\section{Introduction}

The matrix model of uncompactified M theory \cite{dhn}-\cite{bfss}
has been generalized to
arbitrary toroidal compactifications of type IIA and IIB string theory.
These models can be viewed as particular large $M$ limits of the
original matrix model, in the sense that
they may be viewed as the dynamics of a restricted class
of large $M$ matrices, with the original matrix model Lagrangian.

A separate line of reasoning has led to a description of the
Ho\v rava-Witten domain wall in terms of matrix quantum
mechanics \cite{orbi}. Here,
extra degrees of freedom have to be added to the original matrix model.
As we will review below, if these new variables, which tranform in the
vector representation of the gauge group, are not added, then the model
does not live in an eleven dimensional spacetime, but only on its
boundary..  Although it is, by
construction, a unitary quantum mechanics, it probably does not recover
ten dimensional Lorentz invariance in the large $M$ limit.  Its nominal
massless particle content is the ten dimensional $N = 1$ supergravity
(SUGRA) multiplet, which is anomalous.

With the proper number of vector variables added, the theory does have
an eleven dimensional interpretation.  It is possible to speak of states
far from the domain wall and to show that they behave exactly like the
model of \cite{bfss}.  Our purpose in the present paper is to compactify
this model on spaces of the general form $S^1 / Z_2 \times T^d$.  We
begin by reviewing the argument for the single domain wall quantum
mechanics, and generalize it to an $S^1 / Z_2$ compactification.  The
infinite momentum frame Hamiltonian for this system is practically
identical to the static gauge $O(M)$ Super Yang Mills (SYM)
Hamiltonian for $M$ heterotic D strings
in Type I string theory.  They differ only in the boundary conditions
imposed on the fermions which transform in the vector of $O(M)$.  These
fermions are required for $O(M)$ anomaly cancellation in both models,
but the local anomaly does not fix their boundary conditions.
Along the moduli space of the $O(M)$ theory, the model exactly
reproduces the string field theory Fock space of
heterotic string theory.  The inclusion of both Ramond and Neveu Schwarz
boundary conditions for the matter fermions, and the GSO projection, are
simple consequences of the $O(M)$ gauge invariance of the model.

Generalizing to higher dimensions,
we find that the heterotic matrix model on
$S^1 / Z_2 \times T^d$ is represented by a $U(M)$ gauge theory on
$S^1 \times T^d / Z_2$ .  On the orbifold  circles, the gauge
invariance reduces to $O(M)$.  We are able to construct both the
heterotic and open string sectors of the model, which dominate in
different limits of the space of compactifications.

In the conclusions, we discuss the question of whether the heterotic
models which we have constructed are continuously connected to the
original uncompactified eleven dimensional matrix model.  The answer to
this question leads to rather surprising conclusions, which inspire us
to propose a conjecture about the way in which the matrix model solves
the cosmological constant problem.  It also suggests that string vacua
with different numbers of supersymmetries are really states of {\it
different} underlying theories.  They can only be continuously connected
in limiting situations where the degrees of freedom which differentiate
them decouple.

\section{Heterotic Matrix Models in Ten and Eleven Dimensions}

In \cite{evashamit} an $O(M)$ gauged supersymmetric
 matrix model for a single Ho\v rava-Witten domain
wall embedded in eleven dimensions was proposed.  It was based on an
extrapolation of the quantum mechanics describing $D0$ branes near an
orientifold plane in Type IA string theory \cite{df}.  The model was
presented as an orbifold of the original \cite{bfss} matrix model in
\cite{motl}.
 In the Type IA context
it is natural to add degrees of freedom transforming in the vector of
$O(M)$ and corresponding to the existence
of $D8$ branes and the $08$ strings connecting them to the $D0$ branes.
Since $D8$ branes are movable in Type IA theory, there are consistent
theories both with and without these extra degrees of freedom.  That is,
we can consistently give them masses, which represent the distances
between the $D8$ branes and the orientifold.
However,
as first pointed out by \cite{df}, unless the number of $D8$ branes sitting
near the orientifold is exactly $8$, the $D0$ branes feel a linear
potential which either attracts them to or repels them from the
orientifold. This is the expression in the quantum mechanical approximation,
of the
linearly varying dilaton first found by Polchinski and Witten \cite{jopo}.
This system was studied further in \cite{rk} and
\cite{bss}. In the latter work the supersymmetry and gauge structure of
model were clarified, and the linear potential was shown to correspond to
the fact that the ``supersymmetric ground state''
of the model along classical flat directions representing excursions
away from the orientifold was not gauge invariant.

{}From this discussion it is clear that the only way to obtain a model
with an eleven dimensional interpretation is to add sixteen massless
real fermions
transforming in the vector of $O(M)$, which is the model proposed in
\cite{evashamit}.   In this case, $D0$ branes can move freely away from the
wall, and far away from it the theory reduces to the $U([{M\over 2}])$
model of \cite{bfss} \footnote{Actually there
is a highly nontrivial question
which must be answered in order to prove that the effects of the
wall are localized.
In \cite{bss} it was shown that supersymmetry allowed an arbitary
metric for the coordinate representing excursions away from the wall.
In finite orders of perturbation theory the metric falls off with
distance but, as in the discussion of the graviton scattering amplitude
in \cite{bfss}, one might worry that at large $M$ these could sum up to
something with different distance dependence.  In \cite{bfss} a
nonrenormalization theorem was conjectured to protect the relevant term
in the effective action.  This cannot be the case here.}.

Our task now is to construct a model representing two Ho\v rava-Witten
end of the world $9$-branes separated by an interval
of ten dimensional space.
As in \cite{motl} we can approach this task by attempting to mod out the $1+
1$ dimensional field theory \cite{bfss}, \cite{taylor}, \cite{motlb},
\cite{bs}, \cite{dvv} which describes M
theory compactified on a circle.  Following the logic of \cite{motl}, this
leads to an $O(M)$ gauge theory. The $9$-branes are stretched around
the longitudinal direction of the infinite momentum frame (IMF)
and the $2-9$ hyperplane of the transverse
space.  $X^1$ is the differential operator
$${R_1 \over i}{\partial \over \partial\sigma} - A_1$$
where $\sigma$ is in $[0,2\pi ]$, and $A_1$ is
an $O(M)$ vector potential.  The other $X^i$ transform in the ${\bf
{M(M+1) \over 2}}$ of $O(M)$.  There are two kinds of fermion multiplet.
$\theta$ is an ${\bf 8_c}$ of the spacetime $SO(8)$, a symmetric tensor
of $O(M)$ and is the superpartner of $X^i$ under the eight dynamical
and eight kinematical SUSYs which survive the projection.  $\lambda$ is
in the adjoint of $O(M)$, the ${\bf 8_s}$ of $SO(8)$, and is the
superpartner of the gauge potential.  We will call it the gaugino.

As pointed out in \cite{rk} and \cite{bss}, this model is anomalous.
One must add $32$ Majorana-Weyl fermions $\chi$ in the ${\bf M}$ of $O(M)$.
For sufficiently large $M$, this is the only fermion content which can
cancel the anomaly.   The continuous $SO(M)$ anomaly does not fix the
boundary conditions of the $\chi$ fields. There are various consistency
conditions which help to fix them, but in part we must make a choice
which reflects the physics of the situation which we are trying to
model.

The first condition follows from the fact that our gauge group is $O(M)$
rather than $SO(M)$.  That is, it should consist of the subgroup of
$U(M)$ which survives the orbifold projection.  The additional $Z_2$
acts only on the $\chi$ fields, by reflection.  As a consequence, the
general principles of gauge theory  tell us that each $\chi$ field
might appear with either periodic or antiperiodic boundary conditions,
corresponding to a choice of $O(M)$ bundle.  We must also make a
projection by the discrete transformation which reflects all the
$\chi$'s.
 What is left undetermined by these
principles is choice of relative boundary
conditions among the $32$ $\chi$'s.

The Lagrangian for the $\chi$ fields is
\eqn{chilag}{\chi (\partial_t
+ 2\pi R_1\partial_{\sigma} - i A_0 - i A_1) \chi
.}  In the large $R_1$ limit, the volume of the space on which the gauge
theory is compactified is small, and its coupling is weak, so we can
treat it by semiclassical methods.  In particular, the Wilson lines
become classical variables.  We will refer to classical values of the
Wilson lines as expectation values of the gauge potential $A_1$.
(We use the term expectation value loosely, for we are dealing with a
quantum system in finite volume.  What we mean
is that these ``expectation values'' are the slow variables in
a system which is being treated by the Born-Oppenheimer approximation.)
An excitation of the system at some position in the direction tranverse
to the walls is represented by a wave function of
$n \times n$ block matrices in which $A_1$
has an expectation value breaking $O(n)$ to $U(1) \times U([n/2])$.
In the presence of a generic expectation value, in $A_0 = 0$
gauge, the $\chi$ fields will not have any zero frequency modes.
The exceptional positions where zero frequency modes exist are $A_1 = 0$
(for periodic fermions) and $A_1 = \pi R_1$ (for antiperiodic fermions).
These define the positions of the end of the world $9$-branes, which we
call the walls.  When $R_1 \gg l_{11}$, all of the finite wavelength
modes of all of the fields have very high frequencies and can be
integrated out.
In this limit, an excitation
far removed from the walls has precisely the degrees of freedom of a
$U([{n\over2}])$ gauge quantum mechanics.  The entire content of the
theory far from
the walls is $U([{M\over 2}])$ gauge quantum mechanics.  It has no
excitations carrying the quantum numbers created by the $\chi$ fields,
and according to the conjecture of \cite{bfss} it reduces to eleven
dimensional M theory in the large $M$ limit.  This reduction assumes
that there is no longe range interaction between the walls and the rest
of the system.

In order to fulfill this latter condition it must be true that at $A_1 =
0$, and in the large $R_1$ limit, the field theory reproduces
the $O(M)$ quantum mechanics described at the beginning of this section
(and a similar condition near the other boundary).
We should find $16$ $\chi$ zero modes near each wall.  {\it Thus,
the theory must contain a sector in which
 the $32$ $1+1$ dimensional $\chi$ fields are grouped
in groups of $16$ with opposite periodicity}. Half of the fields will
supply the required zero modes near each of the walls.  Of course, the
question of which fields have periodic and which antiperiodic boundary
conditions is a choice of $O(M)$ gauge.  However, in any gauge
only half of the $\chi$ fields will have zero modes located at any
given wall.   We could of course consider sectors of the
fundamental $O(M)$ gauge theory in which there is a different
correlation between boundary conditions of the $\chi$ fields.
However, these would not have an eleven dimensional interpretation at
large $R_1$.  The different sectors are not mixed by the Hamiltonian so
we may as well ignore them.

To summarize, we propose that M theory compactified on $S^1 / Z_2$ is
described by a
$1+1$ dimensional $O(M)$ gauge theory with $(0,8)$ SUSY.  Apart from the
$(A_{\mu}, \lambda )$ gauge multiplet, it contains
a right moving $X^i, \theta$ supermultiplet in the symmetric tensor
of $O(M)$ and 32 left moving fermions, $\chi$, in the vector.  The
allowed gauge bundles for $\chi$ (which transforms under the discrete
reflection which leaves all other multiplets invariant), are those in
which two groups of $16$ fields have opposite periodicities.
In the next section we will generalize this construction to
compactifications on general tori.

First let us see how heterotic strings emerge from this formalism in the
limit of small $R_1$.  It is obvious that in this limit, the string
tension term in the SYM Lagrangian becomes very small.  Let us rescale
our $X^i$ and time variables so that the quadratic part of the
Lagrangian is independent of $R_1$.  Then, as in \cite{bs},
\cite{motlb}, \cite{dvv},
the commutator term involving the $X^i$ gets a coefficient
$R^{-3}$ so that we are forced onto the moduli space in that limit.  In
this $O(M)$ system, this means that the $X^i$ matrices are diagonal,
and the gauge group is completely broken
to a semidirect product of $Z_2$ (or
$O(1)$) subgroups which reflect the individual components of the
vector representation, and an $S_M$ which permutes the eigenvalues of
the $X^i$.  The moduli space of low
energy
fields\footnote{We use the term moduli space to refer to the space of
low energy fields whose effective theory describes the small $R_1$
limit (or to the target space of this effective theory).
These fields are in a Kosterlitz Thouless phase and do not have
expectation values, but the term moduli space is a convenient shorthand
for this subspace of the full space of degrees of freedom.}
consists of diagonal $X^i$ fields, their superpartners $\theta_a$ (also
diagonal matrices), and the $32$ massless left moving $\chi$ fields.
The gauge bosons and their superpartners $\lambda^{\dot{\alpha}}$
decouple in the small $R_1$ limit.  All of the $\chi$ fields are light
in this limit.

\subsection{Screwing Heterotic Strings}

As first explained in \cite{motlb} and elaborated in
\cite{bs}, and \cite{dvv},
twisted sectors under $S_N$ lead to strings of arbitrary
length\footnote{These observations are mathematically identical to
considerations that arose in the counting of BPS states in black hole
physics \cite{BPS} . }.
The
strings of conventional string theory, carrying continuous values of the
longitudinal momentum, are obtained by taking $N$ to infinity
and concentrating on
cycles whose length is a finite fraction of $N$.
The new feature which arises in the heterotic string is that the
boundary conditions of the $\chi$ fields can be twisted by the discrete
group of reflections.

A string configuration of length $2\pi k$, $X_S^i (s)$, $0 \leq s \leq
2\pi k$, is represented by a diagonal matrix:

 \eqn{screw}{
X^i(\sigma)=\tb{{cccc}
X_S^i(\sigma)&&&\\
&X_S^i(\sigma+2\pi)&&\\
&&\ddots&\\
&&&X_S^i(\sigma+2\pi(N-1))}.}
This satisfies the twisted boundary condition
$X^i(\sigma+2\pi)=E_O^{-1}X^i(\sigma) E_O$ with
\eqn{bcx}{
E_O=\tb{{ccccc}
&&&&\epsilon_k\\
\epsilon_1&&&&\\
&\epsilon_2&&&\\
&&\ddots&&\\
&&&\epsilon_{N-1}&},}
and $\epsilon_i = \pm 1$.
The latter represent the $O(1)^k$ transformations,
which of course do not effect $X^i$ at all.

To describe the possible twisted sectors of the matter fermions we
introduce the matrix $r^a_b = diag (1\ldots 1 ,-1 \ldots -1)$, which
acts on the $32$ valued index of the $\chi$ fields.
The sectors are then defined by
\eqn{chibc}{\chi^a (\sigma + 2\pi ) = r^a_b E_O^{-1} \chi^b
(\sigma )}

As usual, inequivalent sectors
correspond to conjugacy classes of the gauge group.
In this case, the classes can be described by a permutation
with a given set of cycle
lengths, corresponding to a collection of
strings with fixed longitudinal momentum fractions,
and the determinants of the $O(1)^k$ matrices inside each cycle.
In order to understand the various gauge bundles,
it is convenient to write the ``screwing
formulae'' which express the
components of the vectors $\chi^a$ in terms of string
fields $\chi_s^a$ defined on the interval $[0, 2\pi k]$.
The defining boundary conditions are
\eqn{bcchi}{\chi_i^a (\sigma + 2\pi ) =
\epsilon_i r^a_b \chi_{i + 1}^b (\sigma )}
where we choose the gauge in which $\epsilon_{i < k} =1$
and $\epsilon_k = \pm 1$ depending
on the sign of the determinant.  The vector index $i$ is counted modulo $k$.
This condition is solved by
\eqn{soln}{\chi_i^a (\sigma ) = (r^{i - 1})^a_b \chi_S^b
(\sigma + 2\pi (i - 1))}
where $\chi_S$ satisfies
\eqn{hetbc}{\chi_S^a (\sigma + 2\pi k)
= (r^k)^a_b \epsilon_k \chi_S^b (\sigma )}
For $k$ even, this gives the PP and AA sectors
of the heterotic string, according to the
sign of the determinant.
Similarly, for $k$ odd, we obtain the AP and PA sectors.

As usual in a gauge theory,
we must project on gauge invariant states.  It turns out that
there are only two independent kinds of
conditions which must be imposed.  In a sector characterized by a
permutation $S$, one can be chosen to be
the overall multiplication of $\chi$ fields associated with a given
cycle of the permutation (a given string)
by $-1$.
This GSO operator anticommuting with all the 32 $\chi$ fields
is represented by the ${\bf -1}$ matrix from the gauge group $O(N)$.
The other is the projection associated with the cyclic
permutations themselves.
It is easy to verify that under the latter transformation
the $\chi_S$ fields transform as
\eqn{chitransf}{\chi_S^a (\sigma )
\rightarrow r^a_b \chi_S^b (\sigma + 2\pi)}
Here $\sigma \in [0,2\pi k]$ and we are
taking the limit $M\rightarrow \infty$,
$k/M$ fixed.  In this limit the $2\pi$
shift in argument on the righthand side of
(\ref{chitransf}) is negligible,
and we obtain the second GSO projection of the heterotic string.

Thus, $1+1$ dimensional $O(M)$
SYM theory with $(0,8)$ SUSY, a left moving
supermultiplet in the symmetric
tensor representation and 32 right moving fermion
multiplets in the vector (half
with P and half with A boundary conditions)
reduces in the weak coupling, small (dual) circle limit to two copies of
the Ho\v rava-Witten domain wall
quantum mechanics, and in the strong coupling large (dual)
circle limit, to the string field theory of the $E_8 \times E_8$
heterotic string.

\section{Multidimensional Cylinders}

The new feature of heterotic compactification on $S^1 /Z_2 \times T^d$
is that the coordinates in the toroidal dimensions are represented by
covariant derivative operators with respect to new world volume
coordinates.  We will reserve $\sigma$ for the periodic coordinate dual
to the interval $S^1 / Z_2$ and denote the other coordinates by
$\sigma^A$.   Then,
\eqn{covder}{X^A = {2\pi R_A \over i}{\partial
\over \partial \sigma^A} - A_A
(\sigma ); \quad A = 2\ldots k+1.}
Derivative operators are antisymmetric, so in order to implement the
orbifold projection, we have to include the transformation $\sigma^A
\rightarrow  - \sigma^A$, for $A = 2\ldots d + 1$,
 in the definition of the orbifold symmetry.
Thus, the space on which SYM is compactified is $S^1 \times (T^d /
Z_2)$.   There are $2^d$ {\it orbifold circles} in this space, which are
the fixed manifolds of the reflection.
Away from these singular loci, the gauge group
is $U(M)$ but it will be restricted to $O(M)$ at the singularities.
We will argue that there must be
a number of $1 + 1$ dimensional fermions living only on these circles.
When $d = 1$ these orbifold lines can be thought of as the boundaries of
a {\it dual cylinder}.
Note that if we take $d = 1$ and rescale the $\sigma^A$ coordinates
so that their lengths are $1/R_A$ then a long thin cylinder in spacetime
maps into a long thin cylinder on the world volume, and a short fat
cylinder maps into a short fat cylinder.  As we will see, this
geometrical fact is responsible for the emergence of Type $IA$
and heterotic strings in the appropriate limits.

The boundary conditions on the world volume fields are
\eqn{bca}{X^i (\sigma ,\sigma^A ) = \bar X^i (\sigma ,  - \sigma^A ),\quad
A_a (\sigma ,\sigma^A ) = \bar A_a (\sigma ,  - \sigma^A ),}
\eqn{bcc}{A_1 (\sigma ,\sigma^A ) = -\bar A_1 (\sigma ,  - \sigma^A)}
\eqn{bcd}{\theta (\sigma ,\sigma^A ) = \bar\theta(\sigma ,-\sigma^A),\quad
\lambda (\sigma ,\sigma^A ) = -\bar\lambda (\sigma ,-\sigma^A )}
All matrices are hermitian,
so transposition is equivalent to complex conjugation.
The right hand side of the boundary condition \ref{bcc} can also be shifted
by $2\pi R_1$, reflecting the fact that $A_1$ is an angle variable.

Let us concentrate on the cylinder case, $d=1$.
In the limits $R_1 \ll l_{11} \ll R_2$ and
 $R_2 \ll l_{11} \ll R_1$, we will find that
the low energy dynamics is completely
described in terms of the moduli space,
which consists of commuting $X^i$ fields.
In the first of these limits,
low energy fields have no $\sigma^2$ dependence, and
the boundary conditions restrict
the gauge group to be $O(M)$, and force
$X^i$ and $\theta$ to be real symmetric matrices.
Anomaly arguments
then inform us of the existence of $32$
fermions living on the boundary circles.
The model reduces to the $E_8 \times E_8$
heterotic matrix model described in the previous
section, which, in the indicated limit,
was shown to be the free string field theory of
heterotic strings.

\subsection{Type IA Strings}

The alternate limit produces something novel.
Now, low energy fields are restricted
to be functions only of $\sigma^2$.
Let us begin with a description of closed strings.
We will exhibit a solution of the boundary conditions for each
closed string field $X_S (\sigma )$ with periodicity $2\pi k$.  Multiple
closed strings are constructed via the usual block diagonal procedure.

\eqn{uopen}{
X^i(\sigma^2)=U(\sigma^2)
D
U^{-1}(\sigma^2),}
\eqn{diagstring}{\begin{array}{c}D=\mbox{diag}(X_s^i(\sigma^2),\epsilon
X_s^i(2\pi-\sigma^2),X_s^i(2\pi+\sigma^2),
\epsilon X_s^i(4\pi-\sigma^2),\\
\dots,
X_s^i(2\pi(N-1)+\sigma^2),\epsilon X_s^i(2\pi N-\sigma^2)).
\end{array}}
where $\epsilon$ is $+1$ for $X^{2\dots 9}$ and $\theta$'s,
$-1$ for $A^1$ and $\lambda$'s.
{}From this form it is clear that the matrices will commute
with each other for any value of $\sigma^2$.
We must obey Neumann boundary conditions
for the real part of matrices and Dirichlet conditions
for the imaginary parts (or for $\epsilon=-1$ vice versa),
so we must use specific values of the unitary matrix
$U(\sigma^2)$ at the points $\sigma^2=0,\pi$.
Let us choose
\eq{U'(0+)=U'(\pi-)=0}
(for instance, put $U$ constant on a neighbourhood
of the points $\sigma^2=0,\pi$)
and for a closed string,
\eq{U(\pi)=\tb{{cccc}
m&&&\\
&m&&\\
&&\ddots &\\
&&&m},\qquad
U(0)=C\cdot
U(\pi)\cdot C^{-1},}
where $C$ is a cyclic permutation matrix
\eq{C=\tb{{ccccc}
&1&&&\\
&&1&&\\
&&&\ddots&\\
&&&&1\\
1&&&&}}
where $m$ are $2\times 2$ blocks (there are $N$ of them)
(while in the second matrix the $1$'s are $1\times 1$ matrices so that
we have a shift of the $U(\pi)$ along the diagonal
by half the size of the Pauli matrices.
The form of these blocks guarantees the conversion of $\tau_3$
to $\tau_2$:
\eq{m=\frac{\tau_2+\tau_3}{\sqrt 2}.}
This $2\times 2$ matrix causes two ends to be connected
on the boundary. It is easy to check that the right
boundary conditions will be obeyed.

To obtain open strings, we just change the
$U(0)$ and $U(\pi )$.
An open string of odd length is obtained by
throwing out the last element in (\ref{diagstring}) and taking
\eq{
U(0)=\tb{{ccccc}
1_{1\times 1}&&&&\\
&m&&&\\
&&m&&\\
&&&\dots &\\
&&&&m
},\quad\,
U(\pi)=\tb{{ccccc}
m&&&\\
&m&&\\
&&m&\\
&&&\dots&\\
&&&&1_{1\times 1}}}
Similarly, an open string of even length  will have one of the matrices
$U(0),U(\pi)$ equal to what it was in the closed string case $m\otimes
1$ while
the other will be equal to
\eq{
U(0)={\small\tb{{cccccc}
1_{1\times 1}&&&&&\\
&m&&&&\\
&&m&&&\\
&&&\dots &&\\
&&&&m&\\
&&&&&1_{1\times 1}
}}}

Similar constructions for the fermionic coordinates are straightforward
to obtain.  We also note that we have worked above with the original
boundary conditions and thus obtain only open strings whose ends are
attached to the wall at $R_1 = 0$.  Shifting the boundary condition
\ref{bcc} by $2\pi R_1$ (either at $\sigma^2 =0$ or $\sigma^2 = \pi$ or
both) we obtain strings attached to the other wall, or with one end on
each wall.  Finally, we note that we can perform the gauge
transformation $M \rightarrow \tau_3 M \tau_3$ on our construction.
This has the effect of reversing the orientation of  the string fields,
$X_S (\sigma^2 ) \rightarrow X_S (- \sigma^2 )$.  Thus we obtain
unoriented strings.

We will end this section with a brief comment about moving $D8$ branes
away from the orientifold wall.  This is achieved by adding explicit
$SO(16) \times SO(16)$ Wilson lines to the Lagrangian of the $\chi^a$
fields.  We are working in the regime $R_2 \ll l_{11} \ll R_1$, and we
take these to be constant gauge potentials of the form $\chi^a {\cal
A}_{ab} \chi^b$, with ${\cal A}$ of order $R_1$.  In the presence of
such terms $\chi^a$ will not have any low frequency modes, unless we
also shift the $O(M)$ gauge potential $A_1$ to give a compensating shift
of the $\chi$ frequency.  In this way we can construct open strings
whose ends lie on $D8$ branes which are not sitting on the orientifold.

In this construction, it is natural to imagine that $16$ of the $\chi$
fields live on each of the boundaries of the dual cylinder.  Similarly,
for larger values of $d$ it is natural to put ${32 \over 2^d}$ fermions
on each orbifold circle, a prescription which clearly runs into problems
when $d > 4$. This is reminiscent of other orbifold constructions in M
theory in which the most symmetrical treatment of fixed points is not
possible (but here our orbifold is in the dual world volume).  It is
clear that our understanding of the heterotic matrix model for general
$d$ is as yet quite incomplete.  We hope to return to it in a future
paper.

\section{Conclusions}

We have described a class of matrix field theories which incorporate the
Fock spaces of the the $E_8 \times E_8$
heterotic/Type $IA$ string field theories
into a unified quantum theory.  The underlying gauge dynamics provides a
prescription for string interactions.  It is natural to ask what the
connection is between this nonperturbatively defined system and previous
descriptions of the nonperturbative dynamics of string theories with
twice as much supersymmetry.  Can these be viewed as two classes of
vacua of a single theory?  Can all of these be obtained as different
large $N$ limits of a quantum system with a finite number of degrees of
freedom?

The necessity of introducing the $\chi$ fields into our model suggests
that the original eleven dimensional system does not have all the
necessary ingredients to be the underlying theory.  Yet we are used to
thinking of obtaining lower dimensional compactifications by restricting
the degrees of freedom of a higher dimensional theory in various ways.
Insight into this puzzle can be gained by considering the limit of
heterotic string theory which, according to string duality, is supposed
to reproduce M theory on $K3$.  The latter theory surely reduces
to eleven dimensional M theory in a continuous manner as the radius of
$K3$ is taken to infinity.
Although we have not yet worked out the details of
heterotic matrix theory on higher dimensional tori, we think that it
is clear that the infinite $K3$
limit will be one in which the $\chi$ degrees
decouple from low energy dynamics.

The lesson we learn from this example is that {\it decompactification of
space time dimensions leads to a reduction in degrees of freedom}.
Indeed, this principle is clearly evident in the prescription for
compactification of M theory on tori in terms of SYM theory.  The more
dimensions we compactify, the higher the dimension of the field theory we
need to describe the compactification.  There has been some discussion
of whether this really corresponds to adding degrees of freedom since
the requisite fields arise as limits of finite matrices.  However there
is a way of stating the principle which is independent of how one
chooses to view these constructions. Consider, for example, a graviton
state in M theory compactified on a circle.  Choose a reference energy
$E$ and ask how the number of degrees of freedom with energy less than
$E$ which are necessary to describe this state, changes with the radius
of compactification.  As the radius is increased, the radius of the dual
torus decreases.  This decreases the number of states in the theory with
energy less than $E$, precisely the opposite of what occurs when we
increase the radius of compactification of a local field theory

\subsection{Cosmological Constant Problem}

It seems natural to speculate that this property, so counterintuitive
from the point of view of local field theory, has something to do with
the cosmological constant problem.  In \cite{tbcosmo}
one of the authors
suggested that any theory which satisfied the 't Hooft-Susskind
holographic principle
would suffer a thinning out of degrees
of freedom as the universe expanded, and that this would lead to an
explanation of the cosmological constant problem.  Although the
speculations there did not quite hit the mark, the present ideas suggest
a similar mechanism.  Consider a hypothetical state of the matrix model
corresponding to a universe with some number of Planck size dimensions
and some other dimensions of a much larger size, $R$.  Suppose also that
SUSY is broken at scale $B$, much less than
the (eleven dimensional) Planck scale.
The degrees of freedom
associated with the compactified dimensions all have energies
much higher than the
SUSY breaking scale. Their zero point fluctuations will lead
to a finite, small (relative to the Planck mass) $R$ independent,
contribution to the total vacuum
energy.
As $R$ increases, the number of degrees of freedom at scales less than
or equal to $B$ will decrease.
Thus, we expect a corresponding decrease in the
total vacuum energy.
The total vacuum energy in the large $R$ limit is thus bounded
by a constant, and is dominated by the
contribution of degrees of freedom associated with
the small, compactified dimensions.
Assuming only the minimal supersymmetric cancellation
in the computation of the vacuum energy,
we expect it to be of order $B^2 l_{11}$.  This
implies a vacuum energy density of order
$B^2 l_{11} / R^3$, which is too small to be of
observational interest for any plausible values of the parameters.
If a calculation of this nature turns
out to be correct, it would constitute a prediction
that the cosmological constant is essentially zero in the matrix model.

It should not be necessary to emphasize how premature
it is to indulge in speculations
of this sort (but we couldn't resist the temptation).
We do not understand supersymmetry
breaking in the matrix model and we are even
further from understanding its cosmology.
Indeed, at the moment we do not even
have a matrix model derivation of the fact\footnote{Indeed
this ``fact'' is derived by rather indirect arguments in perturbative
string theory.} that
parameters like the radius of compactification are dynamical variables.
Perhaps the
most important lacuna in our understanding
is related to the nature of the large $N$
limit.  We know that many states of the system
wander off to infinite energy as $N$
is increased.
Our discussion above was based on extrapolating results of the finite $N$
models, without carefully verifying that the
degrees of freedom involved survive
the limit.  Another disturbing thing about our discussion is the absence
of a connection to Bekenstein's area law for the number of states.  The
Bekenstein law
seems to be an integral part of the physical picture of the matrix model.
Despite these obvious problems, we feel
that it was worthwhile to present this preliminary discussion of
the cosmological constant problem
because it makes clear that the spacetime picture
which will eventually emerge from the matrix model is certain to be
very different from the one implicit in local field theory.

\section{Acknowledgements}
L.Motl is grateful to staff and students of Rutgers University
for their hospitality.

\newpag


\section{Introduction}
The concentration of both dark and baryonic matter in the cores of
clusters of galaxies has many profound implications for our
understanding of cluster growth and cosmology. Firstly, the structure
and evolution of the gravitational potential of a cluster of galaxies
depends on the nature of dark matter and thus allows direct comparison
with predictions from numerical simulations (e.g.\ Navarro, Frenk \&
White 1997).  Secondly, the surface density of mass integrated through
the core of a cluster is often sufficiently high to strongly lens
background galaxies into gravitational arcs (Soucail et al.\ 1987;
Mellier et al.\ 1991; Kneib et al.\ 1996; Smith et
al.\ 2001). Detailed analysis of the location and shape of such arcs,
as well as of lens-generated multiple images, can be used to model the
projected mass in the cluster, and in-depth follow-up of the brightest
lensed features often yields valuable insights into the properties of
distant, faint galaxies (e.g.\ Kneib et al.\ 2004; Smail et
al.\ 2007).  Thirdly, the high density and temperature of the gas in
the core of clusters leads to intense X-ray emission that current
instrumentation can detect out to redshifts well above unity. The
selection of massive clusters through X-ray emission has proved very
successful at providing cosmological constraints (Henry 2000; Borgani
et al.\ 2001; Allen et al.\ 2003; Pierpaoli et al.\ 2003; Allen et
al.\ 2008; Mantz et al.\ 2008) and follow-up observations of X-ray
luminous clusters have revealed many spectacular cases of
gravitational lensing (Gioia \& Luppino 1994; Smith et al.\ 2001;
Dahle et al.\ 2002; Covone et al.\ 2006).

In this paper we present a comprehensive multiwavelength study of a
complex gravitational arc and its host cluster MACS\,J1206.2$-$0847,
an X-ray selected system at intermediate redshift found by the Massive
Cluster Survey, MACS (Ebeling, Edge \& Henry 2001, 2007). We describe
our optical, NIR and X-ray observations in Sections 2 and 3,
investigate the properties of the giant arc, the cluster lens, and the
brightest cluster galaxy in Section 4 to 7, and derive mass estimates
for the cluster core and the entire system in Section 7. We present a
discussion of our results as well as conclusions in Section 8.

Throughout we use a $\Lambda$CDM cosmology ($\Omega_M=0.3$,
$\Omega_{\lambda}=0.7$) and adopt $H_0=$ 70 km s$^{-1}$ Mpc$^{-1}$.

\section{Observations}

The galaxy cluster MACS\,J1206.2$-$0847 was originally discovered in a
short two-minute R-band image taken on June 15, 1999 with the
University of Hawaii's 2.2m telescope (UH2.2m) on Mauna Kea. The
observation, performed as part of the MACS project, was triggered by
the presence of the X-ray source 1RXS\,J120613.0$-$084743 in the ROSAT
Bright Source Catalogue which had no obvious counterpart in the
standard astronomical databases and could also not trivially
be identified by inspection of the respective Digitized Sky Survey
image.

Following the initial, tentative identification of the RASS X-ray
source as a potentially massive galaxy cluster, we conducted a range
of follow-up observations to firmly establish the cluster nature of
this source and to characterize its physical properties.

\subsection{Optical}
\label{optspec}

Spectra of two galaxies in MACS\,J1206.2$-$0847, one of them the BCG,
were taken with the Wide-Field Prism Spectrograph on the UH2.2m on
July 4, 1999, using a 420 l/mm grism, a Tektronix 2048$^2$ CCD
yielding 0.355 arcsec/pixel, and a 1.6 arcsec slit. The two redshifts
were found to be concordant, establishing an approximate cluster
redshift of $z\approx 0.434$.

Moderately deep, multi-passband imaging observations ($3\times 240$s,
dithered by 10 arcsec, in each of the V, R, and I filters) of the
cluster were obtained with the UH2.2m on January 29, 2001, using again
the Tektronix 2048$^2$ CCD which provides a scale of 0.22 arcsec per
pixel and a $7.3\times7.3$ arcminute$^2$ field of view. The seeing was
variable throughout the night; we measure seeing values of 0.85, 1.05,
0.90 arcsec in the V, R, and I passbands, respectively, from the
final, co-added images.

Spectroscopic observations of presumed cluster galaxies as well as of
the giant arc in MACS\,J1206.2$-$0847 were performed with the FORS1
spectrograph in multi-object spectroscopy mode at the UT3 Melipal
telescope of the VLT on April 11, 2002. The G300V grism, an order
sorting filter (GG375), and a 1 arcsec slit were used, yielding a
wavelength coverage from $\sim$4000\AA\ to $\sim$8600\AA\ at a
resolution of $R=500$. The total exposure time was 38 minutes. A
single mask was designed, covering the $\sim$7' field of FORS1 with 19
slitlets of fixed length (22"). Credible redshifts could be measured
for 14 objects. During the exposure the seeing was 0.6 arcsec. Spectra
of the spectrophotometric standard star EG274 were obtained for
calibration.

Additional multi-object spectroscopy of colour-selected galaxies in
the field of MACS\,J1206.2$-$0847 was performed on May 8, 2003 using
the MOS spectrograph on the Canada-France-Hawaii Telescope (CFHT) on
Mauna Kea. We used the B300 grism and the EEV1 CCD, which provides a
resolution of 3.3\AA/pixel, and observed through a broadband filter
(\#4611, General Purpose set) to produce truncated spectra, covering
about 1800\AA\ centred on 6150\AA, such that spectra could be stacked
in three tiers along the dispersion direction. This choice of filter
and grism ensured that, for all cluster members, redshifts could be
obtained from the Ca H+K lines which fall at 5660\AA\ and 5710\AA\ at
the approximate cluster redshift of $z=0.44$. Weather conditions were
poor though, and only 48 of the 67 objects observed (total integration
time: one hour) yielded reliable redshifts.

Finally, MACS\,J1206.2$-$0847 was observed on December 6, 2005 with
the Advanced Camera for Surveys (ACS) aboard the Hubble Space
Telescope as part of program SNAP-10491 (PI Ebeling), for a total of
1200 seconds in the F606W filter, resulting in a high-resolution image
of the cluster core, including the giant arc.

\subsection{Near-infrared}

Near-infrared observations of the core of MACS\,J1206.2$-$0847 were
performed in the J and K bands using the United Kingdom Infra-Red
telescope (UKIRT) on April, 5 2001 using the UFTI imager during a
period of good seeing. The observations consisted of two iterations of
a nine-point dither pattern, each of 60 second exposures for a total
integration time of 1080 seconds. The seeing measured from these
observations was 0.47 and 0.59 arc-seconds in the J and K bands,
respectively.

\subsection{X-ray}

MACS\,J1206.2$-$0847 was observed on December 18, 2002 with the ACIS-I
detector aboard the Chandra X-ray Observatory for a nominal duration
of 23.5 ks, as part of a Chandra Large Programme awarded to the MACS
team. The target was placed about two arcmin off the standard aimpoint
to avoid flux being lost in the chip gaps of the ACIS-I detector, while
still maintaining good (sub-arcsec) angular resolution across the
cluster core. VFAINT mode was used in order to maximize the efficiency
of particle event rejection in the post-observation processing.

\section{Data reduction}

\subsection{Optical imaging}

Standard data-reduction techniques (bias-subtracting, flat-fielding,
image combination and registration) were applied to the V, R and I
band data using the relevant \textsc{iraf} packages. The data were
photometrically calibrated via the observation of Landolt
standard-star fields.

In order to measure aperture magnitudes, seeing-matched frames were
produced by applying a Gaussian smoothing to the V, I, J and K images
such that the seeing in these frames was degraded to match the 1.1
arc-seconds measured in the R band. Using SExtractor, the
seeing-matched (undegraded) images were then used to measure aperture
(total) magnitudes.

Photometry of the giant arc was performed by manually defining an
aperture mask fitted to the arc profile. The area defined by this
aperture was then masked out of the science frame and a background
image produced by median smoothing over the absent arc. This
background image was subtracted from the science frame and VRIJK
photometry obtained by applying the aperture to the resulting
sky-subtracted images.

\subsection{Optical spectroscopy}

For the reduction of our spectroscopic data we applied the same
standard techniques as for the imaging data, followed by straightening
of the skylines, extraction of spectra, and wavelength calibration
using the relevant \textsc{iraf} packages.

Preliminary redshifts were determined via visual inspection, typically
from the calcium H and K lines. Final refined redshifts were found
with a multi-template cross-correlation method using the \textsc{iraf}
task \textsc{fxcor}.

Redshifts with cross-correlation peak heights exceeding 0.70 are
typically accurate to 0.0002; for peak heights between 0.5 and 0.7 the
error is typically 0.0005. All redshifts were converted to the
heliocentric reference frame. Table~\ref{ztbl} lists the coordinates
of all galaxies observed, as well as the measured redshifts with their
uncertainties.

\subsection{Near-infrared imaging}

The J and K imaging data were reduced using the ORAC-DR data-reduction
pipeline and were calibrated through observation of observatory
standard stars.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.vri.eps}
\caption{\label{imvri} Colour image (IVR mapped to RGB) of
  MACSJ1206.2$-$0847 from $3\times 240$s observations (per filter)
  obtained with the UH 2.2m telescope (see text for observational
  details).}
\end{figure}

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{spec_gal_pos.eps}
\caption{\label{imgalpos}Locations of all galaxies for which
  spectroscopic redshifts were measured with the VLT (circles) and
  CFHT (diamonds), overlaid on the UH2.2m R-band image. Bold symbols
  mark galaxies found to be cluster members. See Table~\ref{ztbl} for
  coordinates and redshifts.}
\end{figure}

\subsection{X-ray}

We use {\sc Ciao} (version 3.3), the standard suite of software tools
developed for the analysis of Chandra data at the Chandra Science
Center, as well as the most recent calibration information, to
reprocess the raw ACIS-I data. Our inspection of the lightcurve of the
event count rate in the source-free regions of the ACIS-I detector
finds no significant flaring, leading to an effective (dead-time
corrected) total exposure time of 23.2 ks.

To investigate potential spatial variations in the cluster gas
temperature we define various source and background regions.  For each
of these regions we generate auxiliary response files (ARF) and
response matrix files (RMF) which weigh the position-dependent
instrument characteristics by the observed count distribution in the
respective area. Following Markevitch \& Vikhlinin (2001), only the
0.5--2.0 keV band data are used to create these maps of spatial
weights, since the effects of vignetting are small in this energy
range. We also apply a correction of a factor of 0.93 to the effective
area distribution at energies below 2 keV in all ARFs as suggested by
a comparison of calibration results for the front- and
back-illuminated ACIS chips (Vikhlinin et al.\ 2002). Finally, we use
the {\sc acisabs} package to correct all ARFs for the effects of the
(time-dependent) buildup of a contaminating deposit on the optical
detector window which results in a reduction of the effective area at
low energies.

Background regions are defined by copying the respective source
regions to the same chip-y location on the other three ACIS-I CCDs.
This strategy minimizes the impact of any residual chip-y dependence
of the background on the data analysis, an effect that is unavoidable
if the background is selected as a source-free region on the same CCD
as the cluster.

\section{Cluster galaxy distribution}
\label{galprop}

Figure~\ref{imvri} shows a colour image of the cluster generated from
the V, R, I images obtained with the UH2.2m. MACS\,J1206.2$-$0847 is
found to be an optically very rich system with a single dominant
central galaxy and no obvious subclustering in the apparent, projected
cluster galaxy distribution. A giant arc is clearly visible about 20
arcsec to the West of the BCG. The astrometric solution used is based
on eight stars within the field of view of the V band image that have
accurate celestial coordinates in the Hubble Guide Star Catalogue
(GSC2).

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{z_hist.eps}
\caption{\label{zhist}Histogram of galaxy redshifts in the field of
  MACS\,J1206.2$-$0847 as observed with multi-object spectrographs on
  CFHT and the VLT (cf.\ Table~\ref{ztbl}). The overlaid Gaussian
  curve is characterized by the best-fit values for the systemic
  redshift and cluster velocity dispersion of $z=0.4385$ and
  $\sigma=1575^{+191}_{-190}$ km/s, respectively. }
\end{figure}

Our spectroscopic observations of 85 galaxies in the field of
MACS\,J1206.2$-$0847 (Fig.~\ref{imgalpos}) yield redshifts as listed
in Table~\ref{ztbl}.  Two galaxies were observed with both the VLT and
CFHT; their spectroscopic redshifts agree within the errors. Using
only the most accurate redshifts with correlation peak heights
exceeding 0.7, we apply iterative $3\sigma$ clipping to the redshift
histogram to obtain a systemic cluster redshift of $z=0.4384$ and a
very high velocity dispersion in the cluster rest frame of
$\sigma=1581$ km/s based on 38 redshifts. Entirely consistent values
of $z=0.4385$ and $\sigma=1575^{+191}_{-190}$ km/s are found using the
ROSTAT statistics package (Beers, Flynn \& Gebhardt 1990). The
resulting redshift histogram is shown in Fig.~\ref{zhist}. Despite the
extremely high velocity dispersion of the system we find no obvious
signs of substructure along the line of sight; a one-sided
Kolmogorov-Smirnov test finds the observed redshift distribution to be
only mildly inconsistent with a Gaussian ($2.05\sigma$ significance).

\section{Arc properties}

\begin{figure*}[h]
\vspace*{-0.1cm}
\parbox{0.79\textwidth}{
\epsfxsize=0.2\textwidth
\hspace*{9cm} \epsffile{cimage.hst.noastro.eps}\\
\epsfxsize=0.785\textwidth
\hspace*{5mm}\epsffile{MACSJ1206.2-0847.vik.eps}}
\parbox[c]{0.18\textwidth}{
\vspace*{1.8cm}
\epsfxsize=0.2\textwidth
\epsffile{arc.hst.noastro.eps}}\mbox{}\\*[-0.1cm]
\caption{\label{imvik}Colour image (KIV mapped to RGB) of MACSJ1206.2$-$0847
  from $3\times 240$s observations (I,V) and $18\times 60$s observations (K)
  obtained with the UH2.2m and UKIRT, respectively (see text for observational
  details). An arrow points to the counter image of two of the four background
  galaxies distorted by the cluster's gravitational field to create the
  prominent giant arc 20 arcsec west of the BCG. Smaller panels on top and to
  the right show enlarged high-resolution views of the arc and its counter image
  as observed with HST/ACS in a 1200s snapshot with the F606W filter. }
\end{figure*}

The bright ($V=21.0$) giant arc is centered at $\alpha$ (J2000) =
12$^{\rm h}$ 06$^{\rm m}$ 10.75$^{\rm s}$, $\delta$ (J2000) =
$-$08$^{\circ}$ 48' 04.5'', about 20\arcsec west of the BCG.  Its
unusually red colour is apparent in Fig.~\ref{imvik} which shows a
composite V, I, K image of the cluster core. Also prominent, and
marked by the arrow in Fig.~\ref{imvik}, is the counter image, clearly
identifiable by its colour, at $\alpha$ (J2000) = 12$^{\rm h}$
06$^{\rm m}$ 11.27$^{\rm s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 47'
43.0''. Enlarged high-resolution views of arc and counter image
provided by HST are shown in the margins of Fig.~\ref{imvik}. The
photometric properties of arc and counter image are summarized in
Table~\ref{arctbl}; note however that at the resolution of our
ground-based images both the arc and its counter image is blend of
several objects. Figure~\ref{arcspec} shows the spectrum of the giant
arc in MACS\,J1206.2$-$0847 as observed with FORS1 on the VLT.

\begin{table*}
\centering
\caption{Extinction-corrected magnitudes and colours of the
giant arc in MACSJ1206.2$-$0847 and its counter image
(cf.\ Fig.~\ref{imvik}). \label{arctbl}}
\begin{tabular}{@{}lccccccccccc@{}}
\hline
object & I & R & V & J & K & R$-$I & V$-$I & V$-$R & R$-$J & R$-$K & J$-$K\\
\hline
giant arc    & $19.27\pm 0.07$ & $20.27\pm 0.11$ & $20.99\pm 0.08$ & $17.82\pm 0.06$ & $15.95\pm 0.06$ & $1.00\pm 0.12$ & $1.72\pm 0.10$ & $0.72\pm 0.13$ & $2.45\pm 0.12$ & $4.32\pm 0.12$ & $1.87\pm 0.08$ \\
counter image & $20.41\pm 0.06$ & $21.34\pm 0.07$ & $21.94\pm 0.07$ & $18.88\pm 0.06$ & $17.14\pm 0.06$ & $0.93\pm 0.09$ & $1.52\pm 0.09$ & $0.59\pm 0.10$ & $2.46\pm 0.09$ & $4.20\pm 0.09$ & $1.74\pm 0.09$ \\
\hline
\end{tabular}
\end{table*}

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{arc_spectrum.eps}
\caption{\label{arcspec}Spectrum of the giant gravitational arc in
  MACS\,J1206.2$-$0847 as observed with FORS1 on the VLT. We interpret
  the single emission line observed at 7589\AA\ as being [O II], which
  translates into a redshift of 1.036 for the lensed galaxy. Our
  measurement is confirmed by independent observations conducted
  almost simultaneously by Sand et al.\ (2003) using the ESI echelle
  spectrograph on Keck-II which resolves the [O II] doublet thereby
  making the identification unambiguous.}
\end{figure}

At $R-K=4.3$ the giant arc in MACS\,J1206.2$-$0847 is among the
reddest strongly lensed features currently known. Its colour is
comparable to that of the giant arc in Abell 370 ($R-K=4.1$,
Arag\'{o}n-Salamanca \& Ellis 1990) and only slightly bluer than the
red arc in Abell 2390 ($R-K=4.6$, Smail et al.\ 1993).

Our multi-band photometry allows the classification of the background
galaxy lensed into the giant arc. Standard template spectral energy
distributions (SEDs) of five galaxy types were redshifted to match
that of the giant arc. The filter response curves of each of the V, R,
I, J and K bands were then convolved with these SEDs and normalized to
the R band to produce predicted colours for each of the five galaxy
types, all relative to the R band. The resulting model
m$_{\lambda}$--m$_{R}$ colours, as well as the equivalent observed
colours of the arc, are shown in Fig.~\ref{arcsed}. We find the colour
distribution of the arc in MACS\,J1206.2$-$0847 to be consistent with
the background galaxy being a normal spiral of class Sbc.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{arc_sed.eps}
\caption{\label{arcsed}Broad-band colours of the giant gravitational
  arc in MACS\,J1206.2$-$0847, relative to the R band, compared to
  predicted colours for standard galaxy types at the same
  redshift. The best agreement is found for a spiral of type Sbc to
  Scd.}
\end{figure}

\section{Intra-cluster gas properties}
\label{gas}

Figure~\ref{xcont} shows contours of the adaptively smoothed X-ray
emission from MACS\,J1206.2$-$0847, as observed with Chandra/ACIS-I in
the 0.5--7 keV band, overlaid on the UH2.2m R-band image\footnote{
For this overlay, as well as for any other comparisons of the spatial
appearance of the cluster in the X-ray and optical wavebands, we have
used three X-ray point sources with obvious optical counterparts to
slightly adjust the astrometric solution of the X-ray image, namely by
$-$0.03 seconds in right ascension and $-$0.4 arcseconds in
declination.  We estimate the resulting, relative astrometry between
the optical and X-ray images to be accurate to better than 0.2
arcseconds.}.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.r-x.eps}
\caption{\label{xcont}Iso-intensity contours of the adaptively
  smoothed X-ray emission from MACS\,J1206.2$-$0847 in the 0.5--7 keV
  range as observed with Chandra's ACIS-I detector, overlaid on the
  UH2.2m R-band image. The algorithm used, {\textit Asmooth} (Ebeling,
  White \& Rangarajan 2005), adjusts the size of the smoothing kernel
  across the image such that the signal under the kernel has constant
  significance (here 3$\sigma$) with respect to the local background.
  The lowest contour is placed at twice the value of the X-ray
  background in this observation; all contours are logarithmically
  spaced by factors of 1.5.}
\end{figure}

At large distances from the center, the system's X-ray appearance is
close to spherical in projection. The central region at $r\la 250$
kpc, however, shows a pronounced ellipticity as well as non-concentric
X-ray flux contours in the cluster core. The observed elongation, as
well as the displacement of the innermost contours toward the very
compact X-ray core, are both in the direction of a group of galaxies
in the vicinity of the giant arc.

\subsection{Spatial analysis}
\label{xbeta}

Using {\sc Sherpa}, the fitting package provided with {\sc Ciao}, we
fit the observed X-ray surface brightness distribution within 2.5
arcmin of the cluster core with a two-dimensional spatial model,
consisting of an elliptical $\beta$ model
\[
S(r) = S_0 \left[ 1+\left(\frac{r}{r_0}\right)^2\right]^{-3\beta + \frac{1}{2}},
\]
(where $r=r(\phi)$ is the variable radius of an ellipse with
ellipticity $\epsilon$ and orientation angle $\Theta$), an additional
circular Gaussian component to account for the compact core, and a
constant background.

Point sources with detection significances exceeding 99\% (as measured with the
{\sc celldetect} algorithm) have been excised from the image, and a spectrally
weighted exposure map is used as a two-dimensional response function in the
fit. We use a composite exposure map to account for the differences between
photons of cosmic origin and high-energy particles, the latter being subject
neither to off-axis vignetting nor to variable detection rates due to spatial or
temporal variations in the CCD quantum efficiency (QE). At energies between 0.5
and 2 keV particles account for roughly 2/3 of the observed background; at
higher energies this fraction rises to over 90 per cent. Since, overall, more
than 80\% of the background events registered in the 0.5--2 keV band used here
are caused by particles we ignore the sky contribution altogether and compute a
background exposure map which incorporates the effects of bad pixels and
dithering but not those of a variable CCD QE and vignetting. A second exposure
map is computed using weights based on the spectrum of the target of our
observations. The spectral weights for this `cluster-weighted' exposure map are
created assuming a plasma model with $kT=12$ keV, a metal abundance of 0.3
solar, and the Galactic value of $4.23\times 10^{20}$ cm$^{-2}$ for the
equivalent Hydrogen column density value in the direction of the cluster,
consistent with the results of our spectral fits to the global cluster X-ray
spectrum (see below). The resulting exposure map shows significant vignetting of
more than 20\% across the ACIS-I field of view. The final exposure map used in
the following is then the weighted average of the background and cluster
exposure maps, with the weights being given by the fraction of counts from the
cluster and the background, and with the peak value set to that of the
cluster-weighted map.

Since the image used in the fit is relative coarsely binned ($2\times
2$ arcsec$^2$) and no small-scale spatial components are included in
the model, we do not convolve the model with the telescope
point-spread function (PSF). All parameters of the two-dimensional
spatial model are fit. Because of the low number of counts (zero or
one) in the large majority of image pixels, we use the C statistic
(Cash 1979) during the optimization process.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.2dfit-im.eps}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.2dfit-res.eps}
\caption{\label{2dxfit}Iso-intensity contours (logarithmically spaced)
  of the best-fitting analytic model (elliptical $\beta$ model plus
  circular Gaussian component) of the cluster emission in the 0.5--7
  keV range, overlaid on the observed image (top, logarithmic scaling)
  and the residual image (bottom, linear scaling).  The dashed ellipse
  in the bottom panel highlights the excess emission in the direction
  of the arc and marks the region excluded in the spatial fit of the
  data to the two-dimensional surface-brightness model.}
\end{figure}

Our spatial fit yields best-fit values for the parameters of the $\beta$ model
of $r_0=(23.6\pm 0.8)$ arcsec (corresponding to ($134\pm 5)$ kpc) for the core
radius, $\beta=0.57\pm 0.01$ for the slope parameter, $\epsilon=0.17\pm 0.01$
for the ellipticity, and $\Theta=(56\pm 2)$ degrees (counted North through
West). Our two-dimensional spatial fit also finds coordinates of $\alpha$
(J2000) = 12$^{\rm h}$ 06$^{\rm m}$ 12.10$^{\rm s}$, $\delta$ (J2000) =
$-$08$^{\circ}$ 48' 01.7'' for the centroid of the compact cluster core, offset
by $(8.1\pm 1.7)$ arcsec from the position $\alpha$ (J2000) = 12$^{\rm h}$
06$^{\rm m}$ 12.50$^{\rm s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 48' 07.1''
which marks the center of the elliptical component that describes the shape of
the X-ray emission on larger scales.

Figure~\ref{2dxfit} shows the contours of the best-fitting model
overlaid both on the observed exposure-corrected X-ray image and on
the residuals remaining when the model is subtracted from the
data. The residual image shows a clear excess of emission,
corresponding to about 300 photons, to the West of the cluster
core. Note that the best-fit model parameters quoted above were
obtained with the excess region excluded from the fit.

Since the two-dimensional fitting procedure allows no immediate
assessment of the goodness of fit other than via visual inspection of
the residuals (Fig.~\ref{2dxfit}), we also fit a one-dimensional,
spherical $\beta$ model to the radial X-ray surface brightness
profile. In this fit we adopt the center of the elliptical model
component as the overall center of the X-ray emission, and exclude a
60-degree-wide azimuthal section around the compact core and the
excess emission to the West-North-West. Again we account for
variations in the exposure time across the source and background
regions. Since all annuli contain at least 50 photons we are now
justified in using $\chi^2$ statistics in the fit.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{radprof.eps}
\caption{\label{1dxfit}Radial X-ray surface brightness profile of
  MACS\,J1206.2$-$0847 in the 0.5--7 keV band as observed with
  Chandra/ACIS-I. The profile is centered on the peak of the
  elliptical surface brightness component as determined in our
  two-dimensional fit (see text for details). The solid line shows the
  best-fitting $\beta$ model. The dashed horizontal line marks the
  surface brightness corresponding to twice the background level; the
  vertical dotted line shows the radial limit within which a $\beta$
  model provides an acceptable description of the data.}
\end{figure}

The resulting one-dimensional radial profile as well as the
best-fitting $\beta$ model are shown in Figure~\ref{1dxfit}. When
fitting out to a radius of about 150 arcsec (850 kpc), where the
observed surface brightness profile begins to drop below twice the
background level, we find the $\beta$ model to provide an unacceptable
fit to the data at a reduced $\chi^2$ value of 1.9 (48 data points, 45
degrees of freedom (d.o.f.). However, essentially the same model fits
the data very well ($\chi^2$=1.1 for 19 d.o.f.) within a radius of 66
arcsec (375 kpc).  At larger radii the observed slope varies such that
the best-fitting $\beta$ model first systematically underpredicts, then
systematically exceeds the observed values. The best-fit values
of $S_0=(3.23\pm 0.13)$ ct arcsec$^2$, $r_0=(20.1\pm 1.3)$ arcsec
(corresponding to ($114\pm 7$) kpc), and $\beta=0.57\pm 0.02$, all of
which are consistent with the results from our two-dimensional fit,
thus allow a credible parametrization of the observed emission out to
about 375 kpc, provided the compact core and the excess emission
region to the West are excluded. At larger radii, the X-ray morphology
of MACS\,J1206.2$-$0847 is again too complex to be described by a
simple $\beta$ model.

Although the one-dimensional model is too simplistic to adapt to the spatial
variations in the X-ray emission near the cluster core or at very large radii,
it provides an adequate global description of the cluster. Extrapolating the
model to $r_{\rm 200}$ (see Section~\ref{xmass}) yields values of $(4.4\pm 0.07)
\times 10^{-12}$ erg s$^{-1}$ cm$^{-2}$ and $(24.3\pm 0.5)\times 10^{44}$ erg
s$^{-1}$ for the total X-ray flux and X-ray luminosity of MACS\,J1206.2--0847,
respectively (0.1--2.4 keV). The quoted errors do, however, not account for any
systematic errors which are bound to be present in view of the fact that X-ray
emission is detected only out to about 1 Mpc from the cluster center and that,
at larger radii, the $\beta$ model tends to overpredict the observed X-ray
surface brightness (see Fig.~\ref{1dxfit}). A more robust measurement can be
obtained within $r_{\rm 1000}$ (1.04 Mpc) and yields lower limits to the total
X-ray flux and luminosity of $(3.8\pm 0.06) \times 10^{-12}$ erg s$^{-1}$
cm$^{-2}$ and $(21.0\pm 0.4)\times 10^{44}$ erg s$^{-1}$.

\subsection{Spectral analysis}
\label{xspec}

We measure a global temperature for the intra-cluster medium (ICM) in
MACS\,J1206.2$-$0847 by extracting the X-ray spectrum from $r=70$ kpc
to $r=1$ Mpc (r$_{1000}$) and using \textit{Sherpa} to fit a MEKAL
plasma model (Mewe et al.\ 1985) with the absorption term frozen at
the Galactic value. We find ${\rm k}T= (11.6\pm 0.66)$ keV, a high
value even for extremely X-ray luminous clusters (Chen et
al.\ 2007). Although the relatively high reduced-$\chi^2$ value of
this spectral fit of 1.4 is statistically acceptable, it could be
indicative of systematic effects such as spatial temperature
variations or the presence of multiphase gas. We find only mild
evidence of the former when fitting absorbed plasma models to the
X-ray spectra extracted from five concentric annuli. As shown in
Fig.~\ref{tprof}, the ICM temperature is consistently high ($\sim 12$
keV), with the exception of the core region where, at $r<130$ kpc, a
significant drop to about 9 keV (still a very high value) is
observed. The lower gas temperature measured around the cluster core
could be caused by the presence of a minor cool core, in agreement
with the results of our X-ray imaging analysis.

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{tprof.eps}
\caption{\label{tprof}Radial profile of the ICM temperature in
  MACS\,J1206.2$-$0847. The profile is centered on the peak of the
  elliptical surface brightness component as determined in our
  two-dimensional fit (see text for details). Horizontal bars mark the
  width of the respective annulus, defined such that each region
  contains about 3,000 net photons.}
\end{figure}

\section{Properties of the central galaxy}

The spectrum of the central cluster galaxy taken at CFHT
(Fig.~\ref{cdspec}, see also Section~\ref{optspec}) covers only a
small wavelength range mostly redward of the 4000\AA\ break.  While
the wavelength coverage is thus insufficient to check for the presence
of H$\beta$ and OIII in emission, we do detect faint OII emission
($\lambda_{\rm rest}=3727$ \AA), albeit at a much lower level than
typically observed in the central cluster galaxies of large cool-core
clusters (e.g., Allen et al.\ 1992, Crawford et al.\ 1995).

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{cd_spectrum.eps}
\caption{\label{cdspec}Low-resolution spectrum of the central cluster
galaxy in MACSJ1206.2$-$0847 as observed with CFHT (see text for
details). A tentative detection of OII in emission is highlighted, as
are a series of absorption features characteristic of the old stellar
population dominating cluster ellipticals.}
\end{figure}

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{MACSJ1206.2-0847.r-x.core.hst.eps}
\caption{\label{corealign}As Figure~\ref{xcont} but using the HST
  image and zoomed in to show only the cluster core.}
\end{figure}

Bright NVSS (162 mJy at 1.4~GHz) and Molonglo (900 mJy at 365~MHz)
radio sources are coincident with the central galaxy. The radio source
exhibits a relatively steep spectrum ($\alpha=-1.32\pm0.05$) and is
similar to the one found in MACSJ1621.3+3810 (Edge et al.\ 2003) but
an order of magnitude more powerful (6$\times 10^{26}$ W~Hz$^{-1}$ at
1.4~GHz). The radio source is also detected at 74 MHz in the VLA
Low-frequency Sky Survey (VLSS) with a measured flux of
$(6.67\pm0.7)$ Jy that is in excellent agreement with the power-law
prediction from the detections at 365 MHz and 1.4 GHz.

Although no clear signs of cavities are detected in the cluster core, we can,
from our Chandra data alone, not conclusively assess the degree of interaction
between the BCG and the surrounding intra-cluster medium. High-resolution radio
data would be required to map the radio morphology of the BCG. We do observe
though a slight, but significant displacement of $(1.7\pm 0.4)$ arcsec --
$(9.6\pm 2.3)$ kpc at the cluster redshift -- between the position of the
central cluster galaxy ($\alpha$ (J2000) = 12$^{\rm h}$ 06$^{\rm m}$ 12.14$^{\rm
  s}$, $\delta$ (J2000) = $-$08$^{\circ}$ 48' 03.3'') and the X-ray centroid of
the compact cluster core (Fig.~\ref{corealign}). Having carefully aligned the
optical and X-ray images (see footnote in Section~\ref{gas}) we estimate that at
most 10\% of this offset can reasonably be attributed to residual astrometric
uncertainties; the majority of the misaligned is thus real. Similar offsets
($\sim$ 10 kpc) have been noted in previous Chandra studies of galaxy clusters
(Arabadjis, Bautz \& Garmire 2002) and may also have contributed to minor
optical/X-ray misalignments in the cores of cooling clusters observed with ROSAT
(Peres et al.\ 1998).

In addition, the velocity offset of 550 km s$^{-1}$ between the
central galaxy and the cluster mean velocity is comparable to the
largest peculiar velocities observed in local clusters (Zabludoff et
al.\ 1990; Hill \& Oegerle 1993; Oegerle \& Hill 2001), although we
note that the non-Gaussian velocity distribution in this cluster
(Fig.~\ref{zhist}) complicates the measurement of an accurate systemic
velocity.

Spatial and velocity offsets between central galaxy and X-ray peak are
not expected in the simplest cool-core scenario, although it is worth
noting that an offset of the same size as observed by us here (10 kpc)
has been seen in the best-studied cool-core cluster, Perseus
(B\"ohringer et al.\ 1993), and there has been a widespread
realization that the physics of cool cluster cores are more complex
and the role of AGN feedback more important than previously thought
(Peterson et al.\ 2001; Soker et al.\ 2002; Edge 2001; Mittal et
al.\ 2008). We conclude that MACSJ1206.2$-$0847 is likely to contain a
moderate cool core, as well as an extremely luminous radio galaxy (one
of the most powerful ones known in cluster cores at $z>0.4$), with the
observed disturbances being likely due to a recent or still ongoing
cluster merger.

\section{Mass measurements}

\subsection{Virial mass}

Using the measured redshifts of cluster members and their spatial distribution
as projected on the sky, we determined the virial cluster mass based on the
method of Limber \& Mathews (1960) in which the mass is calculated as
\begin{equation}
M_V = \frac{3\pi}{2} \frac{\sigma_{P}^{2}R_H}{G}.
\label{M_Veqn}
\end{equation}

Here $\sigma_P$ is the one-dimensional (radial) velocity dispersion and $R_H$ is
the projected mean harmonic point-wise separation (projected virial
radius). $R_H$ is defined by
\begin{equation}
R_H^{-1} = \frac{1}{N^2}\sum_{i<j} \frac{1}{|\mathbf{r}_i-\mathbf{r}_j|},
\label{R_Heqn}
\end{equation}

where N is the number of galaxies, $|\mathbf{r}_i-\mathbf{r}_j|$ is the
projected separation of galaxies $i$ and $j$, and the $ij$ sum is over all
pairs.  Being a pairwise estimator this quantity is sensitive to close pairs and
quite noisy (Bahcall \& Tremaine 1981).  It also systematically underestimates
the radius for a rectangular aperture typical of cluster redshift surveys
(Carlberg et al.\ 1996).  Carlberg et al.\ therefore introduce a new radius
estimator, the ringwise projected harmonic mean radius $R_h$, given by
\begin{eqnarray}
R_h^{-1} &=& \frac{1}{N^2} \sum_{i<j}\frac{1}{2\pi} \int_{0}^{2\pi}\frac{d\theta}{\sqrt{r_{i}^{2}+r_{j}^{2}+2r_ir_j\cos\theta}} \nonumber \\
&=& \frac{1}{N^2} \sum_{i<j}\frac{2}{\pi (r_i+r_j) K(k_{ij})}.
\label{R_heqn}
\end{eqnarray}

Here $r_i$ and $r_j$ are the projected distances from the cluster center to
galaxies $i$ and $j$ respectively, $k_{ij}^{2} = 4r_ir_j/(r_i+r_j)^2$, and
$K(k)$ is the complete elliptic integral of the first kind in Legendre's
notation.  This estimator requires an explicit choice of cluster center and
assumes the cluster is spherically symmetric with respect to this center.  It
treats one of the particles in the pairwise potential $|r_i-r_j|^{-1}$ as having
its mass distributed in a ring around the cluster center. $R_h$ is less
sensitive to close pairs, less noisy, and tolerates non-circular apertures
better than $R_H$.  If the cluster is significantly flattened or subclustered,
however, $R_h$ will systematically overestimate the true projected virial
radius.

We calculate both radius estimators in our mass determinations to investigate
the resulting systematic uncertainty in the virial mass. For our sample the
virial radius and mass derived using $R_h$ are 7\% larger than those based on
$R_H$. We choose to use $R_h$ as the more robust estimator in our analysis and
define the three-dimensional (deprojected) virial radius as
\begin{equation}
r_V = \frac{\pi}{2}R_h.
\label{r_Veqn}
\end{equation}

Our determination of the virial radius estimators $R_H$ and $R_h$ was made using
all 62 galaxies with redshifts within $3\sigma$ of the cluster mean (see
Section~\ref{galprop}).  The resulting projected virial radius is $R_h = 1.176$
Mpc ($R_H = 1.096$ Mpc) with a virial mass of $3.861\times10^{15}$ M$_{\odot}$
and a three-dimensional virial radius of $r_V = 1.847$ Mpc.

\subsection{X-ray mass}
\label{xmass}
To estimate the total gravitational mass of the cluster from its X-ray emission,
we need to assume that the cluster is in hydrostatic equilibrium. In addition,
we need a description of the density as well as the temperature of the
intracluster gas, often assumed to be isothermal. For a cluster with a core
region as disturbed as the one of MACSJ1206.2$-$0847 such simplifying
assumptions are unlikely to be justified; however, an isothermal $\beta$ model
should provide an adequate description of the cluster outskirts and allow us to
obtain a crude mass estimate.

From the X-ray temperature (see Section~\ref{xspec}) we estimate the virial
radius $R_{200}$ using the formula of Arnaud et al.\ (2002),
\begin{eqnarray*}
R_{200} & = & 3.80~\beta_T^{1/2}\Delta_z^{-1/2}~(1+z)^{-3/2} \nonumber \\
        &  & \times(\frac{kT}{10{\rm keV}})^{1/2}h_{50}^{-1}~{\rm Mpc} \nonumber
\end{eqnarray*}
with
\begin{eqnarray*}
\Delta_z  =  (200\Omega_0)/(18\pi^2\Omega_z). \nonumber
\end{eqnarray*}
Here $\beta_T=1.05$ (Evrard et al.\ 1996) is the normalization of the virial relation, i.e.,
$GM_v/(2R_{200})= \beta_T~{\rm k}T$.  Then, the total mass of the
cluster within a radius $r$ can be computed with the help of the $\beta$-model profile
discussed in Section~\ref{xbeta}:
\begin{eqnarray}
M(r) & = & 1.13\times10^{14}\beta\frac{T}{\rm{keV}}\frac{r}{\rm{Mpc}}\frac{(r/r_c)^2}{1+
(r/r_c)^2}M_{\odot}
\end{eqnarray}
(Evrard et al.\ 1996). Using the above equations we find $R_{\rm 200}=(2.3\pm
0.1)$ Mpc as an approximate value for the virial radius. The X-ray estimates for
the total mass within $R_{\rm 200}$ and the mass within the core region (defined
as the sphere interior to the giant arc, i.e.\ $r<119$ kpc) are then $(17.1\pm
1.2)\times 10^{14}$ M$_{\sun}$ and $(0.46\pm 0.05)\times 10^{14}$~M$_{\sun}$,
respectively.

\subsection{Lensing mass}

Using the high-resolution HST images, we have identified within the
giant arc two features A \& B that are replicated six times. The same
two features are also identified in the southern part of the counter
image as shown in Figure~\ref{fig:critics_closeup}. The northern part
of the counter image is most likely not multiply imaged. Altogether,
the giant arc and its counter image represent a seven-image multiple
system which we use to constrain a strong-lensing model of the cluster
mass distribution.

To model the cluster core we used \textsc{Lenstool}\footnote{publicly
  available at http://www.oamp.fr/cosmology/lenstool} (Kneib et
al.\ 1996; Jullo et al.\ 2007) which now uses a Bayesian MCMC sampler
to optimize the cluster mass model and generate robust lens results.
We have followed the procedure of Limousin et al.\ (2007) to model the
mass distribution using one cluster-scale dark-matter halo described
by a truncated PIEMD (Pseudo Isothermal Elliptical Mass Distribution),
as well as an additional 84 truncated galaxy-scale PIEMD potentials to
describe the dark-matter halos associated with the brightest cluster
member galaxies selected from the cluster V--K red
sequence. Furthermore, to minimize the number of free parameters, we
assumed that the mass of galaxy-size halos scales with the K-band
luminosity of the associated galaxy (Natarajan \& Kneib 1997). The
obtained critical curves shown in figure~\ref{fig:critics_core}
display a winding shape in between the galaxies (see close-up in
Fig.~\ref{fig:critics_closeup}), which explains the extreme elongation
of the giant arc.

Using the most probably strong-lensing mass model, we estimate the mass
enclosed by the giant arc. We find $M(<21'') = (112.0 \pm 5) \times
10^{12}\ M_{\odot}$ and a mass-to-light ratio interior to the giant
arc of $M/L = 56\pm2.5$.

These numbers are very robust and depend little on the mass profile
assumed for the dark-matter distribution of the cluster. Caution is
advised though when extrapolating to larger radius, as the slope of
the cluster mass profile is not well constrained by only one
(multiple) arc. Deeper, high-resolution imaging (with, e.g., ACS or
WFC3), however, would likely detect a large number of multiple images
as was the case for Abell~1703 (Limousin et al.\ 2008), allowing us to
accurately measure the slope of the cluster dark-matter profile. Note
that the current best estimate of the Einstein radius at $z\sim 7$ is
nearly 45\arcsec, making this cluster a superb cosmological telescope
to probe the first galaxies in the Universe.

The total magnification of the system (both the giant arc and the part
of the counter image that is multiply imaged) is about 80$\pm$10, one
of the largest amplification factors known for a giant arc. A detailed
model of the arc surface brightness is beyond the scope of this study
but will be presented in a future paper (Clement et al., in
preparation).

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{newfinal.eps}
\caption{\label{fig:critics_core} HST/ACS image of the cluster core
  observed through the F606 filter. The critical line computed at the
  redshift of the giant arc ($z=1.036$) is shown in red. The yellow
  curve shows the corresponding caustics in the source plane. }
\end{figure}

\begin{figure}
\epsfxsize=0.5\textwidth
\epsffile{poststamps.eps}
\caption{ \label{fig:critics_closeup} Close-up of the critical lines
  at the positions of the giant arc and the counter image. The yellow
  ellipses and identifiers mark components of the set of multiple
  images used to constrain the lens model. }
\end{figure}

\section{Summary and Conclusions}

We present a comprehensive multi-wavelength analysis of the properties of the
massive galaxy cluster MACS\,J1206.2--0847. At a redshift of $z=0.4385$, the
system acts as a gravitational lens for a background galaxy at $z=1.04$,
resulting in spectacular gravitational arc of high surface brightness, 15\arcsec
in length, a total magnitude of V=21.0, and of unusual, very red colour of
$R-K=4.3$. Our X-ray analysis based on Chandra data yields global X-ray
properties ($L_{\rm X}=2.3\times 10^{45}$ erg s$^{-1}$, 0.1--2.4 keV, and ${\rm
  k}T=11.6\pm 0.7$ keV) that make this cluster one of the most extreme systems
known at any redshift.

Belying its relaxed appearance at optical wavelengths, MACS\,J1206.2--0847
exhibits many signs of ongoing merger activity along the line of sight when
looked at more closely, including a disturbed X-ray morphology in the cluster
core, a small but significant offset of the peak of the X-ray emission from the
brightest cluster galaxy, and a very high velocity dispersion of 1580 km
s$^{-1}$. The strongest indication of recent or ongoing cluster growth, however,
is obtained from a comparison of X-ray, virial, and lensing mass estimates for
this system. A high-resolution image of the giant arc and its counter image
obtained with HST allows us to create a lens model that places tight constraints
on the mass distribution interior to the arc. The strong-lensing value of the
mass of the cluster core of $(11.2 \pm 0.5) \times 10^{13}\ M_{\odot}$ thus
obtained is higher by about a factor of two than the X-ray estimate of $(4\pm
0.4)\times10^{13}$ M$_{\odot}$. A similar discrepancy is found between the X-ray
estimate of the total mass within $r_{\rm 200}$ and the virial mass estimate
derived from radial-velocity measurements for 38 cluster galaxies.

Comparable discrepancies between X-ray and lensing mass estimates, in particular
for cluster cores, have been reported before for other systems, the perhaps most
famous example being A1689 (e.g., Miralda-Escud\'e \& Babul 1995; Xue \& Wu
2002; Limousin et al.\ 2007). In all cases, including the one presented here,
the mass derived using gravitational-lensing features is two to three times
higher than the one obtained by an X-ray analysis assuming hydrostatic
equilibrium. In agreement with simulations (Bartelmann \& Steinmetz 1996),
detailed observational studies of such discrepancies for individual clusters
find deviations from hydrostatic equilibrium and the presence of substructure
along the line of sight to be responsible, tell-tale signs being offsets between
the X-ray peak and the location of the BCG as well as extreme elongations and
structure in radial-velocity space (Allen 1998, Machacek et al.\ 2001). We
conclude that MACS\,J1206.2--0847 is a merging cluster with a merger axis that
is close to aligned with our line of sight. A modest cool core has either
survived the merger or is in the process of formation.

The discovery of a giant arc in this MACS cluster underlines yet again the
efficiency of X-ray luminous clusters as gravitational lenses. Much deeper,
high-resolution images of systems like MACS\,J1206.2--0847 will, owing to the
large number of multiple images detected, allow a detailed mapping of the mass
distribution in the cluster core, and, through the power of gravitational
magnification, provide an ultra-deep look at the very distant Universe.

\section*{Acknowledgments}

We thank Alexey Vikhlinin for very helpful advice and suggestions
concerning systematic effects in the Chandra data analysis.  HE
gratefully acknowledges financial support from grants NAG 5-8253 and
GO2-3168X. ACE thanks the Royal Society for generous support during
the gestation phase of this paper.  JPK acknowledges support from the
{\it Centre National de la Recherche Scientifique} (CNRS), and the ANR
grant 06-BLAN-0067.  EJ acknowledges support from an ESO-studentship
and from CNRS.