\chapter*{ }
\vskip -5truecm

\centerline {\Large\bf Quantization and eigenvalue distribution}
\centerline {\Large\bf of noncommutative scalar field theory}
\vskip 10truemm
\centerline {\bf Harold Steinacker}
\vskip 5truemm
\centerline{Institute for Theoretical Physics, University of Vienna}
\centerline{Boltzmanngasse 5, A-1090 Vienna, Austria}

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\vskip 5truemm

\begin{quote}
\small{\bf Abstract.} {\it

The quantization of noncommutative scalar field theory
is studied from the matrix model point of view, exhibiting
the significance of the eigenvalue distribution.
This provides a new framework to study
renormalization, and predicts a
phase transition in the noncommutative $\phi^4$ model.
In 4-dimensions, the corresponding
critical line is found to terminate at a non-trivial point.
 }
\end{quote}

\begin{quote}
\small{\bf Key words:} {\it Noncommutative field theory, phase transition,
matrix model, renormalization }
\end{quote}
\vskip 5truemm

\centerline {\sc 1. INTRODUCTION: NC FIELD THEORY AND UV/IR MIXING}
\vskip 3truemm

Noncommutative (NC) field theory has been studied intensively in recent years,
see e.g. \cite{douglas, Szabo} and references therein.
Part of the motivation has been the hope that the UV divergences of Quantum Field
Theory should be controlled and perhaps regularized by noncommutativity,
which in turn should be related to quantum fluctuations of geometry
and quantum gravity.
In the simplest case of the quantum plane
${\Bbb R}^{d}_\theta$, the coordinate functions $x_i, i=1,...,d$ satisfiy
the canonical commutation relations
\begin{equation}
[x_i,x_j] = i \theta_{ij}.
\label {CC}
\end{equation}
We assume for simplicity that $\theta_{ij}$ is
nondegenerate and maximally-symmetric, characterized by a single
parameter $\theta$ with dimension $({\rm length})^2$. This
implies uncertainty relations among the coordinates,
\begin{equation}
\Delta x_i \Delta x_j \geq \frac 12 \theta_{ij} \approx \theta.
\label{uncertainty}
\end{equation}
This leads to an energy scale
\begin{equation}
\Lambda_{NC} = \sqrt{\frac 1{\theta}},
\end{equation}
and noncommutativity is expected to be important for energies
beyond $\Lambda_{NC}$.

The most naive guess would be that $\Lambda_{NC}$ plays the role of a UV cutoff
in Quantum Field Theory.
However, this turns out not to be the case.
The quantization of NC field theory
not only suffers from divergences in the UV, but also from new divergencies
in the IR, which are in general not under control up to now.
This type of behavior can to some extent be expected from \eq{CC}.
To understand this, expand a scalar field on ${\Bbb R}^{d}_\theta$
in a basis of plane waves,
$\phi(x) = \int d^d p \,\phi(p) \exp(i p x)$, and
consider the range of momenta $p$. Formally, $p \in {\Bbb R}^d$
has neither an IR nor a UV cutoff.
As usual in field theory, it is necessary however
to impose cutoffs $\Lambda_{IR} \leq p \leq \Lambda_{UV}$, with the hope that the
dependence on the cutoffs can be removed in the end;
this is essentially the defining property of a renormalizable field theory.

In view of \eq{uncertainty}, if $\Delta x_i$ is minimized according to
$\Delta x_i = \Lambda_{UV}^{-1}$,
it implies that some $\Delta x_j \geq \theta \Lambda_{UV}$.
This means that momenta near
$\Lambda_{UV}$ are inseparably linked with momenta in the IR,
and suggests $\Lambda_{UV} \Lambda_{IR} = \Lambda_{NC}^2$.
A consistent way to impose IR and UV cutoffs in NC field theory is to
replace ${\Bbb R}^n_\theta$ by some compact NC spaces such as fuzzy tori, fuzzy
spheres etc. The algebra of functions is then given by a
finite matrix algebra $Mat({\cal N}\times{\cal N},{\Bbb C})$. One then finds
quite generally
\begin{equation}
\Lambda_{IR} =  \sqrt{\frac 1{N\theta}} \quad\ll\quad \Lambda_{NC} =
\sqrt{\frac 1{\theta}} \quad\ll\quad \Lambda_{UV} = \sqrt{\frac N{\theta}}
\end{equation}
 where ${\cal N} \approx N^{d/2}$
(depending on the details of the compact NC space) is the
dimension of the representation space. This shows that
the scales in the UV and IR are intimately related, which is
the origin of $UV/IR$ mixing in NC field theory. It is then
not too surprising that a straightforward Wilsonian renormalization
scheme involving only $\Lambda_{UV}$
will generically fail \cite{SMV}.

The remainder of this paper is a qualitative discussion of a
new approach to quantization of euclidean NC scalar field theory,
which was proposed in \cite{NCscalarFT} with the goal to
take into account more properly the specific features of NC field theory.

\vskip 5truemm
\centerline {\sc 2. NONCOMMUTATIVE SCALAR FIELD THEORY}
\vskip 3truemm

Consider the scalar $\phi^4$ model on a suitable $d$-dimensional NC space,
\begin{equation}
S =  \int d^d x\, \left(\frac 12  \phi \Delta \phi
+ \frac 12 m^2 \phi^2 + \frac{\lambda}{4} \phi^4\right)
 = S_{kin} + \int d^d x\, V(\phi).
\label{Rd-action}
\end{equation}
We assume that the algebra of functions on the NC space is represented
as operator algebra on some {\em finite-dimensional} Hilbert space
(this is the defining property of the so-called fuzzy spaces), so that
$\phi \in Mat({\cal N}\times {\cal N},{\Bbb C})$ is a Hermitian matrix.
The integral is then replaced by the
appropriately normalized trace,  $\int d^d x = (2\pi\theta)^{n/2} Tr()$.

In this finite setup, the quantization is defined by an integral over all
Hermitian matrices, e.g.
\begin{eqnarray}
Z &=& \int [{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] e^{-S},  \label{Z-full}\\
\langle \phi_{i_1 j_1} \cdots \phi_{i_n j_n} \rangle &=&
\frac 1Z \int [{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] e^{-S}\;  \phi_{i_1 j_1} \cdots \phi_{i_n j_n}
\end{eqnarray}
As a warm-up, we recall the analogous but simpler case of

\paragraph{Pure matrix models.}

These are defined by an action of the form
\begin{equation}
S^{M.M.} = {\cal N} Tr(V(\phi))
\end{equation}
for some polynomial $V(\phi)$,
where again $\phi \in Mat({\cal N}\times {\cal N},{\Bbb C})$ is a Hermitian matrix.
These pure matrix models can be solved exactly by the following change
of coordinates in matrix space:
\begin{equation}
\phi = U^{-1} D U
\end{equation}
where $D$ is a diagonal matrix with real eigenvalues $\phi_i$, and $U \in U({\cal N})$.
Using $[{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F} \phi] = \Delta^2(\phi_i) dU d\phi_1 ... d \phi_N$, where $dU$ denotes the
Haar measure for $U({\cal N})$ and $\Delta(\phi_i) = \prod_{i<j}(\phi_i-\phi_j)$
is the Vandermonde-determinant.
The integral over $dU$ can be carried out trivially,
reducing the path integral to an integral over eigenvalues,
\begin{equation}
Z =  \int d\phi_i \Delta^2(\phi_i-\phi_j)\exp(-{\cal N} \sum_i V(\phi_i))
= \int d\phi_i \exp(-S_{eff}^{M.M}(\vec\phi))
\label{Z-MM}
\end{equation}
with an ``effective action'' of the form
\begin{equation}
\exp(-S_{eff}^{M.M.}(\phi_i)) = \exp(\sum_{i\neq j}
\log|\phi_i-\phi_j| -{\cal N} \sum_i V(\phi_i)).
\label{Seff-MM}
\end{equation}
We can assume that the eigenvalues are ordered,
$\phi_1 \leq \phi_2 \leq ... \leq \phi_{\cal N}$, and
denote with $\cE_{\cal N}$
the space of ordered ${\cal N}$-tupels $\vec \phi = (\phi_1, ..., \phi_{\cal N})$.
They will be interpreted as
coordinates of a point in the space of eigenvalues $\cE_{\cal N}$.

The remaining integral over $\phi_i$ can be evaluated with a variety of techniques
 such as orthogonal polynomials, Dyson- Schwinger resp. loop equations,
or the saddle-point method. It turns out that the large ${\cal N}$ limit
is correctly reproduced by the saddle-point method,
because the Vandermone-determinant
$\Delta^2(\phi_i) = \exp(\sum_{i\neq j} \ln|\phi_i-\phi_j|)$ corresponds to a
strongly repulsive potential between the eigenvalues, which
leads to a strong localization in $\cE_{\cal N}$.
This means that the effective action $\exp(-S_{eff}^{M.M.}(\vec\phi))$
is essentially a delta-function in $\cE_{\cal N}$,
\begin{equation}
\exp(-S_{eff}^{M.M.}(\vec \phi)) \approx \exp(-{\cal N} Tr(V(\vec \phi_0))) \,
 \delta(\vec \phi - \vec \phi_0).
\label{localiz-delta}
\end{equation}
One now proceeds to determine the
localization of the maximum $\vec \phi_0 \in \cE_{\cal N}$ by solving
$\frac{\partial}{\partial \phi_i} S_{eff} =0$, which can be written as an
integral equation in the large ${\cal N}$ limit.
Expectation values can then be computed as
\begin{equation}
\langle f(\vec\phi)\rangle =  f(\vec\phi_0)
= f(\langle\vec\phi \rangle)
\end{equation}
which is indeed correct in the large ${\cal N}$ limit of pure matrix models,
consistent with  \eq{localiz-delta}.
This  "factorization'' of
expectation values is characteristic for a delta-function
integral density, and can be used to test the localization hypothesis
\eq{localiz-delta} resp.
the sharpness of the (approximate) delta function.

\paragraph{Strategy for the full model.}

This analogy suggests to apply similar techniques
also for the field theory \eq{Z-full}.
At first sight, this may appear impossible,
because the action is no longer invariant under
$U({\cal N})$. However, let us assume that we could somehow evaluate the
``angular integrals''
\begin{equation}
\int dU \exp(-S_{kin}(U^{-1}(\phi) U))  = : e^{-\cF(\vec\phi)},
\end{equation}
Note that the resulting $\cF(\phi_i)$ is a totally symmetric, analytic function
of the eigenvalues $\phi_i$ of the
field $\phi$, due to the integration over the unitary matrices $U$.
Using this definition, the partition function
can indeed be cast into the same form as \eq{Z-MM},
\begin{equation}
Z  = \int d \phi_i \Delta^2(\phi_i)\exp(-\cF(\vec\phi) -(2\pi\theta)^{d/2}\;Tr V(\vec\phi))
=  \int d\phi_i \exp(-S_{eff}(\vec\phi))
\label{Z-F}
\end{equation}
(with the obvious interpretation of $Tr V(\vec\phi)$), defining
\begin{equation}
\exp(-S_{eff}(\vec\phi)) =\Delta^2(\phi_i)\exp(- \cF(\vec\phi) - (2\pi\theta)^{d/2}\;Tr V(\vec\phi)).
\label{S-eff}
\end{equation}
Of course we are only allowed
to determine observables depending on the eigenvalues with this effective action
(the extension to other observables is discussed in \cite{NCscalarFT}).
These are determined by
$S_{eff}(\vec\phi)$ in the same way as for pure matrix models,
since  the degrees of freedom
related to $U$ are integrated out.
This demonstrates that all thermodynamic properties
of the quantum field theory, in particular
the phase transitions,  are determined
by $S_{eff}(\vec\phi)$  and  the resulting eigenvalue distribution
in the large ${\cal N}$ limit.
The advantage of this
formulation is that it is very well suited to include interactions,
and naturally extends to the non-perturbative domain.

So far, all this is exact but difficult to apply, since we will not be
able to evaluate $\cF(\vec\phi)$ exactly.
However, the crucial point in \eq{S-eff} is the presence of the
Vandermonde-determinant
$\Delta^2(\phi_i) = \exp(\sum_{i\neq j}\ln|\phi_i-\phi_j|)$,
which prevents the eigenvalues from
coinciding and strongly localized them. This effect cannot be
canceled by $\cF(\vec\phi)$, because it ts analytic.
This and the analogous exact analysis of the pure matrix models
suggests the following {\em central hypothesis} of our approach:
\eq{S-eff} is strongly localized, and
the essential features of the full QFT
are reproduced by the following approximation
\begin{equation}
\exp(-S_{eff}(\vec\phi)) \approx \exp(-\cF(\vec\phi_0)
- (2\pi\theta)^{d/2}\;Tr V(\vec\phi_0)) \,
\delta(\vec \phi - \vec \phi_0),
\label{localiz-delta-full}
\end{equation}
where $\vec \phi_0$ denotes the maximum (saddle-point) of $S_{eff}(\vec\phi)$.
The
remaining integral over the eigenvalues $\phi_i$ can
hence be evaluated by the saddle-point method.
This will be justified to some extent and made more precise
below, by studying
$\cF(\vec\phi)$ using the free case.
If $\cF(\vec\phi)$ is
known, the free energy can be found by determining the minimum
$\phi_0$ of $S_{eff}(\vec\phi)$.
However, \eq{localiz-delta-full} strongly suggests that also
correlation functions can be computed by restricting
the full matrix integral to the
dominant eigenvalue distribution  $\vec\phi_0$. This
would provide nonperturbative control over the full NC QFT.

It is important to keep in mind that
a saddle-point approximation {\em before} integrating out $U({\cal N})$ would be
complete nonsense. It is only appropriate for the effective action
\eq{localiz-delta-full}.
The possibility of separating the path integral into
these 2 steps is only available for NC field theory.

\vskip 5truemm
\centerline {\sc 3. THE CASE OF FREE FIELDS}
\vskip 3truemm

For free fields, we can verify the validity of the basic hypothesis
\eq{localiz-delta-full}, and compute the unknown function
$\cF(\vec\phi)$ explicitly at least in some domain. This
will then be applied in the interacting case.

To check localization in the sense of \eq{localiz-delta-full},
we need to show that
the expectation values of all observables which depend only on the eigenvalues
$\vec\phi$ factorize, and can be computed by evaluating the observable at a
specific point $\vec  \phi_0$.
A complete set of such observables is given by the product of traces of
various powers of $\phi$, which in field theory language is
$\langle(\int d^d x \phi(x)^{2n_1})...(\int d^d x \phi(x)^{2n_k}) \rangle$.
It is shown in \cite{NCscalarFT} using Wicks theorem
that  in the large $N$ resp. $\Lambda_{UV}$ limit,
\begin{eqnarray}
\frac{\langle\big( \frac 1V\int d^d x \phi(x)^{2n_1}\big)
\big( \frac 1V\int d^d x \phi(x)^{2n_k}\big) \rangle}
{\langle  \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_1}
\langle  \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_k}}
&=& \frac{\langle \frac 1V\int d^d x \phi(x)^{2n_1}\rangle}
  {\langle  \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_1}}\;
\;\frac{\langle \frac 1V\int d^d x \phi(x)^{2n_k} \rangle}
{\langle  \frac 1V\int d^d x \phi(x)^{2} \rangle^{n_k}} \nonumber\\
&=& N_{Planar}(2n_1)\; ... \;N_{Planar}(2n_k).
\label{cluster}
\end{eqnarray}
Here $N_{Planar}(2n)$ is the number of planar contractions of a
vertex with $2n$ legs.
The non-planar contributions always involve oscillatory
integrals, and do not contribute to the above ratio in the large $N$ limit.

This is exactly the desired factorization property, which implies
that the effective action of the eigenvalue sector localizes
as in \eq{localiz-delta-full}, after suitable rescaling.
This is only true for
noncommutative field theories, where nonplanar (completely contracted) diagrams
are suppressed by their oscillating behavior, so that only
the planar diagrams contribute\footnote{this is only true for observables of
the above type, not e.g. for propagators}.

To complete the analysis of the free case, we need to find the dominant
eigenvalue distribution $\vec\phi_0$. That is an easy task using the known
techniques from pure matrix models described above.
Writing
$\varphi(s) = (\phi_0)_j, \quad   s = \frac j{{\cal N}}$ for $s \in [0,1]$
in the continuum limit,
the saddle-point $\vec\phi_0$ then corresponds to a density of eigenvalues
$\rho(\varphi) = \frac{ds}{d\varphi}$
with $\int_{-\infty}^{\infty} \rho(p) dp =1$. This turns out to be
the famous Wigner semi-circle law
\begin{equation}
\rho(p) = \left\{\begin{array}{ll}\frac 2{\pi} \sqrt{1-p^2} & \;\; p^2<1\\
                           0, & \mbox{otherwise}.
                \end{array}\right.
\label{wigner-law}
\end{equation}
This is the same as for the pure Gaussian matrix model, and it follows that
the effective action for the eigenvalue sector is given by
\begin{equation}
S_{eff}^{free}(\vec\phi)
= f_0(m) -\sum_{i\neq j}\ln|\phi_i-\phi_j| + \frac{2{\cal N}}{\alpha_0^2(m)} (\sum\phi_i^2).
\label{eff-act-eig}
\end{equation}
Here
$f_0(m)$ is some numerical function of $m$ which is not important here,
and
\begin{equation}
\alpha_0^2(m) = \left\{\begin{array}{ll} \frac{1}{4\pi^{2}}\;
\Lambda_{UV}^2 \left(1 - \frac{m^2}{\Lambda_{UV}^2}\ln(1+\frac{\Lambda_{UV}^2}{m^2})\right), & d=4\\
                 \frac{1}{\pi}\ln(1+\frac{\Lambda_{UV}^2}{m^2})  , & d=2
                \end{array}\right.
\label{alpha}
\end{equation}
depending on the dimension of the field theory.

\vskip 5truemm
\centerline {\sc 3. APPLICATION: THE $\phi^4$ MODEL}
\vskip 3truemm

We can now apply the result  \eq{eff-act-eig} to the interacting case.
Consider for example the $\phi^4$ model \eq{Rd-action},
which is obtained from the free case by adding
a potential of the form
$$
S_{int}(\phi) = \frac{\lambda}{4}\int d^d x \phi^4(x)
= \frac{\lambda}{4} (2\pi\theta)^{d/2}
 Tr \phi^4,
$$
The goal is to derive some properties (in particular
thermodynamical, but also others) of the interacting model.
The basic hypothesis is again \eq{localiz-delta-full}, where the
eigenvalue distribution $\vec\phi_0$ in the interacting case
is to be determined.

It is quite easy to understand the main effect of the interaction term
using the above results \eq{eff-act-eig}:
since $S_{int}$ only depends on the eigenvalues, it seems
very natural to simply add $S_{int}$ to \eq{eff-act-eig}, suggesting
\begin{equation}
Z_{int} = \int d\phi_i e^{-S_{eff}^{free}(\vec\phi) - S_{int}(\vec\phi)}.
\label{Z-perturb}
\end{equation}
To justify this, we can expand the
interaction term into a power series in $\lambda$.
Using the factorization property \eq{cluster},
we can immediately compute any expectation value of
observables which depend only on the eigenvalues, and sum up the
expansion in $\lambda$. As in the case of pure matrix models \cite{BIPZ},
the effect of this perturbative computation is reproduced
for ${\cal N} \to \infty$
by the second (nonperturbative) point of view, which is that
\eq{Z-perturb} is localized at a new eigenvalue distribution
$\vec\phi_0^{\lambda}$, which now depends on the interaction.
It can again be found by the saddle-point approximation.

We conclude with some remarks and results of this analysis \cite{NCscalarFT}:

\begin{enumerate}
\item
A caveat: the validity of \eq{eff-act-eig} has been established only
``locally'', i.e. the function $\cF(\vec\phi)$ is really known only
for $\vec \phi \approx\vec \phi_0$. The only free parameter available
is the mass\footnote{this can be enhanced by adding other
  terms such as the one in \cite{wulki}, which will be pursued
  elsewhere},
which enters in \eq{eff-act-eig} through $\alpha_0$ and allows
to test a one-dimensional submanifold of $\cE_{\cal N}$. Therefore we
can trust \eq{eff-act-eig} only if $\vec\phi_0^{\lambda} \approx \vec\phi_0$.
This leads  to the second observation:
\item
The physical properties of the interacting model will be close to those of
the free case only if $\vec\phi_0^{\lambda} \approx \vec\phi_0$.
In particular, the correlation length (i.e. physical mass) of the interacting model
will be reproduced best by the free action whose EV distribution
$\vec\phi_0$ is closest to $\vec\phi_0^{\lambda}$.
Working this out \cite{NCscalarFT} leads to a very simple understanding of mass
renormalization. Namely, it turns out that the bare mass must be
adjusted to
\begin{equation}
m^2  =  m_{phys}^2 - \frac{3}{16\pi^{2}}\;
   \Lambda_{UV}^2 \left(1 -\; \frac{m_{phys}^2}{\Lambda_{UV}^2}\;
\ln(\frac{\Lambda_{UV}^2}{m_{phys}^2})\right)\; \lambda.
\label{mass-ren-4d}
\end{equation}
in the 4-dimensional case. This reproduces precisely the result of
a conventional one-loop computation, however it is now based on
a nonperturbative analysis and not just a formal expansion.
\item
Stretching somewhat the range where \eq{eff-act-eig} has been tested,
one can study the
thermodynamical properties and phase transitions of the interacting model.
Indeed a phase transition is found at the point where the eigenvalue distribution
$\vec\phi_0^\lambda$ breaks up into 2 disjoint pieces ("2 cuts"), due to the
interaction term.
This is expected to be
the transition between the ``striped'' and disordered phase
found in NC scalar field theory \cite{gubser,bieten}.
In the most interesting case of 4 dimension, a critical line
is found which ends at a nontrivial critical point $\lambda_c >0$
\cite{NCscalarFT}.
This is strongly
suggestive for a {\em nontrivial} NC $\phi^4$ model in 4 dimensions,
since the endpoint of the critical line $\lambda_c >0$ should correspond to
a fixed-point under a suitable RG flow.

Since the mechanism of the phase transition is very generic and does
not depend on the details of the potential, the
qualitative features of this result are expected to be correct.

\end{enumerate}
Full details can be found in \cite{NCscalarFT}.

\paragraph{Acknowledgments} It is a pleasure to thank the organizers
of the II. Southeastern European Workshop BW 2005, Vrnjacka Banja,
Serbia
and the XIV.  Workshop of Geometric Methods in Physics,
Bialowieza, Poland
for providing an enjoyable and stimulating
atmosphere. This work was supported by the FWF project
P16779-N02.

\nopagebreak
\renewcommand {\bibname} {\normalsize \sc References}
\nopagebreak


\section{Introduction}
\label{s:intro}

Active galactic nuclei (AGN) that exhibit H$_{2}$O  maser emission at 22~GHz
have been objects of particular interest since discovery that the
maser emission can trace circumnuclear accretion disks, as in NGC~4258
\citep[]{Miyoshi1995,Greenhill1995a,Greenhill1995b}.
The presence of maser emission enables direct mapping of AGN
dynamics within 1 pc of supermassive black holes via Very Long Baseline
Interferometry (VLBI) \citep[see reviews of][]{Greenhill2002,Maloney2002}.
The additional discovery that H$_{2}$O  maser emission can be associated with
shocked gas in jets and outflows has also provided a tracer for a second component
in AGN \citep[][]{Gallimore1996,Claussen1998,Peck2003,Greenhill2003}.

Maser emission at 22~GHz is believed to be predominantly
collisionally-pumped at gas kinetic temperatures of T$_{k}$ =  400 -- 1000~K, hydrogen
densities of n(H$_{2}$) =  10$^{8}$ -- 10$^{10}$ cm$^{-3}$, and dust temperatures
T$_{d}$ $<$ 100 K \citep[e.g.,][]{Deguchi1977,Elitzur1989,Yates1997}.
In AGN accretion disks, gas heating is thought to arise from X-ray irradiation
of the disk surfaces,  whereas masers in jets and outflows may be heated via
shock compressions \citep{Neufeld1994,Kartje1999}.

Since the physical conditions that give rise to emission at 22~GHz also pump other H$_{2}$O
maser transitions,
notably those at 183, 321 and 325~GHz \citep[e.g.,][]{Deguchi1977,Neufeld1991,Yates1997},
22~GHz maser emission is unlikely to be alone in AGN disks, jets and outflows.
We note that, in galactic regions of star formation and evolved stars, inversion of the
22 GHz transition is accompanied by inversion of other rotational transitions from the ground
and vibrationally excited H$_{2}$O  states  \citep[e.g.,][]{Phillips1980,Waters1980,Menten1989,Menten1990a,
Menten1990b,Feldman1991,Menten1991,Melnick1993}.
 In AGN, some of the masers should occur in broadly the same regions as at 22~GHz, but others could
 probe uncharted regions of the central engines of AGN, including regions at smaller radii.
Line ratios of two or more maser transitions originating from the same gas would constrain radiative transfer models
far better than is now possible.
Previous searches for extragalactic millimeter/submillimeter H$_{2}$O  masers have been performed, but
have been hampered by r.m.s. sensitivities worse than 1~Jy (Menten, private communication).

NGC~3079  is an almost edge-on spiral Seyfert 2/LINER galaxy
with optical heliocentric velocity  of 1125 $\pm$ 6~km~s$^{-1}$ \citep[e.g.,][]{Heckman1980,Ho1997},
at a distance of 16~Mpc \citep{Sofue1999}.
It hosts
22~GHz H$_{2}$O  maser emission at velocities between $\sim$940 and 1350
km~s$^{-1}$, which are believed to trace a disk-like structure at radii
of 0.4 -- 1.3 pc from a $2 \times 10^6$ $M_{\odot}$ central engine.
The pc-scale maser disk is aligned with  a dense molecular kpc-scale disk
\citep{Koda2002} and is located at the apex of a kpc-scale superbubble \citep{Kondratko2005,Cecil2002}.
The bubble is believed to be inflated by a wide-angle outflow \citep{Cecil2001} that is also responsible for the radio continuum lobes along
the minor axis of the galaxy \citep[e.g.,][]{Duric1988,Trotter1998,Sawada2000,Yamauchi2004}.
Within a parsec of the nucleus, the relativistic jet appears aligned with the inner wall
of the bubble and may be responsible for weak 22~GHz maser emission away from
the disk \citep[][and references therein]{Kondratko2005}.
All of these emission sources lie within the primary beam of the SMA observations reported here.

\section{Observations and Data Reduction}

\subsection{SMA Observations at 183~GHz}

We observed the ground-state
3$_{13}$ $\rightarrow$ 2$_{20}$ transition of para-H$_{2}$O  (E$_{u}$/$k$ $\sim$200~K)
at a rest frequency of 183.310~GHz ($\lambda$: 1.6~mm)
towards NGC~3079  using the SMA\footnote{The SMA is a
joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute
of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.}
on 2005 March 1.
The compact configuration included six antennas, resulting in projected baselines of 8 -- 72 m.
The SMA receivers operate in double-sideband mode with two 2 GHz sidebands that we tuned to
183.310 and 193.310 GHz (band-center). The correlator provided
continuous coverage of each sideband (in 24 separate chunks of 256 channels) and 0.4 MHz
channel spacing.
The phase center of the observations was
$\alpha_{2000} =10^h01^m58^s.53$,
$\delta_{2000} =55^o40'50\rlap{.}''1$ \citep{Cotton1999}. The zenith opacity measured with the NRAO tipping radiometer
located at the Caltech Submillimeter Observatory was $\tau$(225~GHz)=0.03 - 0.05 throughout.

We calibrated the data within
the IDL superset MIR
developed for the Owens Valley Radio Observatory \citep{Scoville1993} and adapted for the SMA; the imaging was performed in MIRIAD.
Data reduction took into account the effect of the pressure-broadened terrestrial H$_{2}$O
absorption line centered at 183.310~GHz.
Fortunately, the redshift of the galaxy reduces the line frequency of the maser
transition by $\sim$0.7~GHz,  greatly lowering the corresponding
atmospheric opacity.
For bandpass calibration we used Jupiter and Saturn.
Phase calibration was performed via frequent observations
of the quasar 0923+392 about 18$\rm ^o$ from the phase center.
We performed the amplitude calibration separately for upper and lower sideband
data. For the upper sideband, we used 0923+392;
for lower sideband data, we performed amplitude calibration on each of the 24 spectral
chunks separately using Saturn and Jupiter. The calibration of the lower sideband both as a function
of elevation and frequency was designed to reduce the effect of variation
in terrestrial H$_{2}$O  absorption
across the sideband. We derived the flux scale using observations of Callisto with an
estimated accuracy of 20\%.
Single sideband system temperatures corrected to the top of the atmosphere
were between 400 and 850 K.

\subsection{JCMT Observations at 439~GHz}

We observed the ground-state
6$_{43}$ $\rightarrow$ 5$_{50}$ transition of ortho-H$_{2}$O  (E$_{u}$/$k$ $\sim$1100 K)
at a rest frequency of 439.151~GHz ($\lambda$:~0.7~mm)  towards NGC~3079  using the 15 meter JCMT
on 2004 November 18
in $\tau_{225}$=0.041 -- 0.045 weather.
The instantaneous bandwidth was 1840~MHz, corresponding to $\sim$1200~km~s$^{-1}$,
with a channel spacing of 1250 kHz or 0.85~km~s$^{-1}$.
The integration time on source was 2~hrs in beam-switching mode with a throw of  60$''$. The average T$_{sys}$ during the observations
was 2650~K. We reduced the data using the Starlink spectral-line reduction software, {\sc SpecX}.

\section{Results}
\label{s:results}

We detected millimeter (193~GHz) continuum emission with a peak signal to noise ratio (S/N) of
$\sim$10 (Fig.~\ref{f:continuum}) at $\alpha_{2000} =10^h 01^m 57^s .80\pm0.02$,
$\delta_{2000} = 55 \rm ^o 40' 46\rlap{.}''9\pm0.3$ (statistical uncertainties).
Systematic uncertainties of ($0\rlap{.}^{s}02$,$0\rlap{.}''3$) were estimated through measurement of
an apparent position for 1150+497 against the phase calibrator 0923+392. These place the
peak of the millimeter continuum within 1$\sigma$ of the
22 GHz maser position.
We partially resolved the nucleus in the north-south direction and the elongation is approximately
parallel to the position angles of both the pc-scale accretion disk traced by
22 GHz maser emission and the kpc-scale molecular disk detected using  e.g., CO \citep{Koda2002}.

We detected 183~GHz H$_{2}$O  emission with a peak S/N of $\sim$7 in both the
real part of the
amplitude spectrum and the interferometer phase spectrum (Figs.~\ref{f:continuum} and~\ref{f:line}).
The peak flux density is 0.55 Jy at 1017 km~s$^{-1}$.
The line covers 9 frequency channels and overlaps a persistent Doppler component appearing in published 22 GHz maser spectra for dates 1998 May 8, 2000 March 18 where it is especially prominent at $\sim$2.5 Jy \citep[both][]{Hagiwara2002},  2001 March 23 \citep{Kondratko2005} and
 2002 April 12 \citep{Braatz2003}. We note possible detection of additional spectral features at 1208 (5 channels wide)
and 1265 km~s$^{-1}$ (4 channels wide). Both features appear in spectra of real amplitude and phase (near zero degrees, which indicates positions similar to the 1017 km~s$^{-1}$  emission), and both lie within the velocity interval of 22 GHz emission.
Note that the correspondence of features between 22 and 183 GHz
observations is not precise and is also not expected to be true for
all features under certain pumping scenarios.
The peak maser emission is at $\alpha_{2000} =10^h01^m57^s.75\pm 0.03$, $\delta_{2000}=55^o40'46\rlap{.}''7\pm0.4$, which is within 1$\sigma$ of the
193 GHz continuum peak. At our resolution, we cannot determine
whether 183~GHz emission originates from the
disk or outflow, or from both. We note that in addition to
tracing a disk, a subset of 22 GHz
maser emission is associated with the outflow within a few parsecs of
the nucleus \citep{Kondratko2005}. 22~GHz emission at around
 1000 and 1200 km~s$^{-1}$ is present both in the disk and outflow.
Possible angular extension of the 183~GHz    emission to the southeast
is marginally detected (2$\sigma$), which lies along the axis of the relativistic flow \citep[e.g.,][]{Kondratko2005}. We note that \citet{Hagiwara2004} detect OH absorption associated
with the outflow at an overlapping velocity, and that we cannot rule out
shocks in circumnuclear material as an origin for 183~GHz emission.

We obtained a tentative detection (5$\sigma$) of the 439~GHz maser
at 1157 km~s$^{-1}$ (Fig.~\ref{f:439ghz}). With peak S$_{\nu}$ $\sim$2.5 Jy,
439~GHz emission is of comparable strength to
typical values for peak 22~GHz emission towards NGC~3079 and occurs
within the 22~GHz velocity interval.

\section{Discussion}
\label{s:discussion}

\subsection{Millimeter Dust Emission}

The millimeter continuum of NGC~3079 is probably dominated by thermal emission from optically thin dust.
As in the 1.2 mm observations of \citet{Braine1997} at 11$''$ resolution, we find dust concentrated in the nuclear region of NGC~3079 (Fig.~\ref{f:continuum}) and place an upper limit of 8~mJy (3$\sigma$) on emission associated with the radio lobes $\sim$1 kpc from the emission peak.
\citet{Braine1997} estimates the dust temperature to be $\sim$30 K at kpc radii. However, in two other AGN the presence of hotter dust at 100 times smaller radii, of $\sim$300 K, has been inferred from VLT observations
\citep{Jaffe2004,Prieto2004}. Adopting a dust temperature of 100 K for NGC~3079  on
intermediate scales and a dust opacity index $\beta=2$, we estimate the enclosed
dust mass of the inner 200 pc is $M_{dust}$ $\sim$ 10$^{6.3\pm0.8}$ M$_{\odot}$ using the method described in \citet{Hildebrand1983}. Using CO Nobeyama observations by \citet{Koda2002}, who find a gas mass of  $M_{gas}$ $\sim$10$^{8.5}$ M$_{\odot}$ within the central 150 pc, we estimate a gas-to-dust ratio  of $150$, which is comparable to the Galactic value of 160 and
to the ratios derived in bright local galaxies \citep{Dunne2000}.

\subsection{183~GHz Emission}

The 183~GHz emission we detect in the nucleus of NGC~3079 is likely from a maser process. It arises from one, and possibly more, narrow
unresolved features in the same range of velocity as that of known 22~GHz maser
emission. From radiative transfer models we know that the same high densities and
temperature conditions that produce maser emission at 22~GHz
also strongly invert the 183~GHz H$_{2}$O  transition.
Noting that
the 22 GHz feature at 1017 km~s$^{-1}$ varies between 0.4 -- 2.6 Jy (Hagiwara
et al. 2002), and assuming an ortho:para H$_{2}$O  ratio of 3:1, pumping
models could reproduce likely line ratios \citep[e.g.,][]{Yates1997}.
Narrow linewidths ($<$8 km~s$^{-1}$)
at 183~GHz, small compared with the characteristic rotational and apparent random velocities for
molecular gas in this nucleus \citep{Kondratko2005}, also indicate that emitting regions should be
compact.
Our beam size is too large to determine whether emission is thermal
or maser based on brightness temperature arguments. For the emission to be thermal, with a brightness
temperature of $\leq$ 2000 K such that H$_{2}$O  molecules are not dissociated, the emitting region
must be $>$ 0$\rlap{.}''$1 (10 pc).

\subsection{439~GHz Emission}

439 GHz
emission is probably also due to maser amplification.
Emission occurs over a narrow velocity range (Fig.~\ref{f:439ghz}),
and conditions for strong 439 GHz inversion are likely to exist in
the galactic nucleus.
The 22~GHz spectrum peaks to the blue of the v$_{sys}$ of 1125 km~s$^{-1}$, yet
the 439~GHz spectrum peaks slightly above v$_{sys}$ at 1157 km~s$^{-1}$ (Fig.~\ref{f:439ghz}).
We ascribe this difference to the different  pump mechanisms
operating for each maser. Whereas 22~GHz emission is collisionally-pumped
and is increasingly quenched by local dust temperatures $>$100~K,
the 439~GHz maser is predominantly radiatively-pumped
and is most strongly inverted in the presence of $T_{d}=300~K$, $T_{k}=400~K$
and n$(H_{2})$ = 5 x 10$^{9}$ cm$^{-3}$ \citep{Yates1997}.
Dust temperatures of this order
and higher are reasonable in the central parsecs of the nucleus
\citep{Jaffe2004,Prieto2004}.
Peak 439~GHz emission (1157 km~s$^{-1}$) is therefore likely to originate from a hot,
dusty region around
the central engine, possibly at radii $\ll$1 pc, separated spatially from the gas emitting
most strongly at 22~GHz (956 km~s$^{-1}$). At 22 GHz, features at these velocities are indeed
separated spatially by $\sim$1 pc projected \citep{Kondratko2005}.
The observations in Fig.~\ref{f:439ghz} were taken on different dates,
and we note that
the 22~GHz feature near 1157 km~s$^{-1}$ is not present at all epochs (e.g., in
Fig.~\ref{f:line}).
However, the feature line ratio in Fig.~\ref{f:439ghz} of $\sim$100
is accommodated by the vicinity of hot dust \citep{Yates1997}.

\vspace*{-0.55cm}

\acknowledgments
We thank the all the SMA and JCMT staff.
We thank S. Paine for opacity data,   J. Braatz for 22 GHz data
and G. Melnick for useful discussions.
H. B. acknowledges support from the Emmy-Noether Program of the
Deutsche Forschungsgemeinschaft (DFG-BE2578/1).