ \int_G \langle v, \log(ag) - \log(g) \rangle^2 d\mu_t(a) = |v|^2 t + O(\|v\|^2 |\log g|^2 t + \|v\|^2 |\log g|t). $$

We conclude the proof by combining all the terms together. □

**Lemma 8.6.** Recall that $A_k^{(N)} := 1 - \frac{\|M_k^{(N)}\|^2}{2} s_k^{(N)}$. For all $p > 1$, we have

$$ \left| E_{2^{N}-1}^{(N)} - \prod_{i=0}^{2^{N}-1} A_{i}^{(N)} \right| \le C_p \left( s_{N,1}^{\frac{3}{2}} + s_{N,\frac{1}{2}} s_{N,1}^{\frac{p}{2}} + s_{N,p} + s_{N,\frac{3}{2}} + s_{N,1}^{2} \right) $$

where $s_{N,\alpha} := \sum_{i=0}^{2^N-1} (s_i^{(N)})^\alpha$.

*Proof.* We have from Lemmas 8.3 and 8.5,

$$ E_{2^{N}-1}^{(N)} = A_{2^{N}-1}^{(N)} E_{2^{N}-2}^{(N)} + \mathbb{E} \left[ R_{s_{2^{N}-1}^{(N)}} \left( \prod_{i=0}^{2^{N}-2} X_i^{N}, M_{2^{N}-1}^{(N)} \right) \prod_{k=0}^{2^{N}-2} \exp i \left\langle \log \left( \prod_{i=0}^{k+1} X_i^{N} \right) - \log \left( \prod_{i=0}^{k} X_i^{N} \right), M_k^{(N)} \right\rangle \right] $$

Therefore,

$$ |E_{2^{N}-1}^{(N)} - A_{2^{N}-1}^{(N)} E_{2^{N}-2}^{(N)}| \le \mathbb{E} \left[ \left| R_{s_{2^{N}-1}^{(N)}} \left( \prod_{i=0}^{2^{N}-2} X_i^{N}, M_{2^{N}-1}^{(N)} \right) \right| \right]. $$

We apply Lemma 8.2 and the previous lemma, to get

$$ \begin{aligned} |E_{2^{N}-1}^{(N)} - A_{2^{N}-1}^{(N)} E_{2^{N}-2}^{(N)}| & \lesssim s_{2^{N}-1}^{(N)} \cdot (s_{0}^{(N)} + \cdots + s_{2^{N}-1}^{(N)})^{\frac{1}{2}} + (s_{2^{N}-1}^{(N)})^{\frac{1}{2}} \cdot (s_{0}^{(N)} + \cdots + s_{2^{N}-1}^{(N)})^{\frac{p}{2}} \\ & \quad + (s_{2^{N}-1}^{(N)})^{p} + (s_{2^{N}-1}^{(N)})^{\frac{3}{2}} + s_{2^{N}-1}^{(N)} \cdot (s_{0}^{(N)} + \cdots + s_{2^{N}-1}^{(N)}) \end{aligned} $$

where the constant in $C_p(\max(\|v\|^3, 1))$. We get the desired result by triangular inequality. □
## 8.3 Classical lattice gauge actions

In this subsection we discuss some historically important actions that were used in both, physicist and mathematics literature. We will prove that all these actions verify the $(H)$-property. This, together with Section 10 conclude the proof of Theorem 1.1. In practice, the proof consists in reducing the computation to the Gaussian case on the Lie algebra with a small error, thanks to the following lemma.

**Lemma 8.7.** Let $G \subset U(N)$ be a compact Lie group with lie algebra $\mathfrak{g}$. Let $U$ be a neighborhood of $1_G$ of diameter less than 1, for which $\log$ is a diffeomorphism on its image. Then, there exists a constant $C$ such that for any positive bounded measurable function $f: U \subset G \to \mathbb{R}$,

$$ \left| \int_U f(g)dg - \int_{\log U} f(\exp X)dX \right| \le C \int_{\mathfrak{g}} |X|^2 f(\exp X)dX. $$

*Proof.* By a change of variables $g \mapsto \log g$, we have

$$ \int_U f(g)dg = \int_{\log U} f(\exp X)|J(X)|dX, $$

where $|J(X)| = \det(d\exp(g))$. However, we have $|1 - |J(X)|| \le C|X|^2$. This gives

$$ \left| \int_U f(g)dg - \int_{\log U} f(\exp X)dX \right| \le C \int_{\log U} |X|^2 f(\exp X)dX \le C \int_{\mathfrak{g}} |X|^2 f(\exp X)dX. $$

**The Villain action.** The heat kernel. Let for all $t > 0$, $\mu_t(dg) := p_t(g)dg$, where $p_t$ is the heat kernel on $G$. The heat kernel verifies all the assumptions and it was the standard choice use by Lévy for the construction of the measure.

**The Manton action.** Consider $t > 0$ and define

$$ \mu_t(dg) = \frac{1}{Z_m(t)} \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg, \text{ with } Z_m(t) := \int_G \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg. $$

**Proposition 8.8.** We have $Z_m(t) = (2\pi t)^{\frac{\dim(G)}{2}} + O(t^{\frac{\dim(G)}{2}+1})$.

*Proof.* We have

$$ \begin{aligned} Z_m(t) &= \int_G \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg = \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg + \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right)^c \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg \\ &= \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left( -\frac{d(g, 1_G)^2}{2t} \right) dg + O(\exp(-t^{-\epsilon})). \end{aligned} $$

Now using Lemma 8.7,

$$ \begin{aligned} \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left(-\frac{d(g, 1_G)^2}{2t}\right) dg &= \int_{\log B} \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left(-\frac{|X|^2}{2t}\right) |J(X)| dX \\ &= \int_{\log B} \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left(-\frac{|X|^2}{2t}\right) (1 + O(|X|^2)) dX = \int_{\log B} \left(0, t^{\frac{1-\epsilon}{2}}\right) \exp \left(-\frac{|X|^2}{2t}\right) dX + O\left(t^{\frac{\dim(G)}{2}+1}\right) \\ &= \int_{\mathfrak{g}} \exp \left(-\frac{|X|^2}{2t}\right) dX + O\left(\exp(-t^{-\epsilon})\right) + O\left(t^{\frac{\dim(G)}{2}+1}\right) = (2\pi t)^{\frac{\dim(G)}{2}} + O\left(t^{\frac{\dim(G)}{2}+1}\right). \end{aligned} $$
**Proposition 8.9.** For all $v \in \mathfrak{g}$,

$$ \int_G \langle v, \log a \rangle^2 d\mu_t(a) = |v|^2 t + o(t). $$

*Proof.* Let $v \in \mathfrak{g}$, we have

$$ \begin{aligned} \int_G \langle v, \log a \rangle^2 d\mu_t(a) &= \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \langle v, \log a \rangle^2 d\mu_t(a) + \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right)^c \langle v, \log a \rangle^2 d\mu_t(a) \\ &= \frac{1}{Z_m(t)} \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \langle v, X \rangle^2 \exp\left(-\frac{|X|^2}{2t}\right) (1 + O(|X|^2)) dX + O(\exp(-t^{-\epsilon})). \end{aligned} $$

Now,

$$ \begin{aligned} & \frac{1}{Z_m(t)} \int_B \left(0, t^{\frac{1-\epsilon}{2}}\right) \langle v, X \rangle^2 \exp\left(-\frac{|X|^2}{2t}\right) (1 + O(|X|^2)) dX \\ & = \frac{(2\pi t)^{\frac{\dim(G)}{2}}}{(2\pi t)^{\frac{\dim(G)}{2}} + O(t^{\frac{\dim(G)}{2}+1})} (|v|^2 t + o(t)) = (1 - O(t)) (|v|^2 t + o(t)) = |v|^2 t + o(t). \end{aligned} $$

□

Therefore, we get the following.

**Proposition 8.10.** The Manton family has the property (H).

**The Wilson action.** The Wilson action. In this example, we assume $G = U(N)$. In this case, for $g \in U(N)$, we have, when $g$ is close to the identity,

$$ d(g, 1)^2 = |\log g|^2 = \frac{1}{2}\mathrm{Tr}(\log g)(\log g)^* \approx \frac{1}{2}\mathrm{Tr}(1-g)(1-g)^* = -\frac{1}{2}\mathrm{Tr}(1-g^*-g+gg^*), $$

which gives

$$ d(g, 1)^2 \approx \frac{1}{2}\mathrm{Tr}(1-g^*) + \frac{1}{2}\mathrm{Tr}(1-g) = \mathrm{Re}\mathrm{Tr}(1-g). $$

This motivates the Wilson actions which is defined, for all $t > 0$ as

$$ \mu_t(dg) = \frac{1}{Z_w(t)} \exp \left( - \frac{\mathrm{Re} \mathrm{Tr}(1-g)}{t} \right), \quad \text{with } Z_w(t) := \int_G \exp \left( - \frac{\mathrm{Re} \mathrm{Tr}(1-g)}{t} \right) dg. $$

We can show as well the following.

**Proposition 8.11.** We have $Z_w(t) = (2\pi t)^{\frac{\dim(G)}{2}} + O(t^{\frac{\dim(G)}{2}+1})$, and the Wilson family verifies conditions (H).

# 9 Scaling limit of lattice gauge theories on surfaces

Let $\Sigma$ be a surface with one outgoing boundary component, and $G$ be a Lie group. Take a Morse function verifying the Morse–Smale conditions, blow up the maximum, which gives a new surface $\bar{\Sigma}$, and consider the corresponding lattice introduced in Section 6. Fix for the moment an integer $N$ and consider the Morse lattice of resolution $N$. Let us introduce a few notations.

* Call the graph obtained $\Lambda^N$, the set of its edges $E^N$, and the set of its faces $F^N$.
* Denote by $H^N$ the set of its horizontal edges. This is the set of edges that are pieces of level sets of the Morse function.

* Denote by $V^N$ the set of vertical edges, i.e. that are pieces of flow lines of the gradient of the Morse function.

We consider now on $G$ a family of probability measures $(\mu_t := \rho_t dg)_{t>0}$ verifying the condition $(H)$. We consider the following lattice gauge theory, which is a probability measure on $G^{EN}$.

**Definition 9.1 (Discrete Yang–Mills measure).** The discrete Yang–Mills measure on $\Lambda^N$ associated with the family $\mu$ is the probability measure on $G^{EN}$ given by

$$dM_{\mu;N}((g_e)_{e \in E^N}) := \frac{1}{Z_N} \prod_{f \in F^N} \rho_{\sigma(f)}(\mathrm{Hol}(g, \partial F)dg,$$

where $\mathrm{Hol}(g, \partial F) := \prod_{e \in \partial F} g_e$, and $Z_N$ is a normalization constant. Note this product is well-defined and can be taken in any order thanks to the invariance by inversion and conjugation of the measures $\mu_t$.

The goal is to show that for every step $N$, one is able to define a random 1-form in Morse gauge from $M_{\mu;N}$ which

* represents faithfully the lattice gauge measure, meaning that the law of the holonomies for any finite sets of closed loops in $\Lambda^N$ is the same for the two measures;

* show that as $N \to \infty$, this random 1-form converges to a universal limit which is the Morse-gauge fixed, free boundary Yang–Mills measure defined in (5.1).

To do so, we will need a discrete analogue of the Morse gauge that will be introduced in the next section.

## 9.1 A discrete Morse gauge

Let us recall the Morse lattice introduced in Section 6. We will assume for simplicity that the critical sets are at integer height, meaning that in the discretizations $\Lambda^N$ of $\Sigma$, the height $r$ runs over

$$R^N := \left\{ c \frac{k}{2^N}; 0 \le k \le 2^N - 1, c \in \{1, \dots, 2g+1\} \right\},$$

and $\theta$ runs over

$$\Theta^N := \left\{ c \frac{k}{2^N}; 0 \le k \le 2^N - 1, c \in \{1, \dots, 2g\} \right\}.$$

We will also denote by

$$R_{-}^{N} := R^{N} \setminus \{(2g+1)2^{N}-1\} \text{ and } \Theta_{-}^{N} := \Theta^{N} \setminus \{2g \cdot 2^{N}-1\},$$

respectively, the set of all levels except the last one, and all flow lines except the last one. We will denote the vertices by their coordinates $(r, \theta)$, and the horizontal edge between two vertices $(r, \theta_1)$ and $(r, \theta_2)$ by $\theta_1 \xrightarrow{r} \theta_2$. Similarly, vertical edges by $r_1 \xrightarrow{\theta} r_2$. Moreover, for $r \in R_{-}^N$ and $\theta \in \Theta_{-}^N$ we will denote by

$$r_{+}^{(N)} := r + 2^{-N} \text{ and } \theta_{+}^{(N)} := \theta + 2^{-N}.$$

To simplify the notations, when it is clear that we are working in the resolution $N$ we will only denote $r_+^{(N)}$, and $\theta_+^{(N)}$ by $r_+$ and $\theta_+$. Therefore, a configuration, which is an element of $G^{EN}$, can be represented as

$$(H^N, V^N) := \left( \left( (M_{\theta \xrightarrow{r} \theta_+})_{\theta \in \Theta_{-}^N}, (M_{\theta \xrightarrow{r} r_+})_{\theta \in \Theta_{-}^N} \right), \right)$$
<figure><img src="image_27.png" /><figcaption>Figure 24: Conflict in fixing the gauge at saddle points.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>Figure 25: Inserting the bands.</figcaption></figure>

where we see $M$ as a function $E^N \to G$. For example, the discrete Yang-Mills measure on $\Lambda^N$ associated to $\mu$ is the law of $M$ when edge holonomies are sampled under $M_{\mu,N}$.

It would seem natural to say that a configuration on $\Lambda^N$ is in discrete Morse gauge if $V^N \equiv 1_G$. However, this definition is problematic because, for a general configuration, it may be impossible to perform a gauge transformation to reach this condition. The obstruction occurs precisely at the saddle points of the Morse function. There, solving for $V \equiv 1_G$ can give two conflicting solutions for the gauge transform, depending on which path from the bottom to the saddle point is chosen to fix the gauge (see Figure 24). This issue mirrors the one encountered in the continuum case discussed in Section 2.3.

The idea is to ensure that for every point on $\bar{\Sigma}$, there is a unique path connecting the bottom of the surface to the boundary component. This is achieved by cutting the surface along the green line in Figure 24 and inserting a band. In the planar representation, this corresponds to cutting inside the red lines, inserting a small rectangle, and then identifying the blue points to reconstruct the surface, as shown in Figure 25.

This construction results precisely in the Morse lattice on the resolved surface, which we will also call the resolved lattice. For a surface of genus $g$, we add $2g$ bands $(B_i)_{1 \le i \le 2g}$. These bands introduce new bonds into the lattice, which must be accounted for. This leads to a slight modification of a configuration, as described in the following definition.

**Definition 9.2 (Resolved Configuration).** A configuration in the discrete Morse gauge on the resolved lattice is an element of $G^{H_N} \times \prod_{i=1}^{2g} G^{B_i^N}$ where $H_N$ is the set of horizontal edges of the unresolved lattice, and $B_i^N$ is the set of horizontal edges inside the band $B_i$.

Now, the following proposition is straightforward.

**Proposition 9.3 (Discrete Morse Gauge).** For any configuration $M := (M_e)_{e \in E^N}$ on the unresolved Morse lattice, there exists a unique gauge transformation on the resolved surface such that $g(0) = 1$ and
g · M is a configuration in the discrete Morse gauge on the resolved lattice. This gauge transform is given by

$$ \forall r \in \mathbb{R}^N, \theta \in \Theta^N, g(r, \theta) = \left( \prod_{\substack{\rho \in \mathbb{R}^N \\ \rho < r}} M_{\rho \to \rho_+} \right)^{-1}. $$

Now, we define the Morse gauge fixed discrete Yang-Mills measure.

**Definition 9.4.** The Morse-Gauge fixed Yang-Mills measure $M_{\text{Morse};\mu;N}$ associated to the family $\mu$, is the probability measure on $G^{H^N} \times \prod_{i=1}^{2g} G^{B_i^N}$ given by

$$ \begin{aligned} & dM_{\text{Morse};\mu;N} \left( \left( M_{\theta \xrightarrow{r} \theta_+} \right)_{r \in \mathbb{R}^N} \left( \left( (U_b)_{b \in B_i^N} \right)_{1 \le i \le 2g} \right) \right) \\ & := \frac{1}{Z_N} \prod_{\substack{\theta \in \Theta_N \\ r \in \mathbb{R}^N}} \rho_{\sigma(\Box(r,r_+,\theta,\theta_+))} \left( M_{\theta \xrightarrow{r_+} \theta_+}^{-1} M_{\theta \xrightarrow{r_+} \theta_+} \right) dM \prod_{1 \le i \le 2g} \prod_{b \in B_i} \delta_{U_b, u_i} du_i. \end{aligned} $$

Let us comment on this definition.

* In our construction, each band is a narrow region tracing an unstable line. This means that the band should be treated as a region with zero area. Therefore, any closed face lying entirely within a band, has to have a trivial holonomy. Since we are in the Morse gauge, we let the vertical edges delimitating these bands have trivial holonomy, however, We choose to assign one independent uniform random variable to the entire band, and set the value of every horizontal edge within that band to this same variable. Therefore, in a closed loop inside the surface, these uniform variables cancel in pairs, resulting in a total holonomy of 1.

* This definition ensures that the joint law of the holonomies around any collection of finite loops in $\Lambda^N$ under $M_{\text{Morse};\mu;N}$ and $M_{\mu;N}$ are the same.

In the next subsection, we explain how to get at fixed $N$ a random 1-form that faithfully represents $M_{\text{Morse};\mu;N}$.

## 9.2 Fixed N analysis

Let $(M, U)$ be a configuration in Morse gauge on the resolved lattice of resolution $N$. We want to define a 1-form $A_{(M,U)}^{(N)}$ on the surface such that its holonomies agree with the Yang-Mills measure. We will proceed as follows: we will always work on the resolved surface, and then use the analytical results of the earlier sections to pull the results back on the original surface.

We will start by decomposing $A_{(M,U)}^{(N)}$ as the sum of two one forms: one that lives on the actual surface $A_{M,\Sigma}^{(N)}$, and one that lives inside the bands of zero area $A_{U,\Sigma}^{(N)}$ giving

$$ A_{(M,U)}^{(N)} = A_{M}^{(N)} + A_{U}^{(N)}. $$

First, we define

$$ A_{U}^{(N)}(x) = \left( \sum_{i=1}^{2g} \frac{\log(U_i)}{\text{length}(B_i)} \mathbf{1}_{x \in B_i} \right) d\theta, $$

where $\text{length}(B_i)$ is the thickness of the band $B_i$.
Then, we start by defining $A_M$ on the skeleton of the graph by putting

$$A_M^{(N)}(x) := \left( \sum_{e \in \text{edges}} \frac{\log M_e}{\text{length}(e)} 1_{x \in e} \right) \text{d}\theta,$$

This gives, for all $r \in \mathbb{R}^N$, a piecewise constant function $\theta \mapsto A_M^{(N)}(r, \gamma)$ which we interpolate in a piecewise affine function in $r \in \mathbb{R}^N$. This gives

$$A_M^{(N)}(u, \gamma) = 2^N \sum_{r \in \mathbb{R}_-^N} \left( (u-r)A_M^{(N)}(r_+, \gamma) + (r_+ - u)A_M^{(N)}(r, \gamma) \right) 1_{[r, r_+]}(u).$$

It is clear that the holonomy induced by $A_{(M,U)}^{(N)}$ on the horizontal edges is exactly the configuration. Now, consider the mapping

$$D_N : (M, U) \mapsto A_{(M,U)}^{(N)}.$$

The push forward $(D_N)_* \mathbb{M}_{\text{Morse};\mu;N}$ of the discrete Morse gauge fixed Yang-Mills measure gives rise to a random connection in $\Omega^1(\Sigma, \mathfrak{g})$. It is exactly this sequence of random connections for which we would like to show in the next section the convergence in law towards the Yang-Mills measure, i.e. the formula in (5.1).

## 9.3 Limit in law

The idea is to study the limit of $A_{(M,U)}^{(N)} = A_M^{(N)} + A_U^{(N)}$, seen as a sequence of probability measure in the set of $g$-valued currents of degree 1 on $\Sigma$. This set denoted by $\mathcal{D}'_1(\Sigma, \mathfrak{g})$, is the topological dual of the set $\Omega_0^1(\Sigma, \mathfrak{g})$ of smooth $g$-valued 1-forms on $\Sigma$, vanishing up to all orders at $\partial\Sigma$, equipped with the $C^\infty$-topology. Roughly speaking, it has the same properties as the Schwartz distribution, and it is somehow differential forms with distributional coefficients.

Since the two terms of the decomposition are (statistically) independent, they can be studied separately from the point of view of convergence in law.

It is clear that $A_U^{(N)}$ converges almost surely to the unstable currents, the leftmost part of (5.1). The remaining part is to study the convergence of first term. Indeed, we will show the following theorem in what remains of this subsection.

**Theorem 9.5.** The sequence of 1-forms $(A_M^{(N)})_{N \ge 1}$ converges in $\mathcal{M}(\mathcal{D}'_1(\text{Cyl}, \mathfrak{g}))$ to the random 1-form $(\partial_\theta \langle \xi, 1_{\square(r,\theta)} \rangle) \text{d}\theta$, defined in (5.1).

The proof requires several steps. The first one is to note that in the continuum, the gauge fixed Yang-Mills measure was nothing but a De Rham primitive of a white noise. There is a discrete analogue of this fact which will make the computations easier. In fact, let

$$\xi_N(u, \gamma) = 2^{2N} \sum_{\substack{r \in \mathbb{R}_-^N \\ \theta \in \Theta_-^N}} \left( \log \left( M_{\theta \xrightarrow{r_+} \theta_+} \right) - \log \left( M_{\theta \xrightarrow{r} \theta_+} \right) \right) 1_{[\theta, \theta_+]}(\gamma) 1_{[r, r_+]}(u),$$

which is a discrete approximation of the white noise. We have

$$\left( \int_0^t \xi_N(u, \gamma) du \right) d\gamma = A_U^{(N)}(t, \gamma).$$

This means that $A_U^{(N)} = \iota_V \mathcal{L}_V^{-1} (\xi_N du \wedge d\gamma)$, where $\iota_V \mathcal{L}_V^{-1}$ is just integration w.r.t. the $r$ variable in our systems of pseudo-coordinates, and $\mathcal{L}_V^{-1}$ is the inverse of the Lie derivative operator $\mathcal{L}_V$, where $V = \nabla f$ is the gradient of the Morse flow. Thereofre, the problem is reduced to study the convergence of $\xi_N$ and to prove the continuity of $\iota_V \mathcal{L}_V^{-1}$. Let's start with the continuity.
**Lemma 9.6.** Let $V = \partial_r$ be the radial vector field in the cylinder **Cyl**. The linear function

$$ \iota_V \mathcal{L}_V^{-1} : \mathcal{D}'_2(\mathbf{Cyl}, \mathfrak{g}) \to \mathcal{D}'_1(\mathbf{Cyl}, \mathfrak{g}) $$

is continuous, where similarly to $\mathcal{D}'_1(\mathbf{Cyl}, \mathfrak{g})$, the set $\mathcal{D}'_2(\mathbf{Cyl}, \mathfrak{g})$ is the topological dual of smooth 2-forms on **Cyl** vanishing up to all orders at $\partial\mathbf{Cyl}$.

*Proof.* The definition of $\iota_V \mathcal{L}_V^{-1}$ is just by duality, and the continuity is straightforward. □

**Remark 4.** In fact we can tell a bit more. For the moment, the operator $\iota_V \mathcal{L}_V^{-1}$ is defined only on distributions that are dual to test functions that do not see the boundary of the cylinder since they vanish at infinite order. We can extend $\iota_V \mathcal{L}_V^{-1}$ to our anisotropic spaces $W_{\text{loc}}^{\alpha, \alpha-1, p}(\mathbf{Cyl} \setminus \Psi(\text{Crit}(f)_1))$, $\alpha \in (0, \frac{1}{2})$ of distributions that feel the boundary. We prove in the appendix the fact that this space injects continuously in $C_{\text{loc}}^{\beta}(\mathbf{Cyl} \setminus \Psi(\text{Crit}(f)_1))$ for some $\beta > -1$. For any $\beta > -1$, the linear map

$$ \iota_V \mathcal{L}_V^{-1} : C_{\text{loc}}^{\beta}(\mathbf{Cyl} \setminus \Psi(\text{Crit}(f)_1)) \to C_{\text{loc}}^{\beta}(\mathbf{Cyl} \setminus \Psi(\text{Crit}(f)_1)) $$

is linear continuous. Let $T$ be a distribution on $\mathbb{R}$ which is in the Hölder-Besov space $C_{\text{loc}}^{\beta}$ for $\beta > -1$. Then we can define the distribution $T1_{\mathbb{R} \ge 0}(r)$ by restriction to test functions in $C_c^\infty(\mathbb{R} \ge 0)$ then by extension procedure by 0 on $\mathbb{R} > 0$, in such a way that the restriction plus extension map is linear continuous. The proof is just a consequence of Lemma A.1 in the Appendix.

The second step, is to study the convergence of $(\xi_N)_{N \ge 1}$ when seen as a sequence of measures in $\mathcal{D}'_2(\Sigma, \mathfrak{g})$. To show this, we will use Fernique's theorem [40, Théorème III.6.5]. This theorem states that a sequence of random distribution converges in law –in the topology of Shwartz distributions – if and only if the sequence characteristic functions converges to a continuous functional. Therefore, we need to study the convergence of

$$ \mathbb{E}[\exp(i\langle\xi_N, \psi\rangle)], $$

for some test function $\psi$. This is what we will do in the remaining of this section, and we will use the results of Section 8.2. Let us first prove a key lemma explaining the general form of $\mathbb{E}[\exp(i\langle\xi_N, \psi\rangle)]$.

**Lemma 9.7.** Let $\psi \in C^\infty(\mathbf{Cyl}, \mathfrak{g})$. Define

$$ \forall r_1, r_2, \theta_1, \theta_2, \quad K_{r_1, r_2, \theta_1, \theta_2} \psi := \frac{1}{(r_2 - r_1)(\theta_2 - \theta_1)} \int_{\Sigma} \psi(u, \gamma) 1_{[\theta_1, \theta_2]}(\gamma) 1_{[r_1, r_2]}(u) du d\gamma. $$

We have

$$ \mathbb{E}[\exp(i\langle\xi_N, \psi\rangle)] = \prod_{\theta \in \Theta_N} \left\{ \prod_{r \in R_N} \left( 1 - \frac{|K_{r, r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) + O_{N, \theta} \right) + O_{N, \theta} \right\}, $$

where $\sum_{\theta \in \Theta_N} |O_{N, \theta}| \xrightarrow{N \to \infty} 0$.

*Proof.* We have

$$ \langle \xi_N, \psi \rangle = \sum_{\substack{r \in R_N \\ \theta \in \Theta_N}} \left\langle \log \left( M_{\theta \xrightarrow{r \to \theta_+}} \right) - \log \left( M_{\theta \xrightarrow{r \to \theta_+}} \right), K_{r, r_+, \theta, \theta_+} \psi \right\rangle. $$

Let us calculate the characteristic function. We have from the independence in $\theta$, and from Lemma 8.6,

$$ \begin{aligned} \mathbb{E}[\exp(i\langle\xi_N, \psi\rangle)] &= \prod_{\theta \in \Theta_N} \mathbb{E} \left[ \prod_{r \in R_N} \exp \left( i \left\langle \log \left( M_{\theta \xrightarrow{r \to \theta_+}} \right) - \log \left( M_{\theta \xrightarrow{r \to \theta_+}} \right), K_{r, r_+, \theta, \theta_+} \psi \right\rangle \right) \right] \\ &= \prod_{\theta \in \Theta_N} \left\{ \prod_{r \in R_N} \left( 1 - \frac{|K_{r, r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) + O_{N, \theta} \right) + O_{N, \theta} \right\}, \end{aligned} $$
where

$$O_{N,\theta} = O\left\{ \left( \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+}) \right)^{\frac{3}{2}} + \sum_{r \in R_{i}^{N}} \sigma(r, r_{+}, \theta, \theta_{+})^{\frac{1}{2}} \left( \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+}) \right)^{p} \right. \\ \left. + \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+})^{p} + \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+})^{\frac{3}{2}} + \left( \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+}) \right)^{2} \right\}$$

We have

$$O_{N,\theta} = O\left\{ \max_{\theta \in \Theta_{-}^{N}} (\sigma(0, 2g + 2, \theta, \theta_{+}))^{\frac{1}{2}} \left( \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+}) \right) + 2^{-\frac{N}{2}} 2^{-N\beta p} 2^{N} \right. \\ + \left( \sup_{\substack{r \in R_{-}^{N} \\ \theta \in \Theta_{-}^{N}}} \sigma(r, r^{+}, \theta, \theta^{+}) \right)^{p-1} \sigma(0, 2g, \theta, \theta_{+}) + \left( \sup_{\substack{r \in R_{-}^{N} \\ \theta \in \Theta_{-}^{N}}} \sigma(r, r^{+}, \theta, \theta^{+}) \right)^{\frac{1}{2}} \sigma(0, 2g, \theta, \theta_{+}) \\ + \left. \max_{\theta \in \Theta_{-}^{N}} (\sigma(0, 2g + 2, \theta, \theta_{+})) \left( \sum_{r \in R_{-}^{N}} \sigma(r, r_{+}, \theta, \theta_{+}) \right) \right\},$$

where for the second line, we have used Lemma 9.8. Now,

$$\sum_{\theta \in \Theta_{-}^{N}} |O_{N,\theta}| \lesssim \max_{\theta \in \Theta_{-}^{N}} (\sigma(0, 2g + 2, \theta, \theta_{+}))^{\frac{1}{2}} \sigma(\Sigma) + 2^{-\frac{N}{2}} 2^{-N\beta p} 2^{N} + \left( \sup_{\substack{r \in R_{-}^{N} \\ \theta \in \Theta_{-}^{N}}} \sigma(r, r^{+}, \theta, \theta^{+}) \right)^{p-1} \sigma(\Sigma) \\ + \left( \sup_{\substack{r \in R_{-}^{N} \\ \theta \in \Theta_{-}^{N}}} \sigma(r, r^{+}, \theta, \theta^{+}) \right)^{\frac{1}{2}} \sigma(\Sigma) + \max_{\theta \in \Theta_{-}^{N}} (\sigma(0, 2g + 2, \theta, \theta_{+})) \sigma(\Sigma),$$

which for $p$ chosen big enough, and using Lemma 9.8, gives $\sum_{\theta \in \Theta_{-}^{N}} |O_{N,\theta}| \xrightarrow[N \to \infty]{} 0$. $\square$

**Lemma 9.8.** There exists $C, \beta, \gamma > 0$, such that for all $N$,

$$\sup_{\theta \in \Theta_{-}^{N}} \sigma(0, 2g + 2, \theta, \theta_{+}) \le C 2^{-N\beta} \quad \text{and} \quad \sup_{\substack{r \in R_{-}^{N} \\ \theta \in \Theta_{-}^{N}}} \sigma(r, r^{+}, \theta, \theta^{+}) \le C 2^{-N}$$

*Proof.* The second estimate was already proved. The challenge is to give uniform bounds for area of very thin strips of dyadic thickness. If the angle $\theta$ is away from the singular angles then the bound is obvious. The difficult case is when strips intersect saddle points; see Figure 26.

Because of the specific structure of Morse gradient flows, we only stay some finite time outside hyperbolic boxed and everything reduces to bounding areas

$$\int_0^1 \int_{k2^{-N}}^{(k+1)2^{-N}} \frac{\tilde{\sigma}(r, u)}{\sqrt{r^2 + 4u^2}} drdu = \int_0^1 \int_k^{(k+1)} \frac{\tilde{\sigma}(r, 2^{-N}u)}{\sqrt{r^2 + 2^{-2N}4u^2}} 2^{-N} drdu \\ = 2^{-N} \int_0^{2^N} \int_k^{(k+1)} \frac{\tilde{\sigma}(2^{-N}r, 2^{-N}u)}{\sqrt{r^2 + 4u^2}} drdu \le 2^{-N} \int_0^{2^N} \frac{\tilde{\sigma}(2^{-N}r, 2^{-N}u)}{r} dr = \mathcal{O}(N2^{-N}).$$

So this proves the claim for all exponents $\beta < 1$. $\square$

We still need one more lemma before being ready to prove the convergence of $\xi_N$.
<figure><img src="image_29.png" /><figcaption>Figure 26: Strips near saddle points.</figcaption></figure>

**Lemma 9.9.** There exists a constant $C > 0$ that depends on $\psi$ such that for all $N \ge 1$, all $\theta \in \Theta_{-}^{N}$,

$$ \prod_{r \in R_{-}^{N}} \left( 1 - \frac{|K_{r, r_{+}, \theta, \theta_{+}} \psi|^{2}}{2} \sigma(\square(r, r_{+}, \theta, \theta_{+})) \right) \ge C $$

*Proof.* We have $|K_{r, r_{+}, \theta, \theta_{+}} \psi| \le \|\psi\|_{\infty}$, and

$$ \sigma(\square(r, r_{+}, \theta, \theta_{+})) \le \sup_{r \in R_{-}^{N}, \theta \in \Theta_{-}^{N}} \sigma(\square(r, r_{+}, \theta, \theta)) \le C 2^{-N}. $$

Therefore,

$$ \prod_{r \in R_{-}^{N}} \left( 1 - \frac{|K_{r, r_{+}, \theta, \theta_{+}} \psi|^{2}}{2} \sigma(\square(r, r_{+}, \theta, \theta_{+})) \right) \ge \prod_{r \in R_{-}^{N}} \left( 1 - \frac{C \|\psi\|_{\infty}^{2}}{2} 2^{-N} \right). $$

However,

$$ \log \prod_{r \in R_{-}^{N}} \left( 1 - \frac{C \|\psi\|_{\infty}^{2}}{2} 2^{-N} \right) = \sum_{r \in R_{-}^{N}} \log \left( 1 - \frac{C \|\psi\|_{\infty}^{2}}{2} 2^{-N} \right) = (2g+1) 2^{N} \log \left( 1 - \frac{C \|\psi\|_{\infty}^{2}}{2} 2^{-N} \right) \\ \xrightarrow[N \to \infty]{} \exp \left( - \frac{C(2g+1) \|\psi\|_{\infty}^{2}}{2} \right) > 0. \quad \square $$

We are able now to conclude the proof of the convergence.

**Proposition 9.10.** The sequence $(\xi_{N} \mathrm{d}r \wedge \mathrm{d}\theta)_{N \ge 1}$ converges in $\mathcal{M}(\mathcal{D}'_{2}(\mathrm{Cyl}, \mathfrak{g}))$ to the white noise, seen as a 2-form.
Proof. By the previous lemmas,

$$
\begin{align*}
\log \mathbb{E}[\exp(i\langle\xi_N, \psi\rangle)] &= \sum_{\theta \in \Theta_{\underline{N}}} \log \left\{ \prod_{r \in R_{\underline{N}}} \left( 1 - \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) \right) \right\} \\
&\quad + \sum_{\theta \in \Theta_{\underline{N}}} \log \left( 1 + \frac{O_{N,\theta}}{\prod_{r \in R_{\underline{N}}} \left( 1 - \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) \right)} \right) \\
&\xrightarrow[N \to \infty]{} - \frac{\|\psi\|_{L^2(\Sigma, \sigma)}^2}{2}
\end{align*}
$$

Since the second term converges to 0 and

$$
\log \left( 1 - \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) \right) = - \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) + O\left(\sigma(\square(r, r_+, \theta, \theta_+))^2\right)
$$

and therefore,

$$
\lim_{N \to \infty} \sum_{\substack{r \in R_{\underline{N}} \\ \theta \in \Theta_{\underline{N}}}} \log \left( 1 - \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)) \right) = \lim_{N \to \infty} - \sum_{\substack{r \in R_{\underline{N}} \\ \theta \in \Theta_{\underline{N}}}} \frac{|K_{r,r_+, \theta, \theta_+} \psi|^2}{2} \sigma(\square(r, r_+, \theta, \theta_+)),
$$

which by classical martingale convergence arguments, is equal to $-\frac{\|\psi\|^2_{L^2(Cyl, \sigma)}}{2} = -\frac{\|\psi\|^2_{L^2(\Sigma, \sigma)}}{2}$. $\square$

proof of Theorem 9.5. It is a combination of Lemma 9.6, Proposition 9.10 and Fernique’s theorem.

10 Convergence in functional spaces

Consider again a surface Σ with one outgoing boundary component. We will study the convergence of
the free lattice Yang–Mills measures in more tailored functional spaces.

In the previous section, we showed that for each $N$, $A_{(M,U)}^{(N)}$ can be decomposed into the sum of two independent functions $A_M^{(N)}$ and $A_U^{(N)}$: one living on the actual surface, and the other one living on the bands of zero area. We have also seen that the latter converges, as $N$ goes to infinity, almost surely to the unstable currents associated to unstable curves. These currents belong to $H^{-1-\epsilon}$, and the convergence of $A_U^{(N)}$ happens as well in these spaces. Therefore, what we still need to study is the convergence of $(A_M^{(N)})_{N \ge 0}$ in functional spaces. To alleviate the notations in this section, we will denote $A_M^{(N)}$ only by $A_N$.

Since we have already shown convergence in law, i.e. we identified the limit, the only missing argument is tightness. Since usual scales of functional spaces enjoy compact embeddings from smaller into bigger spaces (which we will be showed in Appendix C in our special case), we only need to show that the sequence of the norms of $A_N$ is bounded independently of $N$. It is therefore crucial to identify the functional spaces in which convergence will hold in a way that reflects the anisotropic regularity of the limit connection on the one hand, and enables easy computations on the other hand.

10.1 Discretizing the spaces of distributional connections

In Section 5.3, we introduced some norms on smooth connections, with which we defined spaces of distributional connections by completion. The goal of this section is to explain how can one compute