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\section{Corollaries}\label{sec:9}

In this section we prove the two corollaries of the introduction (more detailed proofs are in Section~\ref{S4-S:S4.9} of the supplement).
The central phases are described by a U(1) Wilson model with configuration-dependent couplings
at most $n\beta$ (Section~\ref{ssec:1.2}), also in the presence of a Higgs term invariant
under the centre. Since the plaquettes of a finite $2$-complex are attached along
closed edge paths, $U_p$ (the ordered product of the links around $p$) and
$\theta_p$ are defined. We write $\Tr$ for the trace of $n\times n$ matrices.

\begin{proposition}[Fr\"ohlich's comparison]\label{prop:9.1}
Let $G\subseteq U(n)$ be a closed subgroup with $\{zI:\abs{z}=1\}\subseteq G$,
$X$ a finite $2$-complex, and
$\mu_{G,\beta}(dU)\propto\prod_p\exp(\beta\ReP\Tr U_p)\prod_edU_e$, $dU_e$ Haar
measure on $G$ (free boundary conditions). For every closed edge path
$\ell$ in $X$,
\begin{equation}\label{eq:9.frohlich}
\Bigl|\Bigl\langle\Tr\prod_{e\in\ell}U_e\Bigr\rangle_{G,\beta}\Bigr|\le2n\,\ex{\cos\theta_\ell}^{W}_{X,2n\beta},
\end{equation}
where the right side is the U(1) Wilson model on the same complex.
\end{proposition}

\begin{proof}
Let $\nu$ be the law on $U(1)^{E(X)}\times G^{E(X)}$ with density proportional to
$\prod_p\exp(\beta\ReP\Tr(\xi_pU_p))$, where $\xi_e=e^{i\phi_e}$ and
$\xi_p=e^{i\phi_p}$ is the product of the $\xi_e^{\pm1}$ around $p$. Since
$\xi_eI\in G$ is central, its image under $(\xi,U)\mapsto\xi U$ is
$\mu_{G,\beta}$, and $\Tr\prod_{e\in\ell}(\xi_eU_e)=e^{i\phi_\ell}\Tr\prod_{e\in\ell}U_e$.
Given $U$, $\xi$ has the law $\ex{\cdot}_U$ with density proportional to
$\prod_p\exp(\beta\abs{\Tr U_p}\cos(\phi_p-\psi_p))$, where $\psi_p:=-\arg\Tr U_p$. Hence
$\abs{\ex{\Tr\prod_{e\in\ell}U_e}_{G,\beta}}\le n\sup_U\sup_\alpha\ex{\cos(\phi_\ell-\alpha)}_U$.
We bound $\ex{\cos(\phi_\ell-\alpha)}_U$ by $\ex{\cos\theta_\ell}^{W}_{X,n\beta}$
using \cite[Proposition~1 and the remark after its proof, p.~21]{MMSP78}, with
couplings $n\beta\ge\beta\abs{\Tr U_p}$ on the plaquette vectors, where terms with the
same vector up to sign are combined. The remark
is stated (p.~21) for $\alpha$ equal to the phase of the interaction, but for each
$\alpha$ it follows from the proof of Proposition~1, since for the edge vector
$M$ of $\ell$ and two copies $\phi,\phi'$ of the angles
$\cos(M\cdot\phi)-\cos(M\cdot\phi'-\alpha)=2\sin(M\cdot\Phi-\frac\alpha2)\sin(M\cdot\Phi'-\frac\alpha2)$,
where $\Phi=\frac12(\phi'+\phi)$ and $\Phi'=\frac12(\phi'-\phi)$. Consequently, the
left side of \eqref{eq:9.frohlich} is at most $n\ex{\cos\theta_\ell}^{W}_{X,n\beta}$,
which is at most the right side, since U(1) Wilson loops are nonnegative and
nondecreasing in the coupling by Ginibre's inequality, on all finite complexes,
as in Theorem~\ref{thm:2.1}(a), (b). The constants
$2n$ and $2n\beta$ in \eqref{eq:9.frohlich} are those of \cite[(23),~(24)]{Fr79}.
\qed
\end{proof}

\begin{corollary}[Corollary~\ref{cor:1.5}]\label{cor:9.5}\label{cor:9.2}
Let $G$ be as in Proposition~\ref{prop:9.1}, with the Wilson action
$\beta\ReP\Tr U_p$ on finite boxes of $\Zthree$, $\rho$ a representation of $G$
with $\rho(zI)=1$ for $\abs{z}=1$, $(\Phi,\rho_0)$ a probability space of Higgs
field values, and $h\ge0$ a measurable function of
$((\rho(U_e))_e,(\varphi_x)_x)$, the Higgs weight, for any values of the Higgs
couplings, with $0<\int h<\infty$ ($h\equiv1$ is the model without Higgs
field). On a finite box $\Lam$ take the measure with density proportional to
$h((\rho(U_e))_e,(\varphi_x)_x)$ times $\prod_p\exp(\beta\ReP\Tr U_p)$ with
respect to $\prod_edU_e\prod_x\rho_0(d\varphi_x)$. Then \eqref{eq:9.frohlich} holds for every closed path $\ell$, and the bound of
Corollary~\ref{cor:1.5} for every rectangle $C$ with
$\min(R,T)\ge\Rone(2n\beta)$ and every finite box $\Lam\supseteq C$, with
$\sigma$, $\Rone$, $c$ of Theorem~\ref{thm:1.2}. In particular the string
tension of $G$ at coupling $\beta$ is at least $\sigma(2n\beta)>0$, also for
$U(2)$ with an adjoint Higgs field, at all gauge and Higgs couplings.
\end{corollary}

\begin{proof}
Since $\rho(\xi_eU_e)=\rho(\xi_eI)\rho(U_e)=\rho(U_e)$, the proof of
Proposition~\ref{prop:9.1} applies to the joint law with the extra factor
$h((\rho(U_e))_e,\varphi)\prod_x\rho_0(d\varphi_x)$, which cancels from the
conditional law of $\xi$. We then apply Theorem~\ref{thm:1.2} at $2n\beta$. The case of
$U(2)$ with an adjoint Higgs field is included, since $\mathrm{Ad}(zU)=\mathrm{Ad}(U)$. (\cite[Corollary~2]{Fr79} states
confinement of static quarks for all $U(n)$ in three dimensions from the
comparison and the results cited there; the area law is new.)
\qed
\end{proof}

\paragraph{The Georgi\textendash Glashow model.} On a finite box $\Lam$ we take
links $U_e\in SU(2)$, $U_{-e}=U_e^{-1}$, a unit adjoint Higgs field
$\mathsf n_x\in S^2$ (the London limit), the adjoint representation
$R:SU(2)\to SO(3)$, $\beta,\kappa>0$, and
\begin{equation}\label{eq:9.GG}
\mu_{\Lam;\beta,\kappa}\propto\exp\Bigl(\beta\sum_p\tfrac12\Tr U_p+\kappa\sum_{e=(x,y)}\mathsf n_x\cdot R(U_e)\mathsf n_y\Bigr)\prod_edU_e\prod_xd\mathsf n_x
\end{equation}
($dU_e$ Haar, $d\mathsf n_x$ uniform on $S^2$). The weight in \eqref{eq:9.GG} is
invariant under $U_e\mapsto g_xU_eg_y^{-1}$, $\mathsf n_x\mapsto R(g_x)\mathsf n_x$
($g\in SU(2)^\Lam$), and $f(U,\mathsf n)$ is \emph{gauge invariant} if it is. Write
$D(\theta)_e:=\mathrm{diag}(e^{i\theta_e},e^{-i\theta_e})$ and $e_3$ for the
constant field $\mathsf n_x=e_3$.

\begin{proposition}[Georgi\textendash Glashow model at infinite Higgs coupling]\label{prop:9.3}
For every finite box $\Lam$, every $\beta>0$ and every bounded continuous gauge
invariant $f$, the limit
$\lim_{\kappa\to\infty}\ex{f}_{\Lam;\beta,\kappa}$ equals
$\ex{f(D(\theta),e_3)}^{W}_{\Lam,\beta}$, the expectation in the U(1) Wilson
model at coupling $\beta$. In particular, for every closed path $C$,
$\ex{\frac12\Tr\prod_{e\in C}U_e}_{\Lam;\beta,\kappa}\to\ex{\cos\theta_C}^{W}_{\Lam,\beta}$.
\end{proposition}

\begin{proof}
We pass to the unitary gauge. Since $R(SU(2))$ is transitive on $S^2$, gauge invariance gives
$\int\!\!\int F\,d\mathsf n\,dU=\int F(U,e_3)\,dU$ for $F=fw$ and $F=w$, where $w$ is the
weight of \eqref{eq:9.GG}. With $U_e$ of diagonal entries $a_e$, $\bar a_e$ and
off-diagonal entries $b_e$, $-\bar b_e$, the hopping factor is
$\prod_ee^{\kappa-2\kappa\abs{b_e}^2}$, Haar measure is $b_e$ uniform on the unit
disc times $\arg a_e=\theta_e$ uniform, and as $\kappa\to\infty$ the weights
$e^{-2\kappa\abs{b_e}^2}$ concentrate at $b=0$, where $U_e=D(\theta)_e$ and
$\frac12\Tr U_p=\cos\theta_p$. The limit is taken in each fixed box, in the
London limit $\abs{\mathsf n_x}=1$, and for $\frac12\Tr$ (the full trace gives
$2\cos\theta_C$).
\qed
\end{proof}

\begin{corollary}[Corollary~\ref{cor:1.4}]\label{cor:9.4}
At $\kappa=\infty$, for every $\beta>0$, every rectangle $C$ with
$\min(R,T)\ge\Rone(\beta)$ and every finite box $\Lam\supseteq C$, the limit
of $\langle\frac12\Tr\prod_{e\in C}U_e\rangle_{\Lam;\beta,\kappa}$ as
$\kappa\to\infty$ has absolute value $\ex{\cos\theta_C}^{W}_{\Lam,\beta}$,
which is at most $\exp(-\sigma(\beta)RT+c(R+T))$.
\end{corollary}

\begin{proof}
This follows from Proposition~\ref{prop:9.3}, Theorem~\ref{thm:2.1}(a) and Theorem~\ref{thm:1.2}.
\qed
\end{proof}

\begin{remark}[Villain action and other models]\label{rem:9.6}\label{rem:9.7}
The theorem of \cite{GM82} holds in the form of Definition~\ref{def:1.1}: for
$\tau\ge K\beta_0(K)$ and $\beta:=\tau/K$, Lemma~\ref{lem:2.3}(c),(d) as in
Proposition~\ref{prop:2.9} give
$\ex{\cos\theta_C}^{V}_{\Lam,\tau}\le\gfix_{\Lamo,\tau}(C;\emptyset)\le\EE_D\gfix_{\Lamo,\tau}(C;D)$,
Steps \eqref{eq:8.2} to \eqref{eq:8.5} give the bound of Theorem~\ref{thm:1.2}
at $\beta$, and Lemma~\ref{lem:2.3}(b), pointwise in $\Lam$, gives smaller
$\tau$. The proof uses $d=3$ throughout. The Georgi\textendash Glashow model at finite Higgs and weak gauge
coupling is not covered: the centre $\ZZ_2$ of $SU(2)$ contains no U(1), and
\cite[Theorem~1]{Fr79} leads to the pure $\ZZ_2$ gauge theory, which confines
only at strong coupling.
\end{remark}