We prove that the Yang–Mills (YM) measure for the trivial principal bundle over the two-dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing, and Bourgain's method for invariant measures. Several corollaries are presented including a gauge-fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder,
and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain, and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.

§ INTRODUCTION

We show that the Euclidean Yang–Mills (YM) measure associated to the trivial principal
$G$-bundle over the two-dimensional (2D) torus $\T^2$ is invariant
for the corresponding renormalised Langevin dynamic,
where $G$ is an arbitrary connected, compact Lie group.
The YM measure is formally given by the Gibbs-type probability measure

μ(A) ∝e^-S(A)A

where $S(A)$ is the YM action

S(A) = ∫_^2 |F(A)|^2  x

and $\mrd A$ is a formal Lebesgue measure on the space of connections on a principal $G$-bundle $P\to \T^2$.
Above, $F(A)$ is the curvature $2$-form of $A$.
In everything that follows, we consider $P$ as the trivial bundle $P=\T^2\times G$.
In this case, after choosing a global section, every connection can be written as a
$\mfg$-valued $1$-form $A=(A_1,A_2) \colon \T^2\to \mfg^2$, where $\mfg$ is the Lie algebra of $G$.
In this case

F_ij(A)= _̣iA_j - _̣jA_i + [A_i,A_j]

and the norm $|F(A)|$ in (<ref>) is understood as $|F_{12}(A)|^2$ where $|X|^2=\scal{X,X}$ and $\scal{\cdot,\cdot}$ is an $\Ad$-invariant inner product on $\mfg$, e.g.
if $G$ and $\mfg$ are subsets of unitary and skew-Hermitian matrices respectively, one can take $\scal{X,Y}=-\Trace(XY)$.

The 2D YM measure $\mu$ is one of the few quantum gauge theories that has been rigorously constructed.
This construction relies on an elegant exact solvability property, known as `invariance under subdivision',
initially observed in the physics literature by Migdal <cit.> (see also the work of Witten <cit.>).
Mathematically, the YM measure on a compact surfaces has been constructed as a stochastic process indexed by Wilson loops in <cit.>, which built on earlier works <cit.> for the plane.
We refer to the surveys <cit.> for further discussions and references.
The two constructions that are most relevant for us are that of Lévy <cit.>,
which identifies the YM measure associated to a given isomorphism class of principal $G$-bundles as the limit of discrete approximations,
and that of <cit.>, which constructs a gauge-fixed representation of
the measure as a random distributional $1$-form (under the assumption that $G$ is simply connected).

A natural approach to construct and study Euclidean quantum field theories (EQFTs) is via stochastic quantisation.
The stochastic quantisation equation, or Langevin dynamic, for any measure of the form (<ref>) is the stochastic partial differential equation (SPDE)

_̣t A = -1/2∇S(A) + ξ ,

where $\xi$ is a space-time white noise and $\nabla S$ is the gradient of $S$,
both taken with respect to some metric on the space of fields $A$.
There has been a lot of progress in recent years in using the Langevin dynamic (<ref>)
to construct and study properties (e.g. Osterwalder–Schrader axioms) of scalar EQFTs, most notably for the $\Phi^4_3$ measure <cit.>
(see also <cit.> for a related approach).

For the YM measure, taking the natural $L^2$-metric on connections, the Langevin dynamic is

_̣t A = -_A^* F(A) + ξ ,

where $\mrd_A^*$ is the adjoint covariant derivative.
This is (the 2D, finite volume version of)
the stochastic quantisation equation introduced by Parisi–Wu <cit.>.
To indicate the type of nonlinearities in (<ref>), we heuristically write the equation as

_̣t A = ΔA + ^̣2 A + AẠ + A^3 + ξ ,

where $\Delta$ is the Laplacian and $\Delta + \d^2$ indicates a second order differential operator that is not elliptic (i.e. has an infinite dimensional kernel).
The lack of ellipticity of $\Delta+\d^2$ is a well-known feature of YM theory
and stems from the invariance of the YM action under the infinite dimensional gauge group of bundle automorphisms,
i.e. the group of functions $g\colon \T^2\to G$ that act on connections by

A↦A^g gAg^-1 - g g^-1 .

Gauge invariance is an important feature of YM theory which indicates that
all physical quantities should depend on the gauge orbit $[A]\eqdef \{A^g\,:\,g\colon \T^2\to G\}$ instead of on $A$.
($A$ can be seen as a coordinate-based representation of $[A]$ coming from our initial choice of the global section of the bundle $P\to\T^2$.)

A celebrated method to bypass the lack of parabolicity of (<ref>) (see <cit.>)
is to consider instead
the equation

_̣t A = -_A^* F(A) - _A^* A + ξ ,

where $-\mrd_A\mrd^* A$ is called the DeTurck term, after the work of DeTurck <cit.> on the Ricci flow,
and has the heuristic form $\d^2A + A\d A$ that precisely cancels the $\d^2 A$ term in (<ref>).
Furthermore, on the level of gauge orbits, (<ref>)
is equivalent to (<ref>) since any term $\mrd_A \omega$ for a $0$-form $\omega\colon \T^2\to \mfg$ is tangent to $[A]$ at $A$.

In coordinates, (<ref>) reads
_̣t A_i = ΔA_i + [A_j, 2_̣jA_i - _̣i A_j+[A_j,A_i]] + ξ_i  ,  i=1,2 ,

with implicit summation over $j=1,2$
and which we write in the sequel simply as

_̣t A = ΔA + AẠ + A^3 + ξ .

We see at this stage that (<ref>) is singular in the sense that we expect the solution $A$
to be a distribution of the same regularity as the linear stochastic heat equation (SHE),
which renders the products $A\d A$ and $A^3$ ill-defined.
It is therefore non-trivial to make rigorous sense of the SPDE (<ref>).

A programme to solve and analyse (<ref>) was initiated in <cit.>.
It was shown therein that, given a space-time mollifier $\moll$ and any operator $C\in L(\mfg^2,\mfg^2)$,
the solution to the regularised and renormalised equation

_̣t A = ΔA + AẠ + A^3 + CA + ^*ξ
converges, as $\e\downarrow0$, locally in time to a Markov process with values in a suitable Banach space of distributional $1$-forms $\Omega^1_\alpha$
(which is similar to but stronger than the space introduced in <cit.>).
Here we write $\moll^\eps(t,x) = \eps^{-4}\moll(\e^{-2}t,\e^{-1}x)$ for the natural approximation of the Dirac delta built from $\moll$.
Furthermore, and more importantly, if $\moll$ is non-anticipative,
then there exists a choice for $C\in L(\mfg^2,\mfg^2)$, which depends explicitly on $\moll$ and takes the form $C(A_1,A_2)=(CA_1,CA_2)$,
such that the limiting dynamic $A$ projects to a Markov process $X\colon t\mapsto [A(t)]$ on the quotient space of gauge orbits $\mfO_\alpha = \Omega^1_\alpha/{\sim}$, where
$\sim$ is a canonical extension of gauge equivalence to $\Omega^1_\alpha$.
For a simple heuristic reason behind this `gauge-covariance' based on Itô isometry, see <cit.> or the survey <cit.>.

It was conjectured in <cit.> that the YM measure for the trivial principal $G$-bundle is invariant for the Markov process $X$.
More precisely, the YM measure $\mu$ as constructed in <cit.> can be seen as a probability measure on the space of functions $G^\mcL/G$, where $\mcL$ is a suitable class of loops $\ell\colon[0,1]\to\T^2$
(for concreteness, taken as all piecewise smooth loops based at $0\in \T^2$)
and $G^\mcL$ carries the diagonal adjoint action $G^\mcL\times G \to G^\mcL$, $(H,g)\mapsto \{\ell\mapsto gH(\ell)g^{-1}\}$.
On the other hand, every $A\in\Omega^1_\alpha$ canonically defines a `holonomy process' $H_A\in G^\mcL$
with the property that $A\sim B$ if and only if $gH_Ag^{-1}=H_B$ for some $g\in G$,
i.e. the orbit space $\Omega^1_\alpha/{\sim}$ from <cit.> can be identified with a subset of $G^\mcL/G$.
The precise conjecture is that the YM measure
$\mu$ gives full measure to  $\Omega^1_\alpha/{\sim}$
and is the unique invariant measure of the Markov process $X$.

The main result of this article answers this conjecture in the affirmative, thereby making the first rigorous link between the YM measure and its Langevin dynamic.
Our argument is based on discrete regularity structures
(for which we build on ideas from <cit.>)

and lattice gauge fixing (for which we use ideas from <cit.> and rough paths theory),

and we combine these in a Bourgain-type method for invariant measures <cit.>.

In our use of Bourgain's method, we approximate the solution to (<ref>) through lattice dynamics,
and one of the novelties in our approach is the way in which we identify the continuum limit of these discrete dynamics.
Our argument for this is geometric, based on preservation of gauge symmetries,
and an important step is to show that there is a unique mass renormalisation of the YM Langevin dynamic (<ref>)
that preserves gauge covariance (see Theorem <ref>).

This step resolves another problem from <cit.>,
where the question of uniqueness of the mass renormalisation constant $C\in L(\mfg^2,\mfg^2)$ was left open.[<cit.> proved gauge covariance via a particular coupling argument, for which
uniqueness of $C$ does hold. See also <cit.>.]
Our proof of this result, which we believe is of independent interest, relies on Euler-type estimates for singular SPDEs and for Young ODEs associated to Wilson loops,
combined with ideas from sub-Riemannian geometry.

We further derive a number of consequences on the YM measure itself based on our main result.
Notably, we prove that
the scaling limit of a broad class of lattice gauge theories on $\T^2$ is the YM measure,
thereby establishing a general universality result for the YM measure.
More precisely, we provide conditions (Assumptions <ref> and <ref>)
on the local action of a lattice gauge theory that ensures its scaling limit is the YM measure for the trivial principal bundle -
see Corollary <ref>.

This universality result covers the Wilson, Manton, and Villain actions (the result for Villain action was already known <cit.>).
A related result on $\R^2$ (or more simply $[0,1]^2$) for the Villain and Wilson actions, was shown in <cit.>
by use of a central limit theorem (CLT) for group-valued random variables
(see also <cit.> for a type of universality related to $\tau$-continuum limits <cit.>);
it appears difficult, however, to extend proofs of universality using the CLT
to surfaces with non-trivial topology, like $\T^2$ that we cover here.
This universality of the measure, which can be seen as a functional CLT for random connections, is derived from a corresponding universality for the dynamic (Theorem <ref>).

Another consequence is a gauge-fixed decomposition of the YM measure $\mu$ (Corollary <ref>):
there exist, on the same probability space, random distributional $1$-forms $\Psi$ and $B$ such that $\Psi$ is a Gaussian free field (GFF) and $|B|_{\CC^{1-\kappa}}$ has moments of all orders for any $\kappa>0$, and such that
$[\Psi+B] \sim \mu$.
This decomposition rigorously establishes the YM measure as a perturbation of the GFF.
It furthermore implies strong regularity properties of the measure, improving the main result of <cit.> in two directions: (i)
it removes the restriction of axis-parallel lines in the norms of <cit.>,
and (ii) it establishes moments bounds for these norms – see Remark <ref> for more details.

We use the construction of the YM measure from the Villain (heat kernel) action in <cit.>
(see also <cit.>) as an important guide, but
we do not explicitly use this construction except to identify the invariant measure of $X$ with the well-known YM measure.
In particular, we show existence and uniqueness of the invariant measure using only SPDE techniques,
and therefore our results can be seen as providing a new construction and characterisation of the YM measure.

§.§ Related works and open problems

Some of the results of <cit.> were recently extended to 3D in <cit.>,
including the construction of a state space of distributional $1$-forms and of a Markov process on gauge orbits corresponding to (<ref>).
See <cit.> for a summary and comparison of the results of <cit.>.
See also <cit.> for a different but related approach to a state space for 3D quantum YM.
In contrast to <cit.>,
the state spaces in <cit.> are based on the deterministic DeTurck–YM heat flow (see <cit.> for related ideas) and are nonlinear,
a property shown in <cit.> to be unavoidable.

The results of this article are restricted to the 2D pure YM theory, and identifying the invariant measure of (<ref>) in 3D remains an important open problem.
In fact, the construction of the 3D YM measure in the continuum is open, even in finite volume,
although progress was made in establishing
a form of ultraviolet stability <cit.>.

The restriction to dimension 2 and to the pure YM theory (i.e. no coupling with matter) arises from our use of Bourgain's invariant measure argument <cit.> (see also <cit.>), which requires moment bounds on the invariant measure.
Improving the results of <cit.>, we derive in this article the necessary moment bounds
for lattice approximations of the 2D (pure) YM measure by exploiting its exact solvability together with ideas from rough paths.
It remains an open problem to extend these bounds to
higher dimensions or to other models, such as the 2D YM–Higgs model that has no known exact solvability property.
(The 2D and 3D Abelian Higgs models have been constructed in <cit.>,
but a construction in the non-Abelian case remains open even in 2D.)
However, see <cit.> for progress on moment estimates for the Abelian Higgs model,
and <cit.> where the stochastic quantisation equation of an Abelian gauge theory is solved using lattice approximations.
The discrete version of (<ref>)
was furthermore used in <cit.> to study lattice YM.

Another problem worth mentioning is the extension of the results of this article to non-trivial principal bundles, which would
require a solution theory for (<ref>) outside the `periodic' setting.
It would furthermore be of interest to extend the results from finite to infinite volume, i.e. from $\T^2$ to $\R^2$.

Finally, we mention recent progress on understanding invariance of EQFTs for corresponding Hamiltonian flows in the singular regime <cit.>, that parallels developments for the Langevin dynamic (<ref>) (see <cit.> for earlier work). In particular <cit.> studies randomness from both initial data and stochastic forcing.
These results have largely been restricted to scalar theories (but see <cit.> for exceptions), and it would be of interest if similar results can be obtained for the YM measure;
Bringmann–Rodnianski <cit.> recently exposed a potential obstruction to such a result.

§.§ Acknowledgements

HS gratefully acknowledges support by NSF via DMS-1954091 and CAREER DMS-2044415.
IC acknowledges support by EPSRC via EP/X015688/1.

§ MAIN RESULTS AND PRELIMINARIES

We state in this section our main results and corresponding notations.
We fix $d\geq 1$
(taken as $d=2$ further down).

§.§ Notation

Let $\T^d=\R^d/\Z^d$ denote the $d$-dimensional torus which we identify as a set with $[0,1)^d$.
We equip $\T^d$ with the geodesic distance inherited from the Euclidean structure of $\R^d$, which we denote as $|x-y|$ for $x,y\in\T^d$.

For $N\geq 1$,
let $\Lambda_N\subset \T^d$ denote the lattice with spacing $2^{-N}$,
which we identify with the set $\{k2^{-N}\,:\,k=0,\ldots, 2^N-1\}^d$.
Let $\bonds_N \eqdef \{(x,y)\in\Lambda_N^2\,:\, |x-y|=2^{-N}\}$,
denote the set of oriented bonds (also called edges) on the lattice.

When clear from the context, we will drop reference to $N$ in our notation and simply write $\Lambda$, $\bonds$, etc. and write $\eps = 2^{-N}$.

We denote $[d]=\{1,\ldots, d\}$ and write $\{e_j\}_{j\in [d]}$ for the
set of unit basis vectors in $\R^d$.
For $j\in [d]$ we will write
$\eps_j$ as shorthand for $\eps e_j$.

Let $\obonds_N\equiv \obonds \eqdef \{(x,x+\eps_j)\,:\, x\in\Lambda\,,\; j\in[d]\}\subset \bonds$,
denote the set of positively oriented bonds.
We also write
$\obonds_j \eqdef \{(x,x+\eps_j)\,:\, x\in\Lambda\}$
so that $\obonds = \cup_{j\in[d]} \obonds_j$.

For $e=(x,y)\in \bonds$ we write $e\pm \e_i \eqdef (x\pm\e_i,y\pm\e_i)$.

An oriented plaquette, or sometimes just a plaquette,
is a tuple $p=(x,\e,\bar\e)$, where $x\in\Lambda$ is a lattice site and $\e,\bar \e \in \CE$ where $\e\neq \pm \bar \e$.
We will also use the alternative notation

$p=(e^{(1)},e^{(2)},e^{(3)},e^{(4)})$ where $e^{(i)}=(x_i,x_{i+1})$ for $i=1,\ldots, 4$ with $x_5\eqdef x_1$,
since this plaquette $p$ can be also thought as the square enclosed by these four points or edges.
We will call $x$ the base point of $p$, and say that $\overleftarrow{p}\eqdef (x,\bar\e,\e)$ is the plaquette with the reversed orientation.
We denote by $\plaq\equiv \plaq_N$ the set of all plaquettes of $\Lambda$.

We fix throughout the paper a connected compact Lie group $G$ and let $\mfg= T_\id G$ be its Lie algebra, which we identify with the space of left invariant vector fields on $G$.
Without loss of generality, we assume there exists $n\geq 1$ such that $G\subset \U(n)$, the $n\times n$ unitary matrices,
and $\mfg\subset \mfu(n)$, the $n\times n$ skew-Hermitian matrices.

Let $\conf\equiv \conf_N$ denote the set of all functions $U\colon\bonds\to G$ with the property that $U(x,y)=U(y,x)^{-1}$
for all $(x,y)\in\bonds$.
Remark that we can identify $\conf \simeq G^{\obonds}$ and view it as the configuration space of discrete gauge fields.

Let further $\mfq\equiv \mfq_N$ denote the space of functions
$A\colon \bonds\to \mfg$ with the property that $A(x,y)=- A(y,x)$ for all $(x,y)\in\bonds$.
Remark that $\mathfrak{q}$ can be identified with $\mfg^{\obonds}$
and thus with the Lie algebra $T_\id G^{\obonds}$ of $G^{\obonds}$.
We call elements of $\mfq$ discrete $\mfg$-valued $1$-forms, or simply $1$-forms.
We further write $\mfq_i = \mfg^{\obonds_i}$ so that $\mfq \simeq \oplus_{i=1}^d \mfq_i$.[The
clash of notation $\obonds_N$ vs. $\obonds_j$ and $\mfq_N$ vs. $\mfq_i$ will not cause confusion.]

Fix sufficiently small neighbourhoods $\mathring V\subset \mfg$ of $0$ and $\mathring W\subset G$ of $\id$
on which $\exp \colon \mathring V\to \mathring W$ is a diffeomorphism.
For $x\in G$, let $\Ad_x\colon\mfg\to\mfg$ denote the adjoint action.
We equip $\mfg$ with an $\Ad$-invariant inner product $\scal{\cdot,\cdot}_\mfg$
and equip $\mfq$ with the corresponding inner product[The factor $\e^{d-2}$ is such that the
right-hand side of (<ref>) is of order $1$, since $A,B$ here will be, in a sense, order $\eps$.]
A,B_= ^d-2∑_e∈ A(e),B(e)_ .

We fix a bounded, measurable map $\log\colon G\to \mfg$ such that (see <cit.> for existence of such a map $\log$)

* $\log = \exp^{-1}$ when restricted to $\mathring W$,

* $\exp(\log x)=x$ for all $x\in G$,

* $\log(yxy^{-1})=\Ad_y(\log x)$ and $|\log x|_\mfg = |\log x^{-1}|_\mfg$ for all $x,y\in G$, and

* $\log x=-\log (x^{-1})$ for all $x\in G$ outside a set with Haar measure zero.

We choose $\log$ to be defined on all of $G$ only for convenience; the more common choice of defining $\log$ only on $\mathring W$ would suffice for our purposes but causes some complications down the line.

Define $W \eqdef \mathring W^{\obonds}\subset\conf$, the subset of functions $U$ on bonds taking values in $\mathring W$,
and similarly $V \eqdef \mathring V^{\obonds} \subset \mfq$.
Then $\exp \colon V\to W$
is still a diffeomorphism.

For a plaquette $p$, which we write as $p=(x_1,x_2,x_3,x_4)$ as in Definition <ref>, and $U\in\conf$, we define the holonomy of $U$ around $p$ as
U(∂p) U(x_1,x_2)U(x_2,x_3)U(x_3,x_4)U(x_4,x_1) .

Let $\mfG\equiv \mfG_N$ denote the group of all functions $g\colon \Lambda\to G$.
Elements of $\mfG$ are called gauge transformation.
There is a left group action of $\mfG$ on $\conf$ which we denote, for $g\in\mfG$ and $U\in\conf$, by
$(g,U) \mapsto g\act U \eqdef U^g$,

U^g(x,y)= g(x)U(x,y)g(y)^-1 .

Remark that $U^g(\partial p)=g(x)U(\partial p)g(x)^{-1}$ for any plaquette $p=(x,\e,\bar\e)$.

For $U\in\conf$,
we write $[U]=\{U^g\,:\,g\in\mfG\}\subset \conf$, which is called the gauge orbit of $U$.
We say that $\bar U$ and $U$ are gauge equivalent and write $\bar U\sim U$, if $\bar U\in [U]$.
We write $\pi\colon Q\to Q/{\sim}$ for the  projection $\pi(U)=[U]$.

§.§ Approximations of the Yang–Mills measure and Langevin dynamic

In this subsection, we describe the types of lattice approximations of the YM measure that we consider together with their Langevin dynamics
that we show convergence to a continuum limit.
Fix again $d\geq 1$ in this subsection.

Let $G$ (resp. $Q$) be endowed with the bi-invariant Riemannian metric $\scal{\act,\act}$ coming from the $\Ad$-invariant inner product $\scal{\act,\act}_\mfg$ on $\mfg$ (resp. $\scal{\act,\act}_{\mfq}$) as fixed in Section <ref>.
(Note that this metric on $Q\simeq G^{\obonds}$ differs from the product metric by a factor $\e^{d-2}$.)
We also equip $G$ and $\conf\simeq G^{\obonds_N}$ with their Haar measures of total mass $1$.
We denote by $\int_G f(x)\mrd x$ the integral of a function $f$ on $G$ against the Haar measure, and similarly for $\conf$.

Let $S\equiv S_N \colon G \to \R\cup\{\infty\}$
be a measurable symmetric class function, i.e. $S(x)=S(x^{-1})$ and $S(yxy^{-1})=S(x)$ for all $x,y\in G$, that is twice differentiable on a fattening of $\mathring W$.
Define $\CS\equiv \CS_N\colon\conf\to \R\cup\{\infty\}$ by

=1/8 ∑_p∈ S[U(∂p)] .

(The factor $1/8$ arises from the fact that $8$ plaquettes in $\plaq$ enclose the same square in $\T^d$.)
Consider the probability measure $\mu_N$ on $\conf$ defined by

μ_N(U) Z^-1 e^-(U) U ,

where $\mrd U$ is the Haar measure on $Q$ and
$Z\equiv Z_N^{-1}$ is the normalisation constant which makes $\mu_N$ a probability measure.
By the remark on $U^g$ below (<ref>), $\mcS$ and $\mu_N$ are invariant under the action
by $\mfG$.

(We allow $S,\CS$ to take values $\infty$ so that the density $e^{-\CS}$ is allowed to be $0$ outside a neighbourhood of identity.)

One should think of the plaquette variable $U(\partial p)$ as approximating (the exponential of) the curvature.
Writing $U=e^{\eps A}$,  the
Baker–Campbell–Hausdorff (BCH) formula (e.g. <cit.>) yields $U(\partial p) \approx
e^{\e^2 F_{ij}(A)(x)}$

for $p=(x,\e_i,\e_j)$ for $i\neq j\in [d]$
where $F(A)$ is as in (<ref>).
This is one of the motivations for viewing
$\e^{d-4} |\log x|_{\mfg}^2$ as the “leading term” in our class function $S_N$ (see (<ref>) below),
since the YM action in the continuum is $|F(A)|^2_{L^2}$.

Recall that $\mfq$ is identified with the left invariant vector fields on $\conf$.
For a differentiable function $f\colon Q\to \R$, the linear map $Df_U\colon T_U Q \to \R$ (i.e. “differential” of $f$ at $U$) is given  at $A\in \mfq \simeq T_U Q$ by

Df_U(A) = /α |_α=0 f(Ue^αA) = lim_α→0 f(Ue^αA) - f(U)/α .

For any $U\in Q$, the gradient $\nabla \mcS(U)$ is the unique element of the tangent space $T_U Q$ such that $\scal{\nabla \mcS(U),X} = D \mcS_U(X)$ for all $X\in T_U Q$.
A natural diffusion to consider, which is a discrete version of (<ref>), is

Ũ_t = -1/2∇(Ũ_t) t + Ũ_t (∘_t) ,

where $\tilde \BM$ is a $\mfq$-valued Brownian motion

and we denote time and stochastic differentials by $\dt$ to distinguish from exterior and covariant derivatives $\mrd$ and $\mrd_A$.
By a $\mfq$-valued Brownian motion we mean a centred Gaussian process with covariance
$\E\scal{\tilde\BM_t,X}_\mfq\scal{\tilde\BM_s,Y}_\mfq = (s\wedge t)\scal{X,Y}_\mfq$ where we recall  (<ref>).

There are two issues with this definition of $\tilde U$:

* the SDE (<ref>) approximates the SPDE (<ref>), which is not parabolic, and thus we have no way to establish its convergence, and

* the function $S_N\colon G\to \R\cup\{\infty\}$ is assumed differentiable only on a neighbourhood of $\id$, so $\nabla \mcS(U)$ may not make sense for $U$ far from $\id$.

To resolve these issues, we instead define a dynamic with an added DeTurck term and which is stopped upon exiting $W$.
Specifically, let $\Log \in \CC^\infty(G,\mfg)$ such that $\Log\restr_{\mathring{W}} = \log\restr_{\mathring{W}}$
and define $\mrd^* \in\CC^\infty(Q, \mfg^\Lambda)$ by

(^* U)(x)
^-1 ∑_j=1^d [ U(x,x+_j) - U(x-_j,x) ]  ,

which is a discrete approximation of the adjoint exterior derivative.
For $U \in \conf$ and $\omega \in \mfg^\Lambda$, define the infinitesimal transformation
(_U ω)(x,y) ^-1(ω(x) U(x,y) - U(x,y)ω(y)) ,

which is a tangent vector at $U$ that
is tangent to the gauge orbit of $U$.

We further consider another class function $\check S\equiv \check S_N \colon G\to \R$ which is symmetric, twice differentiable
everywhere, agrees with $S_N$ on $\mathring W$,
and arbitrary otherwise.
We define $\check\mcS = \sum_{p\in\plaq} \check S[U(\d p)]$
together with the dynamic $\check U\equiv \check U^{(N)}$ that solves

Ǔ_t = -1/2∇(Ǔ_t)t  - _Ǔ_t ^* Ǔ_tt + Ǔ_t (∘_t) ,

where $\BM$ is a $\mfq$-valued Brownian motion as in (<ref>).
We then define the exit time

ϖ= inf{t≥0 :  Ǔ_t ∉W} ,

and the process

U_t ≡U^(N)_t Ǔ_t∧ϖ .

One of our main results (Theorem <ref>) is that $\log U$ converges as $N\to\infty$ to the $\eps\downarrow0$ limit of (<ref>) identified in <cit.>.

The following proposition reveals the basic gauge-covariance properties of the above processes.
For $X\in\mfq$ and $g\in\mfG$, define $X^g\in \mfq$ by $
X^g_{xy} \eqdef g_y X_{xy} g_y^{-1}$ for all
In particular, for a Hölder continuous process $\BM\colon[0,T]\to\mfq$ and a finite-variation process $g\colon[0,T]\to\mfG$, define $\BM^g\colon[0,T]\to\mfq$ by
\dt (\BM^g)_{xy} \eqdef g_y\dt \BM_{xy} g_y^{-1}$
for all $(x,y)\in\obonds$.
If $g$ is adapted and $\BM$ is a $\mfq$-valued Brownian motion, remark that $\BM^g$ is another $\mfq$-valued Brownian motion by Itô isometry.

Suppose $S_N\colon G\to\R$ is twice differentiable on $G$ and $S_N=\check S_N$.

*  The diffusion (<ref>) has invariant measure $\mu_N$ defined in (<ref>).

*  Suppose $\tilde U$ satisfies (<ref>).
Define adapted finite-variation  $g\colon[0,T]\to\mfG$ by
g g^-1 = -^-1 ^*(Ũ^g) t ,   g_0=𝕀 ,

(which admits a unique  solution since $\mrd^*\colon Q \to \mfg^\Lambda$ is smooth).
Then $\check U \eqdef \tilde U^g$ solves (<ref>) with $\BM\eqdef \tilde\BM^g$.

Conversely, given $\check U$ satisfying (<ref>), there exists an adapted finite-variation $g\colon[0,T]\to\mfG$  such that $g_0=\id$ and $\tilde U \eqdef \check U^g$ solves (<ref>) with $\tilde\BM\eqdef \BM^g$.

In particular, $[\check U_t] \eqlaw [\tilde U_t]$ for all $t\geq 0$ whenever $\check U_0=\tilde U_0$.

*  Let $\tilde U^{(u)}$ denote the solution to (<ref>) with $\tilde U_0=u$.
If $v=u^h$ for some $h\in\mfG$, then $(\tilde U^{(u)})^h$ solves (<ref>) with $\tilde\BM$ replaced by $\tilde\BM^h$.

In particular, $[\tilde U^{(u)}_t] \eqlaw [\tilde U^{(v)}_t]$ for all $t\geq 0$ whenever $v\sim u$.

Let $\check U^{(u)}$ denote the solution to (<ref>) with $\check U_0=u$.
If $v=u^h$ for some $h\in\mfG$,
then there exists an adapted finite-variation $g\colon[0,T]\to\mfG$ with $g_0=h$ such that $(\check U^{(u)})^g$ solves (<ref>)
with $\BM$ replaced by $\BM^g$.

In particular, $[\check U^{(u)}_t] \eqlaw [\check U^{(v)}_t]$ for all $t\geq 0$ whenever $v\sim u$.

<ref> is clear from general principles (see also <cit.>)
while <ref> follows from <ref> and <ref>
and the fact that $Q\times\mfG \ni (U,g) \mapsto U^g\in Q$ and $\mfq\times\mfG \ni (X,g) \mapsto X^g\in \mfq$ are left group actions.

We distinguish for the moment between $\mfq$ and $T_UQ$ and write $UX\in T_UQ$ for the left-invariant vector field associated to $X\in\mfq$.
Note that $\nabla \mcS$ is gauge covariant in the sense that,
for any $g\in \mfG$,

∇(U^g)=[∇(U)]^g ∈T_U^gQ ,

by which we mean
$\nabla\mcS (U^g)(x,y) = g(x) \nabla \mcS(U)(x,y) g(y)^{-1}$ for every $(x,y)\in\obonds$.
Indeed, for any $X\in \mfq$ and $\alpha\in\R$, one has $\mcS(U^g e^{\alpha X})=\mcS((U^g e^{\alpha X})^{g^{-1}}) = \mcS(U \exp(\alpha X^{g^{-1}}))$ by gauge-invariance of $\mcS$.

∇(U^g), U^g X

=/α |_α=0 (U^g e^αX)
=/α |_α=0 (U exp(αX^g^-1))

=  ∇(U),  U X^g^-1
= (∇(U))^g, U^g X

where we used the bi-invariance of $\scal{\act,\act}$ in the final step and which proves (<ref>).

Furthermore (recall (<ref>))

(U^g) = (U)^g + _U^g(g g^-1)

(which is valid for any semi-martingale $U$ and finite-variation $g$).

Ǔ    =
(Ũ)^g + _Ǔ (g g^-1)

= -1/2∇( Ǔ)t  +  Ǔ (∘)
- _Ǔ (^* Ǔ)t ,

where we used (<ref>) in the first equality, and (<ref>)-(<ref>)
in the second equality.

This proves the first claim in <ref>.
The second claim follows in the same manner.

This follows from (<ref>), (<ref>),
and the fact that $h$ is constant in time.

§.§ Assumptions on the actions

We will work with a general function $S_N$ satisfying the properties mentioned at the start of Section <ref> and

S_N (x)= ^d-4 |x,𝕀|_G^2+ R_(x) ,

where $R_\e$ is a remainder that we soon discuss
and $|\act ,\act|_G$ is the geodesic distance on $G$.
By standard considerations, note that $|x,\id|_G = |\log x|_\mfg$ for all $x$ in a small neighbourhood of $\id$.

Since our metric on $\mfg$
is $\Ad$-invariant, and we assumed that $S_N$ is a class function invariant under inversion,
all the three terms in (<ref>) are class functions invariant under inversion for all $x$ outside of a null set.

The assumption we make on $R_\e$ is the following. Define the pullback function

R̂_→,  R̂_(X) = R_(e^X) .

There exist linear maps $E^{(1)}_{\e} \in L(\mfg , \mfg^*)$ and $E^{(2)}_{\e}\in L(\mfg^{\otimes 3} , \mfg^*)$,
such that
\sup_{\e\in (0,1]} |E^{(1)}_\e| + |E^{(2)}_\e| < \infty
and, uniformly in $\eps\in (0,1]$ and $X\in\mathring V$,
DR̂_(X) = E^(1)_(X) + ^d-4 E^(2)_(X^3) + O(X^2) + ^d-4O(X^4) .

See Section <ref> and Remark <ref> for the motivation behind this assumption.
We show below that some common actions, namely the Manton, Wilson, and Villain actions, satisfy Assumption <ref>.

In (<ref>) and the rest of the paper,
for a normed space $(\CB,\|\cdot\|)$, $X\in \CB$, and $n\geq 1$, we let
$O(X^n)$ denote $Y\in \CB$ such that $\|Y\|=O(\|X\|^n)$.

It is natural that we assume no term proportional to $X^2$ appears in the expansion for $D\hat R_\eps(X)$
since $\hat R_\eps$ is even in $X$, and therefore $D\hat R_\eps$ is odd.

§.§.§ Examples

The simplest action that satisfies Assumption <ref> is the Manton action <cit.> where $S_N(x)=\e^{d-4}|x,\id|_G^2$, so that $\hat R_\e(x)=0$ and $E^{(1)}_\e=0$ and $E^{(2)}_\e=0$.
Remark that $S_N$ in this case is not differentiable on all of $G$, e.g. for $G=\U(1)$, $S_N$ is not differentiable at $-1$.

Suppose that the inner product on $\mfg\subset \mfu(n)$ is given by $\scal{X,Y}_\mfg = -\frac12\Trace{XY}$.
The Wilson action, which is a smooth function $S_N\colon G\to [0,\infty)$, is

\[
S_N(x) = \eps^{d-4}\Re\Trace (\id-x)\;.
\]

See <cit.> for explicit forms of the Langevin dynamic arising from the Wilson action for $\SO(n)$, $\U(n)$, and $\SU(n)$.
The results in this paper yield convergence of
these dynamics on the two dimensional torus.

We have for $x\in \mathring W$

S(x)    = ^d-4(𝕀-x)
= ^d-4(𝕀- e^logx)

= -^d-4(log(x) + (logx)^2/2! + (logx)^3/3! + (logx)^4/4! + …)

= ^d-4 |logx|_^2 - ^d-4 ∑_k≥2  (logx)^2k/(2k)!
= ^d-4 |logx|_^2 + R_(x)  ,

where we used that $|X|_\mfg^2 = -\frac12\Trace(X^2)$.

Here $R_{\eps}(x)$ is defined in the last line and

R̂_(X) = R_(e^X)
= - ^d-4 ∑_k≥2  1/(2k)!X^2k
= -^d-4(cosh(X) - 1 - X^2/2!) .

It follows that, for all $Y\in\mfg$,

DR̂_(X)(Y) = -^d-4 1/3! (X^3 Y) + ^d-4O(X^5 Y) .

Therefore the Wilson action satisfies Assumption <ref> with $E^{(1)}_\e=0$ and

E^(2)_(X_1⊗X_2 ⊗X_3)(Y)

=- 1/3! (X_1X_2X_3 Y) .

Restricting to $d\leq 3$, the Villain (heat kernel) action $S_N\colon G\to\R$ is

e^-S_N(x)-c_ ∝e^1/4^4-d Δ(x) ,

where $\Delta$ is the Laplace–Beltrami operator on $G$,
$e^{t\Delta}$ is the associated heat kernel at time $t>0$,
and $c_\e$ does not depend on $x$ and is chosen so that $S_N(\id)=0$.
Like the Wilson action, the Villain action is smooth on $G$.
Recall the Varadhan formula

loge^1/4^4-d Δ(x) = -^d-4|logx|_^2 - R̅_(x)

where $\lim_{\e\to0}\e^{4-d} \bar R_{\e}(x) = 0$ uniformly in $x$.
Remark that $\bar R_\e(x) = R_\eps(x) + c_\e$ for $R_\e$ as in (<ref>).
Stroock–Turetsky <cit.> (see also <cit.>)
showed furthermore that the limit $\e^{4-d} \bar R_\e(x) \to 0$ commutes with gradients:
their result implies that, for any collection of smooth vector fields $X_1,\ldots,X_k$ on $G$,

sup_∈(0,1]sup_x∈W|(X_k∘⋯∘X_1) R̅_(x)| <∞ .

Since $\bar R_\e$ and $R_\e$ differ by a constant, (<ref>) holds also for $R_\e$, which implies, for all smooth vector fields $Y_1,\ldots, Y_k$ on $\mfg$,
\sup_{\e\in(0,1]}\sup_{X\in\mathring V}|(Y_k\circ\cdots\circ Y_1)\hat R_\e(X)| <\infty
which in particular implies for every $k\geq 1$

sup_∈(0,1]sup_X∈V|D^k R̂_(X)| <∞ ,

where $D^k \hat R_\e \colon \mfg \to L(\mfg^{\otimes k},\R)$ is the $k$-th derivative.

It follows that

DR̂_(X) = DR̂_(0) + D^2R̂_(0)(X) + O(|X|^2) ,

where the proportionality constant in $O(|X|^2)$ is uniform in $\e\in (0,1]$ and $X\in\mathring V$ due to (<ref>).
Due to the symmetry $\hat R_\e(X)=\hat R_\e(-X)$, which itself follows from $e^{\frac14\e^{4-d}\Delta}(x)=e^{\frac14\e^{4-d}\Delta}(x^{-1})$ and $|\log x|_\mfg=|\log x^{-1}|_\mfg$,
we obtain
$D\hat R_\e(0) = 0$.
Since $\sup_{\e\in (0,1]}|D^2\hat R_\e(0)| < \infty$
we conclude that
$\hat R_{\e}$ satisfies Assumption <ref>
with $E_{\e}^{(1)} \eqdef D^2\hat R_{\e}(0) \in L(\mfg\otimes \mfg,\R)\simeq L(\mfg,\mfg^*)$ and $E^{(2)}_\e=0$.

§.§ Lattice and continuum 1-forms

We recall in this section the norms on lattice and continuum $1$-forms from <cit.>.
Let $d\geq 1$ and define the sets

_N     = {(x,ke_i 2^-N) ∈^d×^d :  x∈Λ_N  ,  k∈{0,…, 2^N-1} ,  i∈[d]} ,

_N    = _N ∪{(x,-h) :  (x,h)∈_N} .

For $\ell=(x,ke_i 2^{-N})\in\olines_N$, we call $|\ell| \eqdef |k|2^{-N}$ the length of $\ell$.
The set $\olines_N$ can be interpreted as the set of (positively oriented) lines on the lattice $\Lambda_N$ of length at most $\frac12$.
We will frequently identify every $\ell=(x,h)\in\olines_N$ with the set $\{x+th\,:\,t\in[0,1]\}\subset \T^d$ (which completely characterises $\ell$).

We say that $\ell,\bar \ell\in\olines_N$ are parallel, and write $\ell\parallel\bar \ell$, whenever $\ell=(x,ke_i2^{-N})$ and $\bar\ell=(x+m e_j 2^{-N}, ke_i2^{-N})$ for some $j\neq i$ and $|m|\leq 2^{N-1}$.
We write $d(\ell,\bar\ell)\eqdef m2^{-N}$, which is the Hausdorff distance between $\ell$ by $\bar \ell$ treated as subsets of $\T^d$.
We further define the quantity

ρ(ℓ,ℓ̅) = |ℓ|^1/2d(ℓ,ℓ̅)^1/2 .

The area of the (smallest) rectangle with two of its sides given by $\ell,\bar\ell$ is $\rho(\ell,\bar\ell)^2$.

Every $A\in \mfq_N$ defines a function still denoted by $A\colon \olines_N \to \mfg$ as

A(x,ke_i2^-N) ∑_j=1^kA(x+(j-1)_i,x+j_i) .

We denote by $\Omega_N$
the set of all functions $A\colon \olines_N \to \mfg$ arising this way, which is clearly in bijection with $\mfq_N\simeq \mfg^{\obonds_N}$.
We extend every $A\in\Omega_N$ to a function $A\colon \lines_N \to \mfg$ given by requiring $A(\overleftarrow{\ell})=-A(\ell)$ where $\overleftarrow{(x,h)} = (x+h,-h) \in\lines_N$ is the reversal of $(x,h)\in\lines_N$.

Recalling $\log\colon G\to \mfg$ from Section <ref>,
every $U\in Q_N$ defines a function $A\in\mfq_N$ by $A(b) = \log U(b)$ for all $b\in\obonds_N$ and thus defines an element of $\Omega_N$.
Conversely, every $A\in\Omega_N$ defines an element $U\in Q_N$ by $U(b)=e^{A(b)}$ for all $b\in\bonds_N$.
We correspondingly write $A\sim B$ for $A,B\in\Omega_N$ whenever $e^A\sim e^B$
and denote $[A] \eqdef \{\omega\in\Omega_N\,:\,\omega\sim A\}$.
For $g\in\mfG_N$ and $A\in\Omega_N$ we further denote $A^g \eqdef \log [(e^A)^g]$.
Remark that $A^g\sim A$, but it is possible that $B\sim A$ with $B \neq A^g$ for any $g\in \mfG_N$.
We denote by $\pi\colon\Omega_N\to\Omega_N/{\sim}$ the projection

Fix $\alpha\in [0,1]$ throughout this subsection.
We define on $\Omega_N$ the Hölder-type norm $|\cdot|_{N;\gr\alpha}$ and semi-norm $|\cdot|_{N;\alpha;\rho}$ by

|A|_N;α    = sup{ |ℓ|^-α|A(ℓ)|  :  ℓ∈_N ,  |ℓ|>0} ,

|A|_N;α;ρ    = sup{ ρ(ℓ,ℓ̅)^-α |A(ℓ)-A(ℓ̅)|  :  ℓ,ℓ̅∈_N ,
ℓ∥ℓ̅} .

We then denote by $\Omega_{N;\alpha}$
the space $\Omega_N$ equipped with the norm
|\cdot|_{N;\alpha} \eqdef |\cdot|_{N;\gr\alpha}+|\cdot|_{N;\alpha;\rho}
For $M\leq N$, we clearly have inclusions $\olines_M\subset \olines_N$, which gives the natural projections $\pi_{N,M}\colon\Omega_{N} \twoheadrightarrow \Omega_{M}$ by restriction.

(π_N,MA)(x,e_i 2^-M) = A(x,e_i 2^-M)

for all $x\in \Lambda_M$ and $i\in[d]$.
Furthermore $\pi_{N,M}$ is clearly a contraction $\Omega_{N;\alpha}\to \Omega_{M;\alpha}$.
This endows $\{\Omega_{N;\alpha}\}_{N\geq 1}$ with the structure of a projective system of Banach spaces.
The projective limit $\varprojlim_{N\to\infty} \Omega_{N;\alpha}$ is naturally described as follows.

We denote by $\olines = \cup_{N=1}^\infty \olines_N$ the set of positively oriented lines in $\T^d$ of length $\leq \frac12$ whose starting and ending points are dyadic rationals.
We say that $\ell,\bar\ell\in\olines$ are joinable if $\ell=(x,e_ih)$ and $\bar\ell = (x+e_ih,e_i\bar h)$ for some $x\in\Lambda_N$, $i\in[d]$ and $h,\bar h\geq 0$ such that, treated as subset of $\T^d$, $\ell\cup\bar\ell\in\olines$,
i.e. if $|h+\bar h| \leq \frac12$.
We say that a function $A\colon \olines \to \mfg$ is additive if $A(\ell\cup\bar\ell)=A(\ell)+A(\bar\ell)$ for all joinable $\ell,\bar\ell\in\olines$.
We denote by $\Omega$ the vector space of additive functions.

With these definitions, it is clear that $\Omega = \varprojlim_{N\to\infty} \Omega_N$ as a projective limit of vector spaces.
We define the (extended) norm $|\cdot|_{\gr\alpha} = \lim_{N\to \infty}|\cdot|_{N;\gr\alpha}$ and (extended) semi-norm $|\cdot|_{\alpha;\rho} = \lim_{N\to \infty}|\cdot|_{N;\alpha;\rho}$ on $\Omega$, which are well-defined since $|\cdot|_{N;\gr\alpha}$ and $|\cdot|_{N;\alpha;\rho}$ are increasing in $N$.
Define the Banach spaces

Ω_α = {A∈Ω :  |A|_α<∞} ,  Ω_α = {A∈Ω :  |A|_α+|A|_α;ρ<∞} .

where $\Omega_{\gr\alpha}$ is equipped with the norm $|\cdot|_{\gr\alpha}$,
and $\Omega_{\alpha}$ is
equipped with the norm
$|\cdot|_{\alpha}\eqdef |\cdot|_{\gr\alpha}+|\cdot|_{\alpha;\rho}$.
Then $\Omega_{\alpha} = \varprojlim_{N\to\infty} \Omega_{N;\alpha}$ as a projective limit of Banach spaces.
We let $\pi_N \colon \Omega_\alpha\to \Omega_{N;\alpha}$ denote the corresponding projection.

For a vector space $\mcE$ of $\mfg$-valued distributions (in the sense of Schwartz),
let $\Omega \mcE$ denote the space of $\mfg$-valued distributional $1$-forms $A=(A_1,\ldots,A_d)$ with $A_i\in \mcE$.
If $\mcE$ carries a norm $|\cdot|_\mcE$, we denote
$|A|_{\mcE}\eqdef \max_{i\in[d]}|A_i|_{\mcE}$.
For  the space $\CC$  of $\mfg$-valued continuous functions
there exists an injection $\imath \colon \Omega\CC\to \Omega$
given for $A=(A_1,\ldots,A_d)\in\Omega\CC$ by
(\imath A)(x,e_ih) = \int_0^{|h|} A_i(x+te_i) \mrd t
It is easy to see that

|A|_α ≲|A|_^α/2 .

Here and below, we let $\CC^\gamma$ for $\gamma\in\R$ denote the Banach space of Hölder–Besov distributions on $\T^d$ as in <cit.>.
Conversely, if $\alpha>0$, then one also has for $A\in\Omega\CC$
|A|_^α-1 ≲|A|_α

(see <cit.>).
It follows from (<ref>) that one can view $\Omega\CC^\infty$ as a subspace of $\Omega_\alpha$ via $\imath\colon \Omega\CC^\infty \to \Omega_\alpha$
and we let $\Omega^1_\alpha$ denote the closure of $\Omega\CC^\infty$ in $\Omega_\alpha$.
Moreover, by (<ref>),
one can view $\Omega^1_\alpha$ as a subspace of $\Omega\CC^{\alpha-1}$ via
In the sequel, we will not write $\imath$ and $\imath^{-1}$ explicitly and will simply treat $\Omega^1_\alpha$
as a space of distributional $1$-forms.

Denote $\mfG^{\alpha}=\CC^{\alpha}(\T^d,G)$
and let $\mfG^{0,\alpha}$ denote the closure of smooth functions in $\mfG^\alpha$.
It follows from the exact same argument
as in <cit.> that, for $\alpha\in(\frac23,1]$, there is a locally uniformly continuous group action $\Omega_\alpha\times \mfG^\alpha \to\Omega_\alpha$, denoted $(A,g)\mapsto A^g$,
which restricts to a group action $\Omega^1_\alpha\times \mfG^{0,\alpha} \to\Omega^1_\alpha$
for which $\mfO_\alpha \eqdef \Omega^1_\alpha/ \mfG^{0,\alpha}$ is a Polish space.
For smooth $A,g$, the $1$-form $A^g$ takes the familiar form $A^g = \Ad_g A - (\mrd g) g^{-1}$.
As usual, for $A,B\in\Omega^1_\alpha$, we write $B\sim A$ if $B=A^g$ for some $g\in\mfG^{0,\alpha}$
and denote $[A] = \{\omega\in\Omega^1_\alpha\,:\,\omega\sim A\}$.
We denote by $\pi\colon\Omega^1_\alpha\to\mfO_\alpha$ the projection

The spaces $\Omega_{\gr\alpha}$, $\Omega_\alpha$, $\Omega^1_\alpha$, $\mfO_\alpha$ differ from
those denoted by the same symbols in <cit.>: our spaces are weaker as we only consider axis-parallel lines (vs. all straight lines in <cit.>).
However, it is straightforward to verify that all results on these spaces shown in <cit.>
remain true for our definitions
and we will use them without constantly highlighting this distinction.

§.§ Main results

We define our $\mfq$-valued Brownian motion $\BM$ and the white noise $\xi$ on the same probability space by setting
for $e\in\obonds_i$, $i\in [d]$,
(e) = ^1-d
⟨ξ_i(t,·),1_B(e,) ⟩
where $B(e,\e)$ is the $d$-cube centred at the midpoint of $e$ with side length $\e$.
The scaling in (<ref>) is such that  $\dt \BM$ is a discrete white noise under the metric (<ref>).

A mollifier is a smooth function $\moll\in\CC^\infty(\R\times\R^d,\R)$ such that $\int\moll = 1$ and with
support contained in $\{z\,:\, |z|<\frac14\}$.
We say that $\moll$ is non-anticipative if the support of $\moll$ is contained in $\{(t,x)\in\R\times\R^d\,:\, t\geq0\}$.

Throughout this subsection, we fix a non-anticipative mollifier $\moll$ and
independent and identically distributed (i.i.d) $\mfg$-valued white noises $\xi=(\xi_1,\ldots,\xi_d)$ on $\R\times \T^d$.
We tacitly identify $L(\mfg,\mfg)$ with a subspace of $L(\mfg^d,\mfg^d)$ by
extending every $C\in L(\mfg,\mfg)$ to a `diagonal' map $C \in L(\mfg^d,\mfg^d)$ by $C(A_1,\ldots,A_d)=(CA_1,\ldots,CA_d)$.

For the remainder of the section, we take $d=2$.
We further fix $\alpha\in (\frac45,1)$.
For $C\in L(\mfg^2,\mfg^2)$ and $a\in\Omega^1_{\alpha}$, let $\SYM_t(C,a)$ denote the solution at time $t>0$ to
the stochastic YM equations (SYM) driven by $\xi$ with mollifier $\moll$, mass renormalisation $C$, and initial condition $a$.
That is,
\SYM(C,a) = \lim_{\e \downarrow 0} A^\e
where $A^\e$ solves (<ref>)
and the limit is taken in the metric space $(\Omega_{\alpha}^1)^\sol$ that allows from blow-up, see <cit.>.
Therefore $\{\SYM_t(C,a)\}_{t\geq 0}$ is a continuous stochastic process with values in $\hat\Omega_{\alpha}^1\eqdef \Omega_{\alpha}^1\sqcup\{\skull\}$.
We recall from <cit.> that, for a metric space $(F,\rho)$, we define $\hat F \eqdef F\sqcup \{\skull\}$
equipped with a metric $\hat\rho$ which allows points to approximate $\skull$ given by

ρ̂(f,g) = ρ(f,g)∧(h(f)+h(g))

where $h\colon \hat F \to [0,1]$, $h(f)\eqdef (1+\rho(f,o))^{-1}$ for an arbitrary fixed $o\in F$.
We further let $\bar C\in L(\mfg,\mfg)$ be the $\moll$-dependent operator as in <cit.>.

Our first main result shows convergence of a class of discrete dynamics on the lattice to $\SYM(\bar C,\cdot)$,
which is a form of universality for the YM Langevin dynamic.
For a normed space $(\CB, \|\cdot \|)$, $T>0$, and $\eta\in\R$, let $\CC_\eta^T(\CB)$ denote the space of functions $f\in\CC((0,T],\CB)$ for which

f__η^T()  sup_t∈(0,T] t^-η/2 f(t)< ∞ .

Suppose Assumption <ref> holds.
Let $\eta<\alpha-1$, $a\in\Omega_{\alpha}^1$, and $a^{(N)} \in \log(\conf_N) \subset \mfq_N$
such that

lim_N →∞|a^(N) - π_N a|_N;α = 0 .

Consider the $\mfq$-valued Markov process $A^{(N)} \eqdef \log U^{(N)}$ with initial condition $a^{(N)}$.
Denote by $T^*>0$ the blow-up time of $\SYM(\bar C,a)$.
Then there exists an increasing sequence of stopping times $T_N$ such that $\lim_{N\to\infty}T_N=T^*$ almost surely and

lim_N→∞A^(N) - π_N (C̅,a)_^T_N_ η-α/2(Ω_N;α) = 0   in probability.

The stopping times $T_N$ can be taken as $T_N=K_N\wedge\inf\{t>0\,:\,|\SYM(\bar C,a)|_\alpha>K_N\}$
for some increasing sequence $K_N$ with $\lim_{N\to\infty} K_N=\infty$.

Assumption <ref> is purely local around $\id\in G$.
The reason such a local assumption on $S_N$ suffices is that the probability of $e^{A^{(N)}}$ leaving the neighbourhood $W$ in a short amount of time becomes small as $N\to\infty$.
Consequently, the only role that the exit time (<ref>) plays is to ensure that $U$ is well-defined
and Theorem <ref> remains true with $U^{(N)}$ replaced by $\check U^{(N)}$.

We give the proof of Theorem <ref> at the end of Section <ref>.
The proof is based on regularity structures and involves several steps.
In Section <ref> we derive the dynamics for $A^{(N)}$ on the lattice.
The main idea is to work directly on the Lie algebra $\mfq$ by pulling back the Riemannian structure of $G$ to $\mfg$ via the derivative of the exponential map.
In Sections <ref> and <ref>, we show local well-posedness of $A^{(N)}$ uniform in $N\geq1$ by means of discrete regularity structures.
The main work in Section <ref> is to show moment bounds on discrete models and that there is a finite mass renormalisation arising from the lattice approximations.
We emphasise that the equation for $A^{(N)}$ is non-polynomial for finite $N$ with a multiplicative noise
– this forces us to work with a substantially larger regularity structure than the one considered in <cit.>, see Section <ref>.
In Section <ref> we analyse the stochastic heat equation, which is the roughest part of $\SYM$.

In Section <ref> we use a diagonal argument to show that $A^{(N)}$ (along subsequences)
converges to $\SYM(C,\cdot)$ for some operator $C\in L(\mfg^2,\mfg^2)$.
Finally, in Section <ref> we show that $C=\bar C$ by arguing that $\bar C$ is the unique operator in $L(\mfg^2,\mfg^2)$ for which $\SYM(C,\cdot)$ is gauge covariant.
This final point is of independent interest,
resolving another problem left open in <cit.>, and can be summarised in the following theorem (see Theorem <ref> for a more precise statement).

Let $C\in L(\mfg^2,\mfg^2)$ for which $C\neq \bar C$.
There exists a loop $\ell\in\CC^\infty(S^1,\T^2)$ with the following property.
For all $t>0$ sufficiently small, there exists $g\in \CC^\infty(\T^2,G)$ such that

|W_ℓ(_t(C,0)) - W_ℓ(_t(C,0^g))| ≳t^2 ,

$W_\ell\colon\Omega_{\alpha}^1\sqcup\{\skull\} \to \C$  is the Wilson loop defined by $W_\ell(\omega) = \Trace\hol(\omega,\ell)$ for $\omega\in\Omega^1_{\alpha}$ and $W_\ell(\skull)=0$.

Here $\hol(\omega,\ell)\in G$ is the holonomy of $\omega$ along $\ell$, see <cit.> or (<ref>) below.
If $G$ is Abelian, a similar result is much simpler to show <cit.>
(see also Remark <ref> that contrasts the Abelian and simply connected cases).

The lower bound $t^2$ in (<ref>) is important as it rules out differing explosion times as the source of non-equality between $\E W_\ell(\SYM_t(C,0))$ and $\E W_\ell(\SYM_t(C,0^{g}))$ for small $t$.
That is, although $\SYM(\bar C,\cdot)$ is `gauge-covariant' in a suitable sense, $\SYM(\bar C,0^{g})$ and $\SYM(\bar C,0)$ may still explode at finite (and different!) times,
so we do not know if $\E W_\ell(\SYM_t(\bar C,0)) = \E W_\ell(\SYM_t(\bar C,0^{g}))$ holds for any $t>0$.
However, the probability that either process explodes before time $t$ is of order $O(t^M)$ for any $M>0$,
so one does have

|W_ℓ(_t(C̅,0)) - W_ℓ(_t(C̅,0^g))|  = O(t^M) .

Therefore, (<ref>) shows that the characterising feature of $\bar C$ is that one can take $M>2$ in (<ref>) while every other choice of $C$ at best allows $M=2$.

As a corollary of Theorem <ref>, we show that $\bar C$ is the unique operator for which `generative probability measures' (in the language of <cit.>) give rise to a Markov process on gauge orbits, see Corollary <ref>.

Our next main result is on the invariant measure of $\SYM(\bar C,\cdot)$ projected to gauge orbits.
To state this result, let $X$ be the time homogenous Markov process
taking values in $\mfO_\alpha$ from <cit.>
(one should think of $X_t=[\SYM_t(\bar C,\cdot)]$, although this is only formal because $\SYM(\bar C,\cdot)$ might blow-up before $X$ does).
Letting $X^x$ denote the law of $X$ with initial condition $x\in\mfO_\alpha$,
recall that $X^x$ has a.s. continuous sample paths with values in $\hat\mfO_\alpha \eqdef \mfO_\alpha\sqcup \{\skull\}$
and its law is $\pi_* \nu$, where
$\nu$ is a generative probability measure with initial condition $x$.

* There exists a unique probability measure $\mu$ on $\mfO_\alpha$ such that its finite dimensional distributions, seen as a process indexed by loops,
agree with those of the YM measure
for the trivial principal $G$-bundle over $\T^2$ in the sense of <cit.>.

* $\mu$ is the unique invariant probability measure on $\mfO_\alpha$ of $X$. Furthermore, $\mu$ has full support in $\mfO_\alpha$ and $X$ is reversible with respect to $\mu$.

The proof of Theorem <ref>
is given in Section <ref>
and employs Bourgain's invariant measure argument, lattice gauge-fixing, and ergodicity properties of $X$ (the latter following from recent ergodicity results in singular SPDE <cit.>).
The first part of Theorem <ref> (for simply connected $G$) is essentially due to <cit.>,
so the main new content is in the second part.
We prove the theorem first for the case that $G$ is simply connected, so that every principal $G$-bundle is trivial,
and then deduce the general result by an appropriate decomposition and projection.

We make no claims on the invariant measure of $\SYM(\bar C,\cdot)$ itself (in fact, without modifications, $\SYM(\bar C,\cdot)$ does not have an invariant probability measure already in the Abelian case, see <cit.>).
We do not even know if $\SYM(\bar C,\cdot)$ has global-in-time solutions.

The construction of the Markov process $X$ is one of the main results of <cit.>
and is shown therein through
a coupling argument with time-dependent gauge-transformations.
We believe our results, with further technical effort,[Specifically, one would need to improve the
$O(1)$ bounds in Sections <ref> and <ref> to convergences.]
can recover independently this construction via lattice approximations;
specifically, one should be able to prove the existence and uniqueness (of pushforwards) of
generative probability measures for some mass renormalisation $C\in L(\mfg,\mfg)$
(see <cit.>).
We do, however, crucially rely on the coupling argument of <cit.> in Section <ref> to prove Theorem <ref>.
which is used to identify the continuum limit of our lattice approximations and, perhaps more importantly,
to show that the Markov process is entirely canonical in that $\SYM$ does not yield
a Markov process on gauge orbits for any other mass renormalisation.

Finally, we turn to several consequences of Theorem <ref>.

The Markov process $X^x$ does not blow up for every initial condition $x\in \mfO_\alpha$.

Since $\mu$ has full support in $\mfO_\alpha$ and is invariant for $X$,
the conclusion follows from the strong Feller property of $X$ (see Proposition <ref> below).

We give the proofs of the next two corollaries in Section <ref>.
The first provides a decomposition of the YM measure into the GFF plus an almost Lipschitz remainder.

A Gaussian free field (GFF) is a random Gaussian  $\mfg$-valued distributional $1$-form $\Psi=(\Psi_1,\Psi_2)$ such that $\scal{\Psi_i,1}=0$ a.s., $\Psi_1,\Psi_2$ are i.i.d, and
\E\scal{\Psi_i,\phi}\scal{\Psi_i,\bar\phi} = \scal{\phi,(-\Delta)^{-1}\bar\phi}
for all $\phi,\bar\phi\in\CC^\infty(\T^2,\mfg)$ with $\int \phi = \int\bar\phi=0$.

There exist, on the same probability space, random distributional $1$-forms $\Psi,B$ such that $\Psi$ is a GFF
and $|B|_{\CC^{1-\kappa}}$ has moments of all order for each $\kappa>0$,
and $\mbox{Law}([\Psi+B]) = \mu$.

By the same (and simpler) arguments as in <cit.>,
a GFF $\Psi$ admits a modification with $|\Psi|_\alpha < \infty$, and thus $[\Psi+B]$ indeed makes sense as an $\mfO_\alpha$-valued random variable.

Furthermore, let  $|\cdot|_{\bullet}$ denote any norm from <cit.>, e.g.
$|\cdot|_{\v\alpha}$ or $|\cdot|_{\gr\alpha}$ (note that the norms $|\cdot|_{\bullet}$ are isotropic, i.e.
not restricted to axis-parallel lines,
in particular $|\cdot|_{\gr\alpha}$ in <cit.> denotes
a stronger norm than $|\cdot|_{\gr\alpha}$ here).
Then $\E|\Psi|^p_{\bullet} < \infty$ for all $p\geq 1$.
Therefore, Corollary <ref> shows that $\mu$ admits a gauge-fixed representation $A=\Psi+B$ with strong regularity properties,
namely with $\E |A|^p_{\bullet} < \infty$ for all $p\geq 1$.
This significantly improves <cit.> which only shows that there exists $A$ with $|A|_\alpha < \infty$ a.s. and $\mbox{Law}([A])= \mu$.

If $\mu$ were the invariant measure of $\SYM(\bar C,\cdot)$ itself, then
Corollary <ref> would be a straightforward consequence of decompositions in regularity structures.

However, because the Markov process with invariant measure $\mu$ is defined by restarting $\SYM(\bar C,\cdot)$ at appropriate stopping times, care is needed in applying this decomposition.

The final corollary that we present is a universality result for discrete approximations of the YM measure.
Since we are working only with trivial principal bundles, stating the result for non-simply connected $G$ requires some preparation.

Recall that $G$ is connected and compact. By the structure Theorem <ref>,  there exists $n\geq 0$ and a simply connected $L$ such that $G\simeq \T^n\times L / Z$ where $Z$ is a finite subgroup of the centre of $\T^n\times L$.
Denote $H\eqdef \T^n\times L$
and let $p\colon H\to G\simeq H/Z$ be the canonical projection.
(We recommend the reader first considers the case that $G$ is simply connected, which is already interesting and in which case $n=0$ and $G\simeq H \simeq L$.)

We consider a family of probability measures $\mu_N$ on gauge fields $G^{\obonds_N}$ as follows.
Let $\mu_{N,\T^n}$ be the discrete YM probability measure on $(\T^n)^{\obonds_N}$
associated to the trivial principal $\T^n$-bundle over $\T^2$ as in <cit.>.
We let $\mu_{N,L}$ be any probability measure on $L^{\obonds_N}$ as in (<ref>) with $S_{N,L}\colon L\to \R$ satisfying Assumption <ref> and
Assumption <ref> below.
Define $\mu_{N,H}=\mu_{N,\T^n}\times \mu_{N,L}$, understood canonically as a probability measure on $H^{\obonds}$.
Finally, let $\mu_N \eqdef p_* \mu_{N,H}$
be the pushforward of $\mu_{N,H}$ under $p\colon H^{\obonds} \to G^{\obonds}$.

The condition that $\mu_{N,\T^n}$ is distributed exactly as the Abelian YM measure for the trivial bundle can be significantly relaxed.
One way to do this is to work with families of actions $S_{N,\R^n}\colon \R^n\to \R$
and condition the corresponding plaquette variables to sum to zero, akin to <cit.>.
Since the interesting component is $\mu_{N,L}$, we do not elaborate on this point.

We say that a function $f\colon G^k\to \R$ is $\Ad$-invariant if, for all $h,g_1,\ldots, g_k\in G$,
f(g_1,\ldots,g_k) = f(hg_1h^{-1},\ldots, hg_kh^{-1})

A path $\ell \in \CC([0,1],\T^2)$ is called a lattice loop if there exist $N\geq 1$ and $x\in\Lambda_N$ such that $\ell(0)=\ell(1)=x$
and its image is contained in $\cup_{b\in\obonds_N}\iota(b)$ where $\iota(x,y) = \{x+t(y-x)\,:\,t\in[0,1]\} \subset \T^2$ is the natural embedding of $(x,y)\in\obonds_N$ into $\T^2$.
We call $x$ the base of $\ell$.

Observe that, for such $\ell$ and any $U\in Q_N$ and $A\in\Omega_{\gr\alpha}$, the holonomies  $\hol(U,\ell)$ and $\hol(A,\ell)$ are well-defined.

For $\Ad$-invariant $f\colon G^k\to\R$ and $k$-tuple of lattice loops $\ell=(\ell_1,\ldots, \ell_k)$,
$f_\ell\colon\Omega_{\gr\alpha}\to \R$ by

f_ℓ(A) = f((A,ℓ_1),…,(A,ℓ_k)) .

By the same expression, we define $f_\ell\colon Q_N \to \R$, which is well-defined provided $N\geq 1$ is sufficiently large (depending on $\ell$).

Observe that for $A,B\in \Omega^1_\alpha$, the following statements are equivalent:

* $A\sim B$

*  $f_\ell(A)=f_\ell(B)$ for all $k\geq 1$, continuous $\Ad$-invariant functions $f\colon G^k\to \R$, and $k$-tuples $\ell=(\ell_1,\ldots,\ell_k)$ of lattice loops with the same base.

In particular, every $f_\ell$ as in <ref> descends to a function $f_\ell\colon\mfO_\alpha \to \R$.
A similar statement holds for $U,V\in Q_N$ (provided we restrict to loops supported on $\obonds_N$).
(For the proof of this equivalence for $\Omega^1_{\alpha}$, see <cit.>; for the case of $Q_N$, the proof is similar and simpler.)
The implication (ii) $\Rightarrow$ (i) implies that, for probability measures $\mu,\nu$ on $\mfO_\alpha$, one has $\mu=\nu \Leftrightarrow \mu(f_\ell)=\nu(f_\ell)$ for all $f,\ell$, see <cit.>.

The following corollary can be interpreted as a functional central limit theorem for random connections on $\T^2$.

Let $k\geq 1$, $f\colon G^k\to \R$ a continuous $\Ad$-invariant function, and $\ell$ an $k$-tuple of lattice loops with a common base.
Then, for $\mu_N$ as above,
\lim_{N\to\infty}\mu_N(f_\ell) = \mu(f_\ell)
where $\mu$ is the YM measure on $\mfO_\alpha$ from Theorem <ref>.

§ LATTICE DYNAMICS IN THE LIE ALGEBRA

In this section, we derive a discrete SPDE
for $A\eqdef\e^{-1}A^{(N)}$
with $A^{(N)}$ as in Theorem <ref>
(i.e. $e^{\eps A}=U$ with $U=U^{(N)}$ defined by (<ref>))
which is in a form that “approximates” the limiting SPDE.
Recall that $\eps=2^{-N}$.
The derivation will be in arbitrary $d\ge 1$, although
we will only prove convergence in $d=2$.

This derivation requires several steps.
We first introduce some further notation to do with the lattice.
Some of the notation (e.g. (<ref>)-(<ref>)) may depend on a fixed plane in our $d$-dimensional lattice that is parallel with the $i$th and $j$th axes for some $i\neq j\in [d]$, but we will often omit the dependence on $i,j$ in our notation.

§.§ More notation

For any edge $e=(x,y)\in \obonds_i$, and $j\neq i$, we write

e^= (y,y+_j) ,

e^= (x,x+_j) ,

e^= (y-_j,y) ,

e^= (x-_j,x) .

One has $\{ e^\northeast ,e^\northwest ,e^\southeast,e^\southwest \}  \subset \obonds_j$,

see Figure <ref>.

Again for $i\neq j\in [d]$,
we write
_×_×^(i,j) {12 (s_1 _i+ s_2 _j) : s_1,s_2∈{±1}} .

If we identify a bond $e\in\obonds$ with its midpoint, then
\{ e^\northeast ,e^\northwest ,e^\southeast,e^\southwest \}
=\{e+\be:\be\in \CE_\times\}
Finally for $e=(x,x+\e_i)\in \obonds_i$, we write $p\succ e$ if $p=(x,\e,\bar\e)\in \plaq$ with $\e=\e_i$. For a given $e$ there are $2(d-1)$ plaquettes $p$ such that $p\succ e$.

[scale = 1.3]
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at (0.15,1.15) $x$;
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[draw, dot, name=d02] at (0,2) ;
[draw, dot, name=d12] at (1,2) ;

[draw,dot,name=L] at (-1,1) ;
[draw,dot,name=R] at (2,1) ;

[-Latex[length=3mm]] (x) – (d11)
node[draw=none, midway, above=0.5]$e$;
[-Latex[length=3mm]] (d11) – (d12) node[draw=none, midway, right=0.5]$e^\northeast$;
[-Latex[length=3mm]] (d02) –(d12) node[draw=none, midway, below=0.5];
[-Latex[length=3mm]] (x) – (d02) node[draw=none, midway, left=0.5]$e^{\northwest}$;

[-Latex[length=3mm]] (d10) – (d11) node[draw=none, midway, right=0.5]$e^{\southeast}$;
[-Latex[length=3mm]] (d00) – (d10) node[draw=none, midway, above=0.5];
[-Latex[length=3mm]] (d00) – (x) node[draw=none, midway, left=0.5]$e^{\southwest}$;

[-Latex[length=3mm]] (L) – (x) node[draw=none, midway, below=0.5];
[-Latex[length=3mm]] (d11) – (R) node[draw=none, midway, below=0.5];

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[draw, dot, name=d13] at (1,3) ;
[draw, dot, name=d22] at (2,2) ;
[draw, dot, name=d21] at (2,1) ;

[-Latex[length=3mm]] (x) – (y)
node[draw=none, midway, left=0.5]$e$;
[-Latex[length=3mm]] (d01) – (x) node[draw=none, midway, below=0.5]$e^\southwest$;
[-Latex[length=3mm]] (d01) – (d02) node[draw=none, midway, right=0.5];
[-Latex[length=3mm]] (d02) – (y) node[draw=none, midway, above=0.5]$e^\southeast$;
[-Latex[length=3mm]] (y) – (d22) node[draw=none, midway, above=0.5]$e^\northeast$;
[-Latex[length=3mm]] (d21) – (d22) node[draw=none, midway, left=0.5];
[-Latex[length=3mm]] (x) – (d21) node[draw=none, midway, below=0.5]$e^\northwest$;

[-Latex[length=3mm]] (d10) – (x) node[draw=none, midway, right=0.5];
[-Latex[length=3mm]] (y) – (d13) node[draw=none, midway, right=0.5];

We have $i=1$, $j=2$   in the left figure,
and $i=2$, $j=1$ in the right figure.
(Here horizontal bonds are in $\obonds_1$ and vertical bonds are in $\obonds_2$.)

§.§.§ Discrete derivatives

We define a collection of discrete differentiation operators. Note that we will omit their dependence on $\e$
in our notation, since it will be clear from the context whether we are dealing with discrete or continuous operators in the sequel.

Consider $A\in\mfq$.
For $j\in[d]$, we define the “forward and backward” difference operator $\nabla^{\pm}_j A\in\mfq$ by
%\nabla^\pm_j A(x,y) &= A(x\pm\e_j,y\pm \e_j) - A(x,y)\;,
%\mbox{i.e.} \quad
\nabla^\pm_j A(e) = A(e\pm\e_j) - A(e)
We define its rescaled version by $\partial^\pm_j A \eqdef \eps^{-1} \nabla^\pm_j A$.
Similarly, for $f\colon\Lambda\to \mfg$, we define $\nabla^{\pm}_j f, \partial^\pm_j f\colon\Lambda\to \mfg$ by the same expression with $e$ replaced by $x\in\Lambda$.

Moreover we define the second order derivative and the discrete Laplacian
∂_j^2 A (e) 1/^2(A(e+_j)+A(e-_j) -2 A(e)),
ΔA (e) 1/d ∑_j=1^d ∂_j^2 A (e).

For $A\in\mfq$, we define the discrete divergence $\nabla\cdot A \colon\Lambda\to\mfg$ by

∇·A (x) ∑_j=1^d ∇^+_j A(x-_j,x)
=∑_j=1^d [ A(x,x+_j) - A(x-_j,x) ] .

§.§.§ Symmetrised derivatives and averaging

Fix  $j\neq i\in[d]$.
For any $A_j\in \mfq_j$ and $e\in \obonds_i$, recall that
$A_j(e)$ is not defined since $j\neq i$, but
we will write[This “averaging” notation is prior to products, for instance,
$(A_j (e^{(a)}))^2$ just means the square of $A_j (e^{(a)})$,
rather than
$\frac14 \sum_{\be \in \CE_\times} A_j(e+\be)^2$.]
A_j (e^(a))
1/4 ∑_∈_× A_j(e+) .

Namely, the script $(a)$  indicates
an “average” of $A_j$ over the four edges in direction $j$
which are neighbours of $e$.
Thus $\obonds_i \ni e \mapsto A_j(e^{(a)}) \in\mfg$ is an element of $\mfq_i$.

We also define a symmetrised difference for  functions on $\Lambda$ or $\obonds$
∇_j 1/2 (∇^+_j - ∇^-_j) ,
∂_j ^-1 ∇_j  .

with $i\neq j$, for a function $A_j \in\mfq_j$ and edge $e \in \obonds_i$, we define additional symmetrised
differences as follows:

(∇̅_i A_j) (e)
1/2 (  A_j (e^)+A_j (e^)
- A_j (e^)-A_j (e^)) ,
∂̅_i = ^-1∇̅_i ,

and[(<ref>) will only be used
in the “remainder” terms of our discrete equation.]
(∇̅_j A_j) (e)
1/2 (  A_j (e^)+A_j (e^)
- A_j (e^)- A_j (e^)
) ,
∂̅_j = ^-1∇̅_j .

The assumption $j\neq i$ in the two definitions is important.
If $A\in\mfq$, we sometimes omit the index in $A_j$, e.g. write
$(\bar\partial_j A) (e) \equiv (\bar\partial_j A_j) (e)$ for (<ref>).

The `bar' in the notation $\bar\partial_i$ indicates a “switch” of bond sets,
i.e. $\bar\partial_i, \bar\partial_j \colon \mfq_j \to\mfq_i$ for $i\neq j$.

We remark that if $K$ is an even (resp. odd) function
in $x_j$ (the $j$-th coordinate of space variable),
then $\partial_j K$ and  $\bar \partial_j K$ are odd  (resp. even) function in $x_j$,
but this is not true for $\partial^+_j K$.
Such `parity' considerations on symmetrised v.s. unsymmetrised derivatives
will be crucial when deriving the form of the renormalisations in later sections.

§.§.§ Plaquettes and 1-forms

Recall from Section <ref> that we have fixed neighbourhoods $W = \mathring W^{\obonds_N}\subset\conf$
and $V = \mathring V^{\obonds_N} \subset \mfq$
for which $\exp \colon V\to W$
is a diffeomorphism.
Throughout this section, we only care about our actions on $W$ and $V$.

For the rest of this section, for  $U\in W$, we write $\A\eqdef \log U\in V$, and $A\eqdef \eps^{-1}\A \in \mfq$.
Consider a bond $e=(x,y)$ where $y=x+\eps_i$ for some $i\in [d]$.

U(x,y)= e^(x,y) ,      (x,y) = A(x,y) .

When we fix $i\neq j\in [d]$, there are two plaquettes $p\succ e$ for a given $e\in\obonds_i$, which we often write $p$ and $\bar p$, namely
p=(x,\e_i,\e_j)$ and  $\bar p=(x,\e_i, - \e_j)$.
To simplify notation, we use the conventions $\A_{1,2,\cdots,9}$ for the values of $\A$ at the bonds around $p$ and $\bar p$ as in Figure <ref>.
(This should be distinguished from the notation $A_i$
where $i\in [d]$ is the spatial index.) For instance, for $i\neq j \in [d]$,
\A_1 = \A(x,x+\eps_i)$,
$\A_2 = \A(x+\eps_i,x+\eps_i+\eps_j)$,
$\A_3 = \A(x+\eps_i+\eps_j,x+\eps_j)$, etc.

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[-Latex[length=3mm]] (d11) – (d10) node[draw=none, midway, right=0.5]$\A_5$;
[-Latex[length=3mm]] (d10) – (d00) node[draw=none, midway, below=0.5]$\A_6$;
[-Latex[length=3mm]] (d00) – (x) node[draw=none, midway, left=0.5]$\A_7$;

[-Latex[length=3mm]] (L) – (x) node[draw=none, midway, below=0.5]$\A_8$;
[-Latex[length=3mm]] (d11) – (R) node[draw=none, midway, below=0.5]$\A_9$;

(bp) at (0.5,0.5) $\bar p$;
(p) at (0.5,1.5) $p$;

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[-Latex[length=3mm]] (y) – (d02) node[draw=none, midway, above=0.5]$\A_5$;
[-Latex[length=3mm]] (y) – (d22) node[draw=none, midway, above=0.5]$\A_2$;
[-Latex[length=3mm]] (d22) – (d21) node[draw=none, midway, right=0.5]$\A_3$;
[-Latex[length=3mm]] (d21) – (x) node[draw=none, midway, below=0.5]$\A_4$;

[-Latex[length=3mm]] (d10) – (x) node[draw=none, midway, right=0.5]$\A_8$;
[-Latex[length=3mm]] (y) – (d13) node[draw=none, midway, right=0.5]$\A_9$;

(bp) at (0.5,1.5) $\bar p$;
(p) at (1.5,1.5) $p$;

The plaquettes $p,\bar p$ with conventions for $\A_{1,2,\cdots,9}$,

where $(i,j)$ in the two figures are as in Fig. <ref>.
We omit the dependence of $\A_{1,2,\cdots,9}$ on $i,j$ in our notation.

In calculating the cubic terms in the equation for $A$, it will be important to keep track of polynomials that contain factors $\A_1+\A_3$ and $\A_2+\A_4$ and other sums of oppositely pointing parallel bonds.
To this end, let $\CL(\R^m)$ denote the free Lie algebra generated by $\{\A_i\}_{i=1}^m$, where we view $\{\A_i\}_{i=1}^m$ as the canonical basis of $\R^m$.
We next define a class of polynomials called $\mcI_3$, which will allow us to treat error terms of the form $\e A^2\partial A$
(after rescaling) all together in Section <ref> without calculating their precise forms.

Let $\mcI$ denote the Lie ideal in $\CL(\R^9)$ generated by $\{\A_i+\A_j\,:\,i\in \{1,8,9\}, j\in \{3,6\}\}$ and $\{\A_i+\A_j\,:\,i \in \{2,7\},j\in \{4,5\}\}$.
Let $\mcI_3$ denote the homogenous degree 3 polynomials in $\mcI$.

Given $i\neq j$, $e\in\obonds_i$ and  $Q\in\mcL(\R^9)$, we denote by the same symbol $Q\colon \mfq\to\mfg$ the map $Q(\A)=Q(\A_1,\ldots,\A_9)$ with $\A_{1,2,\ldots,9}$ related to $e$ and $(i,j)$ as in Fig. <ref>.

We also say that a polynomial $I\colon \mfq\to\mfq$ is in $\mcI_3$ to mean that there exist $I_{ij}\in \mcL(\R^9)$ with $I_{ij}\in \mcI_3$
for $i\neq j$ such that $I(\A)(e)=\sum_{j\neq i}I_{ij}(\A_1,\ldots\A_9)$ for all $e\in\obonds_i$, where $\A_{1,2,\ldots,9}$ are related to $e$ and $(i,j)$ as in Fig. <ref>.

The following lemma is helpful in determining when Lie polynomials are in $\mcI$.

Suppose a Lie polynomial $Q(\A_1,\ldots,\A_9)$ vanishes upon the substitutions $\A_1,-\A_3,-\A_6,\A_8,\A_9\mapsto B_1$ and
$\A_2,-\A_4,-\A_5,\A_7,\mapsto B_2$.
Then $Q\in \mcI$.

Let $M\colon \R^m \to \R^n$ be linear and denote by $\hat M\colon \CL(\R^m)\to\CL(\R^n)$ its extension to a Lie algebra morphism.
We claim that $\Ker(\hat M)$ is the Lie ideal generated by $\Ker(M)$.
Indeed, by changing coordinates,
we may suppose $M(X_i)=Y_i$ for $1\leq i\leq m'$ and $M(X_i)=0$ for $m'<i\leq m$ for some $0\leq m'\leq m$,
where $\{X_i\}_{i=1}^{m}$, $\{Y_i\}_{i=1}^n$ are canonical bases for $\R^{m},\R^n$.
By freeness of $\CL(\R^n)$, $\Ker(\hat M)$ consists of all Lie polynomials in which $X_i$ occurs for $m'<i\leq m$, which is the ideal generated by
$\Ker(M)$, thus proving the claim.

To conclude the proof, it suffices to apply the claim to the linear map $M\colon\R^9\to\R^2$ defined by the mappings in the statement.

§.§ Pull-back of Riemannian metric to Lie algebra

Consider $U \in W$ and $B\in T_{U}W$.
For $\A\eqdef \log U\in V$
we say that $\fancyC\in T_\A V \simeq \mfq$
is the pull-back of $B$ if $D \exp_\A (\fancyC)=B$.
Similarly, denoting by $\eps\colon\mfq\to\mfq$ the map $A\mapsto \eps A$,
for $A\eqdef \eps^{-1} \log U$
we say that $C\in T_A\mfq\simeq\mfq$ is the rescaled pull-back of $B$ if $D( \exp\circ\eps)_A(C) =B$,
i.e. $C=\e^{-1}\fancyC$.

To find the discrete SPDE for $A=\e^{-1}A^{(N)}$, for $A^{(N)}$ as in Theorem <ref>,
we need to find the rescaled pull-back of the right-hand side of (<ref>).
This can, in principle, be done directly, by computing the right-hand side in $T_U \conf$ and finding its pull-back.
This ends up being a rather lengthy
calculation, so we opt for a somewhat different way.
Namely, we first pull back the Riemannian metric on $W$, coming from the bi-invariant Riemannian metric $\scal{\act,\act}$ on $G$,
to $V$ via the
Lie exponential map,
so that $\exp\colon V\to W$ becomes an isometry of Riemannian manifolds
(not just a diffeomorphism).
This endows $V$ with a Riemannian metric $g$
(that differs from its Euclidean metric)
and we work with $V$ instead of $W$ right from the start.

For the rest of this subsection, $A,B,C$ denote generic elements in the Lie algebra which are not necessarily related to our equation.

To write down the Riemannian metric $g$ on
$V$, recall the derivative of the exponential map <cit.>
/τ e^A(τ) =  e^A(τ) Φ(A(τ)) /τ A(τ) ,

Φ(A) = 1-e^-_A/_A
= ∑_k ≥0 (-1)^k/(k+1)!(_A)^k
∈L(,) .

Therefore, given tangent vectors $B,C\in T_A\mfq \simeq \mfq$,
the required inner product is
g_A(B,C) = e^A Φ(A) B, e^A Φ(A) C_e^A = Φ(A) B, Φ(A) C_ ,

where in the final equality we used the identification of $T_{e^A} \conf \simeq \mfq$ as left invariant vector fields.
We have thus put all the structure from $W$ that we require onto $V$ and can take gradients of actions on $V$ with respective to the metric $g$.
The picture we keep in mind is Figure <ref>.
The right-hand side of (<ref>) essentially contains three (unrelated) terms: the DeTurck term, the gradient flow, and the noise.
We derive each of these separately.

[smooth cycle, tension=0, fill=white] plot coordinates(1.5,2) (-0.5,-1) (3,-2) (5,1) node at (0.1,0.5) $\mfq$;

[smooth cycle, tension=0.8, fill=white] plot coordinates(2,1) (1,-0.2) (3,-1.3) (4,0.7) node at (1.5,-1.2) $V$;

[draw,dot,name=zero,label=180:$0$] at (2.25,0) ;

[draw,dot,name=A,label=-90:$A$] at (3,-0.5) ;

[-Latex[length=1.5mm]] (A) – (4.2,1.8) node[draw=none, midway, left=0.5]$B$;

[-Latex[length=1.5mm]] (A) – (1.8,1.5) node[draw=none, midway, right=0.5]$C$;

[smooth cycle, tension=0.4, fill=white] plot coordinates(8,2) (5.5,0) (9,-2) (11,1) node at (6.2,1.4) $Q$;

[smooth cycle, tension=0.8, fill=white] plot coordinates(7.6,1.3) (6.6,0) (9,-1) (10,0.7) node at (8,-1.2) $W$;

[draw,dot,name=id,label=180:$\id$] at (8.25,0) ;

[draw,dot,name=eA,label=-90:$e^A$] at (8.8,-0.2) ;

[-Latex[length=1.5mm]] (eA) – (11.2,0.5) node[draw=none, midway, below,xshift=3mm,yshift=0.5mm]$e^A\Phi(A)B$;

[-Latex[length=1.5mm]] (eA) – (9.4,2) node[draw=none, midway, left=0.5]$e^A\Phi(A)C$;

[->] (4, -1) edge [bend right] node[below] $\exp$ (6.5, -1);

Illustration of sets $V$, $W$ and tangent vectors $B,C\in T_A\mfq\simeq \mfq$.

§.§ DeTurck term

The DeTurck term in (<ref>) is $-\mrd_U\mrd^* U\in T_U\conf$.
Recall our notation for discrete derivatives from Section <ref>.
By (<ref>) one has $(\mrd^* e^\A)(x)= \eps^{-1} (\nabla \cdot \A) (x)$.

Together with (<ref>)
one has
(_e^ ^* e^)(x,y)
= ^-2
(∇·) (x)  e^(x,y)
- e^(x,y)  (∇·) (y)

Since $-\mrd_U\mrd^* U$ is in the tangent space $T_U \conf$,
we need to pull it back to $\A$ via the $\exp$ map.
More precisely, recalling (<ref>),
the pull back of the DeTurck term to $\A$ is the element  $h \in T_{\A}\mfq$ such that
$-\mrd_{e^{\A}} \mrd^* e^{\A} = e^{\A} \Phi(\A) h$, namely
-h = Φ()^-1e^-(_e^ ^* e^) .

To obtain a series representation for $\Phi(\A)^{-1}$, note that

Φ()^-1 = (1+ψ())^-1 = 1+∑_n≥1(-1)^nψ()^n ,

$\psi(\A) = \sum_{k \geq 1} \frac{(-1)^k}{(k+1)!}(\ad_{\A})^k
=-\frac12\ad_{\A} + \frac16(\ad_{\A})^2 + O(\A^3)$.

= 1+1/2_ + 1/12(_)^2
+O(^3) .

We now list the contributions to $-\eps^2 h(e)$ for $e=(x,y)$ of the first three orders, using formulas (<ref>), (<ref>), and  (<ref>)

Order 1:
Taking $e^{\A} = 1 + O(\A)$ and $\Phi(\A)^{-1}=1+O(\A)$, we find

(∇·)(x) - (∇·)(y) = -∇^+_i (∇·)(x) .

Order 2:
From the series expansion of $\Phi(\A)^{-1}$,
$e^{-\A}$ and $e^{\A}$, it is easy to see that the 2nd order terms are

Δ_2 (e)

-(e) Δ_1(e)
+(∇·)(x)(e) - (e)  (∇·)(y)

-1/2[(e),(∇·)(x)+(∇·)(y)] .

Order 3: Due to $(\nabla \cdot \A) (x)$ and $(\nabla \cdot \A) (y)$ in (<ref>), clearly the order 3 part of $\e^2h$ is in $\mcI_3$ (in the sense of Definition <ref>).
(One can verify that the order 3 term is $\Delta_3(e)
\eqdef
\frac1{12}[\A (e),[ \A (e),\Delta_1(e)]]$,
but we do not need its exact expression.)

There exists $\Delta_3\colon \mfq\to\mfq$ in $\mcI_3$

and analytic $r_h\colon V \to\mfq$ such that $r_h(\A) = O(\A^4)$
and for every $e\in \obonds_i$,

^2 h (e)    = -Δ_1(e)-Δ_2(e)-Δ_3()(e)+r_h()(e) .

The leading terms are obtained by the above derivation.
The remainder
$r_h(\A)$ is $O(\A^4)$ (recall Remark <ref>)
by analyticity of the functions $x\mapsto e^x, 1/(1+x), (1-e^{-x})/x$
on $\R$ in a neighborhood of the origin.

§.§ Gradient term

We now turn to the gradient term in (<ref>).
In view of Sections <ref> and <ref>,
the action
$\mcS \colon W \to \R$  is  given by
(U) = 1/8∑_p∈ {^d-4 l^p(U) + R^p_(U)}

where (recalling $R_\e$ in (<ref>) and $U(\partial p)$ in (<ref>))

l^p(U) |logU(∂p)|_^2 ,     R^p_(U)R_(U(p̣)) .

For a function $f\colon W\to \R$ we write $\hat f \colon V\to \R$ for the pullback $\hat f(\A) \eqdef f(e^\A)$.
By construction, the gradient $\nabla \hat f(\A)\in T_\A V\simeq \mfq$ with respect to the Riemannian metric $g_\A$ is the pull-back of $\nabla f(e^\A)\in T_{e^\A}W$.

For any bond $e\in\obonds$ there are $8$ plaquettes containing $e$ and which enclose the same square in $\T^d$. Therefore

= -^d-4/2 ∑_p≻e ∇l̂^p()(e)
- 1/2 ∑_p≻e∇R̂^p_()(e)  .

Focusing first on $
-\frac12{\eps^{d-4}} \sum_{p\succ e} \nabla \hat l^p(\A)
\in T_{\A} V\simeq \mfq
$, which is the
pull-back of $-\frac12\eps^{d-4} \sum_{p\succ e} \nabla l^p(U) \in T_U W$,
we compute in this subsection its linear, quadratic, and cubic terms.
The terms with $R^p_{\eps}(U)$  in (<ref>) will be dealt with in Section <ref>.

We can write
$\hat l^p(\A) = |P(\A)|_\mfg^2$,
where $P\colon V\to\mfg$ is the analytic function
$P(\A) = \log (e^{\A}(\partial p))$.
To find the gradient $\nabla \hat l^p(\A) \in T_\A \mfq$, we compute for $B\in T_\A\mfq \simeq \mfq$,

D l̂^p() (B) = 2DP()(B), P()_= B, 2DP()^* P()_
where $DP(\A) \in L(\mfq,\mfg)$ with adjoint $DP(\A)^* \in L(\mfg^*,\mfq^*) \simeq L(\mfg,\mfq)$
and where we used the inner products on $\mfg$ and $\mfq$ to identify them with their duals.

On the other hand by (<ref>)

g_(B,∇l̂^p()) = Φ() B,Φ()∇l̂^p()_= B,Φ()^* Φ()∇l̂^p()_
where $\Phi(\A)^*\in L(\mfq^*,\mfq^*) \simeq L(\mfq,\mfq)$.
Hence $2DP(\A)^* P(\A) = \Phi(\A)^* \Phi(\A)\nabla \hat l^p(\A)$
and therefore:

$\frac12\nabla \hat l^p(\A) = (\Phi(\A)^* \Phi(\A))^{-1} DP(\A)^* (P(\A))$.

We proceed to calculate the series expansion of $\frac12\nabla \hat l^p(\A)(e)$.
For this we need to have
more explicit expressions for $(\Phi(\A)^* \Phi(\A))^{-1}$
and $P(\A)$.

Since $\ad_\A$ is skew-Hermitian, i.e. $\ad_\A^*=-\ad_\A$, one has $(e^{-\ad_\A})^*=e^{\ad_\A}$ and
%= 1+\sum_{k\geq 1} \frac{1}{(k+1)!}(\ad_\A)^k
= \frac{e^{\ad_\A}-1}{\ad_\A}$.

Φ()^* Φ()
1 + ϕ() ,

where $
\phi(\A) \eqdef
\sum_{k \geq 1}\frac{2}{(2k+2)!} (\ad_\A)^{2k}
Hence expanding $(1+\phi(\A))^{-1}$ in a Neumann series one has
(Φ()^* Φ())^-1

= 1 + ∑_k≥1 (-1)^k ϕ()^k =
1-1/12_^2+O(^3) .

We calculate $P(\A)$ using
the following
Baker-Campbell-Hausdorff (BCH) formula.

For any $A_1,A_2,A_3,A_4\in \mfg$ we have

= exp(∑_a A_a + 1/2 ∑_a<b [A_a,A_b] + 1/12 ∑_a,b[A_a,[A_a,A_b]]

+ 1/6 ∑_a<b<c ([A_a,[A_b,A_c]] + [A_c,[A_b,A_a]]) +O(A^4) )	.

Here $a,b,c \in \{1,2,3,4\}$
and $O(A^4) $ is a remainder in the sense of Remark <ref>
and is analytic in $\{A_i\}_{i=1}^4$ in a neighbourhood of $0$.

This can be checked via direct calculation, for instance starting from the well-known formula for two matrices, so we omit the details.

<cit.>, the series in the parenthesis on the right-hand side is absolutely convergent for $A_i$ sufficiently small and
the truncated error is $O(A^4)$ in the sense of Remark <ref>.

Since $(\Phi(\A)^* \Phi(\A))^{-1}$ acts on the “diagonal”, to calculate $\frac12\nabla \hat l^p(\A)(e)$
we need to find $DP(\A)^*(X)(e)$ for an element $X\in\mfg$
(which is a placeholder for $P(\A)$).
Consider $\bar{\A}\in \mfq \cong\mfg^{\obonds_N}$ such that $\supp \bar{\A}= \{e\}$.
Then, recalling (<ref>),

^d-2 DP()^*(X)(e),(e)_=DP()^*(X),_= X,DP()()_ .

If $p$ does not contain $e$, one has $DP(\A)(\bar \A)=0$ and thus $\nabla \hat l^p(\A)(e)=0$.
It therefore suffices to consider one plaquette $p$ as in Figure <ref>, and the derivation for the other $p\succ e$ (such as  $\bar p$  Figure <ref>) simply follows by suitable flipping of components.
Recall our notation $\A_{1,\cdots,4}$ as in Figure <ref>.
Since $P(\A) = \log (e^{\A_1}e^{\A_2}e^{\A_3}e^{\A_4})$, by the BCH formula (Lemma <ref>),

P() =
∑_a _a +1/2 ∑_a<b[_a,_b]+ O(^3)	 .

DP() ()
+ 1/2∑_b=2^4 [(e),_b]
+O(^2(e)) .

Recalling Definition <ref>, there exist $R^p_3 \in\mcI_3$ and analytic $r^p_l\colon V \to\mfq$ such that $r^p_l(\A)=O(\A^4)$

- ^d-2/2 ∇l̂^p() (e)
-∑_a=1^4 _a - 1/2([_2,_3] + [_2,_4]+[_3,_4])

+ 1/2 [_2,[_2,_1]]
+ R^p_3()(e) + r^p_l()(e) .

By (<ref>), we obtain

^d-2 DP()^*     (X)(e)
- 1/2∑_b=2^4 [X,_b] + O(^2 X) .

Orders 1 and 2:
Lemma <ref> and (<ref>)+(<ref>)+(<ref>)
immediately imply that
^d-2/2 ∇l̂^p()(e)

= ∑_a _a + 1/2∑_a<b[_a,_b] - 1/2∑_a∑_b=2^4 [_a,_b]

=  ∑_a _a + 1/2([_2,_3]+ [_2,_4]+[_3,_4])
+ O(^3) .

Order 3:
The terms at order 3 are obtained as follows:

* The $\A^2$ term of $(\Phi(\A)^* \Phi(\A))^{-1}$ with the $\A$ part (i.e. linear part) of $P(\A)$.

* The quadratic part of $P(\A)$ with the linear part of $DP(\A)^*$.

* The $\A$ part of $P(\A)$ with the $\A^2$ part of $DP(\A)^*$.

* The $\A^3$ part of $P(\A)$.

Since the linear part of $P(\A)$ is in $\mcI$, the 1st and 3rd cases yield terms in $\mcI_3$.
The 4th term gives
$ \frac12\big([\A_1,[\A_1,\A_2]] -  [\A_2,[\A_2,\A_1]]\big) + \mcI_3$ as can be easily checked by Lemmas <ref> and <ref>.
The 2nd case gives $- \frac14\sum_{b=2}^4 [ [\A_2,\A_3]+ [\A_2+\A_3,\A_4],\A_b]$,
which, by Lemma <ref>, equals $- \frac14[ [\A_2,\A_3]+ [\A_3,\A_4],\A_3]+\mcI_3=-\frac12 [\A_1,[\A_1,\A_2]]+\mcI_3$,
which cancels the first term in the contribution from the 4th case modulo $\mcI_3$.

We thus obtain the desired equality in which the error bound $r^p_l(\A)=O(\A^4)$
follows from Lemma <ref>
and analyticity of $(e^{-x}+e^x-2)/x^2$ and $1/(1+x)$.

§.§ Gradient of the remainder

We now turn to the term $R^p_{\eps}(U)$ in the action (<ref>).
Recall the pullback  to $V$

R̂_^p V→,   R̂_^p()= R^p_(e^) = R_(e^(p̣)) .

To complete the computation of the gradient term, we find $\nabla \hat R_\e^p(\A) \in T_\A V \simeq \mfq$.

Suppose that Assumption <ref> holds and
let $E_{\eps}^{(1)}\in L(\mfg,\mfg^*)$ and $E_{\eps}^{(2)}\in L(\mfg^{\otimes 3},\mfg^*)$ be the corresponding linear operators.
For $p \succ e$ as in Fig. <ref>,
=^2-d E_^(1)(∑_a=1^4 _a)^*
+ ^-2 E_^(2)[(∑_a=1^4 _a)^⊗3]^*

+ O(^2-d^2)
+ ^-2O(^4)

where  $F^*\in\mfg$ is the dual of $F\in \mfg^*$, i.e. $\scal{F^*,Y}_\mfg=F(Y)$.

By a similar derivation as above Lemma <ref>, for $\bar \A\in T_\A\mfq \simeq \mfq$,

= g_(,∇R̂_^p())
=,Φ()^*Φ()∇R̂_^p()_ ,

and therefore

∇R̂^p_() = [Φ()^*Φ()]^-1[DR̂_^p()]^* ∈ ,

where $F^*\in\mfq$ is the dual of $F\in\mfq^*$.
Since $[\Phi(\A)^*\Phi(\A)]^{-1}$ acts on the diagonal,
it suffices to find $[D\hat R^p_\e(\A)]^*(e)\in\mfg$, which is characterised by

^d-2 [DR̂^p_()]^*(e), (e)_= DR̂^p_()()

for all $\bar \A\in\mfq$ supported on $\{e\}$.
Fixing $\bar \A$ supported on $\{e\}$,
by the chain rule

DR̂^p_()() = [DR̂_(P())](DP()())

where $P(\A) = \log e^\A (\d p)$ and $\hat R_\e(X) = R_{\eps}(e^X)$ as before.
By (<ref>), we have

DP()() = (e) + O() .

Furthermore, by Assumption <ref>, we have that $D\hat R_\e(P(\A))](Y)$ for $Y\in\mfg$ equals

E_^(1)(P())(Y) + ^d-4E_^(2)(P()^3)(Y)
+ O(P()^2Y) + ^d-4O(P()^4Y)

= E_^(1)(∑_a=1^4 _a)(Y)
+^d-4 E_^(2)[(∑_a=1^4 _a)^⊗3](Y)
+ O(^2Y)
+ ^d-4O(^4Y)

where we used $P(\A)=\sum_{a=1}^4 \A_a + O(\A^2)$ in the final equality.
Combining with (<ref>) and (<ref>), we obtain
= E_^(1)(∑_a=1^4 _a)((e))    +
^d-4 E_^(2)[(∑_a=1^4 _a)^⊗3]((e))

+^d-4O(^4) .

Finally, recall from (<ref>) that
$[\Phi(\A)^*\Phi(\A)]^{-1} = 1+ O(\A^2)$.
The desired expansion (<ref>)
thus follows from (<ref>)
and (<ref>).

§.§ Noise term

The final term in (<ref>)
is the noise term $U \circ \dt \BM$.

Let $\xi^\e= \eps^{-1}\dt \BM$ (cf. (<ref>)).

Below we omit $\circ$ in Stratonovich products.
As with the DeTurck term, we need to pull the vector $U\eps \xi^\e\in T_U \conf$ back to $\A$ via the $\exp$ map.
More precisely, recalling (<ref>),
the pull back of the noise term to $\A$ is the element  $z \in T_{\A}\mfq$ such that
$e^{\A}\eps\xi^\e= e^{\A} \Phi(\A) z$, namely
$z = \eps\Phi(\A)^{-1}\xi^\e$.
We rewrite this as

where (after possibly shrinking $\mathring V\subset\mfg$), $F\colon \mathring V\to L(\mfg,\mfg)$ is the analytic function

F(X) = Φ(X)^-1-1/_X = 1/2 + 1/12_X + O(X^2) ,

where we recall the series expansion for $\Phi(X)^{-1}$ from (<ref>).
Below, whenever clear from the context, we often write $\xi$ for $\xi^\e$.

Recalling further that $\eps A=\A$ and that $Z\eqdef\eps^{-1}z$ is the rescaled pull-back of $U \circ \dt \BM$,
we obtain
Z(e)    = ξ(e) +  F(A(e))[A(e),ξ(e)]

= ξ(e)+ /2[A(e),ξ(e)] + ^2/12[A(e),[A(e),ξ(e)]] + O((A)^3ξ) .

Note that the second and third terms in the 2nd line of (<ref>) have negative power counting
and that even $O((\eps A)^3\xi)$ is classically ill-defined (requiring the Stratonovich product in time to make sense).

To handle this `multiplicative noise term', we will introduce a new `abstract' noise modelling $\eps^{1-\kappa} \xi$ and a `multiplication by $\eps$' map sending $A$ to a function-like modelled distribution $\e A$,
and show that $\e F(\e A(e))[A(e),\xi(e)]$ vanishes on the level of modelled distributions.

§.§ Rescaled dynamics in Lie algebra

We collect all the terms obtained from previous subsections.
In the sequel, we let $\A = \log \check U^{(N)}$ for $\check U^{(N)}$ as in (<ref>).
Recall the exit time $\varpi$ of $\check U$ from $W$.

We rewrite our equation in terms of the “macroscopic” variable $A(t,e)= \eps^{-1}\A (t, e)$

which will have a form that at least formally “approximates” the limiting SPDE,
modulo various “remainder” terms.
Recall our notation $\Delta$ for discrete Laplacian in Section <ref> and
various notation
in Section <ref>.

We further define

R̂^∇_V→     R̂^∇_(A)(e)
∑_p≻e  ∇R̂^p_(A)(e)  .

For $i\in [d]$ and every $e\in \obonds_i$,
one has on the interval $[0,\varpi]$
∂_t A_i  (e)
= ΔA_i (e)
+ ∑_j≠i
[A_j (e^(a)),
(2∂_j A_i)(e)  - (∂̅_i A_j) (e) ]

+ [A_i (e), (∂_i A_i)(e) ]
+  ∑_j≠i[A_j(e^(a)) ,[A_j(e^(a)) ,A_i(e) ]]
+ ξ_i (e)

+ F(A_i(e))[A_i(e),ξ_i(e)]

+ ^-1 r(A)(e) ,

where $\tilde{R}_2(A)$ is defined
in (<ref>)
$I_3\colon\mfq\to\mfq$ is in $\mcI_3$ (in the sense of Definition <ref>).

Moreover, with

$r_h$ as in (<ref>) and $r_l = \sum_{p\succ e} r_l^p$ for $r_l$ as in Lemma <ref>,
^-1 r(A) (e)
-^-1/2  R̂^∇_(A)(e)
+  ^-3 r_h ( A)(e)
+  ^-3  r_l(A)  (e) .

For most of the proof
we omit dependence on $e$
and write e.g. $r(\A)=r(\A)(e)$.
Following (<ref>),
\partial_t \A(e)
= - \frac{\e^{d-4}}{2} \sum_{p\succ e} \nabla \hat l^p(\A)
-\frac12 \hat R^{\nabla}_\e(\A)
+ h(e) + z(e)
We now add up the linear, quadratic and cubic terms.
We  claim that
∂_t (e) =
+ Q ()+ T() +r() + ξ+ F((e))[(e),ξ]

where $F$ is given by (<ref>).
$Q(\A) = \sum_{j\neq i}Q_j(\A)$ where for each $j\neq i$

Q_j()    ^-2 [ 1/4  (_2-_4 -_5+_7)  ,    (_6 - _3)  -  1/2 (_2 + _4 - _5 -_7)]

+^-2/2(d-1)[_1,_9 - _8]

+^-2/4 [2_1 + _3 + _6 , _2+_5-_4-_7]
- ^-2/4 [_2 +_5, _4+_7]

as in Fig. <ref> (understood as in the $(i,j)$ plane),
and $T(\A)$ is equal to $\big(\sum_{j\neq i}T_j(\A)\big) -\e^{-2} \big( \sum_{p\succ e}R_3^p(\A) + \Delta_3(\A)\big)$ where $R^p_3$ and $\Delta_3$ are as in Lemmas <ref>
and <ref> respectively, and
[_2, [_2, _1]]
[_5 , [_5 , _1]]
) .

$ r (\A)\eqdef
-\frac{1}{2} \hat R^{\nabla}_\e(\A)
+   \e^{-2} r_h (\A)+  \e^{-2}  r_l(\A) $.

To prove the claim,
the linear terms in $- \frac{\e^{d-4}}{2} \sum_{p\succ e} \nabla\hat l^p(\A)$ in Lemma <ref>

plus the 1st order DeTurck term  $-\e^{-2}\Delta_1$
from Lemma <ref> or (<ref>)

clearly gives the discrete Laplacian $\Delta \A (e)$.

For $j\neq i$,
the quadratic terms from
$- \frac{\e^{d-4}}{2} \sum_{p\succ e} \nabla\hat l^p(\A)$
in the $(i,j)$ plane are

[_2,_3] + [_2, _4] + [_3,_4] + [_5,_6] + [_5,_7]  + [_6,_7]
) .

The 2nd order DeTurck term $-\e^{-2}\Delta_2$ in Lemma <ref> is the sum over $j\neq i$ of

[_1 ,  1/d-1_9+_2+_5 - (_4+1/d-1_8 + _7)] .

By straightforward computation,
the sum of (<ref>) and (<ref>) is equal to $Q_j(\A)$ as in (<ref>). Here, note that
the term $[\A_1,\A_9-\A_8]$ is shared
among all choices of $j\neq i$,
which is responsible for the factor $1/(d-1)$.

The form of the cubic term  $T(\A)$
in (<ref>)
follows from Lemma <ref>.
The noise terms and the term
also clearly follow from the above calculations.

This proves the claim.

Next, by Lemma <ref>, we can
rewrite $T_j(\A)$ in (<ref>)

^2 T_j () =
1/16 [_2-_4+_7-_5 , [_2-_4+_7-_5,_1]]+R̃_3,j ,

$\tilde{R}_{3,j}\in \mcI_3$.
Remark that $\frac14(\A_2-\A_4+\A_7-\A_5) = \e A_j(e^{(a)})$.

Let $j\neq i$ for the rest of the proof.

We rewrite the above equation

with $A= \eps^{-1}\A $.
The linear term remains as the discrete Laplacian.
By (<ref>) and recalling the notation
$A_j (e^{(a)})$, $\partial$, $\bar\partial$, $\d^2$, from Section <ref>,
the term $\e^{-1}Q(\e A)$ is equal to
the 2nd and 3rd term on the right-hand side of (<ref>) plus
R̃_2(A)(e) ∑_j≠i
- ^2/2 [∂_j^2 A_i(e), ∂̅_j A_j(e)]
+ /4
[ (∂^+_j A_j) (e^)   ,
(∂^+_j A_j)(e^)    ]

(which arises from the last line in  (<ref>)).

Moreover, since all the terms on the right-hand side of (<ref>) are cubic in $\A$,
we see that $\e^{-1} T(\e A) $ is equal to the cubic terms in
(<ref>) with $I_3(A)(e) =  \sum_{p\succ e} R_3^p(A) - \Delta_3(A)(e) +\sum_{j\neq i}\tilde{R}_{3,j}(A)(e)$.

Finally the expression for $\e^{-1} r(\e A) $
follows obviously
and the terms with $\xi$ follow from (<ref>) (or from (<ref>)).

The terms in (<ref>) all contain a power of $\e$.
this is the case for $\e^{-3} r_h ( \e A)$ and  $\e^{-3}  r_l(\e A)$
since $r_h(\A)$ in  (<ref>) and $r_l(\A)$ in  Lemma <ref> are both $O(\A^4)$.
Regarding $\e^{-1}  \hat R^{\nabla}_{\eps}(\e A)$, under Assumption <ref> and by
Lemma <ref>,
$ \hat R^{\nabla}_{\eps}(\A)$ is of the form
So assuming $d=2$,
the terms  $O(\A^2)$ and $\e^{-2}O(\A^4)$ in (<ref>) contribute a term $\eps O(A^2)$ and $\eps O(A^4)$
to $\e^{-1} r(\e A) $,
which contains a power of $\eps$.
For the other terms in (<ref>),
upon identifying $\mfg^*$ with $\mfg$,
the term $E^{(1)}_{\eps}(\sum_{a=1}^4 \A_a)$ contributes to $\e^{-1} r(\e A) $ a term
$\e E^{(1)}_{\eps}(\d_i A_j - \d_j A_i)$
which does have a factor $\e$.
Lastly, the term $\e^{-2} E^{(2)}_{\eps}(\sum_{a=1}^4 \A_a)^{\otimes 3}$ contributes to $\e^{-1} r(\e A) $ a term
\e E^{(2)}_{\eps}\big[(\d_i A_j - \d_j A_i)\otimes A\otimes A\big]
again containing a power of $\e$.

§ DISCRETE REGULARITY STRUCTURES

§.§ Discrete function spaces

Denote by $\T_\e^d \subset \T^d$ a generic square lattice with lattice spacing $\e$ which is embedded in $\T^d$.
Note that $\Lambda_N$ (where $\e=2^{-N}$) and $\obonds_j$
for $j\in [d]$ are all incidences of
$\T_\e^d$, which are embedded in $\T^d$ in the natural way (i.e. identify midpoint of a bond with a point in $\T^d$).
We will write $A\asymp B$ if $A$ is bounded from above and below by $B$ up to proportionality constants uniform in $\e$ (or equivalently, in $N$).

We will write $\int_{\T_\e^d}$ for $\e^d \sum_{\T_\e^d}$.
Let $(E, |\cdot |)$ be a Banach space.
Given a function $f^\e\in E^{\T_\e^d}$ and  a smooth function $\phi\colon \T^d\to \R$,
we write $ \langle f^\eps, \varphi \rangle_\eps $ or $f^\e(\varphi)$
for $ \int_{\T_\eps^d} f^\eps(y) \varphi(y)\mrd y$.
Whenever it is clear from the context,
for space-time functions $f^\e$ on $[0,T]\times \T_\e^d$ and smooth $\phi$ on $[0,T]\times \T^d$
we also write pairings as $ \langle f^\eps, \varphi \rangle_\eps $ or $f^\e(\varphi)$
with an additional integral in time.

Define the parabolic scaling $\s=(\s_0,\cdots,\s_d)=(2,1,\cdots,1)$ and write $|\s|=d+1$.
For a multi-index $k=(k_0,\ldots,k_d)\in \N^{d+1}$,
we write $|k|_\s = \sum_{i=0}^{d}\s_ik_i$.
For $z=(z_0,\cdots,z_d)\in\R^{d+1}$, let

\[
\|z\|_\s = \sup\{ |z_i|^{1/\s_i}:0\le i\le d\}\;,
\qquad
\CS_\s^\lambda z = (\lambda^{-\s_0} z_0,\cdots, \lambda^{-\s_d} z_d)\;.
\]

For $z=(z_0,\cdots,z_d)\in\R\times \T^d$, we define $\|z\|_\s$ in the same way.

For $\phi:\R^{d+1}\to \R$, we write $\phi^\lambda_z (y) = \lambda^{-|\s|} \phi(\CS_\s^\lambda(y-z))$.
We will also use rescaled test functions in time or space only:

(1) For $\phi:\R\to \R$, recalling $\s_0=2$, we write
$\CS_{2,t}^{\lambda}\phi = \lambda^{-2} \phi(\frac{\cdot - t}{\lambda^2})$
for a rescaled temporal test function centred at $t$.

(2) For $\phi:\R^d\to \R$, we write
$\CS_{x}^{\lambda}\phi = \lambda^{-d} \phi(\frac{\cdot - x}{\lambda})$
for a rescaled spatial test function centred at $x$,
and with a slight abuse of notation we also write it as $\phi^\lambda_x$.

Now we define some norms and distances
for functions over space or over space-time.
Let $\alpha\in [0,1]$.
To compare functions $f\in\CC_\s^\alpha ([0,T]\times \T^d,E)$
and $f^\e\colon [0,T]\times \T_\e^d \to E$, we define
+ sup_z, w∈[0,T]×^d

+ sup_z, w∈[0,T]×_^d
|(f(z)-f(w))-(f^(z)-f^(w))|/z-w_^α .

We then define $\|f^\e\|_{\CC_\e^{\alpha,T}} = \|0;f^\e\|_{\CC_\e^{\alpha,T}}$
and spaces $\CC_\e^{\alpha,T}$ with this norm.

For any $f^\e\in E^{\T_\e^d}$ and $f\in\CC^\alpha(\T^d,E)$ in the usual Hölder space,
we define $\|f;f^\e\|_{\CC^\alpha_\e}$ as in (<ref>)
but with $\|z-w\|_\s$ replaced by $|z-w|$ and with supremums over $\T^d_\e$ and $\T^d$.

We then define $\|f^\e\|_{\CC_\e^\alpha}=\|0;f^\e\|_{\CC_\e^\alpha}$
and define discrete Hölder spaces $\CC_\e^\alpha(\T_\e^d,E)$ as the space of all elements $f^\e\in E^{\T_\e^d}$ with this norm.

Note that $\|f^\e\|_{\CC_\e^{0,T}} \asymp \|f^\e \|_{L^\infty([0,T]\times \T^d_\e)}$,
and likewise for the spatial norms.
We sometimes write $L^\infty_\e$ for $\CC^0_\e$.

Although our time is continuous, the norms on $\CC_\e^{\alpha,T}$ do not measure oscillations when
$z=(t,x)$ and $w=(s,x)$
get very close (i.e. when $|t-s|<\e^2$).
These oscillations in very small time scales will be measured by
another collection of norms $\|\cdot \|_{\alpha;\K_\eps;z;\eps}$
defined in (<ref>).

Now let $\alpha<0$.
For discrete distributions $f^\e$ on $[-1,T+1]\times\T^d_\e$
and distributions $f$ on $[-1,T+1]\times\T^d$
we define

‖f^;f ‖_^α,T_
sup_z ∈[0,T]×_^d
|⟨f^, φ_z^λ⟩_-⟨f, φ_z^λ⟩|  ,

where $r$ is the smallest integer such that $r>-\alpha$ and $\CB^r_0$ is the set of smooth functions $\phi\in \CC^\infty(\R\times \T^d)$ with $|\phi|_{\CC^r}\leq 1$
and $\supp(\phi) \subset \{z\in\R\times\T^d\, :\, \|z\|_\s\leq 1/4\}$.
We define $\Vert f^\e \Vert_{\CC^{\alpha,T}_\e} =\Vert f^\e;0 \Vert_{\CC^{\alpha,T}_\e} $.

For discrete functions $f^\e$ on $\T^d_\e$ and distributions $f$ on $\T^d$
we define

$\Vert f^\e ;f \Vert_{\CC^{\alpha}_\e}$ as in (<ref>)
but with $z\in \T^d_\e$ and $\phi \in \CB^r_{\T^d}$,
$\CB^r_{\T^d}$ is the set of smooth functions $\phi\in \CC^\infty(\T^d)$ with $|\phi|_{\CC^r}\leq 1$
and $\supp(\phi) \subset \{z\in\T^d\, :\, |z|\leq 1/4\}$.
We define $\Vert f^\e \Vert_{\CC^{\alpha}_\e} =\Vert f^\e;0 \Vert_{\CC^{\alpha}_\e} $.
Note that if we take $\alpha=0$ in this definition, it defines a norm equivalent to $\|f^\eps\|_{L^\infty(\T^d_\e)}$ uniformly in $\eps$ (which is then also equivalent with $\CC_\e^0$ defined above).
In contrast, if we take $\alpha=0$ and $f=0$ in (<ref>), this does not define a norm equivalent with $\|f\|_{L^\infty([0,T]\times \T^d_\e)}$.

We write $\CC_\e^\alpha(\Lambda_N)$ or $\CC_\e^\alpha(\obonds_i)$
when specifically referring to a particular lattice.

The following simple facts will be helpful in the bounds for
the “remainder terms” showing up in our discrete equation. We will assume that $f$ is either a function on $\T^d_\e$ or a distribution on $[0,T]\times \T^d_\e$.

*  There exists $C>0$ such that, for all $\alpha \leq \bar\alpha\leq  0$,
one has $\|f\|_{\CC^{\bar \alpha}_\eps} \leq C \eps^{\alpha-\bar\alpha}\|f\|_{\CC^\alpha_\eps}$
(if $\bar\alpha<0$, we can take $C=1$).
The same holds for the space-time norm $\|f\|_{\CC^{\alpha,T}_\e}$ provided that $\alpha,\bar\alpha<0$.

*  For $\alpha,\bar\alpha\in[0,1]$
and any $\partial\in \{\partial_j^\pm,\partial_j,\bar\partial_j\}$,
one has $\| \e \partial f \|_{\CC^{\bar\alpha}_\e}
\leq \e^{\alpha-\bar\alpha}\|f\|_{\CC^\alpha_\eps}$.
The same holds for the space-time norm $\|f\|_{\CC^{\alpha,T}_\e}$.

This essentially follow from definitions; one needs only recall that $\lambda\in[\e,1]$ in the definition of $\|\cdot\|_{\CC^\alpha}$ for $\alpha<0$
and, for <ref>, remark that $\|\e\d f\|_{L^\infty(\T^d_\e)} \leq \e^\alpha \|f\|_{\CC^\alpha_\e}$
and that $|x-y|\geq \e$ for all $x\neq y\in\T^d_\e$.

We will also need “inhomogeneous” norms which distinguish time and space and allow possible blow-up at $t=0$.
For $f^\e\colon [0,T]\times \T^d_\e \to E$, $\alpha\leq 1$, $\eta \le 0$, define
f^_^α,T_η,  sup_t ∈(0, T]
(t^1/2 ∧1)^-η f^(t)_^α_  .

We remark that Young's theorem holds
for our discrete spaces on $\T^d_\e$.

Let $\alpha<0$ and $\beta \in (0,1)$ with  $\alpha+\beta>0$.
Let $f^\e,g^\e\colon \T^d_\e\to \R$.
Then $\|f^\eps g^\eps\|_{\CC^\alpha_\e} \lesssim \|f^\e \|_{\CC^\alpha_\e} \|g^\e\|_{\CC^\beta_\e} $ uniformly in $\e$.

The proof follows in a straightforward way by extending $f^\e,g^\e$ to the continuum via piecewise constant and bilinear interpolation respectively and applying (the continuum) Young's product theorem.

§.§.§ Comparison with line integral spaces

Recall the discrete spaces $\Omega_{N;\alpha}$ with projective limit $\Omega_\alpha$ from Section <ref>.
We will identify every $A\in\Omega_N$ with the family of functions $(A_i)_{i\in[d]}$ with $A_i\colon  \obonds_i\to \mfg$ given by $A_i(b) = \e^{-1} A(b)$.
Note that $A_i$ differs from $A\restr_{\obonds_i}$ for $A\in\mfg^{\obonds}\simeq \Omega_N$ (see under (<ref>))
by a factor of $\e^{-1}$; we will always denote this difference through the subscript $A_i$.

We also denote $\|A\|_{\CC^\eta_\e}=\max_{i\in[d]}\|A_i\|_{\CC^\eta_\e}$ for $A\in\Omega_{N}$,

for which we have the bounds  $|A|_{N;\alpha} \lesssim \|A\|_{\CC_\e^{\alpha/2}}$
and $\|A\|_{\CC^{\alpha-1}_\e} \lesssim  |A|_{N;\alpha}$
uniformly in $\e=2^{-N}$, which are the discrete analogues of (<ref>) and (<ref>) respectively
(the latter of these is proved in the same way as <cit.>, see also the proof of Lemma <ref> below).
We therefore have the uniform in $\eps$ embeddings
^α/2_↪Ω_N;α ↪^α-1_ .

The following lemma
compares $A\in\Omega_\alpha^1$ with its projection $\pi_N A\in\Omega_N$.

For $\alpha\in (0,1]$, $\eta\in [\frac\alpha2-1,\alpha-1]$, $i\in[d]$,
and $A=(A_1,\ldots,A_d)\in\Omega_\alpha^1$, we have $\|A_i;\pi_N A_i\|_{\CC^\eta_\e} \lesssim \e^{\alpha-1-\eta}|A|_\alpha$ uniformly in $\e$.

For the proof, let $\CD^{\var p}([a,b],E)$ denote the space of càdlàg (right-continuous with left-limits) functions $X\colon[a,b]\to E$ with finite $p$-variation
|X|_p;[a.b] sup_P⊂[a,b] (∑_[s,t]∈P |X_t - X_s|^p)^1/p

for $p\in[1,\infty)$ and where the $\sup$ is over all partitions $P$ of $[a,b]$ into disjoint (modulo endpoints) interval.
We also denote $|X|_{\var\infty}= \sup_{s,t\in[a,b]} |X_t-X_s|$.
Denote by $\CC^{\var p}\subset \CD^{\var p}$ the subspace of continuous functions.

Consider $1\leq p,q  \leq \infty$
such that $\frac1p+\frac1q\geq 1$ where equality is allowed only if $p,q\in\{1,\infty\}$.
For $Y\in\CD^{\var q}([a,b],\R)$ and $X\in\CD^{\var p}([a,b],E)$,
recall Young's estimate for integrals
(see <cit.> and also <cit.>)
|∫_a^·Y_tX_t|_p;[a,b] ≤C_p,q (|Y_0|+|Y|_q;[a,b])|X|_p;[a,b] ,

where the integral is understood as the limit of Riemann sums
with $Y$ evaluated at the left ends of the subintervals.

For $\phi\in \CB^1(\T^d)$, $\lambda\in[\eps,1]$, and $x\in \T^d$,

we can write $\scal{A_i,\phi^\lambda_x} = \int_{\bar B}\bar L(\bar z) \bar \nu(\mrd \bar z)$
where $\bar B\subset \T^d$ is the $(d-1)$-dimensional hypercube of side-length $\asymp \lambda$ centred at $x$ and perpendicular to $e_i$ and $\bar \nu$ is the $(d-1)$-dimensional Lebesgue measure.
Here $\bar L(\bar z) = \int_{-1}^1 \bar Y^{\bar z}_t\mrd  \bar X^{\bar z}_t$ where
$\bar Y^{\bar z} \in \CC^{\var 1}([-1,1],\R)$
and $\bar X^{\bar z} \in \CC^{\var{1/\alpha}}([-1,1],\mfg)$
are defined by

Y̅^z̅_t = ϕ^λ_x(z̅+tλe_i) ,
X̅^z̅_t = ∫_-1^t λA_i(z+tλe_i)t ,

where the second term is well-defined since $A\in\Omega^1_\alpha$.
It is simple to verify that
|Y̅^z̅|_1 ≲λ^-d|ϕ|_^1 ,
|X̅^z̅|_1/α≲λ^α|A|_α .

We can likewise write $\scal{\pi_N A_i,\phi^\lambda_x}_\e=\int_{B} L(z) \nu(\mrd z)$, where $B$ is the intersection of $\bar B$ with $\obonds_i$ (recall we identify bonds with their midpoints)
$\nu$ is $\eps^{d-1}$ times the counting measure on $B$.
Here $L(z)=\int_{-1}^1 Y^z_t\mrd X^z_t$ where
$Y^z \in \CD^{\var 1}([-1,1],\R)$
and $X^z \in \CD^{\var{1/\alpha}}([-1,1],\mfg)$
are defined by

Y_t^z = ϕ^λ_x(z + t 2^N λ_i) ,
X^z_t = ∑_j=-2^Nλ^t 2^Nλ (π_N A)_i(z+(j-1)_i) ,

and we have the bounds as (<ref>) for $Y^z,X^z$.

Furthermore, we can take $B,\bar B$ such that
nearest neighbour approximation partitions $\bar B$ into subsets $\{P_z\}_{z\in B}$,
each with volume $\bar\nu (P_z) = \e^{d-1}$.
Then for every $z\in B$ and $\bar z\in P_z$,

|Y^z-Y̅^z̅|_∞≲λ^-d-1|ϕ|_^1 ,    |X^z-X̅^z̅|_∞≲(λ)^α/2|A|_α .

Recall the interpolation estimate for $1\leq\gamma\leq\beta\leq \infty$
≲|Z|_γ^γ/β|Z|^1-γ/β_∞ .

Applying (<ref>)
with $\gamma=1$,
we obtain, for any $\beta\in[1,\infty]$ and $z\in B,\bar z\in P_z$

≲λ^-d/β ^1-1/βλ^(-d-1)(1-1/β)|ϕ|_^1
= ^1-1/βλ^-d - (1-1/β)|ϕ|_^1 .

Furthermore, applying (<ref>)
with $\gamma=1/\alpha$, we obtain for any $\beta\in[\frac1\alpha,\infty]$

≲λ^1/β (λ)^α/2(1-1/αβ)
|A|_α= ^α/2-1/2β
λ^α/2 +1/2β|A|_α .

Therefore, taking $1/\beta=1-\alpha+\kappa$ for $\kappa\in(0,\alpha)$, we obtain from (<ref>)

|∫_-1^1(Y̅^z̅_t -Y^z_t) X^z_t| ≲^α-κλ^-d+κ|ϕ|_^1|A|_α ,

while taking $\beta\geq 1/\alpha$, we obtain from (<ref>) for any $\kappa = \frac1{2\beta}\in[0,\frac\alpha2]$

|∫_0^1 Y̅^z̅_t (X^z_t-X̅^z̅_t)|
≲^α/2-κλ^-d+α/2+κ|ϕ|_^1|A|_α .

Since $\frac\alpha2-\kappa$ is the smaller exponent of $\eps$ and $\lambda\geq \eps$,
we obtain for any $z\in B$ and $\bar z \in P_z$ that $|L(z) - \bar L(\bar z)|\lesssim \e^{\frac\alpha2-\kappa}\lambda^{-d+\frac\alpha2+\kappa}$.
Therefore, since $|B|\asymp (\lambda/\e)^{d-1}$, we obtain for $\eta \in [\frac\alpha2-1,\alpha-1]$ (playing the role of $\frac\alpha2+\kappa-1$)

|π_N A_i,ϕ^λ_x_-A_i,ϕ^λ_x|
≤∑_z∈B ∫_P_z |L(z) -L̅(z̅)| ν̅(z)

^α-1-η λ^η .

§.§ Trees and regularity structure

In the rest of the article, we consider only $d=2$.
In this subsection, we define the regularity structure used to analyse the discrete equation (<ref>).
Our construction uses ideas from the continuum 2D YM in <cit.> but with a number of significant differences.
We first recall briefly the important steps in the construction in <cit.>.
In <cit.>, we define the label set $\mfL^\YM=\mfL_-^\YM\sqcup\mfL_+$ with noise labels
$\mfL^\YM_-=\{\mfl_1,\mfl_2\}$ and solution/kernel labels $\mfL_+=\{\mfa_1,\mfa_2\}$,
and consider trees associated with these labels.
We use a rule coming from the equation (<ref>) to build an “abstract” regularity structure $\ST$ consisting of trees.
The trees with negative degree are of the following form (the degree of a tree is given below it).[With
the formulation in <cit.> one would also have $\<I'XXi_notriangle>$, $\<IXiI'XXi_notriangle>$, $\<IXXiI'Xi_notriangle>$ where crossed circles $\<XXi>$ denote
$\mbX \Xi$. Here, since we essentially build the trees by hand as in <cit.>, we drop these trees.]

\<IXi^3_notriangle>\ \<IXiI'[IXiI'Xi]_notriangle> \  \<I[IXiI'Xi]I'Xi_notriangle>
\end{array}$

\<I'[IXiI'Xi]_notriangle>\  \<I[I'Xi]I'Xi_notriangle>\ \<IXiI'[I'Xi]_notriangle> \ \<IXi^2>
\end{array}$

\<IXi> \; \<I'[I'Xi]_notriangle> \;\; {\bf X}\<I'Xi_notriangle>
\end{array}$

Here, the noises which will be denoted by $\Xi_{i}$ are circles $\<Xi>$,
edges $\mcb{I}_{i}$ and $\partial_j\mcb{I}_{i}$
denoting abstract integration operators
are thin and thick grey lines respectively,
and $\kappa>0$ is sufficiently small and fixed, e.g. $\kappa=\frac1{100}$ suffices.
We follow standard conventions that joining the roots of two trees denotes multiplication which is commutative, i.e. $\<IXiI'Xi_notriangle>=\<I'XiIXi_notriangle>$.
Noises are treated as edges that are terminal, so we draw them as simply as $\<Xi>$ (and later as other symbols, e.g. $\<Xi1>$).
See <cit.> for the formal correspondence between graphical trees and symbols.
Recall that each symbol above actually corresponds to a family of trees, determined by assigning indices in a way that conforms to the rule that generates the trees.
For instance, when we say that $\tau$ is of the form $\<I[IXiI'Xi]I'Xi_notriangle> $, then $\tau$ could be
the following tree
I_1(I_2(Ξ_2) ∂_1I_2(Ξ_2))
∂_1I_1(Ξ_1) ,

or other choices of the indices conforming to the structure of the quadratic nonlinear terms in our equation.

One then defines target space assignment  $W_\mft$ and corresponding space assignment $V_\mft$
(in the terminologies as in <cit.>) by
W_= ,  (∈^) ,
V_= ,   (∈_+) ,
V_=^* ,    (∈^_-) .

Recall that the “concrete” regularity structure $\CT$ is then given by a functor $\Func_V$ applied to $\ST$ (likewise for all other algebraic objects).
The concrete regularity structure is a direct sum $\bigoplus_\tau\CT[\tau]$ of vector spaces indexed by trees where $\CT[\tau]=\Func_V(\tau)$.
For example, $\Func_V( \<IXi^2> )\simeq (\mfg^*)^{\otimes_s 2}$, the space of symmetric $2$-tensors of $\mfg^*$.
We write $\CT_\alpha = \bigoplus_{|\tau|=\alpha}\CT[\tau]$ for the subspace of degree $\alpha$.

There are, however, several reasons that
our regularity structure will actually be an enlargement of the above one used in <cit.> for the continuum YM,
which we describe now.

our regularity structure $\CT$ will have a splitting

$\CT=\CT^{(1)}\oplus \CT^{(2)}$.

This splitting encodes the information on whether the trees will be realised as functions on horizontal or vertical bonds,
which will be important for defining models. In particular we will have two copies of abstract polynomials,
written as $\bone^{(1)},\X_1^{(1)},\X_2^{(1)} \in \CT^{(1)}$, and
$\bone^{(2)},\X_1^{(2)},\X_2^{(2)}\in \CT^{(2)}$.   (We will only need abstract polynomials of degree at most $1$ in this paper.)
We recall at this stage that each polynomial encodes a tree $\tau$ with a single vertex and that $\CT[\tau]\simeq \R$.
We will not distinguish between $\R \tau$ and $\CT[\tau]$, i.e. we treat $\bone^{(1)}, \X_1^{(1)}$, etc.
both as trees and as basis vectors for $\CT[\bone^{(1)}]$, $\CT[\X_1^{(1)}]$ and so on.

More importantly, we will need to include more trees due to the “error” terms
F(A)[A,ξ]        (<ref>)
R̃_2(A)        (<ref>) .

The quadratic terms in (<ref>) are the most singular “error” terms, which have scaling dimension $-1$.

Since the choice of the discrete DeTurck term $-\mrd_U \mrd^* U$ in (<ref>) is not unique,

one may wonder if a different choice would simplify  the  terms in (<ref>).
In any case, the general form the DeTurck term is $-\mrd_U (\mrd^* U + \omega)$ for a function $\omega\in\mfg^\Lambda$,
which, up to 2nd order, would amount to replacing $\nabla \cdot \A$ by $\nabla \cdot \A + \omega$ in (<ref>)-(<ref>).
Considering that (<ref>) and (<ref>) consist of terms of the form $\e A_i\xi_i$, $\e(\partial_j A_j)^2$, and $\e^2\d_j^2A_i\d_jA_j$ with $j\neq i$,
it seems unlikely that a significant simplification is possible.

Moreover, our discrete SPDE has nonlocal nonlinearities. To encode this, we introduce shifting operators $\CS_{\be}$, indexed by $\be\in\CE_\times$ (recall the notation in Section <ref>), that will be defined on sectors $V_{\be}$ of our regularity structure.
These maps will satisfy the following properties:

* for all $i\neq j\in\{1,2\}$ and $\alpha\in\R$,
$\CS_{\be}$ maps $\CT_\alpha^{(i)}\cap V_{\be}$ to $\CT_\alpha^{(j)}$,

* $\CS_{\be}$ is norm preserving,

* for each $i\neq j\in \{1,2\}$,
$\CS_{\be}$ maps $\bone^{(i)}$  to $\bone^{(j)}$ and
$ {\bf X}^{(i)}_k$ to ${\bf X}^{(j)}_k$.

Finally, we have multiple types of derivatives appearing in our discrete equation which will be encoded by the set of abstract differentiation

{ _1,_2,_1,_2 , _1^+ , _2^+, _1^-, _2^-} ,

which are defined on sectors of $\CT$ that we soon specify.
We impose that, for every $\CD\in \Diff\backslash \{\bar\CD_1,\bar\CD_2\}$ with domain $V_\CD$,
$\CD\colon V_\CD\cap \CT_\alpha^{(i)} \to \CT_{\alpha-1}^{(i)}$,
while for $\CD\in \{\bar\CD_1,\bar\CD_2\}$
we have
$\CD\colon V_\CD\cap \CT_\alpha^{(i)} \to \CT_{\alpha-1}^{(3-i)}$.
In particular,
_i^±X_j^(k)  = δ_ij ^(k) ,
_i X_j^(k)  = δ_ij ^(k) ,
_i X_j^(k)  = δ_ij ^(3-k) ,

for $i,j,k\in \{1,2\}$.  See Remark <ref> for motivation, as well as Remark <ref>.

The picture one should keep in mind is that $\CT^{(1)}$ and $\CT^{(2)}$ are isomorphic but distinct regularity structures, with isomorphic structure groups,
that communicate only through the shifts $\CS_\be$ and through the “switching” derivatives $\bar\CD_i$.
In particular, products of the form $\tau_1\tau_2$ with $\tau_1\in\CT^{(1)}$, $\tau_2\in\CT^{(2)}$ never appear,
and the polynomials $\bone^{(1)}$ and $\bone^{(2)}$ are units for multiplication in $\CT^{(1)}$ and $\CT^{(2)}$ respectively (i.e. $\bone^{(i)}\tau=\tau$ for $\tau\in\CT^{(i)}$).

We next describe the full regularity structure $\CT$ for our discrete equation.
We use a recursive construction akin to <cit.> (though we could have
equivalently used the
formalism of rules <cit.>).
For clarity, we proceed in three steps
(it is possible to combine these steps, but we
separate them for clarity).

§.§.§ Step 1: continuum YM trees

We recall the usual noise labels

corresponding to the continuum 2D YM (we will later add more noise labels to form $\mfL_- = \mfL^\YM_- \sqcup\mfL^\rem_-$),
as well as the solution/kernel labels $\mfL_+=\{\mfa_1,\mfa_2\}$.

In the first step towards defining $\CT$, we describe a set of trees $\mfT^\YM=\mfT_1^\YM\sqcup \mfT_2^\YM$
which are similar to the trees for the continuum 2D YM.
For motivation behind the construction below, we recommend the reader reviews the nonlinearities in Proposition <ref> (see also the `abstract' fixed-point problem (<ref>) below).

Base Case (Step 1).
For each $i \in \{1,2\}$, we let
_i {^(i), X^(i)_1, X^(i)_2 } ⊂_i^ ,       Ξ_i Ξ__i ∈_i^ .

Integration (Step 1).
Let $\mcb{I}_1$, $\mcb{I}_2$ be abstract integration operators.
For all $i\neq j\in\{1,2\}$, $\be\in\CE_\times$ and $\tau \in \mfT^\YM_i$ with $|\tau|<-1$, we enforce that,
I_i (τ) ∈^_i  ,
_I_i (τ) ∈^_j  .

Nonlinearity (Step 1).
For all

* $i \neq j\in \{1,2\}$, $\be,\be'\in\CE_\times$, and $\bw\in\{\Northwest,\Southwest\}$,

* $\tau_i,\sigma_i\in \mfT^\YM_i$ of the form $\tau_i=\mcb{I}_i(\bar\tau_i),\sigma_i=\mcb{I}_i(\bar\sigma_i)$ or $\tau_i,\sigma_i \in \bar{\mfT}_i$, and

* $\tau_j,\sigma_j,\omega_j\in \mfT^\YM_j$ of the form
where $\bar\omega_j\neq \Xi_j$, or $\tau_j,\sigma_j,\omega_j \in \bar{\mfT}_j$,

we enforce that every
σ∈   {
(_ τ_j )(_j τ_i)  ,
(_ τ_j )(_i _jΞ_j)  ,
(_ τ_j )(_i^+ _ω_j)  ,

τ_i _i σ_i ,
(_ τ_j )(_' σ_j )τ_i }

belongs to $\mfT_i^\YM$ provided that $|\sigma| \leq 0$.

Above, we assign degrees using the rules

(τ) = |τ|+2 ,  |τ|=|τ|-1 ,   |_τ|=|τ| ,
|τ_1τ_2| = |τ_1|+|τ_2| ,

along with the base cases $|\Xi|=-2-\kappa$, $|\bone|=0$, and $|\X|=1$.

The terms in (<ref>) arise from the main nonlinearities in (<ref>).
Since $\tau_i,\sigma_i,\tau_j,\sigma_j$ can be any polynomial and $\CS_\be\colon\bone^{(j)} \mapsto \bone^{(i)}, \X^{(j)}\mapsto \X^{(i)}$, the right-hand side of (<ref>) includes $\CD_i \tau_i$,
$\CD_j \tau_i$,
$\bar\CD_i \CI_j\Xi_j$, $(\CS_\be\tau_j)\tau_i$, $\X^{(i)}\CD_j\tau_i$, and so on.

Since $\bar\d_iA_j = \frac12(\d_i^+A_j(e^{\northwest})+\d_i^+A_j(e^{\southwest}))$,
the average of the 3rd term in $\{\ldots\}$ in (<ref>) over $\bw\in\{\Northwest,\Southwest\}$
is formally $(\CS_{\be}\tau_j) (\bar\CD_i \CI_j\omega_j)$, which would be a more natural term to include (cf. the 2nd term);
our current formulation of the 3rd term,
however, assists in deriving the renormalised equation in Section <ref> (see in particular (<ref>)-(<ref>)).

Integration (Step 1) and Nonlinearity (Step 1), i.e. (<ref>)-(<ref>),
are implemented recursively with (<ref>) as the base case.
One can check that the recursion terminates after applying (<ref>)-(<ref>) two times and that
every symbol in $\mfT^\YM_i$ corresponds to either a polynomial, a tree of the form given in (<ref>), or a tree of the form

I(cherry)> ,    IDPsi> ,

with degrees $1-2\kappa$, $1-\kappa$ respectively.
We use the same graphic notation as  in (<ref>) to describe the form of the trees (in particular the  graphic notation
does not indicate insertions of shifting operator or choice of differentiation operator).

The set of trees obtained this way
is strictly larger than that for the continuum YM because we have trees of the form  $\CS_\be\mcb{I}(\cdots)$, four types of directional derivatives $\CD_i,\bar \CD_i,\CD_i^{\pm}$,
and the abstract monomials are doubled.
But otherwise,  the trees obtained here are just the trees for continuum YM in <cit.> with proper insertions of shifting operators
and choice of differentiation operators.
For instance,
(<ref>) may now become
\mcb{I}_{1}\big(\CS_\be \mcb{I}_{2}(\Xi_{2}) \bar\CD_{1}\mcb{I}_{2}(\Xi_{2})\big)
\CD_{1}\mcb{I}_{1}(\Xi_{1})
We define the negative trees $\mfT^\YM_- = \{\tau\in\mfT^\YM\,:\,|\tau|<0\}$.

§.§.§ Step 2: remainder trees

In our second step towards defining $\CT$, we add more trees which will allow us to handle the “remainder terms” in (<ref>).
We introduce additional noise labels

^_-={_1,_2}⊔{ ^=_i,^+_i,^-_i}_i=1^2 .

In what follows, we assign degree to symbols using using (<ref>) together with $|\Xi_{\bar\mfl_i}| = -1-2\kappa$
$|\Xi_{\mfl^{=,\pm}_i}| = -1-3\kappa$.

The label $\bar\mfl_i$
corresponds to the noise $\e^{1-\kappa} \xi_i$.
The motivation is that we can write $ \eps [ A_i,\xi_i]  = \eps^\kappa [A_i, \eps^{1-\kappa}\xi_i] $
in the term $\eps F(\e A)[A,\xi]$ in (<ref>).

The labels $\mfl^{=,\pm}_{1,2}$
are motivated by the terms in $\tilde{R}_2$ in (<ref>), the first of which we can write as

- ^2/2 [∂_j^2 A_i(e), ∂̅_j A_j(e)]
- /2 [(∂_j^+  +∂_j^-)A_i(e), ∂̅_j A_j(e)]  ,

so $\tilde{R}_2$ has a form $\e \partial A \partial A$.
With $A=\Psi+v$ where $\Psi$ is SHE,
we can then write

∂A ∂A = ^κ∂(^1-κ/2Ψ) ∂(^1-κ/2Ψ)
+ ^κ∂(^1-κΨ) ∂v
+ ^κ∂v ∂(^1-κΨ)
+ ∂v∂v .

The 6 noises $\mfl^{=,\pm}_{1,2}$ arise from  $\d(\e^{\frac{1-\kappa}{2}}\Psi) \partial (\e^{\frac{1-\kappa}{2}}\Psi)$
with the different choices for
$\d$ and $Ψ$.
(We will treat the last term $ ∂v∂v$ in the ``classical'' way and do not lift it to the space of modelled distributions.
See also~\eqref{e:fix-pt-Ae}-\eqref{e:def-tildeCR2} below for this decomposition at the level of modelled distributions.)

With these motivations, we now describe the full set of trees $= _1⊔_2$.

\smallskip
\noindent
\textbf{Base case (Step 2).}
We enforce $^_i⊂_i$, and
for each $i≠j∈{1,2}$
\begin{equ}\label{eq:base_case_2}
\bar\Xi_i \eqdef \Xi_{\bar\mfl_i}\;,
\;\;
\big(\CD_j^\pm \mcb{I}\tilde\Xi_i \big)  \big(\bar{\CD}_j \mcb{I}\tilde\Xi_j\big) \eqdef \Xi_{\mfl^\pm_i} \;,
\;\;
\big(\CD_j^+ \CS_{\southeast} \mcb{I}\tilde\Xi_j \big) \big( \CD_j^+ \CS_{\southwest} \mcb{I}\tilde\Xi_j \big) \eqdef \Xi_{\mfl^=_i}
\end{equ}
all belong to $_i$.
Here $Ξ̃_i$, which formally models $^1/2-κ/2ξ_i$, is purely notational and we do \textit{not} have a tree $Ξ̃_i$ in $_i$.
When drawing trees, we will denote $Ξ̅_i$ by $Xi1>$ and $Ξ̃_i$ by $[xi2] at (0,0) ;$,
so that $Ξ__1,2^=,±$ are all drawn as $R2-1new>$ (understood as a single terminal edge).

\smallskip
\noindent
\textbf{Integration (Step 2).}
For each $i≠j∈{1,2}$, if $τ∈_i$ with $|τ|<-1$, we enforce that~\eqref{e:rule-main3} holds with $^$ replaced by $$.

\smallskip
\noindent
\textbf{Nonlinearity (Step 2).}
To encode the main nonlinearities,
we enforce that \textit{Nonlinearity (Step 1)} holds with $^$ replaced by $$.

Next, to encode the term $R̃_2$ in~\eqref{e:tildeR2},
for each $i≠j∈{1,2}$ and $τ_i∈_i$, $τ_j∈_j$ of the form $τ_i=_i(τ̅_i)$ and $τ_j=_j(τ̅_j)$
we enforce that every
\begin{equ}[e:rule-more3]
\sigma \in \{\;\bar\CD_j \CI_j\Xi_j \;,\;
\CD_j^+ \CS_{\southwest} \tau_j \;, \;
\CD_j^+ \CS_{\southeast} \tau_j \;,\;
\CD_j^\pm \tau_i\;\}
\end{equ}
belongs to $_i$ provided that $|σ|≤0$.
Furthermore, in view of \eqref{e:tildeR2}, \eqref{eq:R_2_other_form}, and the 2nd and 3rd terms on the right-hand side of~\eqref{eq:edAdA},
for all
\begin{itemize}
\item $i\neq j \in \{1,2\}$ and $\bs\in\{\Southeast,\Southwest\}$,
\item $\tau_i=\mcb{I}_i (\bar\tau_i) \in \mfT_i \backslash \{\mcb{I}_i (\Xi_i)\}$  or $\tau_i \in \bar{\mfT}_i$,
\item $\tau_j=\mcb{I}_j (\bar\tau_j)\in \mfT_j  \backslash \{\mcb{I}_j (\Xi_j)\}$   or $\tau_j \in \bar{\mfT}_j$,
\end{itemize}
we enforce that every
\begin{equs}[e:rule-more2]
\sigma \in \big\{\;
&(\CD_j^\pm \mcb{I}_i{\bar\Xi_i}) (\CD_j^+ \CS_{\bs} \tau_j)\;,
\;\;
(\CD_j^\pm \tau_i )(\bar\CD_j  \mcb{I}_j{\bar\Xi_j})\;,
\\
&( \CD_j^+ \CS_{\southeast} \mcb{I}_j{\bar\Xi_j}) ( \CD_j^+ \CS_{\southwest} \tau_j) \;,\;\;
(\CD_j^+ \CS_{\southeast} \tau_j  ) ( \CD_j^+ \CS_{\southwest} \mcb{I}_j{\bar\Xi_j})\;
\big\}
\end{equs}
belongs to $_i$ provided that $|σ|≤0$.

%The terms in~\eqref{e:rule-more2} arise from the 2nd and 3rd terms on the right-hand side of~\eqref{eq:edAdA} (recall also~\eqref{e:tildeR2}).
%Note that $\tau=\CI(\Xi)$ is excluded from~\eqref{e:rule-more2} because it corresponds to the 1st term on the right-hand side in~\eqref{eq:edAdA}
%and is taken care of by~\eqref{eq:base_case_2}.
%Likewise, we do not include terms like
%$(\CD_j^\pm \tau_i )(\bar\CD_j  \tau_j)$ or $(\CD_j^+ \CS_{\southeast} \tau_j  ) (\CD_j^+ \CS_{\southwest} \sigma_j)$
%as these would correspond to the final term $\e\d v \d v$ in~\eqref{eq:edAdA}, which we instead deal with classically.

\begin{remark}\label{rem:barCD_term2}
Like in Remark~\ref{rem:barCD_term1},
the average of the 1st term in $\{\ldots\}$ in~\eqref{e:rule-more2} over $\bs\in\{\Southeast,\Southwest\}$
is \textit{formally} $(\CD_j^\pm \mcb{I}_i{\bar\Xi_i}) (\bar\CD_j \tau_j)$, which may appear more natural;
the current choice, however, helps in deriving the renormalised equation.
\end{remark}
To encode the nonlinearity $F(A)[A,ξ]$,
for each $i ∈{1,2}$,
$χ∈_i$, and $τ,ω∈_i$ of the form
$τ=_i(τ̅),ω=_i(ω̅)∈_i$  or
we enforce that every
\begin{equ}\label{eq:F_trees}
\sigma \in \{\;\tau\bar \Xi_i\;,
\;
\tau\chi\bar\Xi_i\;,
\;
\tau  \CI_i (\omega\bar \Xi_i) \bar \Xi_i
\;
\}
\end{equ}
belongs to $_i$ provided that $|σ|≤0$.

\begin{remark}\label{rem:F_trees}
The final definition \eqref{eq:F_trees} requires some explanation.
Recalling the expansion~\eqref{e:Z-terms},
the term $\tau\bar \Xi_i$ is due to the product $\e^\kappa[A,\e^{1-\kappa}\xi]$.
The next order term, $\e^{1-\kappa}A^2\e^{1-\kappa}\xi$, should naively
entail a general term of the form $\CI_i(\bar\tau)\CI_i (\omega)\bar \Xi_i$.
However, by defining  a `multiplication by $\eps$' map on our regularity structure (see Section~\ref{subsubsec:mult_eps_models})
we can use the remaining $\e^{1-\kappa}$ to increase the degree of $\omega$ by $1-\kappa$
\textit{unless} $\omega$ has $\bar\Xi_i$ at the root --
this is because the degree $|\bar\Xi| = -1-2\kappa$ is limited by the temporal regularity of $\e^{1-\kappa}\xi$ (white-in-time)
and further multiplication by $\eps$ will not increase this small-scale regularity (see also Remark~\ref{rem:cE_motivation}).
The higher powers $O(\e^3A^3\xi)$ in~\eqref{e:Z-terms} do not contribute any new trees for the same reason.
\end{remark}
We implement recursively \textit{Integration (Step 2)}
and \textit{Nonlinearity (Step 2)},
i.e. \eqref{e:rule-main3}+\eqref{e:rule-main2}+\eqref{e:rule-more3}+\eqref{e:rule-more2}+\eqref{eq:F_trees}
together with \textit{Base case (Step 2)}.
One can check that the recursion terminates after applying \textit{Integration (Step 2)}
and \textit{Nonlinearity (Step 2)}
two times each.
The new trees with degrees $-1-$ (i.e. just below $-1$) are of the following form (degrees are in parentheses):\footnote{The noises in  $\<R2-1new>$  are $\tikz \node[xi2] at (0,0) {};$, {\it not} $\<Xi1>$ !
In other words in our graphic notation, rotating lines and thick lines by $\frac{\pi}{4}$ does not change their meaning, but we never rotate the notation for noises.}
\begin{equ}[e:trees-deg-1-rem]
\<Xi1>\quad(-1-2\kappa)\;,\qquad
\<PsiXi1>\quad (-1-3\kappa)\;,\qquad
\<R2-1new> \quad (-1-3\kappa)\;.
\end{equ}
Here the second tree arises from $A ξ$, i.e.~\eqref{eq:F_trees}, and the third one
arises from $∂A ∂A$, i.e.~\eqref{eq:base_case_2}.
The new trees with degrees $0-$ are of the following form:
\begin{equ}[e:trees-deg-0-rem]
\begin{tabular}{cc}
\toprule
\hypertarget{(1-2)}{(1-2)}:
$ \<I'XiI'[IXiI'Xi]less>$
\qquad
\qquad
$\<I'(R2-1)new> $
\qquad
\qquad
$ \<I'XiI(R2-1)new>$
\\
\midrule
\hypertarget{(1-3)}{(1-3)}:
\quad
\quad
\quad
\quad
\quad
\quad
\quad
\quad
$ \X^{(i)} \bar\Xi_i$
\quad
$\X^{(i)} \<PsiXi1>$
\\
\midrule
\hypertarget{(2-2)}{(2-2)}:
$ \<I'Xi1I'(R2-1)new>$
\quad
\hypertarget{(2-3)}{(2-3)}:
\quad
\quad
\quad
\quad
\hypertarget{(3-3)}{(3-3)}:
\quad
\quad
\quad
\quad
\\
\bottomrule
\end{tabular}
\end{equ}
In plain words, these trees are obtained by substitutions among three types of terms:
\begin{equ}
\text{(1)}\;\; A\partial A\;,\quad \text{(2)}\;\; \e \partial A \partial A\;,\quad
\text{(3)}\;\;
\e^\kappa F(\e A) [A,\e^{1-\kappa} \xi]\;,
\end{equ}
subject to the conditions in~\eqref{e:rule-more2} and~\eqref{eq:F_trees}
%(see also Remarks~\ref{rem:dVdV_trees} and~\ref{rem:F_trees}).
(see also Remark~\ref{rem:F_trees}).
We classified them into 5 groups, with group labels indicating how we obtained the trees in the group,
for instance trees in group \hyperlink{(1-3)}{(1-3)} are obtained by
substituting between (1) $A∂A$ and (3) $^κF(A) [A,^1-κ ξ]$,
and group \hyperlink{(3-3)}{(3-3)} are obtained by
substituting  (3) $^κF(A) [A,^1-κ ξ]$ into itself.
%Remark that some trees appearing from naively substituting the non-linearities into each other, like $\<r_z_avoid1>$, do not appear
%due to the particularity of~\eqref{eq:F_trees}.

%(see Remarks~\ref{rem:F_trees} and~\ref{rem:cE_motivation}).

Remark also that, due to \eqref{e:rule-more3},
we should in principle also include $I'[IXiI'Xi]_notriangle>$, $I'[I'Xi]_notriangle>$ in group  \hyperlink{(1-2)}{(1-2)},
but these graphical trees are already in \eqref{e:tree-list-YM},
except that the differentiation at the bottom of the trees  should be the differentiations appearing in \eqref{e:rule-more3},
so we do not duplicately list them in  \hyperlink{(1-2)}{(1-2)}.
Similarly, $I'(R2-1)new> $ in  \hyperlink{(1-2)}{(1-2)}
should also be  in \hyperlink{(2-2)}{(2-2)},
and $I'Xibar>$ and $supercherry1less>$ in \hyperlink{(1-3)}{(1-3)} should also be in \hyperlink{(2-3)}{(2-3)}.

\subsubsection{Step 3: two more trees}
\label{subsubsec:Step_3}

Finally, we add two more types of trees to $_i$ for reasons apart from the non-linearity.

The first is due to (negative) renormalisation.
There is one tree in $_i$ of the form $r_z_large>$, namely $Ψ_iΞ̅_i_i[Ψ_iΞ̅_i]$ where we write $Ψ_i_iΞ_i∈_i^$.
These trees have a sub-divergence $Ξ̅_i_i[Ξ̅_i]$ of the form $I(PsiXi1)Xi1less>$ that we will be forced to renormalise (see Section~\ref{subsec:renorm_group}) and which produces a contracted tree $Ψ_iΨ_i=Ψ_i^2$ of the form $IXi^2>$, which we now add to $_i$ for $i∈{1,2}$.
(We already added trees of the form $IXi^2>$ to $_i$ due to~\eqref{e:rule-main2}, but these are of the form $_Ψ_j_'Ψ_j$ or $_Ψ_j Ψ_i$ for $j≠i$.)

The second type of tree we add is of the form $IXiI'Xibar>$
which we obtain by replacing the factor $Ξ$ `on the right' in all trees of the form $IXiI'Xi_notriangle>$ in $^$ by $Ξ̅$.
recalling \eqref{e:rule-main2}, the new trees appearing in $_i$
are $_ Ψ_j _j Ψ̅_i$,
$_ Ψ_j  _i Ψ̅_j$,
$Ψ_i _i Ψ̅_i$,
where $Ψ̅_i = _iΞ̅_i$,
$i≠j$, and $∈_×$.
Note $IXiI'Xibar>$ already appears in \eqref{e:trees-deg-0-rem} but these new trees did not arise in Step 2.
We add these trees to make a multiplication by $$ map in Section \ref{subsubsec:mult_eps_models} well-defined (see also Remark \ref{rem:F_trees}).

This concludes our definition of $= _1⊔_2$.
We denote $_- = {τ∈ : |τ|<0}$,
$^= {Ψ_i^2∈ : i∈{1,2}}⊂_-$,
$^= ∖(^⊔^)$ and $^_- = _- ∖(^_-⊔^)$.
Note that the trees of the form $IXiI'Xibar>$ added in Step 3 are in $^$
and we will treat them as belonging to group \hyperlink{(1-3)}{(1-3)}.
The positive trees in $^$ are of the form (recall that we enforce $|σ|≤0$ in \eqref{e:rule-more2})
\begin{equ}\label{eq:pos_trees_rem}
\<IXibar>\;,\qquad \<I(R2-1new)>\;, \qquad \<IPsiXi1>\;.
\end{equ}
%and the positive non-planted trees in $\mfT^\rem$ are of the form

\subsubsection{Regularity structure}
\label{subsubsec:RS}

To construct our regularity structure, we denote $_- = ^_-⊔^_-$,
%with $\mfL^\YM_-$ and $\mfL^\rem_-$\label{page:mfT_rem_-}
%defined in~\eqref{eq:mfL_YM} and~\eqref{eq:mfL_rem},
with $^_-$\label{page:mfT_rem_-}
defined in~\eqref{eq:mfL_rem},
and extend the space assignment~\eqref{e:space-assig}
W__i = $ and  $W_^=,±_i = ⊗$
so that, in the notation of~\cite[Sec.~5.6]{CCHS_2D},
V__i = ^*$ and  $V_^=,±_i = ^* ⊗^*

The regularity structure is then constructed by applying the functor $_V$, i.e.
$⊕_τ∈ [τ]$
More precisely, we consider symmetric sets built from finite labelled rooted trees exactly as in~\cite[Sec.~5.4]{CCHS_2D} with the addition that kernel edges may have an additional decoration $_$ and several types of derivatives from $$ (recall~\eqref{eq:Diff_def}),
and that isomorphisms between trees are required to preserve these additional decorations (in addition to the usual node and edge decorations).
For example,
[ _j^+ _ _j(σ_j)^+_j__jΞ̅_j] ≃^*⊗^* ⊗^*
where $σ_j = Ξ__j^== (_i^+ _ IΞ̃_i ) ( _i^+ _ IΞ̃_i ) ∈_j$.
(The reader may take these outputs of \cite[Sec.~5]{CCHS_2D}
for granted since the precise definition of $[τ]$ is seldom used in this article.)
For $L∈{,,}$,
we define
\begin{equ}
\CT^{L} = \bigoplus_{\tau\in\mfT^L}\CT[\tau]\;,\quad
\CT^{L}_- = \bigoplus_{\tau\in\mfT^L_-}\CT[\tau]\;,\quad
\CT_- = \CT^{\YM}_-\oplus \CT^{\rem}_-\oplus \CT^{\ren}_-\;.
\end{equ}
The splitting discussed in the beginning of Sec.~\ref{sec:tr-reg} %\eqref{e:T1T2_heuristic}
is given by
\begin{equ}[e:T1T2]
\CT=\CT^{(1)}\oplus\CT^{(2)}\;,\qquad
\CT^{(i)} = \bigoplus_{\tau\in\mfT_i} \CT[\tau]\;.
\end{equ}
Recalling $_i$ from~\eqref{e:rule-main1},
the polynomial regularity structure is denoted by
= ^(1)⊕^(2)
$^(i)= ⊕_τ∈_i[τ] ≃^3 .
For $H ∈{_i,,_}$, where $i∈{1,2}$, $∈$ and $∈_×$,
we view $H$ as a map $D_H→$ defined on the subset $D_H{τ∈ : Hτ∈}$.
Then $H$ lifts to an operator, denoted by the same symbol, $HV_H →$
via the functor $_V$ as in~\cite[Sec.~5.2.1]{CCHS_2D},
where $V_H⊕_τ∈D_H[τ]$.
We use the shorthand $V_= V__$.

The structure group $$ is determined by the usual Connes--Kreimer-type procedure as in~\cite[Sec.~8.1]{Hairer14} or~\cite[Sec.~2]{ESI_lectures}
upon treating $_i$, $__i$, and $__i$ for $∈$, $∈_×$, and $i∈{1,2}$ as normal kernel edge decorations
(recall also Remark~\ref{rem:splitting}).

For example, denoting $Ψ_i=_iΞ_i$,
for $i≠j∈{1,2}$ and $τ∈[_Ψ_i] ⊂^(j)$ we have $Γτ=τ$ for all $Γ∈$,
while for $σ=  __i[(_'Ψ_j)(_jΨ_i)]_jΨ_i ∈_j$
(which is of the form $I[IXiI'Xi]I'Xi_notriangle>$), and
\begin{equ}
\CT^{(j)}\ni\tau = \tau_1\otimes\tau_2\otimes \tau_3\in \CT[\sigma] \simeq (\mfg^*)^{\otimes 3}
\end{equ}
we have
Γσ= σ+ h ⊗τ_3 ∈[σ]⊕[_jΨ_i]
where $h∈[^(j)] ≃$ and $τ_3 ∈[_jΨ_i ]⊂^(j)$.

One can readily check that the domain $V_H$ of $H∈{_i,_,}$ with $∈$ are all sectors (i.e. subspace closed under the action of $$).
Furthermore, for all $Γ∈$, $∈$, and $τ∈V_$, we have
\begin{equ}\label{eq:Gamma_Diff_commute}
\Gamma\CD\tau = \CD\Gamma\tau\;.
\end{equ}
The following is an elementary but useful lemma.

\begin{lemma}\label{lem:Gamma_simple}
Let $\tau\in\mfT\setminus\bar\mfT$ with $|\tau|>0$.
Then $
\Gamma\sigma=\sigma + \CT[\bone]
$ for all
$\Gamma\in\CG$ and $\sigma\in\CT[\tau]$.
\end{lemma}

\begin{proof}
All non-polynomial positive trees are of the form~\eqref{eq:pos_trees-YM} or~\eqref{eq:pos_trees_rem},
and the proof follows from the fact that $\Gamma \bar\tau = \bar\tau$ for all $\bar\tau\in\CT_\alpha$ with $\alpha<-1$.
\end{proof}

\subsection{Discrete framework}
\label{sec:Discrete framework}

In  \cite{EH19}, given a regularity structure, a discretization is a collection of data
\begin{equ}[e:framework]
(\CX_\eps,\|\cdot \|_{\alpha;\K_\eps;z;\eps}, \$\cdot\$_{\gamma;\eta;\K;\eps})
\end{equ}
which we now specify.
We set
$_^(i)= ^-1+κ(,^_i)$ for $i∈{1,2}$.
%where we recall that $\obonds_i$ is the set of positively oriented bonds in the $i$-th direction.
%We view $\obonds_i$ as a copy of $\T^2_\e$.
Write $__^(1)⊕_^(2)$.
As in \cite[Sec.~2]{EH19} we view $_^(i)$ as a subspace
of the space of distributions,
namely we define an injection map $ι__^(i) →'(×^2)$ by setting
\begin{equ}
\label{eq:iota}
(\iota_\eps f)(\varphi)
= \eps^2\sum_{e\in\obonds_i}\int_{\R} f(t,e)\varphi(t, e)\mrd t\;,
\qquad
\forall \phi\in\CC^\infty_c(\R\times \T^2)
\end{equ}
in distributional sense.
We will often omit $ι_$