\section{Introduction}
Let $\theta \in \mathsf{M}_n(]-1;1[)$ be an antisymmetric matrix and let
$\rho_\theta$ be the bicharacter on $\mathbb{Z}^N$ given by
$\rho_\theta(x,y)=\exp(2\pi i x^t \theta y)$. The twisted group $C^*$-algebra
$\mathcal{A}_\theta := C^*(\mathbb{Z}^N,\rho)$ is called a non-commutative
$N$-torus due to its being isomorphic to $C(\mathbb{T}^N)$ in the
commutative case, $\theta=0$.

During the eighties the non-commutative tori were studied extensively,
most notably by Rieffel, see \cite{Rieffel-CaseStudy} for an overview,
and the structure of the algebra itself and the category of projective
modules over it are now well-known, see
e.g. \cite{Rieffel-Rotations,Rieffel-Cancellation,Rieffel-Projective}. The
non-commutative torus $\mathcal{A}_\theta$
possesses a high degree of symmetry, in fact the $N$-torus $\mathbb{T}^N =
\hat{\mathbb{Z}}^N$ acts as a group of automorphisms, $\alpha: \mathbb{T}^N
\rightarrow \operatorname{Aut}(\mathcal{A}_\theta)$, forming a $C^*$-dynamical system,
and this gives rise to a natural smooth sub-$*$-algebra
$\mathcal{A}_\theta^\infty$ consisting of the elements upon which the
$N$-torus acts smoothly. Said smooth part is an important and
tractable example of a non-commutative differentiable manifold
\cite{Connes-NoncommutativeGeometry}.
Lately the interest in the non-commutative tori has been revived as
they seem to appear naturally in string theory \cite{ConnesDouglasSchwarz}. One
subject of interest in that context is the classification of flat
connections on projective $\mathcal{A}_\theta$-modules as these determine
the minima of the Yang-Mills functional. In
the case $N=2$ the flat connections were completely classified by
Connes and Rieffel in 1987 \cite{ConnesRieffel}. In this paper we
extend their results to the higher-dimensional case. In fact the
following holds:

\begin{theorem}
Let $\mathsf{X}$ be a full, finitely generated, projective right
$\mathcal{A}_\theta$-module admitting an integrable connection. Then the
moduli space of $\mathsf{X}^\infty$ is homeomorphic to $(\mathbb{T}^n/
\sigma_n)^N$ for some $n \in \mathbb{N}$.
\end{theorem}
The notion of integrable connections is introduced below and these do
in fact exist in the generic case.

The approach below differs from \cite{ConnesRieffel} in as much as the
analysis of the structure of the moduli space of flat connections is
first done on the free $\mathcal{A}_\theta$-module,
$L^2(\mathcal{A}_\theta)$, and then a trick of Connes and Rieffel
\cite[Theorem 5.5]{ConnesRieffel} is used to transfer it to the
case of a general projective module admitting an integrable connection.

The authors would like to thank Hanfeng Li for kindly reading through an earlier version of the manuscript, pointing out some mistakes and for helpfully suggesting alternate proofs.

\section{Preliminaries}
\subsection{The Smooth Non-Commutative Torus}
The non-commutative torus $\mathcal{A}_\theta$, defined in the
introduction, is the universal
$C^*$-algebra generated by $N$ unitaries $u_1,\ldots,u_N$ satisfying
the commutation relations
\begin{equation*}
u_k u_l = \exp(2\pi i \theta_{kl}) u_l u_k
\end{equation*}
and every element of $\mathcal{A}_\theta$ can be written uniquely in the form
\begin{equation*}
\sum_{\alpha \in \mathbb{Z}^N} c_\alpha u^\alpha
\end{equation*}
where $\alpha = (\alpha_1,\ldots,\alpha_N)$ is a multi-index,
$u^\alpha = u_1^{\alpha_1} \cdots u_N^{\alpha_N}$, $c_\alpha \in \mathbb{C}$
and the sum converges in norm in $\mathcal{A}_\theta$. Note that there is
no canonical choice of the $N$ unitary generators and henceforth we
shall consider a generating set fixed. Since
$\mathcal{A}_\theta$ is the twisted group $C^*$-algebra,
$C^*(\mathbb{Z}^N,\rho)$, the dual group $\mathbb{T}^N$ of $\mathbb{Z}^N$ acts
pointwise continuously on $\mathcal{A}_\theta$ by $\sigma_z(u^\alpha) := z^\alpha
u^\alpha = \prod_{k=1}^n z_k^{\alpha_k} u_k^{\alpha_k}$. The smooth non-commutative torus $\mathcal{A}_\theta^\infty$
is defined as the sub-$*$-algebra of elements  $\mathcal{A}_\theta$ for
which the map
\begin{equation*}
\mathbb{T}^N \ni z \mapsto \sigma_z (a) \in \mathcal{A}_\theta
\end{equation*}
is smooth. The infinitesimal action of $\mathbb{R}^N$, the Lie algebra
associated with
$\mathbb{T}^N$, on $\mathcal{A}_\theta^\infty$ is given by derivations
\begin{equation*}
\delta_k u_l = i \delta_{kl} u_l \quad (1 \leq k,l \leq N) .
\end{equation*}

\subsection{The Canonical Trace}
The non-commutative torus admits a unique normalized $\mathbb{T}^N$-invariant
trace, namely
\begin{equation*}
\tau : \sum_{\alpha \in \mathbb{Z}^N} c_\alpha u^\alpha \mapsto c_0 \, ,
\end{equation*}
and the map $u^\alpha \mapsto z^\alpha$ from $\mathcal{A}_\theta$ to
$C(\mathbb{T}^N)$ extends to an isomorphism between the Hilbert spaces
$L^2(\mathcal{A}_\theta,\tau)$ and $L^2(\mathbb{T}^N)$
when $\mathbb{T}^N$ is equipped with the normalized Haar measure. This
isomorphism maps $\mathcal{A}_\theta^\infty$ onto $C^\infty(\mathbb{T}^N)$
such that the $\delta_k$'s are identified with the usual partial derivatives
on $\mathbb{T}^N$.

\subsection{The Yang-Mills Functional}
For each finitely generated, projective (right) module $\mathsf{X}$ of
$\mathcal{A}_\theta$ there exists a smooth version, an
$\mathcal{A}_\theta^\infty$-module $\mathsf{X}^\infty$ unique up to
algebraic isomorphism, satisfying
\begin{equation*}
\mathsf{X}^\infty \otimes_{\mathcal{A}^\infty}^{\text{alg}} \mathcal{A}_\theta \cong
\mathsf{X} \, .
\end{equation*}
A connection on $\mathsf{X}^\infty$ is a map from $\mathsf{Lie}(\mathbb{T}^N)$ to
$\mathsf{End}(\mathsf{X}^\infty)$, $x \mapsto \nabla_x$, implementing the action
of the $\delta_k$'s on $\mathcal{A}_\theta$, i.e.,
\begin{equation*}
[\nabla_k, a] \xi = \delta_k(a) \cdot \xi \quad (\xi \in
\mathsf{X}^\infty) .
\end{equation*}
Since $\mathsf{Lie}(\mathbb{T}^N)$ is an abelian Lie algebra, the curvature of
$\nabla$, $\Theta_\nabla(x,y) := [\nabla_x,\nabla_y]$, is an element of
$(\mathcal{A}_\theta^\infty)' \cap \mathsf{End}(\mathsf{X}^\infty)$.

We will always assume $\mathsf{X}$ to be equipped with a right $\mathcal{A}_\theta$-valued
inner product, $\rip{\cdot}{\cdot}_{\mathcal{A}_\theta}$, and any connection on
$\mathsf{X}^\infty$ to be compatible
with this, i.e., $\rip{\nabla_x \xi}{\eta}_{\mathcal{A}_\theta} + \rip{\xi}{\nabla_x \eta}_{\mathcal{A}_\theta} =
\delta_x \rip{\xi}{\eta}_{\mathcal{A}_\theta}$.

The Yang-Mills functional on the space of connections on
$\mathsf{X}^\infty$ is given as follows.
Firstly, we denote by $\mathcal{L}(\mathsf{X})$ the adjointable operators on $\mathsf{X}$. Moreover, we denote by $\lip{\mathcal{L}(X)}{\cdot}{\cdot}$ the left $\mathcal{L}(\mathsf{X})$-valued inner product on $\mathsf{X}$ given by $\lip{\mathcal{L}(\mathsf{X})}{x}{y}(z) := x \cdot \rip{y}{z}_{\mathcal{A}_\theta}$, and we denote by $\hat{\tau}$ the trace on the
adjointable operators satisfying
$\hat{\tau}(\lip{\mathcal{L}(\mathsf{X})}{x}{y})=\tau(\rip{y}{x}_{\mathcal{A}_\theta})$,
see \cite[Proposition 2.2]{Rieffel-Rotations}. Finally, choosing a basis $\{e_i\}$ for $\mathsf{Lie}(\mathbb{T}^N) = \mathbb{R}^N$ and using $\Theta_{ij}$ as a convenient shorthand for $\Theta_\nabla(e_i,e_j)$, the Yang-Mills functional, $\operatorname{YM}$, is defined by
\begin{equation*}
\operatorname{YM}(\nabla) := \sum_{i<j} \hat{\tau}(\Theta_{ij}^* \Theta_{ij}).
\end{equation*}
If $\mathsf{X}^\infty$ admits flat connections, i.e., connections with scalar
curvature, then the Yang-Mills functional attains its minimum exactly
on these connections \cite[Theorem 2.1]{ConnesRieffel}. We denote the
collection of flat connections by
$\mathsf{MC}(\mathsf{X}^\infty)$. The group $\mathcal{U}(\mathsf{X}^\infty)$ of unitary
operators, also called the gauge group, acts by
conjugation on $\mathsf{MC}(\mathsf{X}^\infty)$,
\begin{equation*}
(\gamma_u (\nabla))_x \xi := u \cdot (\nabla_x (u^* \cdot \xi)),
\end{equation*}
and connections in the same orbit are said to be gauge equivalent. The
Yang-Mills functional $\operatorname{YM}$ is easily seen to be invariant under the
gauge group \cite{ConnesRieffel}. The quotient $\mathsf{MC}(\mathsf{X}^\infty) /
\mathcal{U}(\mathsf{X}^\infty)$ is called the moduli space of flat connections on
$\mathsf{X}^\infty$ and it is this space that we wish to determine.

\section{The Free Module Case}
We start out by considering connections on $\mathsf{M}_n(\mathcal{A}_\theta)$ as a
right module over itself. Ultimately we will be able to reduce the
general case to this.

Let $\operatorname{tr}$ be the normalized trace on the matrix algebra $\mathsf{M}_n(\mathbb{C})$ and denote by
$\mathfrak{H}_n$ the Hilbert space $L^2(\mathcal{A}_\theta \otimes \mathsf{M}_n(\mathbb{C}),
\tau \otimes \operatorname{tr})$. The aforementioned isomorphism
$L^2(\mathcal{A}_\theta,\tau) \cong L^2(\mathbb{T}^N)$ extends to an
isomorphism $\mathfrak{H}_n \cong L^2(\mathbb{T}^N) \otimes \mathsf{M}_n(\mathbb{C})$ when
$\mathsf{M}_n(\mathbb{C})$ is equipped with the inner product $\hip{x}{y} :=
\operatorname{tr}(y^*x)$. Denote by $\eta$ the canonical embedding $\eta :
\mathsf{M}_n(\mathcal{A}_\theta) \rightarrow \mathfrak{H}_n$ and define $\mathfrak{H}_n^\infty
:= \eta(\mathsf{M}_n(\mathcal{A}_\theta^\infty))$. Define on $\mathfrak{H}_n^\infty$ the
derivations $D_1,\ldots,D_N$ by $D_k \eta(x) := \eta(\delta_k
x)$. These are the usual derivations on $C^\infty(\mathbb{T}^N) \otimes
\mathsf{M}_n(\mathbb{C})$ and each $D_k$ may be extended to its usual domain
\begin{equation*}
\mathcal{D}(D_k) = \operatorname{span}\left\{f \otimes m \in L^2(\mathbb{T}^N) \otimes \mathsf{M}_n(\mathbb{C}) \, \big|\,
  (\alpha_k \mathcal{F}(f)(\alpha))_{\alpha \in \mathbb{Z}^N} \in \ell^2(\mathbb{Z}^N)
\right\} ,
\end{equation*}
where $\mathcal{F}$ denotes the Fourier transform from $L^2(\mathbb{T}^N)$ to $\ell^2(\mathbb{Z}^N)$. On this domain $D_k$
is known to be skew-adjoint, i.e., $iD_k$ is self-adjoint in the sense
of unbounded operators. Obviously, the
connection $D = (D_k)_k$ has curvature zero.

Using the connection $D=(D_1,\ldots,D_N)$ we define the analogue of
the standard Laplacian.
\begin{definition}[Laplacian]
\index{Laplacian on $\mathfrak{H}_n$}
The \emph{Laplacian} on $\mathfrak{H}_n$ is the operator $\Delta := - \sum_k
D_k^2$.
\end{definition}

By the usual rules for addition and composition of unbounded
operators, the domain of the Laplacian is $\mathcal{D}(\Delta) = \bigcap_k
( \mathcal{D}(D_k) \cap D_k^{-1} \mathcal{D}(D_k))$, i.e.,
\begin{equation*}
\mathcal{D}(\Delta) = \operatorname{span} \left\{ f \otimes m \mid \prt{\norm{\alpha}_2^2
    \mathcal{F}(f)(\alpha)}_{\alpha \in \mathbb{Z}^N} \in \ell^2\left(\mathbb{Z}^N\right)
\right\}.
\end{equation*}
Written in terms of $\sum c_\alpha u^\alpha \otimes m$ the domain of
$\Delta$ is thus
\begin{equation} \label{Equation: Domain of Delta}
\mathcal{D}(\Delta) = \operatorname{span} \left\{ \eta \prt{ \sum_{\alpha \in \mathbb{Z}^N} c_\alpha
    u^\alpha \otimes m } \mid \prt{ \|\alpha\|_2^2 c_\alpha }_{\alpha \in \mathbb{Z}^N}
  \in \ell^2 \prt{ \mathbb{Z}^N } \right\} .
\end{equation}
Note that as $\|\alpha\|_2 \leq \|\alpha\|_1 \leq N^{1/2} \|\alpha\|_2$,
the $\|\alpha\|_2^2$ appearing in (\ref{Equation: Domain of Delta})
may be replaced with $\|\alpha\|_1^2$.
It is well-known that $\Delta$ is a positive self-adjoint operator.

It turns out to be useful to have a symbol for the left module action
of $\mathsf{M}_n(\mathcal{A}_\theta)$ on $\mathfrak{H}_n$:

\begin{definition}
Let $\pi : \mathsf{M}_n(\mathcal{A}_\theta) \rightarrow \mathbb{B}(\mathfrak{H}_n)$ be the map given by $\pi(a) \eta(x) := \eta(ax)$ for $a,x \in
\mathsf{M}_n(\mathcal{A}_\theta)$.
\end{definition}
Note that $\pi(a)$ is bounded as claimed as
\begin{equation*}
\norm{\pi(a)\eta(x)}_{\tau_\theta \otimes \operatorname{tr}}^2 = \tau_\theta
\otimes \operatorname{tr} \prt{ x^*a^*ax } \leq \norm{a^*a} \norm{x}_{\tau_\theta
  \otimes \operatorname{tr}}^2 .
\end{equation*}

We now turn to the case where an arbitrary connection $\nabla$ on $\mathfrak{H}_n$
is given.

\begin{lemma}
For every connection $\nabla$ on $\mathfrak{H}_n^\infty$ there exists
unique skew-adjoint elements $h_k \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$,
$k=1,\ldots,N$, such that $\nabla_k = D_k + \pi(h_k)$ on
$\mathfrak{H}_n^\infty$ for $k=1,\ldots,N$.
\end{lemma}

\begin{proof}
For any $\eta(a) \in \mathfrak{H}_n^\infty$ we have
\begin{equation*}
\nabla_k(\eta(a)) = \nabla_k(\eta(\boldsymbol{1}) \cdot a) = (\nabla_k
\eta(\boldsymbol{1})) \cdot a + D_k \eta(a).
\end{equation*}
Choosing $h_k \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$ such that $\eta(h_k) = \nabla_k
\eta(\boldsymbol{1})$ it follows that $\eta(h_k) \cdot a = \eta(h_k a) =
\pi(h_k) \eta(a)$, wherefore $\nabla_k - D_k =
\pi(h_k)$. Uniqueness of $h_k$ follows from the faithfulness of
$\tau_\theta$ and skew-adjointness follows from the skew-symmetry of
$\nabla_k$ and $D_k$.
\end{proof}

By boundedness of $\pi(h_k)$ we obtain the following corollary.

\begin{corollary}
For every connection $\nabla$ on $\mathfrak{H}_n^\infty$, the derivations
$\nabla_1,\ldots,\nabla_N$ may be extended to skew-adjoint operators on the domains
$\mathcal{D}(D_1),\ldots,\mathcal{D}(D_N)$, respectively.
\end{corollary}

For the remainder of this section we fix a connection $\nabla$ and
the corresponding skew-adjoint elements $h_1,\ldots,h_N \in
\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ that satisfy $\nabla_k = D_k + \pi(h_k)$.

\begin{lemma}
\label{Lemma: Small Wonder}
If $h \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$ then $\pi(h)$ maps
$\mathcal{D}(\Delta)$ into $\mathcal{D}(\Delta)$.
\end{lemma}
\begin{proof}
Let $h \otimes m_1$ be an element of $\mathsf{M}_n(\mathcal{A}_\theta^\infty) \subseteq L^2(\mathcal{A}_\theta) \otimes \mathsf{M}_n(\mathbb{C}) \cong L^2(\mathbb{T}^N) \otimes \mathsf{M}_n(\mathbb{C})$ and let $f \otimes m_2$ be an element of $\mathcal{D}(\Delta) \subseteq L^2(\mathcal{A}_\theta) \otimes \mathsf{M}_n(\mathbb{C}) \cong L^2(\mathbb{T}^N) \otimes \mathsf{M}_n(\mathbb{C})$. Denoting by $\hat h$ and $\hat f$ denote the Fourier transform of $h$ and $f$, respectively, we note that $(\|\alpha\|_2^n \hat{h}(\alpha))_\alpha$ and $(\|\alpha\|_2^2 \hat{f}(\alpha))_\alpha$ belong to $\ell^2(\mathbb{Z}^N)$ for any $n \in \mathbb{N}$. To prove the lemma we must show that $(\|\alpha\|_2^2 |\hat h \times \hat f (\alpha)|)_\alpha$ is likewise in $\ell^2(\mathbb{Z}^N)$.

We first note the following inequality for $\alpha, \beta \in \mathbb{Z}^N$:
\begin{equation*}
\|\alpha\|_2^2 \leq \|\alpha - \beta\|_2^2 + 2 \|\alpha\|_2 \|\beta\|_2 + \|\beta\|_2^2 \leq \|\alpha - \beta\|_2^2 + 2 \|\alpha - \beta\|_2 \|\beta\|_2 + 2 \|\beta\|_2^2 .
\end{equation*}
A direct computation then yields the lemma:
\begin{equation*}
\begin{split}
\|\alpha\|_2^2 |\hat h \times \hat f (\alpha)| & = \|\alpha\|_2^2 \sum_\beta |\hat h (\beta)| |\hat f (\alpha - \beta)|
\\
& \leq \sum_\beta |\hat h (\beta)| (\|\alpha - \beta\|_2^2 |\hat f (\alpha - \beta)|) + 2 \sum_\beta (\|\alpha\|_2 |\hat h (\alpha)|) (\|\alpha - \beta\|_2 |\hat f (\alpha - \beta)|)
\\
& \quad
+ 2 \sum_\beta (\|\beta\|_2^2 |\hat h (\beta)|) |\hat f (\alpha - \beta)| .
\end{split}
\end{equation*}
As the convolution product of two $\ell^2$-functions is itself a function in $\ell^2$, the above expression is finite.
\end{proof}

As a consequence of the previous lemma $\sum (D_k + \pi(h_k))^2$ is well-defined
on $\mathcal{D}(\Delta)$.

\begin{definition}
We denote by $H$ the operator $-\sum_k (D_k + \pi(h_k))^2$ with
domain $\mathcal{D}(H) := \mathcal{D}(\Delta)$.
\end{definition}

Like the Laplacian, $\Delta$, the operator $H$ is self-adjoint.

\begin{proposition} \label{Proposition: H Self-Adjoint}

The unbounded operator $H$ is self-adjoint.
\end{proposition}
\begin{proof}
First note that $H$ can be written as
\begin{equation}
\label{Equation: H = Delta - V}
\begin{split}
H & = - \sum_k (\partial_k + \pi(h_k))^2 = \Delta - 2 \sum_k \pi(h_k) \partial_k - \sum_k \pi(\partial_k h_k) - \pi(\sum_k h_k^2)
\\
& = \Delta - 2 \sum_k \pi(h_k) \partial_k - \pi(h),
\end{split}
\end{equation}
where $h := \sum_k \partial_k h_k + h_k^2$ belongs to $\mathsf{M}_n(\mathcal{A}_\theta^\infty)$. For simplicity of notation we will assume that $n=1$ in the following. The case $n>1$ follows from similar argumentation.

As $H$ is symmetric it is enough to show that $\mathcal{D}(H^*) \subseteq \mathcal{D}(H)$. Assume that $f \in \mathcal{D}(H^*)$ and let $g = H^* f$. The aim is to show that $f$ belongs to $\mathcal{D}(H)$, i.e., that $(\|\alpha\|_2^2 \hat f(\alpha) )_\alpha$ belongs to $\ell^2(\mathbb{Z}^N)$. For that purpose let $\phi$ be an arbitrary element of $\mathcal{D}(\Delta)$. Then
\begin{equation*}
\ip{\phi}{g} = \ip{H\phi}{f} = \ip{\Delta \phi}{f} - 2 \sum_k \ip{\partial_k \phi}{\pi(h_k^*) f} - \ip{\phi}{\pi(h^*)f} .
\end{equation*}
Let $f_k =\pi(h_k^*) f$ and $f_0 = \pi(h^*) f$ and identify for ease of notation each involved function with its Fourier transform. Then the above equation can be rewritten as follows.
\begin{equation*}
\sum_\alpha \phi(\alpha) \overline{g(\alpha)} = \sum_\alpha \|\alpha\|_2^2 \phi(\alpha) \overline{f(\alpha)} - 2 \sum_{\alpha, k} \alpha_k \phi(\alpha) \overline{f_k(\alpha)} - \sum_\alpha \phi(\alpha) \overline{f_0(\alpha)}.
\end{equation*}
Rearranging the equation we get:
\begin{equation*}
\sum_\alpha \phi(\alpha) \prt{\|\alpha\|_2^2 f(\alpha) - g(\alpha) - 2 \sum_k \alpha_k f_k(\alpha) - f_0(\alpha) }^- = 0.
\end{equation*}
Using Lemma \ref{Lemma: Small Wonder} and the fact that $\mathcal{D}(\Delta)$ is dense in $\mathfrak{H}_n$, we conclude that $(\|\alpha\|_2^2 f(\alpha))_\alpha$ is in $\ell^2(\mathbb{Z}^N)$ as desired.

\end{proof}

\begin{lemma}
\label{Lemma: H-Eigenvectors Are Smooth}
All eigenvectors of $H$ are smooth, i.e., if $H \xi = \lambda \xi$ then $\xi$ belongs to $\mathcal{D}^\infty(\Delta)$.
\end{lemma}
\begin{proof}
As in the previous proof, see Eq. (\ref{Equation: H = Delta - V}), we can write $H = \Delta + V$ where $V$ is a first order differential operator. Note that $[V,H]$ is likewise of first order, that is $HV = VH + (\text{first order})$. Thus,
\begin{equation*}
\Delta^n = (H - V)^n = H^n + \sum_{k=1}^n H^{n-k}VH^{k-1} + (\text{order $2n-2$}) .
\end{equation*}
Hence,
\begin{equation}
\label{Equation: Deltan}
\Delta^n = H^n + VH^{n-1} + (\text{order $2n-2$}).
\end{equation}
Now, assuming that $\xi$ is an eigenvector for $H$ then equation (\ref{Equation: Deltan}) implies that if $\xi \in \mathcal{D}(\Delta^{n-1})$ (i.e., $\xi$ is $2n-2$ times differentiable by regularity of $\Delta$) then $\xi \in \mathcal{D}(\Delta^n)$.

The lemma now follows by induction.
\end{proof}

The operator $H$ has much in common with the Laplacian $\Delta$. In
fact, the compactness of the resolvent of $\Delta$ implies by the
variational formulae compactness of the resolvent of $H$:

\begin{lemma}
The resolvent of $H$ is compact.
\end{lemma}
\begin{proof}
For a positive, self-adjoint operator $T$ and any finite-dimensional
subspace $L$ of $\mathcal{D}(T)$ we define
\begin{equation*}
\lambda(L) := \sup \left\{ \hip{Tf}{f} \mid f \in L , \, \|f\|=1
\right\}.
\end{equation*}
These are then used to define for each $k \in \mathbb{N}$:
\begin{equation*}
\lambda_k := \inf \left\{ \lambda(L) \mid \text{$L$ $k$-dimensional
    subspace of $\mathcal{D}(T)$} \right\}.
\end{equation*}
We denote by $\lambda_k^H$ and $\lambda_k^\Delta$ the corresponding
numbers for the positive self-adjoint operators $H$ and $\Delta$.

As $\Delta$ has compact resolvent it follows by \cite[Theorem
4.5.1]{Davies} that $\lambda_k^\Delta \nearrow \infty$.
The inequality
\begin{equation*}
\begin{split}
\ip{\Delta \xi}{\xi} & = \sum_k \|D_k \xi\|^2 = \sum_k \| (\nabla_k - \pi(h_k))\xi\|^2 \leq 2 \sum_k \|\nabla_k \xi\|^2 + 2 \sum_k \|\pi(h_k) \xi\|^2
\\
& \leq 2 \ip{H \xi}{\xi} + 2 \sum_k \ip{\pi(h_k^* h_k) \xi}{\xi}
\end{split}
\end{equation*}
shows that $\Delta \leq C (H+1)$ for some $C \geq 0$, and so in particular $\hip{\Delta f}{f} \leq
C \hip{Hf}{f} + C$ for $\|f\|=1$, whence we conclude that
$\lambda_k^\Delta \leq C (\lambda_k^H + 1)$. Since $\lambda_k^\Delta
\nearrow \infty$ it follows that $\lambda_k^H \nearrow \infty$. By
\cite[Theorem 4.5.2]{Davies} this implies that $H$ has empty essential
spectrum which by \cite[Corollary 4.2.3]{Davies} further implies that
$H$ has compact resolvent as desired.
\end{proof}

\begin{proposition} \label{Proposition: Connection up to Gauge Equivalence}
Suppose that $\nabla$ has constant curvature. Then there exists a unitary $u \in
\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ and a commuting family of skew-adjoint
elements $\Lambda_k \in \mathsf{M}_n(\mathbb{C})$, $k=1,\ldots,N$,  such that
\begin{equation} \label{Equation: Connections Almost Classified}
\pi(u^*) \nabla_k \pi(u) = D_k + \pi(\Lambda_k) , \quad k=1,\ldots,N
\, ,
\end{equation}
when restricted to the domain $\mathfrak{H}_n^\infty$.
\end{proposition}
\begin{proof}
As $H$ has compact resolvent we can find a finite-dimensional
eigenspace $\mathfrak{K}$ of $H$. By Lemma \ref{Lemma: H-Eigenvectors Are Smooth}, $\mathfrak{K} \subseteq \mathfrak{H}_n^\infty.$
The connections $\nabla$ and $D$ on $\mathfrak{H}_n^\infty$ have constant
curvature and are compatible with the inner product structure on
$\mathfrak{H}_n^\infty$ when this is considered as a right $\mathbb{C}$-module and
$\mathbb{C}$ is equipped with the trivial derivation, $\delta_\mathbb{C} (z) = 0$
for all $z \in \mathbb{C}$. Hence, by equality of the curvatures of flat
connections, \cite[Theorem 2.1]{ConnesRieffel}, the curvature of
$\nabla$ is identical to the curvature
of $D$, i.e., zero. Thus $\nabla_1,\ldots,\nabla_N$ commute pairwise
and therefore also with $H$.
For this reason $\mathfrak{K}$ is invariant under $\nabla_1,\ldots,\nabla_N$
 and we may find a common eigenvector $\xi \in \mathfrak{K}$
for $\nabla_1,\ldots,\nabla_N$ and $H$:
\begin{equation} \label{Equation: Eigenvalue}
\nabla_k \xi = \lambda_k \xi , \quad k=1,\ldots,N .
\end{equation}
By skew-symmetry of $\nabla_k$, $\lambda_k$ is imaginary.

As $\xi$ is an eigenvector of $\nabla_k$ we have $D_k
\xi = \lambda_k \xi - \pi(h_k) \xi$ which as $h_k \in
\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ belongs to
$\mathcal{D}(\Delta)$. Hence, $\xi \in
\mathcal{D}^\infty(\Delta) = \mathfrak{H}_n^\infty$, and so we may choose
$x \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$ such that $\eta(x) = \xi$. The
eigenvalue equation (\ref{Equation: Eigenvalue}) then assumes the
form: $\lambda_k x = \delta_k^{(n)} x + h_k x$ for each
$k=1,\ldots,N$. Since $\delta_k^{(n)}$ is a
$*$-derivation, $\delta_k^{(n)}(x^*x)=0$ wherefore $x^*x \in
\mathsf{M}_n(\mathbb{C})$. By positivity of $x^*x$ we must have $x^*x = \sum
\epsilon_i p_i$ for some $\epsilon_i >0$ and some set $\{p_i\}$ of
pairwise orthogonal projections in $\mathsf{M}_n(\mathbb{C})$. Let $p = \sum p_i$ and
$y = x^*x+(\boldsymbol{1} - p)$. Then $y$ is positive and invertible and $x^*x
= py = yp$ wherefore
\begin{equation*}
p = y^{-1/2} x^*x y^{-1/2} = (xy^{-1/2})^*(xy^{-1/2}).
\end{equation*}
Thus, $v := xy^{-1/2}$ is a partial isometry and as $y^{-1/2}$ belongs
to $\mathsf{M}_n(\mathbb{C})$ on which $\delta_k^{(n)}$ vanishes, we have
\begin{equation*}
\delta_k^{(n)}(v) + h_k v = \delta_k^{(n)}(x) y^{-1/2} + h_k x y^{-1/2} =
\lambda_k v.
\end{equation*}
Hence,
\begin{equation} \label{Equation: Commutation with Nabla}
\nabla_k \pi(v) = (D_k + \pi(h_k))\pi(v) = \pi(v) (D_k + \lambda_k).
\end{equation}
Taking the adjoint of Eq. (\ref{Equation: Commutation with Nabla}) and restricting
to $\mathfrak{H}_n^\infty$ we find that
\begin{equation} \label{Equation: Commutation again}
\pi(v^*) \nabla_k = (D_k +\lambda_k) \pi(v^*).
\end{equation}
Combining Eq. (\ref{Equation: Commutation with Nabla}) and Eq. (\ref{Equation:
  Commutation again}) shows that $\pi(vv^*)$ commutes with $\nabla_k$
(on $\mathfrak{H}_n^\infty$).

If $v^*v = \boldsymbol{1}$ then $v$ is unitary since the
faithfulness of the trace implies $vv^*=\boldsymbol{1}$ as well, and so we have found a unitary $v$
satisfying $\pi(v^*) \nabla_k \pi(v) = D_k + \lambda_k \pi(\boldsymbol{1})$ and
we are done. If $v^*v \neq \boldsymbol{1}$ we proceed by induction on
$\operatorname{tr}(v^*v)$ in the following way.

Denote the above $v$ and $\lambda_k$ by $v_1$ and $\lambda_k^1$,
respectively. Since $\pi(v_1v_1^*)$ commutes with $H$ and
$\nabla_1,\ldots,\nabla_N$ we can, just as above, find a common
eigenvector $\xi_2 = \eta(x_2)$ for $H$ and
$\nabla_1,\ldots,\nabla_N$, $\nabla_k \xi_2 = \lambda_k^2 \xi_2$, such
that $\pi(v_1v_1^*) \xi_2 = 0$. As above we obtain a partial isometry
$w_2 = x_2 y_2^{-1/2}$.

Since $\pi(v_1v_1^*)\xi_2 = 0$ then $(v_1v_1^*x_2)y_2^{-1/2} = 0$
whence $v_1^* w_2 = 0$. Considering $v_1^*v_1$ and $w_2^*w_2$ as
operators on $\mathbb{C}^n$ we can as $\operatorname{tr}(v_1^*v_1) < n$ find a
one-dimensional subspace $\mathfrak{L}$ of $\operatorname{im}(v_1^*v_1)^\perp \subseteq
\mathbb{C}^n$. Choose a partial isometry $\phi \in \mathsf{M}_n(\mathbb{C})$ mapping $\mathfrak{L}$ onto a
one-dimensional subspace of $\operatorname{im}(w_2^*w_2) \subseteq \mathbb{C}^n$ and
which vanishes on the orthogonal complement of $\mathfrak{L}$. Letting $v_2 := w_2
\phi$ we obtain a partial isometry as $v_2^*v_2 = \phi^* w_2^* w_2
\phi = \phi^* \phi$. For this we have
\begin{equation*}
\nabla_k \pi(v_2) = \pi(w_2) \prt{D_k + \lambda_k^2 } \pi(\phi) =
\pi(v_2) \prt{ D_k + \lambda_k^2 },
\end{equation*}
$v_1^* v_2 = (v_1^* w_2) \phi = 0$ and $v_2 v_1^* = w_2 (v_1 \phi^*)^*
= 0$. Thus $(v_1+v_2)^*(v_1+v_2) = v_1^* v_1 + v_2^*v_2$ and
$v_1^*v_1 \perp v_2^*v_2$ wherefore $v_1+v_2$ is a partial isometry
satisfying
\begin{equation*}
\nabla_k \pi(v_1+v_2) = \pi(v_1+v_2) D_k + \lambda_k^1 \pi(v_1) +
\lambda_k^2 \pi(v_2).
\end{equation*}
Continuing by induction over the trace until $(v_1+ \cdots +
v_m)^*(v_1 + \cdots + v_m) = \boldsymbol{1}$ we get a unitary $u := v_1 + \cdots
+ v_m$ such that
\begin{equation*}
\pi(u^*) \nabla_k \pi(u) = D_k + \sum_{i=1}^m \lambda_k^i \pi(v_i^*
v_i).
\end{equation*}
Defining $\Lambda_k := \sum \lambda_k^i v_i^* v_i$ gives us the desired
skew-adjoint matrix in $\mathsf{M}_n(\mathbb{C})$.
\end{proof}

The above theorem reduces the task of determining the moduli space to the task of
classifying the families $(\Lambda_k)_k$ of skew-adjoint matrices
satisfying Eq. (\ref{Equation: Connections Almost Classified}). Before
doing so we turn to a practical lemma.

In the following $\sigma_n$ denotes the permutation group of
$\{1,\ldots,n\}$.

\begin{lemma} \label{Lemma: Marriage Lemma}
For every unitary $U \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$ there exists a
permutation $\rho \in \sigma_n$ such that $U_{i\rho(i)}$ is non-zero
for all $i=1,\ldots, n$.
\end{lemma}
\begin{proof}
Define $x_{ij} := \tau_\theta(U_{ij} U_{ij}^*) \geq 0$ for $i,j
=1,\ldots,n$ and note that $\sum_j x_{ij} = \tau_\theta\prt{
  (UU^*)_{ii}} = 1$ for $i=1,\ldots,n$ and likewise $\sum_i x_{ij} =
\tau_\theta\prt{ (U^*U)_{jj}}=1$ for $j=1,\ldots,n$.
Hence, $(x_{ij})$ forms a matrix of non-negative real numbers such
that every row and every column sums to one.

We define for every subset $S \subseteq \{1,\ldots,n\}$ the set
$\Gamma(S) := \{j \in \{1,\ldots,n\} \mid \exists i \in S : x_{ij} > 0
\}$. That is, for each set of rows $S$ we denote by $\Gamma(S)$ the
set of columns which have a non-zero entry in at least one of the
rows. We wish to show that Hall's condition is satisfied, namely that
for every $S \subseteq \{1,\ldots,n\}$ the cardinality of $\Gamma(S)$
is greater than or equal to the cardinality of $S$. Let $S$ be given.

Obviously, $\sum_{i \in S} \sum_{j=1}^n x_{ij} = \sum_{i \in S} 1 =
\mathsf{Card}(S)$. However, we can estimate the series upwards as follows:
\begin{equation*}
\sum_{i \in S} \sum_{j=1}^n x_{ij} = \sum_{i \in S} \sum_{j \in
  \Gamma(S)} x_{ij} \leq \sum_{j \in \Gamma(S)} \sum_{i=1}^n x_{ij} =
\mathsf{Card}(\Gamma(S)),
\end{equation*}
which gives the desired inequality. Thus, $\mathsf{Card}(S) \leq \mathsf{Card}(\Gamma(S))$
for all subsets $S \subseteq \{1,\ldots,n\}$. It follows by Hall's
marriage theorem \cite[Theorem 5.1]{LintWilson} that there exists a
permutation $\rho$ of $\{1,\ldots,n\}$ such that $x_{i\rho(i)} \neq 0$
for all $i=1,\ldots,n$.
\end{proof}

We are now ready to prove the main theorem of the section which
characterizes the moduli space of $\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ up to
homeomorphism.

\begin{theorem} \label{Theorem: Free Module Case}
The moduli space of $\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ as a right module over
itself is homeomorphic to $(\mathbb{T}^N)^n/{\sigma_n}$.
\end{theorem}
\begin{proof}
We start by establishing a map from $(\mathbb{T}^N)^n/{\sigma_n}$ to the
moduli space of $\mathsf{M}_n(\mathcal{A}_\theta^\infty)$.

Let $\nabla$ be an arbitrarily given compatible connection on
$\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ with constant curvature. By Proposition
\ref{Proposition: Connection up to Gauge Equivalence} there exists a
family of pairwise commuting, skew-adjoint scalar matrices
$\{\Lambda_k\}_{k=1}^N$ such that $\pi(u)\nabla_k \pi(u^*) = D_k +
\pi(\Lambda_k)$,$k=1,\ldots,N$ for some unitary $u \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$. As the $\Lambda_k$ commute pairwise and are
skew-adjoint, they are simultaneously unitarily diagonalizable, i.e.,
there exists a unitary $\hat{u} \in \mathsf{M}_n(\mathbb{C})$ and scalars $\lambda_j^k \in i \mathbb{R}$ such that $\hat{u}
\Lambda_k \hat{u}^* = \operatorname{diag}(\lambda_1^k,\ldots, \lambda_n^k)$ for
$k=1,\ldots,N$. Thus, conjugating
$\pi(u)\nabla\pi(u^*)$ with $\pi(\hat{u})$ yields
\begin{equation*}
\pi(\hat{u}u) \nabla_k \pi(\hat{u}u)^* = \pi(\hat{u}) \prt{ D_k +
  \pi(\Lambda_k) }
\pi(\hat{u})^* = D_k + \pi\prt{\operatorname{diag}\prt{\lambda_1^k,\ldots,\lambda_n^k}} \quad \quad (k=1,\ldots,N),
\end{equation*}
as $D_k$ is invariant under conjugation with scalar matrices.

Hence, $\nabla$ is up to gauge equivalence uniquely given by the
ordered $nN$-tuple
\begin{equation*}
\prt{\lambda_1^1,\ldots,\lambda_n^1,\lambda_1^2,\ldots,\lambda_n^2,\ldots,\lambda_1^N,\ldots,\lambda_n^N
  }.
\end{equation*}
As conjugating $\pi(\hat{u}u) \nabla \pi(\hat{u}u)^*$ with a scalar
permutation
matrix, which interchanges the diagonal elements, does not change the
equivalence class of $\pi(\hat{u}u)\nabla \pi(\hat{u}u)^*$, we conclude that
$\nabla$ is up to gauge equivalence uniquely given by the unordered
$n$-tuple (written with set notation) of ordered $N$-tuples
\begin{equation*}
\left\{ \prt{\lambda_1^1,\ldots,\lambda_1^N} ,\ldots, \prt{
    \lambda_n^1,\ldots, \lambda_n^N} \right\} .
\end{equation*}
Now, define the element $v := \operatorname{diag}(\boldsymbol{1},\ldots,\boldsymbol{1},u^\alpha,\boldsymbol{1},\ldots,\boldsymbol{1})$
where the $u^\alpha$ appears as the $l$th entry. Then $v$ is
unitary and by conjugating $(D_k +
\pi(\operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)))_k$ with $\pi(v)$ we obtain
\begin{equation*}
\begin{split}
\pi(v) \prt{ D_k + \pi\prt{ \operatorname{diag} \prt{ \lambda_1^k,\ldots,\lambda_n^k
      }}} \pi(v)^* & = \pi(v) \pi \prt{\delta_k^{(n)}v} + D_k + \pi
\prt{ \operatorname{diag} \prt{ \lambda_1^k,\ldots,\lambda_n^k}}
\\
& = D_k + \pi \prt{
  \operatorname{diag}\prt{\lambda_1^k,\ldots,\lambda_l^k + i \alpha_k , \ldots,
  \lambda_n^k }} .
\end{split}
\end{equation*}
Choosing $\alpha_m=0$ for $m \neq k$ we see that that we can add any
integer to any entry of $(\lambda_1^1,\ldots,\lambda_n^1,\ldots,
\lambda_1^N,\ldots, \lambda_n^N)$ while leaving the other
entries untouched. Hence, letting $[\cdot] : \mathbb{R} \rightarrow \mathbb{R} / \mathbb{Z} \cong \mathbb{T}$ denote the quotient map, $\nabla$ is up to gauge equivalence uniquely given by
the unordered $n$-tuple of ordered $N$-tuples with values in $\mathbb{T}$,
\begin{equation*}
\left\{ \prt{ [i\lambda_1^1],\ldots,[i\lambda_1^N]} , \ldots, \prt{
    [i\lambda_n^1], \ldots, [i \lambda_n^N] } \right\} \, .
\end{equation*}

Thus, any compatible connection $\nabla$ on
$\mathsf{M}_n(\mathcal{A}_\theta^\infty)$ with constant curvature is up to gauge
equivalence uniquely given by an element of $(\mathbb{T}^N)^n/{\sigma_n}$
as described above.
Summing up: We have a well-defined map
\begin{equation*}
\Phi : \prt{\mathbb{T}^N}^n /{\sigma_n} \rightarrow
\mathsf{MC}\prt{\mathsf{M}_n(\mathcal{A}_\theta^\infty)} /
\mathcal{U}\prt{\mathsf{M}_n(\mathcal{A}_\theta^\infty)}
\end{equation*}
given by
\begin{equation*}
\Phi \prt{ \left\{ \prt{ [\lambda_1^1],\ldots,[\lambda_1^N]} , \ldots
    , \prt{ [\lambda_n^1],\ldots, [\lambda_n^N]} \right\} } := \left[ \prt{
  D_k + i \pi\prt{ \operatorname{diag} \prt{ \lambda_1^k , \ldots,
      \lambda_n^k}}}_{k=1,\ldots,N} \right]_0 \, ,
\end{equation*}
where $[\nabla]_0$ denotes the equivalence class of $\nabla$ in
$\mathsf{MC}\prt{\mathsf{M}_n(\mathcal{A}_\theta^\infty)} /
\mathcal{U}\prt{\mathsf{M}_n(\mathcal{A}_\theta^\infty)}$.
By construction, $\Phi$ is surjective.

Next we show injectivity of $\Phi$. Assume that
\begin{equation*}
\Phi \prt{ \left\{ \prt{ [\lambda_1^1],\ldots,[\lambda_1^N]} , \ldots
    , \prt{ [\lambda_n^1],\ldots, [\lambda_n^N]} \right\} }
=
\Phi \prt{ \left\{ \prt{ [\mu_1^1],\ldots,[\mu_1^N]} , \ldots
    , \prt{ [\mu_n^1],\ldots, [\mu_n^N]} \right\} }
\end{equation*}
and let $\Lambda_k := i \operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)$ and
$M_k := i \operatorname{diag}( \mu_1^k,\ldots, \mu_n^k)$ for $k=1,\ldots,N$. By
Proposition \ref{Proposition: Connection up to Gauge Equivalence}
there exists a unitary $U \in \mathsf{M}_n(\mathcal{A}_\theta^\infty)$ such that
\begin{equation*}
\pi(U) \prt{D_k + \pi(\Lambda_k)} \pi(U)^* = D_k + \pi(M_k) \text{
  for } k=1,\ldots, N \,.
\end{equation*}
This implies that $U \delta_k^{(n)}(U)^* + U \Lambda_k U^* = M_k$
and thus $-\delta_k^{(n)}(U) + U \Lambda_k = M_k U$ for
$k=1,\ldots,N$. Considering the $lm$-component we find that
$-\delta_k(U_{lm})+U_{lm} i \lambda_m^k = i \mu_l^k U_{lm}$. That is,
$\delta_k(U_{lm}) = i (\lambda_m^k - \mu_l^k) U_{lm}$ for any $l,m =
1,\ldots,n$ and $k=1,\ldots,N$. Writing $U_{lm} = \sum c_\alpha
u^\alpha$ we see that $\delta_k(U_{lm}) = \sum c_\alpha i \alpha_k
u^\alpha$ equals $z U_{lm}$ for some $z \in \mathbb{C}$ only if $z \in i
\mathbb{Z}$. By Lemma \ref{Lemma: Marriage Lemma} we may choose a permutation
$\rho \in \sigma_n$ such that $U_{l\rho(l)} \neq 0$ for all
$l=1,\ldots,n$. It follows that
$\lambda_l^k - \mu_{\rho(l)}^k \in \mathbb{Z}$ for all $l=1,\ldots,n$ and
$k=1,\ldots,N$. We conclude that
\begin{equation*}
\left\{ \prt{ [\lambda_1^1],\ldots, [\lambda_1^N]} , \ldots , \prt{
    [\lambda_n^1], \ldots, [\lambda_n^N] } \right\} =
\left\{ \prt{ [\mu_1^1],\ldots, [\mu_1^N]} , \ldots , \prt{
    [\mu_n^1], \ldots, [\mu_n^N] } \right\} ,
\end{equation*}
whereby injectivity of $\Phi$ has been shown.

Lastly we show that $\Phi$ is a homeomorphism. As
$(\mathbb{T}^N)^n/\sigma_n$ is compact, it suffices to show continuity of
$\Phi$. Consider the following commutative diagram,
\begin{equation*}
\begin{CD}
\mathbb{R}^{Nn} @>\psi>> \mathsf{MC}(\mathsf{M}_n(\mathcal{A}_\theta^\infty)) \\
@V p VV  @VV\kappa V \\
\prt{\mathbb{T}^N}^n/\sigma_n @>\Phi>> \mathsf{MC}(\mathsf{M}_n(\mathcal{A}_\theta^\infty)) /
\mathcal{U}(\mathsf{M}_n(\mathcal{A}_\theta^\infty))
\end{CD}
\end{equation*}
where $\kappa$ and $p$ are the canonical quotient maps and $\psi$ is
given by
\begin{equation*}
\psi \prt{ \lambda_1^1,\ldots,\lambda_1^N,\ldots, \lambda_n^1,\ldots,
  \lambda_n^N } := \prt{ D_k + i \pi \prt{ \operatorname{diag} \prt{ \lambda_1^k ,
      \ldots, \lambda_n^k }}}_{k=1,\ldots,N} .
\end{equation*}
As $\kappa$ is continuous and $\psi$ is trivially seen to be so as
well, it follows by the definition of the quotient topology that $\Phi$ is
continuous. This concludes the proof.
\end{proof}

\section{The Yang-Mills Problem for Non-Commutative Tori}
\label{Section: The Yang-Mills Problem for Non-Commutative Tori}

This section sees the convergence of the previous results into the
 main theorem stating that the moduli space of a
finitely generated, projective module over a non-commutative torus is
homeomorphic to $(\mathbb{T}^N)^n/\sigma_n$ as long as the module has a
sufficiently well-behaved connection.

For this section we fix a non-commutative torus $\mathcal{A}_\theta$ and
a smooth part $\mathcal{A}_\theta^\infty$ thereof. Moreover we fix a
finitely generated, projective Hilbert $\mathcal{A}_\theta$-module
$\mathsf{X}$ and a smooth part $\mathsf{X}^\infty$ thereof.

The sense of well-behavedness of connections is defined next.

\begin{definition}[Integrable Connection]
\index{Integrable connection}
A connection $\nabla$ on $\mathsf{X}^\infty$ is \emph{integrable} if it
has constant curvature and there exists a non-commutative torus $\mathcal{A}_\omega$
 such that $\mathcal{L}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty) \cong
\mathsf{M}_n(\mathcal{A}_\omega^\infty)$ and such that $\prt{\sum_j \epsilon_{ij} \nabla_j
  }_i$ is a connection on $\mathsf{X}^\infty$ considered as a left
$\mathsf{M}_n(\mathcal{A}_\omega^\infty)$-module for some $n \in \mathbb{N}$ and some
matrix $\epsilon \in \mathsf{GL}_N(\mathbb{R})$.
\end{definition}

We may also write that a connection is \emph{$n$-integrable} in order to
emphasize the $n$ appearing in the definition above.

Clearly, the existence of an integrable connection on $\mathsf{X}^\infty$
necessitates that the $C^*$-algebra of adjointable operators on
$\mathsf{X}$ (or compact operators for that matter) is isomorphic to a
matrix algebra over another non-commutative torus. For non-commutative
two-tori this is always the case \cite[Corollary
2.6]{Rieffel-Cancellation}. The situation is less clear in the higher-dimensional case. When it comes to
non-commutative two-tori, it can moreover be shown using Hochschild
homology that if $\theta$, represented as a scalar, is sufficiently
irrational, then $\mathsf{X}^\infty$ automatically possess integrable
connections. The condition on the irrationality is called
\emph{generic irrationality}
\index{Generic irrationality}
and can be stated as $\forall n \in \mathbb{N} \, \exists m \in \mathbb{Z} : 0 < |\theta -
m/n| < n^{-2}$. We discuss the existence of integrable connections in the next section.

We call a map $\Phi: \mathsf{MC}(\mathsf{X}^\infty) \rightarrow
\mathsf{MC}(\mathsf{Y}^\infty)$ \emph{equivariant} if $\Phi$ is invertible and
both $\Phi$ and $\Phi^{-1}$ preserve gauge equivalence.
\index{Equivariance}

\begin{theorem}
\label{Theorem: Moduli Space for Non-Commutative Tori}
If $\mathsf{X}$ is a full, finitely generated, projective right
$\mathcal{A}_\theta$-module admitting an $n$-integrable connection, then the
moduli space of $\mathsf{X}^\infty$ is homeomorphic to $(\mathbb{T}^N)^n/
\sigma_n$.
\end{theorem}
\begin{proof}
Let $\tilde{\mathsf{X}}$ denote the dual module of $\mathsf{X}$ and fix a compatible connection $\tilde{\nabla}$ with constant curvature thereon.
Using the methods of \cite[Section 5]{ConnesRieffel}, see especially
Theorem 5.5, we conclude that the map
\begin{equation*}
\mathsf{MC}(\mathsf{X}^\infty,\delta) \rightarrow \mathsf{MC}(\mathsf{X}^\infty
\otimes_{\mathcal{A}_\theta^\infty}^{\text{alg}} \tilde{\mathsf{X}}^\infty,
\hat{\delta}) , \quad \nabla \mapsto \nabla \otimes \operatorname{id}_{\tilde{\mathsf{X}}^\infty} + \operatorname{id}_{\mathsf{X}^\infty} \otimes \tilde{\nabla} ,
\end{equation*}
is in fact an equivariant homeomorphism.
Furthermore, identifying $\mathsf{X}^\infty \otimes_{\mathcal{A}_\theta^\infty}^{\text{alg}} \tilde{\mathsf{X}}^\infty$ with
$\mathcal{L}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty)$ through the map $\Gamma$, $\Gamma(\xi \otimes \tilde{\eta})(\zeta) = \xi \rip{\eta}{\zeta}_{\mathcal{A}_\theta}$,
yields a homeomorphism between $\mathsf{MC}(\mathsf{X}^\infty
\otimes_{\mathcal{A}_\theta^\infty}^{\text{alg}} \tilde{\mathsf{X}}^\infty,
\hat{\delta})$ and
$\mathsf{MC}(\mathcal{L}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty),\hat{\delta})$ which
is equivariant.

By assumption an $n$-integrable connection on $\mathsf{X}^\infty$ exists,
i.e., there exists a connection $\nabla'$ on $\mathsf{X}^\infty$ with
constant curvature, a non-commutative torus $\mathcal{A}_\omega$ and an
invertible matrix $\epsilon \in \mathsf{GL}_N(\mathbb{R})$ such that
$\mathcal{L}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty) \cong
\mathsf{M}_n(\mathcal{A}_\omega)$ and $\epsilon \nabla$ is a connection on
$\mathsf{X}^\infty$ as a left $\mathsf{M}_n(\mathcal{A}_\omega^\infty)$-module. The
derivation on $\mathsf{M}_n(\mathcal{A}_\omega^\infty)$ is the canonical
derivation stemming from the torus action; we denote it
$\delta^\omega$. In the following we tacitly identify
$\mathcal{L}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty)$ and
$\mathsf{M}_n(\mathcal{A}_\omega^\infty)$.
As
\begin{equation*}
\pi(\delta_i^\omega(a)) = \left[ \sum_j \epsilon_{ij} \nabla',\pi(a) \right] =
\sum_j \epsilon_{ij}
\left[ \nabla_j',\pi(a) \right] = \pi \left( \sum_j \epsilon_{ij}
  \hat{\delta}_j(a) \right)
\end{equation*}
we have $\delta^\omega = \epsilon \hat{\delta}$. If $\nabla$ is a
connection with constant curvature on
$\mathcal{K}_{\mathcal{A}_\theta^\infty}(\mathsf{X}^\infty)$ as a
right module over itself then $(\sum_j \epsilon_{ij} \nabla_j)_i$ is a
connection with constant curvature on
$\mathsf{M}_n(\mathcal{A}_\omega^\infty)$ over itself. Obviously, $\nabla \mapsto
\epsilon \nabla$ then establishes a bijection between
$\mathsf{MC}(\mathcal{L}_{\mathcal{A}_\theta^\infty} (\mathsf{X}^\infty), \hat{\delta})$
and $\mathsf{MC}(\mathsf{M}_n(\mathcal{A}_\omega^\infty), \delta^\omega)$. The map is
furthermore clearly a homeomorphism and equivariant whence
the moduli space of $\mathsf{M}_n(\mathcal{A}_\omega^\infty)$ is homeomorphic
to that of $\mathsf{X}^\infty$. The conclusion now follows from Theorem
\ref{Theorem: Free Module Case}.
\end{proof}

This solves the Yang-Mills problem as it was stated earlier for
finitely generated,
projective modules admitting integrable connections and gives, in
light of the earlier discussion of existence of integrable
connections for the case of non-commutative two-tori, the main result
of \cite{ConnesRieffel} as a special case when $\theta$ is generically
irrational.

\section{Existence of Integrable Connections}

The requirement of the existence of an integrable connection on the module seems to be a very weak requirement. In fact, requiring $n = 1$ in the definition of an integrable connection corresponds to the existence of a non-commutative torus with which the non-commutative torus under consideration is completely Morita equivalent in the sense of Schwarz \cite[Section 4]{Schwarz}. Schwarz gives a large class of modules over $\mathcal{A}_\theta$ satisfying this \cite[Section 3]{Schwarz}.

In the following we shall, following Rieffel \cite{Rieffel-Projective}, characterize all projective modules over $\mathcal{A}_\theta$ up to isomorphism when $\theta$ is not rational and indicate how integrable connections may be found on these. The quintessential object in the construction is the Heisenberg bimodule which we now consider.

\begin{example}[Construction of the Heisenberg Bimodule]
The details of the following overview can be found in \cite[Section 3]{Rieffel-Projective}.

Let $M$ be the abelian group of the form $\mathbb{R}^p \times {\hat{\mathbb{R}}}^q \times F$ for some $p,q \in \mathbb{N}_0$ and a finite abelian group $F$. Let $G := M \times \hat{M}$ and let $\beta$ be the continuous bicharacter on $G$ given by
\begin{equation*}
\beta\prt{ (m,\phi) , (n,\psi) } := \ip{m}{\psi} .
\end{equation*}
Denote for any subgroup $H \leq G$ by $\mathcal{S}(H,\beta)$ the $*$-algebra of Schwartz functions on $H$ with product and involution given by
\begin{equation*}
\Phi_1 * \Phi_2 (h_0) = \int_H \Phi_1(h) \Phi_2(h_0 - h) \beta(h,h_0-h) \,dh \quad \text{and} \quad
\Phi^*(h_0) = \beta(h_0,h_0) \overline{\Phi(-h_0)} .
\end{equation*}

Define unitaries on $\mathcal{S}(M)$ by $(u_{(n,\psi)}f)(m) := \ip{m}{\psi}f(m+n)$ for $(n,\psi) \in G$. These integrate up to give a left action of $\mathcal{S}(H,\beta)$ and a right action of $\mathcal{S}(H^\bot,\bar\beta)$ on $\mathcal{S}(M)$ by
\begin{equation*}
\Phi \cdot f := \int_H \Phi(h) u_h f \,dh \quad \text{and} \quad f \cdot \Omega := \int_{H^\bot} \Omega(k) u_k^* f \,dk ,
\end{equation*}
where $H^\bot := \{g \in G \mid \forall h \in H : \beta(g,h) \overline{\beta(h,g)} = 1 \}$.

Introducing $\mathcal{S}(H,\beta)$- and $\mathcal{S}(H^\bot,\bar{\beta})$-valued inner products,
\begin{equation*}
\lip{H}{f}{g}(x) := \hip{f}{u_x g}_2 \quad \text{and} \quad
\rip{f}{g}_{H^\bot}(y)  := \hip{u_y g}{f}_2 ,
\end{equation*}
we obtain the Heisenberg bimodule which, when completed, yields a Morita equivalence bimodule between the twisted group $C^*$-algebras $C^*(H,\beta)$ and $C^*(H^\bot,\bar\beta)$ \cite[Theorem 2.15]{Rieffel-Projective}.
\end{example}

Rieffel has shown that any projective $\mathcal{A}_\theta$-module (when $\theta \notin \mathsf{M}_N(\mathbb{Q})$) is isomorphic to a direct sum of modules obtained as above (with the restriction $2p + q = N$) \cite[Cor. 7.2 + Thm. 7.3]{Rieffel-Projective}.

We now indicate how to obtain integrable connections by explicitly constructing such in a special case of the above, namely $M = \mathbb{R}^p$.

\begin{example}[The Case $M = \mathbb{R}^p$]
Let $H \leq \mathbb{R}^p \times \hat{\mathbb{R}}^p$ be a lattice (i.e., $H$ is a discrete subgroup and $(\mathbb{R}^p \times \hat{\mathbb{R}}^p) / H$ is compact). Thus, $H \cong \mathbb{Z}^{2p}$ and $C^*(H,\beta)$ is a non-commutative torus. Choose generators $(r^k,\phi^k) \in \mathbb{R}^p \times \hat{\mathbb{R}}^p$, $k=1,\ldots,2p$, and define real $p \times 2p$-matrices $R_{lk} := r^k_l$ and $\Phi_{lk} := \phi^k_l$ (here and in the following we identify $\mathbb{R}$ with its dual group $\hat\mathbb{R}$ through the isomorphism $r \mapsto \exp(ir \cdot)$). The lattice condition ensures that $\begin{pmatrix} R \\ \Phi \end{pmatrix}$ is invertible. We define unitaries, $u_k := u_{(r^k,\phi^k)}$, $k=1, \ldots, 2p$.

Let $s_l$, $l=1,\ldots, p$, be the multiplication operators on $\mathcal{S}(\mathbb{R}^p)$ given by $(s_l f)(x) := - i x_l f(x)$.

Let $C$ and $D$ be arbitrary matrices in $\mathsf{M}_{2p,p}(\mathbb{R})$ and define
\begin{equation} \label{Equation: Nabla Example}
\nabla_k := \sum_{l=1}^p C_{kl} s_l + \sum_{l=1}^p D_{kl} \frac{\partial}{\partial x_l} , \quad k=1, \ldots, 2p .
\end{equation}
Note that
\begin{equation*}
[\nabla_k, u_{(r,\phi)}] = i \prt{ \sum_{l=1}^p C_{kl} r_l + \sum_{l=1}^p D_{kl} \phi_l } u_{(r,\phi)}
\end{equation*}
and thus
\begin{equation*}
[\nabla_k, u_l] = i (CR + D\Phi)_{kl} u_l .
\end{equation*}
Thus $(\nabla_k)$ is a connection on the left $\mathcal{S}(H,\beta)$-module $\mathcal{S}(\mathbb{R}^p)$ if $i(CR + D\Phi) = \boldsymbol{1}_{2p}$. By the lattice condition on $H$, $\begin{pmatrix} R \\ \Phi \end{pmatrix}$ is invertible, whence we can choose $\begin{pmatrix} C & D \end{pmatrix} := -i \begin{pmatrix} R \\ \Phi \end{pmatrix}^{-1}$, making $(\nabla_k)$ given by Eq. (\ref{Equation: Nabla Example}) a connection. It is easy to see that this connection is flat.

It remains to be seen whether $(\nabla_k)$ is compatible with the inner product, i.e., whether
\begin{equation*}
\lip{D}{\nabla_k f}{g} + \lip{D}{f}{\nabla_k g} = \delta_k (\lip{D}{f}{g}) \quad\quad (k=1, \ldots, 2p) \, .
\end{equation*}
Recall that $\delta_k u_l = i \delta_{kl} u_l$ and that left multiplication with $\begin{pmatrix} R \\ \Phi \end{pmatrix}$ yields an isomorphism $\mathbb{Z}^{2p} \cong H$. Any $\Psi \in \mathcal{S}(H,\beta)$ can be written
\begin{equation*}
\Psi = \sum_{h \in H} \Psi(h) c_h u^{\begin{pmatrix} C & D \end{pmatrix} h}
\end{equation*}
where $u^\alpha := u_1^{\alpha_1} \cdots u_{2p}^{\alpha_{2p}}$ and $c_h$ is a phase factor, $c_h \in \mathbb{T}$. Hence,
\begin{equation*}
\delta_k \Psi  = \sum_{h \in H} \Psi(h) c_h i \prt{ \begin{pmatrix} C & D \end{pmatrix} h }_k u^{\begin{pmatrix} C & D \end{pmatrix} h} .
\end{equation*}
Thus,
\begin{equation*}
(\delta_k \Psi) (r, \phi) = i \Psi(r,\phi) \prt{ \begin{pmatrix} C & D \end{pmatrix} \begin{pmatrix} r \\ \phi \end{pmatrix} }_k .
\end{equation*}

Now note that $\nabla_k$ is skew-adjoint wrt. $\hip{\cdot}{\cdot}_2$. Hence, for $(r,\phi) \in H$,
\begin{equation*}
\begin{split}
\prt{ \lip{D}{\nabla_k f}{g} + \lip{D}{f}{\nabla_k g} }(r,\phi) & = \hip{\nabla_k f}{u_{(r,\phi)} g}_2 + \hip{f}{u_{(r,\phi)}\nabla_k g}_2
\\
& = \hip{f}{[u_{(r,\phi)}, \nabla_k]g}_2
\\
& =  i \prt{ \begin{pmatrix} C & D \end{pmatrix} \begin{pmatrix} r \\ \phi \end{pmatrix} }_k \lip{D}{f}{g}(r,\phi) .
\end{split}
\end{equation*}
Hence, $(\nabla_k)$ is compatible with the inner product.

Lastly, we must show that $C^*(H^\bot,\bar\beta)$ is a non-commutative torus and that for some $\epsilon \in \mathsf{GL}_{2p}(\mathbb{R})$, $(\sum_l \epsilon_{kl} \nabla_l)$ is a compatible connection on $\mathcal{S}(\mathbb{R}^{2p})$ as a right $\mathcal{S}(H^\bot, \bar\beta)$-module.

As $H$ is a lattice, so is $H^\bot$ \cite[Lemma 3.1]{Rieffel-Projective}, hence $H^\bot \cong \mathbb{Z}^{2p}$ and $C^*(H^\bot,\bar\beta)$ is a non-commutative torus. As for the left module we can find a connection compatible with the inner product given by
\begin{equation*}
\tilde{\nabla}_k := \sum_{l=1}^p \tilde{C}_{kl} s_l + \sum_{l=1}^p \tilde{D}_{kl} \frac{\partial}{\partial x_l}
\end{equation*}
where $\begin{pmatrix} \tilde{C} & \tilde{D} \end{pmatrix}$ is invertible. Thus, letting $\epsilon := \begin{pmatrix} \tilde{C} & \tilde{D} \end{pmatrix} \begin{pmatrix} C & D \end{pmatrix}^{-1} \in \mathsf{GL}_{2p}(\mathbb{R})$ we see that $(\sum_l \epsilon_{kl} \nabla_l) = (\tilde{\nabla}_k)$. Hence, $(\nabla_k)$ is an integrable connection on $\mathcal{S}(\mathbb{R}^p)$ as a left $\mathcal{S}(H,\beta)$-module.
\end{example}


\section{Introduction}
The Solar and Heliospheric Observatory (SoHO) has established coronal
mass ejections (CMEs) as an integral part of solar eruptions.
Today we know that there exist slow and fast CMEs associated with eruptive
quiet-Sun filaments and solar flares, respectively,
with a wide velocity distribution
ranging between a few tens of $km/s$ to $\sim 3000\;km/s$
(St. Cyr et al. 2000; Yashiro et
al. 2004; Yurchyshyn et al. 2005). Solar flares are an exclusive
characteristic of solar active regions (ARs), so fast CMEs, typically
with velocities $\ge 750\;km/s$, are active-region CMEs (Sheeley et
al. 1999). Furthermore, the soft X-ray rise phase of the eruptive
flares almost coincides with the main
acceleration phase of the resulting CMEs (Zhang et al. 2001; Zhang and
Dere 2006).

Observations and models have been utilized to
relate specific AR characteristics with CME speeds.
Following a ``big AR syndrome'',
extraordinary ARs can trigger superfast CMEs (Gopalswamy et al. 2005),
but a more quantitative connection is lacking.
Only recently, Qiu and Yurchyshyn (2005)
reported a strong correlation between CME speeds and the
reconnected magnetic flux in two-ribbon flares, Su et al. (2007)
combined magnetic flux and shear to improve correlations with flare
magnitudes and CME speeds, and T\"{o}r\"{o}k and Kliem (2007)
concluded that increased magnetic complexity, reflected on steep
magnetic gradients in the source ARs' corona, tends
to produce faster CMEs.

The above studies suggest that complex (multipolar and/or with
pronounced magnetic polarity inversion lines [PILs]) ARs
tend to produce faster CMEs. Here, we investigate this effect further,
reporting on work done in the framework of the
Living With a Star Coordinated Data Analysis Workshops (LWS/CDAW).
We identified 23 source ARs with eruptions
{\it unambiguously} traced from the solar surface to 1 AU. For these
ARs, we correlate the {\it peak} photospheric complexity prior to an
eruption with various eruption parameters,
including flare magnitudes, plane-of-sky
CME velocities, and (assumed constant) CME acceleration magnitudes.
Our analysis involves full-halo CMEs with source ARs located close to
disk center. Therefore, it should be kept in mind that
significant discrepancies
may exist between the measured (plane-of-sky) and the true
(unprojected) CME velocities (Schwenn et al. 2005), that may have an impact on
correlations. For this and other reasons, as discussed below,
we emphasize the statistical aspect of the reported correlations,
trying to avoid strong quantitative conclusions.
\section{Magnetic Complexity Analysis and Data Selection}
Georgoulis and Rust (2007) defined the {\it effective connected
magnetic field strength}, $B_{eff}$, that characterizes
a given AR at a given time. The larger the value of $B_{eff}$, the
more intense the PIL(s) present in the AR. The intensity of a PIL
increases when massive amounts of bipolar magnetic flux are
tightly concentrated around it. To calculate
$B_{eff}$ for the $p$ positive-polarity and $n$ negative-polarity flux
concentrations that comprise an AR, we calculate two $p \times n$
connectivity matrices: $\Phi _{i,j}$, containing the magnetic fluxes
that connect a positive-polarity
concentration $i$ ($i \in [1,p]$) to a negative-polarity
concentration $j$ ($j \in [1,n]$), and $L _{i,j}$, containing the
respective separation lengths of the connections.
Then, $B_{eff}$ is the sum of all finite
elements $(\Phi _{i,j} / L_{i,j}^2)$.

Georgoulis and Rust (2007) calculated $B_{eff}$ for $\sim 2140$ SoHO/MDI
magnetograms corresponding to 298 ARs. To avoid severe projection
effects, each AR was required to be located
within $41^o$ EW from solar disk
center. Each AR needed 6-7 days to traverse this $82^o$-zone, and this
determined the typical observing period.
To further account for projection effects, we
(i) estimated the normal AR magnetic field by dividing the
line-of-sight field by $cos \theta$, where $\theta$ is the angular
distance of each location from disk center,
(ii) constructed the local, heliographic, plane, and
(iii) interpolated the normal field
on the heliographic plane. We found that $B_{eff}$
efficiently distinguishes eruptive from non-eruptive ARs
{\footnote{In passing, we note that other magnetic complexity
    measures introduced for this purpose include, e.g., the fractal
    dimension of ARs (McAteer et al. 2005), the length of the main
    PIL (Falconer et al. 2006 and references therein), and the magnetic
    flux along the PIL (Schrijver 2007).}}
and that a well-defined probability for major (X- and M-class, although
{\it not} their exact magnitudes) flares depends solely on $B_{eff}$.
An example of how an increase in
$B_{eff}$ translates to stronger PILs that, in turn, give rise to
repeated flaring activity, is shown in Figure \ref{f1}.

Zhang et al. (2007), on the other hand, identified the
solar sources of 88 geoeffective ($D_{st} \le -100\;nT$)
solar eruptions. Each of these eruptions was traced from solar
source to geomagnetic effect. An AR source could be identified
for 54 of the above 88 events, with 23 different source ARs present.

Combining the above two works, we calculated the peak
  pre-eruption (12 hr, at most, before each flare's onset)
  $B_{eff}$-values for these 23 source ARs. The AR and eruption data
  are summarized in Table \ref{tb1}.
\section{Results}
We seek a possible link between the peak AR photospheric complexity,
quantified by the peak $B_{eff}$, and the flare
magnitude or CME kinematics.
The flare magnitudes are related to the peak
preflare $B_{eff}$-values in Figure \ref{f3}a. To calculate the
logarithmic flare magnitude we arbitrarily assign a magnitude of 1 to
a C1.0 flare.
This implies a magnitude of 10, 100, and 1000, for a M1.0, X1.0, and
X10, flares, respectively (dashed lines).
Notice the significant correlation coefficients (cc)
between the flare magnitude and $B_{eff}$, despite the
scatter. The correlation clearly implies an increasing flare
magnitude for an increasing preflare $B_{eff}$.
However, predictions of the flare magnitude using the shown scaling
are not recommended. This is because an AR may not yield its strongest
flare within the typical 6-7 day observing period despite showing a
high (close to peak) $B_{eff}$-value.
The scatter in Figure \ref{f3}a might also highlight
the probabilistic (or even stochastic) nature of flare triggering,
with the flare magnitude depending in part
on the local magnetic conditions and
their synergy. The goodness of fit in Figure \ref{f3}a has
a confidence level of $\sim 98.6$\%. {\it In brief,
increasing $B_{eff}$ in an AR statistically implies
stronger flares triggered in the AR}.

In Figure \ref{f3}b we correlate the plane-of-sky
CME velocities $V_{CME}$ with the
peak preflare $B_{eff}$-values. Although the small dynamical range of
$V_{CME}$ gives rise to lower and more fragile
correlation coefficients,
the scatter around the least-squares best fit appears smaller in
this case. This allows the introduction of a scaling relation between
$V_{CME}$ and peak preflare $B_{eff}$ that reads
\begin{equation}
V_{CME} (km/s) \simeq 87.3 \times B_{eff}^{0.38 \pm 0.12} (G)\;\;.
\label{eq1}
\end{equation}
Equation (\ref{eq1}) is, again, not recommended for accurate
predictions of $V_{CME}$ but it suggests that {\it more complex ARs},
with larger and more intense PILs, statistically give rise to
{\it faster} CMEs. For the smallest considered $B_{eff}$-value
($\sim 500\;G$) we anticipate $V_{CME} \simeq 920\;km/s$,
in good agreement with the slower end of AR CMEs
($\sim 750\;km/s$). For our largest $B_{eff}$ ($\sim 5000\;G$),
we expect $V_{CME} \simeq 2200\;km/s$, that is relatively close to
(but somewhat smaller than) the largest measured CME velocities
($\sim 2800\;km/s$). The difference between the
two extreme velocities is
$\sim 21$\%, with a typical $\sim 10$\%-uncertainty for the measured
plane-of-sky CME speed (J. Zhang, private communication).
The goodness of fit in Figure \ref{f3}b has
a confidence level of $\sim 91$\%.
{\it Therefore, it is clear that increasing $B_{eff}$ in an AR
statistically implies faster CMEs triggered in the AR.}

In Figure \ref{f3}c we correlate the CME acceleration magnitude
$\gamma _{CME}$ with the peak preflare $B_{eff}$.
We have estimated a constant
$\gamma _{CME}$ by the ratio $(V_{CME}/T_{flare})$,
i.e., by following the ``flare-proxy'' approach (Zhang and Dere 2006)
implying that $T_{flare}$ also reflects the main CME acceleration
phase. Although the trend in Figure \ref{f3}c is to attain stronger
$\gamma _{CME}$ with increasing $B_{eff}$, the correlation is not as
appreciable as in Figures \ref{f3}a, \ref{f3}b. The goodness of the fit
is also lower (confidence level $\sim 80$\%). Besides numerical effects
(e.g., discrepancies between plane-of-sky and true CME velocity), the
weaker correlation between $\gamma _{CME}$ and $B_{eff}$ may be due
to (i) the assumption of a constant CME acceleration magnitude,
and/or (ii) the very weak anti-correlation between $T_{flare}$ and $B_{eff}$
(cc $\in (-0.14, -0.18)$ - not shown). The
loose association between $T_{flare}$ and $B_{eff}$ means that
the intensity of PILs in ARs correlates only weakly
with the impulsiveness of the flares triggered in these ARs.
{\it Nevertheless, Figure \ref{f3}c suggests
that increasing $B_{eff}$ in an AR statistically implies more
impulsively accelerated CMEs triggered in the AR.}

We did not find significant correlations (maximum cc $\in (0.2, 0.5)$ and
maximum confidence levels $\sim 70$\%)
when correlating $B_{eff}$ with (i) the ICME transit time
$T_{ICME}$ and (ii) the peak absolute $D_{st}$ index of the resulting
geomagnetic storms. Neither result is surprising:
Schwenn et al. (2005) and Manoharan (2006) report some correlation
between $T_{ICME}$ and $V_{CME}$ but with substantial scatter. Other
heliospheric effects may also impact the velocity profile , and hence the
arrival time, of ICMEs (Chen 1996; Cargill 2004; Tappin 2006).
Regarding $D_{st}$, many factors, besides the
source AR's complexity, affect the ICME geoeffectiveness.
These factors include, but are not
limited to, the CME's source location in the solar disk,
the orientation of the possible post-eruption flux rope,
in-situ heliospheric distortions, turbulence, interactions
with other heliospheric transients, and the ICME velocity profile.
\section{Conclusions and Discussion}
In a previous work (Georgoulis and Rust 2007) we
quantified the photospheric magnetic complexity in solar ARs, where
complexity reflects the strength of PILs in these
ARs. In this work we combined our sample of ARs with the
sample of AR sources that triggered major geomagnetic eruptions
(Zhang et al. 2007). We identified 23 source ARs for which
we have preflare AR magnetograms, and we correlated the
peak pre-eruption AR complexity with various eruption parameters.

Significant correlations were uncovered when plotting the peak value of the AR
complexity index, $B_{eff}$, vs. (from stronger to weaker correlation)
(i) the flare magnitude, (ii) the CME velocity, and (iii) an assumed constant
CME acceleration magnitude. Though not ideal for predictive purposes, these
correlations clearly imply that more complex ARs, with intense PILs,
can produce statistically stronger flares and
faster, more impulsively accelerated, CMEs. Our finding
goes a step further than the usual ``big AR syndrome'': big
(flux-massive) ARs are not necessarily ARs with intense PILs,
and hence with large $B_{eff}$-values, although
the opposite is almost always true.

At this point, we can only speculate on a possible
interpretation of our results,
making it clear that only further work can validate or
rule out our scenario. Given that the main
CME acceleration phase nearly coincides with the flare impulsive
phase, the initial ascending structure evolving into a CME
(hereafter CME precursor)
should start expanding {\it before} the flare. As it expands, the
CME precursor interacts with the surrounding PIL-supported magnetic
structure causing magnetic
reconnection. Reconnection in stronger magnetic fields organized along
more intense PILs statistically leads to stronger flares (Figure
\ref{f3}a). Stronger flares imply larger amounts of
released magnetic energy that, in turn, probably destabilize larger
parts of the PIL-sustained structure and/or accelerate the unstable
magnetic fields to higher speeds, thus giving rise to statistically
faster (Figure \ref{f3}b) and more impulsive (Figure \ref{f3}c) CMEs.
Again, the above indicate only statistical trends - the
local field conditions at the flare location along the PIL as
well as the overlaying solar magnetic fields (steep magnetic
gradients [T\"{o}r\"{o}k and Kliem 2007],
coronal null points [Antiochos et al. 1999], etc.)
affect both flare magnitudes and CME velocity profiles.
Sometimes, fast CMEs are associated with relatively weak
flares or vice-versa (Vr\v{s}nak et al. 2005).

The above idea might also help understand the difference
between (i) confined and eruptive flares and (ii) fast active-region
CMEs and slow quiet-Sun CMEs. In case the PIL-supported structure,
despite the reconnection,
survives the perturbation applied by the CME precursor,
a confined flare occurs{\footnote{Nindos and
Andrews (2004) suggest that the preflare magnetic helicity of the AR
may determine whether a confined flare may occur.}}. If there is no
intense PIL, the CME precursor
might easily destabilize the surrounding magnetic structure
(especially in the presence of ``open'' overlaying fields, such as
a streamer in the high corona) but, since only minor reconnection is
expected, no major flare and a rather slow CME will occur.
This might be the case for some quiet-Sun CMEs.

While our statistical results appear solid, only
further work can validate the above physical scenario.
It would be essential to (i) uncover the physical mechanism(s)
responsible for the possible
CME precursor (e.g., small-scale helical kink
instability, magnetic flux cancellation on the PIL, etc.),
and (ii) find the relation between
the plane-of-sky velocity $V_{CME}$ and the true CME velocity and
determine whether this improves the correlation with $B_{eff}$ or the
source AR's complexity in general. The Solar Terrestrial Relations
Observatory (STEREO) can be instrumental in
revealing the true CME velocities to be used for
definitive conclusions in our quest to understand the erupting Sun.
\begin{acknowledgments}
I am grateful to J. Zhang for many clarifying discussions and for his
help in handling and utilizing the data of Zhang et al. (2007). I also
thank A. Vourlidas for enlightening discussions in the physics of
CMEs, the organizers of the LWS/CDAW meetings for their sustained and
successful efforts, and the two anonymous referees, whose
thoughtful comments contributed significantly to this paper.
This work has received partial support from NASA grant NNG05-GM47G.
\end{acknowledgments}


\section{Introduction}
\label{intr}

$\alpha$ Cir (HD 128898) is the prototype of a class of Ap stars that oscillate in high frequencies
(roAp, rapidly oscillating Ap, \cite{kurtz82}). Besides knowing its effective temperature and parallaxe, to accurately place $\alpha$ Cir in the HR diagram it is necessary to know also its bolometric correction ($B.C.$). To determine the latter, the method proposed by \cite{north81} was applied.

\section{Bolometric Correction determination and discussion}

The $B.C.$ is given by the equation,

\begin{equation}
B.C.=-2.5\log\int\limits_0^\infty\,F(\lambda)\mathrm{d}\lambda-m_V-11.492,
\label{eq:BC}
\end{equation}
where $m_V$ is the apparent visual magnitude and $F(\lambda)$ is the flux at a given wavelength.
Two values for the integrated flux of $\alpha$ Cir were obtained. The first was determined by combining the observed ultraviolet flux of $\alpha$ Cir retrieved from IUE Newly Extracted Spectra (INES) data archive, with the theoretical flux obtained from the Kurucz model (with IDL routine KURGET1) that best fitted the optical photometry (\cite{rufener89}) for the star. The second was obtained using the same method, but substituting the Kurucz synthetic spectrum by the mean of two low resolution spectra of $\alpha$ Cir calibrated in flux (\cite{aleks96}; \cite{burnashev85}). The two values obtained for the $B.C.$ were $0.15 \pm 0.02$ and $0.23 \pm 0.02$, respectively.

To place $\alpha$ Cir in the H.R. diagram, the results obtained for the bolometric fluxes were combined with the Hipparcos parallaxe. For the effective temperature we adopted the value from \cite{Kupka96}, $T_{\rm eff} = 7900 \pm 200$ K.
\begin{figure}
\centerline{\includegraphics[width=7cm]{hr_vienna2007.ps}}
\caption{The position of $\alpha$ Cir in the HR diagram with four evolution tracks for models with masses of 1.70, 1.73, 1.75 and 1.80 M$_\odot$ for comparison. The two error boxes correspond to two values of the luminosity (and associated error) derived from two different bolometric fluxes (see text for details).}
\label{fig_HR}
\end{figure}

Both values derived for the $B.C.$ have uncertainties that go beyond their formal errors. In the first case, uncertainties are likely associated with the use of Kurucz models which are not apropriate for peculiar stars. In the second case, additional uncertainties are also expected because the errors associated with the calibration in flux of the low resolution spectra were not provided in the catalogues. Hence, we took a conservative approach and considered both values when placing the star in the HR diagram.

\acknowledgements
MSC is supported by the EC's FP6, FCT and FEDER (POCI2010) through the HELAS international collaboration and through the projects POCI/CTE-AST/57610/2004 and PTDC/CTE-AST/66181/2006 FCT-Portugal.


\section{Introduction}

The flavor problem can be divided in several subproblems~\cite{volkas}: Why do weak
isospin partners have different masses? Why are quark and lepton masses split? Why is
there a mass hierarchy between generations? Presently we know that neutrino oscillation
data are well described by massive neutrinos~\cite{concha} and the flavor problem becomes
more interesting:
why is there a mixing angle hierarchy in the quark sector but not in the lepton sector?
Usually mass matrices have the form $M_{\alpha\beta}=\sum_i(\Gamma_i)_{\alpha\beta} \langle
\Phi^0_i\rangle$, where the $\Gamma_i$'s are, for Dirac fermions, arbitrary complex
dimensionless $3\times 3$ matrices, and $\langle\Phi^0_i\rangle$ denotes one or a set of vacuum
expectation values (VEVs) of the neutral scalar fields in the model. For Majorana fermions,
the $\Gamma_i$'s are complex symmetric matrices. The mixing matrix and the mass pattern of each
charged sector depend on the structure of the respective $\Gamma_i$'s. It is well known that
explicit and predictive forms of these matrices can be obtained by imposing flavor
symmetries. For instance, it has been considered global unitary (horizontal) symmetries like
$U(1)$'s, $SU(2)_H$~\cite{zeewilczek} or $SU(3)_H$~\cite{koide}. More recently symmetries
like $S_n$, $Z_n$~\cite{zhang}; $A_4$~\cite{a4}, $^{(d)}T$~\cite{phf1,tprime}, the double
covering of $A_4$; $Q_6$, the double dihedral group~\cite{babu}, and other discrete
symmetries~\cite{phf2} have been also considered.

Here we will turn the problem upside down. Mass matrices will be of the form $M_{\alpha\beta}=
f(\overline{\langle\Phi^0\rangle})_{\alpha\beta}$ in which $f$ denotes a few dimensionless
($\mathcal{O}(1)$) parameters (one at leading order), and $\overline{\langle\Phi^0\rangle}$ is
a matrix built with the VEVs of several scalar fields. At first sight, there is no gain in
predictive power, we are just changing a dimensionless general matrix $\Gamma_{\alpha\beta}$ by
another one with mass dimension $(\overline{\langle\Phi^0\rangle})_{\alpha\beta}$. However, it
seems easier, at least in principle, to explain patterns of dynamical variables like VEVs, than
dimensionless numbers. The value of the former can be explained by the dynamics (for instance by
studying the scalar potential) and extra flavor symmetries that we can impose to the model.

In this Letter we will consider an extension  of the standard model in which the charged
fermion masses, that arise from renormalizable interactions, are diagonal. However,
dimension 5 operators~\cite{dim5} give important non-diagonal contributions to quarks but not
to the charged leptons. On the other hand, the neutrinos mass matrix is generated mainly from
dimension 5 and 6 operators, with the former having zero trace as in the Zee's mode ~\cite{zee1}.
The dimension 6 operators induce non-negligible diagonal terms and a general $3 \times
3$ symmetric mass matrix for those neutrinos is obtained. The non-diagonal elements are
important in the quark sector because they involve also the VEV of the singlet which it is not
constrained by the electroweak scale, and also in the neutrino sector because theses entries
involve ratios of doublets' VEVs that produce an enhancement. This enhancement does not
occurs in the charged lepton sector because in this case the non-diagonal entries depend on the
same doublet's VEVs that appear in the diagonal ones but are suppressed by $\Lambda^{-1}$.
No right-handed neutrinos are introduced.
In summary, we show that all the questions mentioned above have a natural answer: fermion masses
and mixing arise from different scalar sectors and from different effective interactions. The
predictive power is a consequence of the discrete symmetries imposed to the model.

Simple non-Abelian discrete groups have been considered for the first time as candidates for
flavor symmetries by Frampton and Kephart~\cite{phf1}. Later, $A_4$ symmetry has been used to
obtain realistic mixing matrices in the lepton sector~\cite{ma1}, and then also in the quark
sectors~\cite{ma2}. More recently~\cite{a4tribi}, by using these non-Abelian discrete symmetries
it was possible to obtain a tribimaximal neutrino mixing matrix~\cite{tribi}.  The basic idea is
to consider $SU(2)_{L}$ fermion doublets and singlets in the triplet representation of the $A_4$
symmetry and Higgs multiplets transforming as triplet or singlet of $A_4$.

Let us briefly review some properties of the $A_4$ representations.
Denoting the vector basis ($\textbf{3}$) as $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$ we have
$\textbf{3}\otimes \textbf{3}=\textbf{3}_A\oplus \textbf{3}_B\oplus \textbf{1}
\oplus \textbf{1}^\prime\oplus \textbf{1}^{\prime\prime}$. Explicitly~\cite{he},
\begin{eqnarray}
&&[\textbf{3}\otimes \textbf{3}]_A\equiv\textbf{3}_A =(a_2b_3, a_3b_1,a_1b_2),\quad
[\textbf{3}\otimes \textbf{3}]_B\equiv \textbf{3}_B =(a_3b_2, a_1b_3,a_2b_1), \nonumber \\
&&[\textbf{3}\otimes \textbf{3}]_1=a_1b_1+a_2b_2+a_3b_3,\quad\quad\;\;
[\textbf{3}\otimes \textbf{3}]_{1^\prime}=a_1b_1+\omega a_2b_2+\omega^2a_3b_3,\nonumber \\ &&
[\textbf{3}\otimes \textbf{3}]_{1^{\prime\prime}}=a_1b_1+\omega^2a_2b_2+\omega a_3b_3,
\label{produtos}
\end{eqnarray}
and $\textbf{1}\otimes \textbf{1}=\textbf{1}$, $\textbf{1}^\prime\otimes
\textbf{1}^{\prime\prime}=\textbf{1}$, $\textbf{1}^\prime\otimes
\textbf{1}^\prime=\textbf{1}^{\prime\prime}$, $\textbf{1}^{\prime\prime}\otimes
\textbf{1}^{\prime\prime}=\textbf{1}^\prime$.

We consider a model with $G_{SM}\otimes A_4 \otimes Z_3 \otimes Z^{\prime}_3 \otimes
Z^{\prime\prime}_3$ symmetry, where $G_{SM}$ is the gauge symmetry $SU(3)_C\otimes
SU(2)_L\otimes U(1)_Y$. The matter fields transform as usual under $G_{SM}$, left-handed lepton
and quark doublets $L_a,\,a=e, \mu,\tau$; $Q_i,\,i=1,2,3$, and their respective right-handed
singlets $l_{aR}$, $u_{iR}$ and $d_{iR}$. In the scalar sector, we introduce scalar doublets with
$Y=+1$:  $H_i,\hat{H}_i,H^{\prime}_{i}, H^{\prime\prime}_i$, $\Phi_{i}$, $\Phi^{\prime}_{i}$,
$\Phi^{\prime\prime}_i$ and $\chi$; three scalar triplets with $Y=+2$, denoted by
$\mathcal{T}_i$~\cite{chengli}, and finally one complex (with $Y=0$) scalar singlet $\zeta$.
Fermion weak eigenstates fields transform under $(A_4,Z_3,Z^{\prime}_3,Z^{\prime\prime}_3)$ as
follows:
\begin{eqnarray}
&&L\equiv (L_e,L_\mu,L_\tau) \sim (\textbf{3}, \omega, \omega^2, 1);\quad
\;\;l_R\equiv(\mu_R,\tau_R,e_R) \sim (\textbf{3},1, 1, \omega),\nonumber \\ &&
Q_{L} \equiv (Q_{1L},Q_{2L},Q_{3L}) \sim (\textbf{3}, \omega^2, 1, 1);\;\;\quad
 u_R\equiv (c_R,t_R,u_R) \sim (\textbf{3}, 1, 1, \omega); \nonumber \\ &&
d_R\equiv (s_R,b_R,d_R) \sim (\textbf{3},1, \omega, \omega),
\label{fermions}
\end{eqnarray}
in which $\omega=e^{2\pi i/3}$.  Notice that all fermion fields are still symmetry eigenstates
but we do not use an special notation. On the other hand, scalar fields transform under
$(A_4,Z_3,Z^\prime_3,Z^{\prime\prime}_3)$ as:
\begin{eqnarray}
&&H\equiv (H_1,H_2,H_3)\sim (\textbf{3},\omega, \omega^2, 1),\;\; \hat{H}\equiv
(\hat{H}_1,\hat{H}_2,\hat{H}_3)\sim (\textbf{3},\omega, \omega^2,  \omega^2),
\nonumber \\ &&
H^\prime\equiv (H^\prime_1,H^\prime_2,H^\prime_3)\sim(\textbf{3}, \omega, 1,  \omega), \;\;
H^{\prime\prime}\equiv (H^{\prime\prime}_1,H^{\prime\prime}_2,
H^{\prime\prime}_3)\sim(\textbf{3}, \omega^{2}, \omega^{2}, \omega^2),\;
\nonumber \\ &&
\Phi\equiv (\Phi_1,\Phi_2,\Phi_3) \sim (\textbf{3}, 1, 1, 1),\;
\Phi^{\prime} \equiv (\Phi^\prime_1,\Phi^\prime_2,\Phi^\prime_3)\sim (\textbf{3},\omega, \omega,
\omega), \nonumber \\ && \;
\Phi^{\prime \prime}\equiv (\Phi^{\prime\prime}_1,\Phi^{\prime\prime}_2,
\Phi^{\prime\prime}_3) \sim (\textbf{3}, \omega^{2}, 1,  \omega^2),\;
 \chi\sim (\textbf{1}, \omega^{2}, \omega, 1),\nonumber \\ &&
 \mathcal{T}\equiv (\mathcal{T}_1,\mathcal{T}_2,\mathcal{T}_3)\sim (\textbf{3},
 \omega^{2}, \omega, 1), \;
  \zeta \sim (\textbf{1}, 1, \omega,1).
 \label{scalars}
\end{eqnarray}

With the fields in (\ref{fermions}) and (\ref{scalars}) we have the leading contributions to the
lepton and quark Yukawa interactions:
\begin{eqnarray}
\mathcal{L}& = &\frac{1}{\Lambda}\left( f_\nu\,[(\overline{L^c}\epsilon H)]_A
[( L\epsilon \Phi)]_B
+ \frac{f^\prime_\nu}{\Lambda}
[\overline{L^c} \varepsilon\vec{\sigma}
\cdot \vec{\mathcal{T}}]_B[L\Phi^\dagger ]_B\chi+\dots \right)+ H.c.
\nonumber \\ &+&
\left( g_l [\overline{L} \hat{H}]_A + \frac{g^\prime_l}{\Lambda^2} [\overline{L} \hat{H}]_A
\vert\zeta\vert^2 + \cdots \right)
l_R+H.c.
\nonumber \\  &+&
\left( h_u [\overline{Q_L} \widetilde{H}^\prime]_A  + \frac{h^\prime_u}{\Lambda}
[\overline{Q_L}
\widetilde{\Phi}^{\prime} ]_B \zeta +\cdots\right)u_R +H.c.
\nonumber \\  &+& \left(h_d [\overline{Q_L} H^{\prime\prime}]_A    +
\frac{h^\prime_d}{\Lambda}
 [\overline{Q_L} \Phi^{\prime\prime}]_B \zeta^* +\cdots\right)d_R+H.c.,
\label{lagrangiana}
\end{eqnarray}
where $\Lambda$ is an energy scale characterizing an unknown physics; and
$\varepsilon=i\sigma_2$ ($\sigma_2$ being the Pauli matrix), $[~]_{A,B}$ means the appropriate
product defined in Eq.~(\ref{produtos}) and $\cdots$ denotes higher-dimensional operators,
we assume that $\vert V\vert\ll \Lambda$, where $V$ denotes the VEV of  any of the scalar of
the model.

The contributions to the quark mass matrices obtained from renormalizable interactions
are diagonal, but the non-renormalizable dimension 5 interactions
contribute with important and not negligible non-diagonal elements, on
the other hand, for the charged leptons this corrections come from
dimension 6 operators are negligible.
We obtain diagonal entries to the mass matrices as a consequence of the definition of the basis
in which the product $[[~]_A\cdot]_1$ gives the diagonal contribution while  the product
$[[~]_B\cdot]_1$ gives the non-diagonal contributions to the mass matrix. For instance, for the
$u$-quarks, the leading diagonal mass matrix arises through the interaction $[\bar{Q}H^\prime]_A
=(\bar{Q}_2 H^\prime_3,\bar{Q}_3 H^\prime_1,\bar{Q}_1H^\prime_2)$ and then $[[\bar{Q}H^\prime]_A
(c_R,t_R,u_R)]_1=\bar{Q}_2 H^\prime_3c_R+\bar{Q}_3 H^\prime_1t_R+\bar{Q}_1H^\prime_2u_R$.
Similarly, using the $[~]_B$ product, we obtain that $[[\bar{Q}\Phi^\prime]_B(c_R,t_R,u_R)]_1=
\bar{Q}_3 \Phi^\prime_2c_R+\bar{Q}_1\Phi^\prime_3t_R+\bar{Q}_2 \Phi^\prime_1 u_R$, which gives
the non-diagonal contributions to the charged lepton mass matrix.

Hence, from the Yukawa interactions in Eq.~(\ref{lagrangiana}), the mass matrices obtained are
\begin{equation}
M^l \approx g_l\left(
\begin{array}{ccc}
\hat{v}_2 & 0 & 0\\
0&\hat{v}_3 & 0 \\
0& 0& \hat{v}_1\end{array}
\right) + H.c.,
\label{leptons}
\end{equation}
for the charged leptons. We have denoted $\langle\hat{H}_i\rangle=\hat{v}_i$. Notice that the
renormalizable interactions are the dominant one, i.e., $g_l\hat{v_1}\simeq m_\tau$,
$g_l\hat{v}_3~\simeq~m_\mu$ and $g_l\hat{v}_2\simeq~m_e$. In the quark sector we have
\begin{eqnarray}
M^u \approx h_u\,\left(
\begin{array}{ccc}
v^\prime_2 & 0& a_{u} v^{\prime}_{\phi_3} \\
a_{u} v^{\prime}_{\phi_1} & v^\prime_3 & 0\\
0& a_{u} v^{\prime}_{\phi_2}& v^\prime_1\end{array}\right)
+H.c.,
\label{us}
\end{eqnarray}
and
\begin{equation}
M^d \approx h_d\,\left(\begin{array}{ccc} v^{\prime\prime}_2 &
0& a_{d} v^{\prime \prime}_{\phi_3} \\ a_{d} v^{\prime \prime}_{\phi_1} &
v^{\prime\prime}_3 & 0 \\ 0& a_{d} v^{\prime \prime}_{\phi_2}&
v^{\prime\prime}_1\end{array}\right)
+ H.c.,
\label{ds}
\end{equation}
in which $a_{u} = \frac{h^\prime_u}{h_{u}} \frac{v_{\zeta}}{\Lambda}$ and $a_{d} =
\frac{h^\prime_d}{h_{d}} \frac{v_{\zeta}}{\Lambda}$,  for the $2/3$ and $-1/3$ charged
quarks, respectively. We have denoted $\langle H^0_i\rangle=v_i$, $\langle H^{\prime0}_i
\rangle=v^\prime_i$, $\langle H^{\prime\prime0}_i\rangle=v^{\prime\prime}_i$,
$\langle\Phi_{i}\rangle=v_{\phi_{i}}$, $\langle\Phi^\prime_i\rangle=v^\prime_{\phi_{i}}$,
$\langle\Phi^{\prime\prime}_{i}\rangle=v^{\prime\prime}_{\phi_{i}}$,
and $\langle\zeta \rangle=v_\zeta$. Notice, as well, that each charged sector has its
private VEVs, thus all these matrices are independent from each other. In the case of
neutrinos we have the mass matrix
\begin{equation}
M^\nu \approx \left[\left(
            \begin{array}{ccc}
             0&
             \frac{v_2}{v_1} & \frac{v_{\phi_3} }{ v_{\phi_1} } \\
              \frac{v_2}{v_1}  &   0 &
              \frac{v_3}{v_1}
              \frac{v_{\phi_2} }{ v_{\phi_1} }  \\
               \frac{v_{\phi_3} }{ v_{\phi_1} } & \frac{v_3}{v_1}
               \frac{v_{\phi_2} }{ v_{\phi_1} }
               & 0 \\
            \end{array}
          \right)+\left(\begin{array}{ccc}
          \delta_1 &
             0 & 0 \\
             0  &   \delta_2 & 0  \\
             0 & 0 & \delta_3 \\
          \end{array}\right)\right]\frac{f_\nu v_{\phi_1} }{ \Lambda } \,v_1,
          \label{neutrinos}
\end{equation}
where the $\delta_i$ are given by
$\delta_1=\frac{f^\prime_\nu}{f_\nu}\,(v_{_{T3}}v_{\phi_3}v_\chi/\Lambda v_1v_{\phi_1})$,
$\delta_2=\frac{f^\prime_\nu}{f_\nu}\,(v_{_{T1}} v_\chi/ \Lambda v_1)$, and $\delta_3=
\frac{f^\prime_\nu}{f_\nu}\,(v_{_{T2}} v_{\phi_2} v_\chi/ \Lambda v_1v_{\phi_1})$,
where $\langle \chi\rangle=v_\chi$, and $\langle\Delta^0_i\rangle=v_{_{Ti}}$.

It is interesting that the discrete symmetries of the model implies that the
charged lepton mass matrix, Eq.~(\ref{leptons}), is practically diagonal, and that the
quark mass matrices have three zeros, see Eqs.~(\ref{us}) and (\ref{ds}).
To the best of our knowledge, this mass matrices have not been considered in
literature~\cite{xing}. Notice that in the case of the neutrino we have already taken
into account the hermitian conjugate.

Since the only constraint on the VEVs of the doublets is that $\sum v^2<
(\textrm{174}\textrm{GeV})^2$ (there are triplets as well, see below),
a possibility is that only $v^{\prime\,2}_1\lesssim (174\, \textrm{GeV})^2$,
while all the other VEVs are $\ll 174$ GeV. In this case $v^\prime_1$ give the main
contribution to the mass of the $t$ quark.  On the other hand, the sum of the VEVs of the triplets has
to have a upper limit of the order of a few GeV to not spoil the observed value of the $\rho$
parameter~\cite{triplet}. Hence, $\vert V\vert/\Lambda$ are well-defined
expansion parameters. The present model is natural in the Yukawa interactions i.e.,
unlike the standard model, there is no a hierarchy among the Yukawa couplings. In practice, in the
analysis below we will consider that all Yukawa coupling are in the interval $0.1-3$.
The VEV of the singlet $v_\zeta$ is only constrained by the condition
$v_\zeta/\Lambda< 1$. Below we will use always, just as an illustration: $\Lambda=1$ TeV and
$v_\zeta=140$ GeV. However, recall that the singlet $\zeta$ may be related to a new energy scale different
from the electroweak scale.

As a first illustration, let us consider the matrix in Eq.~(\ref{neutrinos}).
We will assume, without losing generality, that this matrix
is a traceless matrix i.e., that the condition $\sum_i\delta_i=0$  is valid~\cite{zee1}.
This is obtained by imposing (using $\tilde{v}_x~=~v_xe^{i\theta_x}$, where $v_x$
and $\theta_x$ are real):
\begin{eqnarray}
&&\theta_{_{T3}}+\theta_{\phi_3}-\theta_1-\theta_{\phi_1}=\pi,\;\; \theta_{_{T1}}-
\theta_1=0;
\;\; \theta_{_{T2}}+\theta_{\phi_1}
-\theta_1-\theta_{\phi_1}=0,\nonumber \\ &&
(1/2)\left\vert\frac{f^\prime_\nu}{f_\nu}\,\frac{ v_{_{T_3}}
v_\chi v_{\phi_3} } {\Lambda v_1 v_{\phi_1}}\right\vert=
\left\vert\frac{f^\prime_\nu}{f_\nu}\,\frac{v_{_{T_1}} v_\chi }
{\Lambda v_1}\right\vert =
\left\vert\frac{f^\prime_\nu}{f_\nu}\,\frac{v_{_{T_2}} v_\chi  v_{\phi_2} }
{\Lambda v_1 v_{\phi_1} }\right\vert \equiv \delta.
\label{ufa}
\end{eqnarray}
Hence, the neutrino mass matrix is of the form~\cite{brama1,brama2}
\begin{equation}
M^\nu\approx m_0\left( \begin{array}{ccc}
-2\delta & \sin\theta & \cos\theta \\
\sin\theta & \delta & \epsilon \\
\cos\theta & \epsilon & \delta
\end{array}\right),
\label{bra1}
\end{equation}
where $m_0=(f_\nu/\Lambda)\vert v_{\phi_1} v_1\vert,\, \vert v_2/v_1 \vert=
\sin\theta, \, \vert v_{\phi_3}/v_{\phi_1} \vert =\cos\theta, \, \epsilon= \vert
(v_3v_{\phi_2}/v_1v_{\phi_1}) \vert$.  From Table~I of Ref.~\cite{brama2} we can see that
neutrinos masses and mixing angles are in agreement with all experimental data if, for
instance, in  Eq.~(\ref{bra1}), $m^2_0=\Delta m^2_{13}\equiv \Delta m^2_{atm}=(0.039\,
\textrm{eV})^2$, $\epsilon=0.21$, $\delta=0.236$, and $\theta=0.84$. In our model, these
values for the above parameters are obtained by assuming: $\Lambda=1$ TeV, $v_1\sim1$ MeV,
$v_{\phi_1}=v_{\phi_2}$, $2v_{T_1}=2v_{T_2}=v_{T_3}$. with $v_{T_1}=1.25$ GeV; then
we obtain: $f_\nu v_{\phi_1}\sim40$ keV, $v_{\phi_3}=40/f_\nu$ keV, $v_2=0.1$ keV, $v_3=0.2$
MeV, $(f_\nu/f^\prime_\nu)v_\chi=0.2$ GeV. We also obtain with these VEVs, $\Delta
m^2_{12}=9\times10^{-5}\,\textrm{eV}^2$,
$\sin^22\theta_\odot=0.82$, $\sin^2 2\theta_{atm}=0.99$, and $\sin\theta_{13}=0.019$ a
value that is in agreement with the CHOOZ limit $V_{e3}=\sin\theta_{13}<0.07$~\cite{chooz}.
We see that a realistic leptonic mixing matrix arises since the mass matrix of the charged leptons
is almost diagonal. If $f_\nu, f^\prime_\nu\sim\mathcal{O}(1)$, it implies that some VEVs are of
the order of MeV or even less depending on the values of $v_\zeta$ and $\Lambda$.
This does not necessarily implies the existence of light scalars since
for having them in a model, there must be some symmetries that allow them to appear in the mass
spectra: The symmetries in the present model do not play that role and $\mu^2_{ij}$ terms
survive in the scalar mass matrices (see Ref.~\cite{ma3}).

On the other hand, with $M^{u,d}$ given by Eq.~(\ref{us}) and (\ref{ds}) we see that $M^{u(d)}M^{u(d)\dagger}
\not=M^{u(d)\dagger}M^{u(d)}$. Hence we need four matrices, $V^{U,D}_{L,R}$, for diagonalizing
these mass matrices by biunitary transformations and the constraint
$V_{CKM}=V^U_LV^{D\dagger}_L$. Besides, the matrices $V^{U,D}_{L,R}$ must satisfy the constraints
$V^D_LM^dM^{d\dagger}V^{D\dagger}_L=diag(m^2_d,m^2_s,m^2_b)=(\hat{M}^d)^2$ and
$V^D_RM^{d\dagger}M^{d}V^{D\dagger}_R=(\hat{M}^d)^2$; similarly for the $V^U_{L,R}$ matrices.
The quark masses and the $V_{CKM}$ matrix arise, for instance, using for the $d$-type quarks
$a_d=0.14$, $h_d= h^\prime_d=0.1$, and (all VEVs and masses below are in
GeV) $v^{\prime\prime}_1=42$, $v^{\prime\prime}_2=1.4$, $v^{\prime\prime}_3=0.062$,
$v^{\prime\prime}_{\phi_1}=2.552$, $v^{\prime\prime}_{\phi_2}=0.0074$, and $v^{\prime\prime}_{\phi_3}=
11.83$ we obtain the masses: $m_d=0.006,\,m_s=0.1,m_b=4.2$. For the $u$-type quarks we use, $a_u=a_d$, but now
$h_u=h^\prime_u=3$, and $v^\prime_1=53.8,\,v^\prime_2=0.4,\,v^\prime_3=0.0001$ and
$v^\prime_{\phi_1}=0.1,\,v^\prime_{\phi_2}=124,\,v^\prime_{\phi_3}=0.082$, which give $m_u=0.0025$,
$m_c=1.27$, and $m_t=171.2$. With these VEVs and parameters we obtain $V^{D,U}_L$ matrices given,
according with the definition above, the mixing matrix
\begin{equation}
\vert V_{CKM}\vert =\left(\begin{array}{ccc}
0.97747\, & 0.21105& 0.00148\\
0.21083&\, 0.97673 &0.03923\\
0.00972\, & 0.03803 & 0.99923\\
\end{array}\right).
\label{ckm}
\end{equation}
Comparing the elements of the matrix above with experimental data in Section~11.2 of
PDG~\cite{pdg}, we see that all the matrix elements, but $us$ and $ub$ entries, are within the
level of confidence (1-2)$\sigma$. However, this is enough to show that a realistic mixing matrix
may arise from the quark mass matrices given in Eqs.~(\ref{us}) and (\ref{ds}). A more rigorous and
detailed analysis is being done and will be presented later.

The suppression of the FCNCs coupled to neutral Higgs bosons can be obtained not naturally
in the sense of Ref.~\cite{gw}, but at least as a reasonable fine tuning among the mixing matrix
in the neutral scalar sector, $U_{ij}$, or by the mass of heavy Higgs scalars. For instance, consider
the neutral kaons mass difference, $\Delta M_K \sim \zeta^i_{sd}f^2_Km_K(\sum_j\vert U_{ij}
\vert^2/m^2_j)$ where $\zeta^i_{sd}=h^2_d\,\vert\sum_a(V^D_L)^*_{as}(V^D_R)_{ad}\vert^2\,,$
with  symmetry eigenstates, say $H^{\prime\prime}_i$ ($i$ fixed), and physical scalars
$h_j$ with mass $m_j$. In order to be consistent with data $\zeta^i_{sd}$ must be smaller
than the contribution of the SM: i.e., smaller that $G^2_Fm^2_c (V^*_{cd}V_{cs})^2/16\pi^2=0.5\times
10^{-8}\textrm{GeV}^{-2}$ (for simplicity we have taken into account only
the $c$ quark contribution and neglect the QCD and $m^2_c/M^2_W$ corrections). Notice that the sum over
the contributions of each $i$ is made coherently. The expression above is valid for any scalar heavier
than $m_K$. For instance, $h^2_d\vert\sum_a(V^D_L)^*_{as}(V^D_R)_{ad}\vert^2\sim10^{-4}-10^{-6}$ and
$\vert U_{ij}\vert^2/m^2_j\sim(10^{-2}-10^{-4})\textrm{GeV}^{-2}$ we have the appropriate FCNC suppression.
These constraints are not a large fine tunings since it can be satisfied with a large set of unitary
matrices and/or the neutral scalar mass spectrum and mixing.
Recall that there is no more freedom to redefine the quark phases since they all have already
been used to put the CKM matrix in the usual form, hence $V^D_R$ has five physical phases.
The case of FCNCs in the $u$-quark sector involve the matrices $V^U_L$ and $V^U_R$.

On the other hand, it would be interesting if we can show that appropriate hierarchies among
the VEVs arises from an analysis of the scalar potential~\cite{zee0}. We have verified that,
under reasonable assumptions on the parameters in the scalar potential, this is indeed the case.
For the sake of simplicity we will consider the scalar potential involving only the scalar
doublets that couple to quarks: the $A_4$ triplets $H^\prime,\Phi^\prime$ for $u$-like quarks, and
$H^{\prime\prime},\Phi^{\prime\prime}$ for $d$-like quarks. Denoting $X_u=H^{\prime\dagger}H^\prime$
and $Y_u~=~\Phi^{\prime\dagger}\Phi^\prime$, $X_d=H^{\prime\prime\dagger}H^{\prime\prime}$ and
$Y_d~=~\Phi^{\prime\prime\dagger}\Phi^{\prime\prime}$ we write the most general scalar
potential invariant under the SM and $A_4\otimes Z_3\otimes Z^{\prime}_3 \otimes Z^{\prime\prime}_3$
symmetries as:
\begin{eqnarray}
\mathcal{V}&=&\mu^2_p[X_p]_{1} \!+\! \mu^{\prime\,2}_p[Y_p]_{1} \!+\!
\mu^{2}_{\zeta}\vert\zeta\vert^2 \!+\! \lambda_{\zeta}\vert\zeta\vert^4 \!+\!
(\mu^2_{H^\prime\Phi^{\prime\prime}}[H^\prime\epsilon\Phi^{\prime\prime}]_{1}
\!+\! \mu^{2}_{H^{\prime \prime}\Phi^{\prime}}
[H^{\prime \prime} \varepsilon \Phi^{\prime}]_{1}
\nonumber \\ &+&
 f [\Phi^{\prime \dag} H^{\prime}]_{1}\zeta \!+\!
 f^\prime [\Phi^{\prime \prime \dag} H^{\prime \prime}]_{1}\zeta^{*}
\!+\! f_{H}^{\prime}[H^{\prime \prime} \varepsilon H^{\prime}]_{1} \zeta \!+\!
 f_{\Phi}^{\prime}[\Phi^{\prime \prime} \epsilon \Phi^{\prime}]_{1}\zeta^{*} \nonumber \\ &+&
\lambda^{pp^\prime}_{mn}[X_p]_m[X_{p^\prime}]_n+
K^{pp^\prime}_{kl}[Y_p]_k[Y_{p^\prime}]_l
\!+\! h^{pp^\prime}_{\alpha\beta}[X_p]_\alpha[Y_{p^\prime}]_\beta
+g_{st}[H^\prime\epsilon\Phi^{\prime\prime}]_s[H^\prime\epsilon
\Phi^{\prime\prime}]_t \nonumber \\ &+& g^{\prime}_{st}[H^{\prime \prime} \varepsilon
\Phi^{ \prime}]_{s}[H^{\prime \prime} \varepsilon \Phi^{ \prime}]_{t}+
g_{st}^{\prime \prime}([H^\prime\epsilon\Phi^{\prime\prime}]_{s}[H^{\prime \prime}
 \varepsilon \Phi^{\prime}]_{t} )
+ H. c.).
\label{potential}
\end{eqnarray}

We have omitted summation symbols and used the notation $p,p^\prime=u,d$;
$mn=11,1^\prime1^{\prime\prime},AA$; $kl=11,1^\prime1^{\prime\prime},BB$;
$\alpha\beta=11,1^\prime1^{\prime\prime},AB$; and
$st=11,1^\prime1^{\prime\prime},AA,BB,AB$. For example,
$\lambda^{pp^\prime}_{mn}[X_p]_m[X_{p^\prime}]_n =
\lambda^{pp^\prime}_{11}[X_p]_1[X_{p^\prime}]_1 +
\lambda^{pp^\prime}_{1^{\prime} 1^{\prime \prime}}[X_p]_{1^{\prime}}
[X_{p^\prime}]_{1^{\prime \prime}} +  \lambda^{pp^\prime}_{AA}[X_p]_A
[X_{p^\prime}]_A$ $+ \cdots$, and each term having contributions
$p, p^{\prime} = u,d$. Considering the Eq.~(\ref{potential}) we obtain the
constraint equations arising from the conditions $\partial \mathcal{V}/\partial
v^{\prime\prime}_{i} = 0, \dots$. For simplicity, in the following analysis it will
be also assumed that the VEVs are all real. In this case the condition
$1+\omega+\omega^2=0$ implies that in the constraint equations the coupling
constants will appear as the sum of some of the coupling constants in
Eq.~(\ref{potential}). The inclusion of all the scalar fields of the model
and also the constraints coming from the condition  $\partial^2 \mathcal{V}/\partial
v^{\prime\prime}_i\partial v^{\prime\prime}_j>0,\dots$ will be shown elsewhere. We have
obtained the constraint equations under the following conditions:
$v^{\prime\prime}_i=v^{\prime\prime}_{\phi_i},\;\forall i$;  and $\lambda^{dd}_{mn}
=K^{dd}_{kl}$, $\forall mn;kl$; and verified that, among the solutions to those
equations, it is possible to have the following hierarchy: $v^{\prime\prime}_1\gg
v^{\prime\prime}_3 \gg v^{\prime\prime}_2$. However for generating the quark masses, as
we have shown above, it is necessary that
$v^{\prime(\prime\prime)}_1\gg v^{\prime(\prime\prime)}_{\phi_1}$. Notwithstanding,
it is interesting that such hierarchies can arise from the model, and we hope that a more
detailed treatment of the scalar potential will produce a more realistic hierarchies among
the VEVs.

As in any model with broken discrete symmetries there are potential troubles
with domain walls~\cite{walls}. However, this issue can be solved by allowing
soft terms in the scalar potential violating these symmetries.

In conclusion, the mass matrices obtained, Eqs.~(\ref{leptons}), (\ref{us}), (\ref{ds}),
and (\ref{neutrinos}), arising because of the symmetries of the model, give appropriate
insight concerning the solution of the flavor problem generating at the same time a realistic
$V_{CKM}$ matrix. Of course, it is necessary to
explain how these symmetries are realized from a more fundamental theory.
Our scheme has some similarities with the ``private Higgs" of Porto and Zee~\cite{porto}
in which there is one Higgs per fermion and a general mass matrices. However, we have more than
one Higgs per fermion and the mass matrices have a specific texture of zeros.

We thank an anonymous referee for useful comments. ACBM was fully supported and VP partially
supported by CNPq.