\section{Introduction}

Very High Energy (VHE) $\gamma$-ray astronomy \cite{ong} is still
in its infancy. Operating in the energy
range from $\sim$\,100 GeV to 30 TeV and beyond, this subfield of
astronomy represents an exciting,
relatively unsampled region of the electromagnetic spectrum, and a
tremendous challenge. To develop more
fully this field needs an instrument capable of performing
continuous systematic sky surveys and
detecting transient sources on short timescales without
a priori knowledge of their location. These primary
science goals require a telescope with a wide field of view
and high duty cycle, excellent
source location and background rejection capabilities -
an instrument that complements both existing and future ground-
and space-based $\gamma$-ray telescopes.

To be viable VHE astronomy must overcome a number of fundamental
difficulties. Since the flux of VHE photons is small, telescopes
with large collecting areas ($\>10^3\,\mathrm{m}^2$)
are required to obtain statistically significant photon samples;
telescopes of this size can, currently,
only be located on the Earth's surface. However,
VHE photons do not readily penetrate the $\sim\,28$
radiation lengths of the Earth's atmosphere
($1030\,\mathrm{g}/\mathrm{cm}^2$ thick at sea-level) but
instead interact with air molecules to produce secondary
particle cascades, or extensive air showers.
Another difficulty of VHE astronomy is the large background
of hadronic air showers, induced by
cosmic-ray primaries (primarily protons), that cannot be vetoed.

In this paper we describe the conceptual design of an instrument
that builds upon traditional extensive air shower methods;
however, unlike typical extensive air shower arrays the detector
design utilizes unique imaging capabilities
and fast timing to identify (and reject) hadronic cosmic-ray
backgrounds and achieve excellent angular
resolution, both of which lead to improved sensitivity.
In the following sections we briefly motivate the need for such
an instrument
(Section 2), discuss in detail telescope design parameters with
emphasis on their optimization (Section 3),
describe the conceptual design of a VHE telescope and the
simulations used in this study (Section 4), and
evaluate the capabilities of such a detector in terms of source
sensitivity (Section 5). Finally, the results
of this study are summarized and compared to both current and
future VHE telescopes.

\section{Motivation}

VHE $\gamma$-ray astronomy has evolved dramatically in the
last decade with the initial detections of steady
and transient sources, galactic and extragalactic sources.
To date 7 VHE $\gamma$-ray sources have been
unambiguously detected
\cite{crab,mrk421,mrk501,1ES2344,p1706,vela,sn1006};
this contrasts dramatically with
the number of sources detected in the more traditional
regime of $\gamma$-ray astronomy at energies below
$\sim$\,20 GeV. The EGRET instrument aboard the
Compton Gamma-Ray Observatory, for example, has detected
pulsars, supernova remnants, gamma-ray bursts, active
galactic nuclei (AGN), and approximately 50 unidentified
sources in the 100 MeV-20 GeV range \cite{agn1,agn2}.
The power-law spectra of many EGRET sources show no sign
of an energy cutoff, suggesting that they may be observable
at VHE energies.

The 4 Galactic VHE objects, all supernova remnants, appear
to have $\gamma$-ray emission that is constant in both
intensity and spectrum. The 3 extragalactic VHE sources are
AGN of the blazar class. Although AGN have been
detected during both quiescent and flaring states, it is the
latter that produce the most statistically
significant detections. During these flaring states the
VHE $\gamma$-ray flux has been
observed to be as much as 10 times that of the Crab nebula -
the standard candle of TeV astronomy \cite{hegra_flare}.

Although observed seasonally since their initial detection,
long term continuous monitoring of the TeV sources
detected to date has never been possible, nor has there ever
been a systematic survey of the VHE sky. This is
primarily due to the fact that all VHE source detections to
date have been obtained with air-Cherenkov telescopes.
Because they are optical instruments, air-Cherenkov telescopes
only operate on dark, clear, moonless nights -
a $\sim\,5-10\,\%$ duty cycle for observations; these
telescopes also have relatively narrow fields of view
($\sim\,10^{-2}\,\mathrm{sr}$). Although they are likely
to remain unsurpassed in sensitivity for detailed
source observations these telescopes have limited usefulness
as transient monitors and would require over a
century to complete a systematic sky survey.

The identification of additional VHE sources would contribute
to our understanding of a range of unsolved
astrophysical problems such as the origin of cosmic rays,
the cosmological infrared background, and the nature
of supermassive black holes. Unfortunately, the field of VHE
astronomy is data starved; new instruments
capable of providing continuous observations and all-sky
monitoring with a sensitivity approaching that
of the air-Cherenkov telescopes are therefore required. A VHE
telescope with a wide field of view and high duty
cycle could also serve as a high-energy early warning system,
notifying space- and ground-based instruments of
transient detections quickly for detailed multi-wavelength
follow-up observations. Its operation should coincide
with the launch of the next-generation space-based instrument
such as GLAST \cite{glast}.

\section{Figure of Merit Parameters}

A conceptualized figure of merit is used to identify the
relevant telescope design parameters. This
figure of merit, also called the signal to noise ratio,
can be written as
\begin{equation}
\left(\frac{signal}{noise}\right) \propto
\frac{R_\gamma Q \sqrt{A^{eff}~T}}{\sigma_\theta}
\label{equation1}
\end{equation}

\begin{table}
\caption{\label{table1} Figure of merit parameter definitions.}
\vskip 0.5cm
\small
\centerline{
\begin{tabular}{|l l l|}
\hline
{\em Parameter} & {\em Units}    & {\em Definition} \\
\hline
\hline
$A_{eff}$       & $\mathrm{m}^2$ & (effective) detector area \\
$T$             & sec            & exposure \\
$\sigma_\theta$ & $^o$           & angular resolution \\
$R_\gamma$      & -              & $\gamma$/hadron
relative trigger efficiency \\
$Q$             & -              & $\gamma$/hadron
identification efficiency \\
\hline
\end{tabular}}
\end{table}
\normalsize
where the various parameters are defined in
Table\,\ref{table1}. Ultimately, source sensitivity is
the combination of these design parameters.
Although a more quantitative
form of the figure of merit is used to estimate the
performance of the conceptual telescope design (see
Equation\,\ref{equation4}), we use Equation\,\ref{equation1}
to address specific design requirements.

\subsection{$R_\gamma$}
\label{sec_alt}

Air showers induced by primary particles in the
100\,GeV to 10\,TeV range
reach their maximum particle\footnote{Throughout the rest
of the paper the generic term ``particles'' will
refer to $\gamma, e^{\pm}, \mu^{\pm}$, and hadrons unless
otherwise noted.} number typically at altitudes
between 10 and 15\,km above sea level (a.s.l.).
An earth-bound detector therefore samples the cascade at
a very late stage of its
development, when the number of shower particles
has already dropped by an
order of magnitude from its value at shower maximum.

Figure\,\ref{longi} shows the result of computer
simulations\footnote {Here and in the following analysis,
the CORSIKA 5.61 \cite{corsika} code is used for air-shower
simulation in the atmosphere. It is briefly
described in the next section.} of the longitudinal profile
of air showers induced in the Earth's
atmosphere by proton and $\gamma$-primaries with fixed
energies and zenith angles
$0^{\mathrm o}\le\theta\le45^{\mathrm o}$. The small number
of particles reaching 2500m detector altitude
sets severe limits for observations at these altitudes.
In addition, the number of particles in proton
showers actually exceeds the number of particles in
$\gamma$-showers at low altitudes (Figure\,\ref{r_gamma}).
This implies that the trigger probability, and thus the
effective area, of the detector is larger for
proton than for $\gamma$-showers, an unfavorable situation
which leads to an $R_{\gamma}$ (the ratio of
$\gamma$-ray to proton trigger efficiency) less than 1.
At 4500\,m, however, the mean number of particles
exceeds the number at 2500\,m by almost an order of magnitude
at all energies. Therefore, telescope location
at an altitude $\geq$\,4000\,m is important for an air
shower array operating at VHE energies not only
because of the larger number of particles, and hence the
lower energy threshold, but also because of the
intrinsic $\gamma$/hadron-separation available, due to
the relative trigger probabilities, that exists at higher
altitudes.

\begin{figure}
\epsfig{file=rmiller_fig1.eps,width=14.0cm}
\caption{\label{longi}
Mean number of particles
($\gamma, e^{\pm}, \mu^{\pm}$, hadrons) vs. altitude for proton-
and $\gamma$-induced air showers with primary
energies 100\,GeV, 500\,GeV, 1\,TeV, and 10\,TeV.
The low energy cutoff of the particle kinetic energy is
100\,keV ($\gamma, e^{\pm}$), 0.1\,GeV ($\mu^{\pm}$),
and 0.3\,GeV (hadrons).}
\end{figure}

\begin{figure}
\epsfig{file=rmiller_fig2.eps,width=14.0cm}
\caption{\label{r_gamma}
Ratio of particle numbers in $\gamma$- and proton-induced
showers vs. altitude.}
\end{figure}

\begin{figure}
\epsfig{file=rmiller_fig3.eps,width=14.0cm}
\caption{\label{n_muon}
For a $150\times 150\,\mathrm{m}^2$ detector area and
cores randomly distributed over the detector area,
(a) shows the mean number of $\mu^{\pm}$ in proton-induced showers
as a function of the energy of the primary particle,
and (b) shows the fraction $f$ of proton showers
without $\mu^{\pm}$ as a function of the energy of the primary
particle. The solid line is the actual value
of $f$, the dashed line is the expected value assuming the
number of $\mu^{\pm}$ follows a
Poisson distribution.}
\end{figure}

\subsection{$Q$}

\begin{figure}
\epsfig{file=rmiller_fig4.eps,width=14.0cm}
\caption{\label{wavelet} Shower image (spatial particle
distribution reaching ground level) for a typical
TeV $\gamma$- and proton shower (top). Event image after
convolution with ``Urban Sombrero'' smoothing function
(middle), and after significance thresholding (bottom).
(0,0) is the center of the detector.}
\end{figure}

The rate of VHE $\gamma$-ray induced showers is significantly
smaller than those produced by
hadronic cosmic-rays\footnote{At 1 TeV the ratio of
proton- to $\gamma$-induced showers
from the Crab Nebula is approximately 10$^4$, assuming an
angular resolution of 0.5 degrees.}.
Therefore rejecting this hadronic background and improving
the signal to noise ratio is crucial to
the success of any VHE $\gamma$-ray telescope. The effectiveness
of a background rejection technique is typically expressed as a
quality factor $Q$ defined as

\begin{equation}
Q = \frac{\epsilon_\gamma}{\sqrt{1-\epsilon_p}}
\label{equation2}
\end{equation}

where $\epsilon_\gamma$ and $\epsilon_p$ are the efficiencies
for {\em identifying} $\gamma$-induced and
proton-induced showers, respectively. Traditional extensive
air-shower experiments have addressed
$\gamma$/hadron-separation (i.\,e. background rejection)
by identifying the penetrating particle
component of air showers (see e.\,g. \cite{hegra_gh,casa_gh}),
particularly muons. Although valid at
energies exceeding 50 TeV, the number of muons detectable
by a telescope of realistic effective area is
small at TeV energies (see Figure\,\ref{n_muon}\,(a)).
In addition, the $N_{\mu}$-distribution deviates
from a Poisson distribution, implying that the fraction of
proton showers {\em without} any muon is larger than
$e^{-\overline{N}_{\mu}}$ (Figure\,\ref{n_muon}\,(b)).
Relying on muon detection for an effective
$\gamma$/hadron-separation requires efficient muon detection
over a large area. A fine-grained absorption
calorimeter to detect muons and perform air shower calorimetry
can, in principle, lead to an effective rejection
factor; however, the costs associated with such a detector
are prohibitive.

In contrast to air-shower experiments, imaging air-Cherenkov
telescopes have achieved quality factors
$Q>$\,7 by performing a {\em shape analysis} on the observed
image \cite{whipple_q}. Non-uniformity of hadronic
images arises from the development of localized regions of
high particle density generated by small
sub-showers. Although some of the background rejection
capability of air-Cherenkov telescopes is a result of their
angular resolution, rejection of hadronic events by
identifying the differences between $\gamma$- and
proton-induced images considerably increases source sensitivity.

Although air-Cherenkov telescopes image the shower as it
(primarily) appears at shower maximum, these
differences should also be evident in the particle distributions
reaching the ground. Figure\,\ref{wavelet}\,(top) shows
the particle distributions reaching ground level for
typical TeV $\gamma$-ray and proton-induced
showers. This figure illustrates the key differences:
the spatial distribution of particles in $\gamma$-ray
showers tends to be compact and smooth, while in proton
showers the distributions are clustered and uneven.
Mapping the spatial distribution of shower particles
(imaging), and identifying/quantifying shower features
such as these should yield improved telescope sensitivity.

\subsection{$\sigma_\theta$}

Shower particles reach the ground as a thin disk of diameter
approximately 100\,m. To first order, the disk
can be approximated as a plane defined by the arrival times
of the leading shower front particles.
The initiating primary's direction is assumed to be a
perpendicular to this plane. Ultimately, the accuracy with which
the primary particle's direction can be determined is related
to the accuracy and total number of the relative arrival
time measurements of the shower particles,
\begin{equation}
\sigma_{\theta} \propto \frac{\sigma_{t}}{\sqrt{\rho}}~,
\end{equation}
where $\sigma_t$ is the time resolution and $\rho$
is the density of independent detector elements
sampling the shower front. The telescope must, therefore,
be composed of elements that have fast timing $\sigma_t$
and a minimum of cross-talk since this can affect the
shower front arrival time
determinations. Once the detector area is larger than the
typical lateral extent of air showers, thus providing
an optimal lever arm, the angular resolution can be further
improved by increasing the sampling density.

To achieve ``shower limited'' resolution, individual
detector elements should have a time response
no larger than the fluctuations inherent in shower particle
arrival times ($\leq10$ ns, see
Figure\,\ref{n_muon}\,(c)); on the other hand, there is no
gain if $\sigma_t$ is significantly smaller than the
shower front fluctuations.

In practice fitting the shower plane is complicated by
the fact that the shower particles undergo
multiple scattering as they propagate to the ground leading
to a curvature of the shower front. This scattering
delays the particle arrival time by $\cal{O}$(ns)/100\,m,
however the actual magnitude of curvature is a function
of the particle's distance from the core. Determination
of the core position, and the subsequent application of a
{\em curvature correction} considerably improves the angular
resolution by returning the lateral particle
distribution to a plane which can then be reconstructed.
Core location accuracy can be improved by increasing
the sampling density of detector elements and the overall
size of the detector itself.

\section{Conceptual Design}

To summarize the previous sections, an all-sky VHE telescope
should satisfy the following design considerations:
\begin{itemize}
\item{$\sim\,100\,\%$ duty cycle ($T$)}
\item{large effective area ($A_{eff}$)}
\item{high altitude ($>$\,4000\,m)}
\item{high sampling density}
\item{fast timing}
\item{imaging capability}
\end{itemize}

In the sections that follow, we study how a pixellated
{\em scintillator-based} large-area
detector with 100$\%$ active sampling performs as an
all-sky monitor and survey instrument. Scintillator is
used since it can provide excellent time resolution and has
high sensitivity to charged particles, ultimately
leading to improvements in angular resolution, energy threshold,
and background rejection. To reduce detector
cross-talk, improve timing, and enhance the imaging
capabilities the detector should be segmented into
optically isolated pixels . This type of detector is easier
to construct, operate, and maintain compared to
other large-area instruments such as water- or gas-based
telescopes - advantageous since the high altitude
constraint is likely to limit potential telescope sites
to remote locations.

Many of the design goals are most effectively achieved by
maximizing the number of detected air-shower
particles. As discussed in Section\,\ref{sec_alt}, detector
altitude is of primary importance; however, at the energies
of interest here only about $10\,\%$
(Figure\,\ref{converter}\,(a,b)) of the particles reaching
the detector level are charged. Thus, the number of detected
particles can be increased dramatically by improving the
sensitivity to the $\gamma$-ray component of showers.
A converter material (e.\,g. lead)
on top of the scintillator converts photons into charged
particles via Compton scattering and pair production,
and, in addition, reduces the time spread of the shower
front by filtering low energy particles
which tend to trail the prompt shower front
(Figure\,\ref{converter}\,(c)) and thus
deteriorate the angular resolution.
Figure\,\ref{converter}\,(d) shows the charged particle
gain expected as a function of the converter thickness
for lead, tin, and iron converters.
The maximum gain is for a lead converter at $\sim$\,2
radiation lengths
($1\,{\mathrm r.\,l.}=0.56\,{\mathrm cm}$), but the gain
function is rather steep below 1\,r.\,l. and flattens
above. Because of the spectrum of secondary $\gamma$-rays
reaching the detector pair production is the most
dominant process contributing to the charged particle gain
(Figure\,\ref{converter}\,(d)).

\begin{figure}
\epsfig{file=rmiller_fig5.eps,width=14.0cm}
\caption{\label{converter}
Mean number vs. primary energy (a) and energy distribution
(b) of secondary $\gamma$'s and $e^{\pm}$
reaching 4500\,m observation altitude. (c) Integral
shower particle arrival time distribution
for particles within 40\,m distance to the core and various
cuts on the particle energy (100\,keV, 1\,MeV, 10\,MeV).
(d) Charged particle gain as a function of the converter
thickness for lead, tin, and iron converters.}
\end{figure}

Techniques for reading out the light produced in
scintillator-based detector elements have progressed
in recent years with the development of large-area sampling
calorimeters. Of particular interest
is the work by the CDF collaboration on scintillating
tiles \cite{bodek}; this technique utilizes fibers,
doped with a wavelength shifter and embedded directly in
the scintillator, to absorb the scintillation light
and subsequently re-emit it at a longer wavelength.
This re-emitted light is then coupled to photomultiplier
tubes either directly or using a separate clear fiber-optic
light guide. This highly efficient configuration
is ideal for detecting minimum ionizing particles (MIPs),
and produces 4 photoelectrons/MIP on average
in a 5\,mm-thick scintillator tile.

Using an array of tile/fiber detector elements one can now
consider a large-area detector that counts
particles and is $\sim\,100\,\%$ active, and it is this
paradigm that we discuss in more detail in the
following sections. It should be noted that a
scintillator-based air-shower detector is not a new idea;
however, the use of the efficient tile/fiber configuration
in a detector whose physical area is fully active
pushes the traditional concept of an air-shower array
to the extreme.

\subsection{Air Shower and Detector Simulation}

The backbone of a conceptual design study is the simulation
code. Here the complete simulation of
the detector response to air showers is done in two steps:
1) initial interaction of the primary particle
(both $\gamma$-ray and proton primaries) with the atmosphere
and the subsequent development
of the air shower, and 2) detector response to air-shower
particles reaching the detector level.

The CORSIKA \cite{corsika} air shower simulation code,
developed by the KASCADE \cite{kascade} group,
provides a sophisticated simulation of the shower development
in the Earth's atmosphere. In CORSIKA,
electromagnetic interactions are simulated using the
EGS\,4 \cite{egs} code. For hadronic interactions,
several options are available. A detailed study of the hadronic
part and comparisons to existing data has
been carried out by the CORSIKA group and is documented in
\cite{corsika}. For the simulations discussed here,
we use the VENUS \cite{venus} code for high energy hadronic
interactions and GHEISHA \cite{gheisha} to treat low energy
($\le\,80\,{\mathrm GeV}$) hadronic interactions.

The simulation of the detector itself is based on the
GEANT \cite{geant321} package. The light yield of
0.5\,cm tile/fiber assemblies has been studied in detail
in \cite{bodek} and \cite{barbaro}, and we adopt
an average light yield of 4 photoelectrons per minimum
ionizing particle. This includes attenuation losses
in the optical fibers and the efficiency of the photomultiplier.
Simulation parameters are summarized in
Table\,\ref{tab_sim}. It should be noted that wavelength
dependencies of the fiber attenuation length and
of the photomultiplier quantum efficiency have not been included.

\begin{table}
\caption{\label{tab_sim} Basic parameters of the shower and
detector simulation.}
\vskip 0.5cm
\small
\centerline{
\begin{tabular}{|l|l|}
\hline
zenith angle range  & $0^{\mathrm o}\le\theta\le45^{\mathrm o}$ \\
lower kinetic energy cuts & 0.1\,MeV ($e^{\pm},\,\gamma$)    \\
                          & 0.1\,GeV ($\mu^{\pm}$)           \\
                          & 0.3\,GeV (hadrons)               \\
scintillator thickness    & 0.5\,cm                          \\
lead converter thickness  & 0.5\,cm                          \\
PMT transit time spread   & 1\,ns (FWHM)                     \\
average light yield/MIP   & 4 photoelectrons                 \\
\hline
\end{tabular}}
\end{table}
\normalsize

As shown in Section\,\ref{sec_alt} only a detector at an
altitude above 4000\,m can be expected to give the desired
performance; however, we study the effect of three
detector altitudes, 2500\,m
($764.3\,{\mathrm g}\,{\mathrm cm}^{-2}$), 3500\,m
($673.3\,{\mathrm g}\,{\mathrm cm}^{-2}$), and
4500\,m ($591.0\,{\mathrm g}\,{\mathrm cm}^{-2}$), for the
purpose of completeness. These altitudes span
the range of both existing and planned all-sky VHE telescopes
such as the Milagro detector \cite{milagro} near Los Alamos,
New Mexico (2630\,m a.s.l.), and the ARGO-YBJ \cite{argo} detector
proposed for the Yanbajing Cosmic Ray Laboratory in Tibet
(4300\,m a.s.l.).

\section{Detector Performance}

In the remainder of this paper we study the expected performance
of a detector based on the conceptual design
discussed above. Although the canonical design is a detector
with a geometric area $150\times 150\,\mathrm{m}^{2}$, a detector
design incorporating a $200\times 200\,\mathrm{m}^{2}$ area has
also been analyzed in order to understand how telescope
performance scales with area. Pixellation is achieved
by covering the physical area of the detector with a mosaic
of 5\,mm thick scintillator tiles each covering
an area of $1\times 1\,\mathrm{m}^{2}$.

\subsection{Energy Threshold}

The energy threshold of air-shower detectors is not well-defined.
The trigger probability for a shower
induced by a primary of fixed energy is not a step-function but
instead rises rather slowly due
to fluctuations in the first interaction height, shower
development, core positions, and incident angles.
Figure\,\ref{energy}\,(a) shows the trigger probability as a
function of the primary $\gamma$-ray energy
for three trigger conditions.

Typically, the primary energy where the trigger probability
reaches either $10\,\%$ or $50\,\%$ is defined as the
energy threshold (see Figure\,\ref{energy}\,(a)).
A large fraction of air showers that fulfill the trigger
condition will have lower energies since
VHE source spectra appear to be power-laws, $E^{-\alpha}$. A
more meaningful indication of energy threshold
then is the {\it median} energy $E_{med}$; however, this
measure depends on the spectral index of the source.
For a source with spectral index $\alpha=2.49$ (Crab),
Figure\,\ref{energy}\,(b, top) shows $E_{med}$ as a
function of detector altitude (for fixed detector size).
The median energy increases as altitude decreases
since the number of particles reaching the detector level
is reduced at lower altitudes. For a detector at
4500\,m a.s.l., $E_{med}$ is about 500\,GeV after imposing
a 40 pixel trigger criterion.

$E_{med}$ is not a strong function of the detector size
(Figure\,\ref{energy}\,(b, bottom)). It is
also noteworthy that a larger pixel size of
$2\times 2\,\mathrm{m}^2$ instead of $1\times 1\,\mathrm{m}^2$
only slightly increases $E_{med}$; due to the lateral extent
of air showers and the large average distances
between particles, nearly $95\,\%$ of all showers with more
than 50 $1\,\mathrm{m}^2$-pixels also have more
than 50 $4\,\mathrm{m}^2$-pixels.

\begin{figure}
\epsfig{file=rmiller_fig6.eps,width=14.0cm}
\caption{\label{energy}
(a) Trigger efficiency as a function of the primary
particles' energy for three trigger conditions
(10, 40, 400 pixels).
(b) Median energy of detected $\gamma$-showers as a function
of the trigger condition (number
of pixels) for three detector altitudes (top) and as a
function of the detector size for
a fixed altitude (4500\,m) (bottom).}
\end{figure}

\subsection{Background Rejection and Core Location}
\label{sec_bg}

Due to its pixellation and $100\%$ active area the telescope
described here can provide true images of
the spatial distribution of secondary particles reaching the
detector. Image analysis can take many forms;
the method of wavelet transforms\,\cite{kaiser} is well
suited for identifying and extracting localized
image features. To identify localized high-density regions
of particles an image analysis technique that utilizes digital
filters is used; the procedure is briefly summarized below
while details are given in \cite{miller}.

Proton- and $\gamma$-induced showers can be identified by
counting the number of ``hot spots'', or peaks, in a
shower image in an automated, unbiased way. These peaks are
due to small sub-showers created by secondary particles and
are more prevalent in hadronic showers than in $\gamma$-induced
showers. To begin, the shower image (i.\,e. the
spatial distribution of detected secondary particles) is
convolved with a function that smooths the image over a
predefined region or length scale
(see Figure\,\ref{wavelet}\,(middle)).
The smoothing function used in this
analysis is the so-called ``Urban Sombrero''\footnote{Also
known as the ``Mexican Hat'' function.} function:
\begin{equation}
g\left(\frac{r}{a}\right)
= \left(2 - \frac{r^2}{a^2}\right) e^{-\frac{r^2}{2\,a^2}}
\end{equation}
where $r$ is the radial distance between the origin of
the region being smoothed and an image pixel, and $a$ is the
length scale over which the image is to be smoothed.
This function is well suited for this analysis since it is a
localized function having zero mean; therefore, image
features analyzed at multiple scales $a$ will maintain their
location in image space. A peak's maximum amplitude is found
on length scale $a$ corresponding to the actual
spatial extent of the ``hot spot.''

Many peaks exist in these images; the key is to tag
statistically significant peaks. To do this the probability
distribution of peak amplitudes must be derived from a random
distribution of pixels. Using $2\times10^4$ events,
each with a random spatial particle distribution,
the probability of observing a given amplitude is computed. This
is done for events with different pixel multiplicities and
for different scale sizes. Results using only a
single scale size of 8\,m are presented here. This scale size
represents the optimum for the ensemble of showers;
scale size dependence as a function of pixel multiplicity is
studied in \cite{miller}. In order to identify
statistically significant peaks a threshold is
applied to the smoothed image. Peaks are eliminated if the
amplitude was more probable than $6.3\times10^{-5}$
corresponding to a significance of $<4\,\sigma$; the value
of the threshold is chosen to maximize the background
rejection. Figure\,\ref{wavelet}\,(bottom) shows the result
of thresholding. After applying thresholding
the number of significant peaks is counted; if the number of
peaks exceeds the mean number of peaks expected from
$\gamma$-induced showers then the event is tagged as
a ``proton-like'' shower and rejected. The number of
expected peaks is energy dependent starting at 1 peak
(i.\,e. the shower core), on average, for $\gamma$-showers
with less than 100 pixels and increasing with pixel
multiplicity; proton-induced showers show a similar
behavior except
that the number of peaks rises faster with the number of
pixels (see Figure\,\ref{peaks}).

\begin{figure}
\epsfig{file=rmiller_fig7.eps,width=14.0cm}
\caption{\label{peaks}
(a) Number of significant peaks for $\gamma-$ (top) and
proton showers (bottom) as a function of pixel multiplicity.
A random probability of $<\,6.3\times10^{-5}$ is used to
define a significant peak.
(b) Mean number of significant peaks for $\gamma-$ and
proton showers as a function of pixel multiplicity.}
\end{figure}

Additional background rejection may be possible by using
information such as the spatial distribution of peaks and
the actual shape of individual peak regions. This is
currently being investigated. The ability to map
and analyze the spatial distributions of air shower particles
implies that, in conjunction with analysis
techniques such as the one described here, the large cosmic-ray
induced backgrounds can be suppressed thereby
improving the sensitivity of a ground-based air-shower array;
quantitative results on the use of this image analysis technique
are described below and summarized in Figure\,\ref{quality}.

Image analysis can also be used to identify and locate the
shower core. Here the core is identified as the
peak with the largest amplitude; this is reasonable assuming
that typically the core represents a relatively
large region of high particle density.
Figure\,\ref{angle_res}\,(a) shows the accuracy of the core fit;
other methods are less accurate and more susceptible
to detector edge effects
and local particle density fluctuations.

The core location is used to correct for the curvature of the
shower front and to veto events with cores outside the active
detector area. Rejecting ``external'' events is beneficial
as both angular resolution and background rejection
capability are worse for events with cores off the detector.
We define the outer 10\,m of the detector as a veto ring
and restrict the analysis to events with
fitted cores inside the remaining fiducial area
($130\times 130\,\mathrm{m}^{2}$ or
$180\times 180\,\mathrm{m}^{2}$ for the
$200\times 200\,\mathrm{m}^{2}$ design). This cut
identifies and keeps $94\,\%$ of the $\gamma$-showers
with {\em true} cores within the fiducial area while
vetoing $64\,\%$ of events with cores outside. It is
important to note that the non-vetoed events are generally
of higher quality (better angular resolution,
improved $\gamma$/hadron-separation). In addition, $R_{\gamma}$ is
smaller for external events than for internal ones
due to the larger lateral spread of particles in proton
showers; thus the veto cut actually improves overall
sensitivity even though total effective area is decreased.

\begin{figure}
\epsfig{file=rmiller_fig8.eps,width=14.0cm}
\caption{\label{angle_res}
(a) Mean distance between reconstructed and true shower
core location.
(b) Mean angle between reconstructed and true shower direction for
a detector without lead converter and with 0.5\,cm and
1.0\,cm lead.}
\end{figure}

\subsection{Angular Resolution}

The shower direction is reconstructed using an iterative
procedure that fits a plane to the arrival times of the pixels and
minimizes $\chi^{2}$.
Before fitting, the reconstructed core position
(see Section\,\ref{sec_bg}) is used
to apply a curvature correction to the shower front.
In the fit, the pixels are weighted with $w(p)=1/\sigma^{2}(p)$,
where $\sigma^{2}(p)$ is the RMS of the time residuals
$t_{pixel}-t_{fit}$ for pixels with $p$ photoelectrons.
$t_{fit}$ is the expected time according to the previous iteration.

In order to minimize the effect of large time fluctuations
in the shower particle arrival times, we reject pixels with times
$t_{pixel}$ with
$\left|t_{pixel}-t_{fit}\right|\ge\,10\,\mathrm{ns}$.
In addition, only pixels within 80\,m distance to the
shower core are included in the fit.

Figure\,\ref{angle_res}\,(b) shows the mean difference
between the fitted and the true shower direction as a
function of the number of pixels for a
detector with and without 0.5\,cm and 1.0\,cm of lead,
again indicating the benefits of the
converter. The angular resolution does not improve considerably
when the converter thickness is increased from 0.5\,cm
to 1.0\,cm, thus 0.5\,cm is a reasonable
compromise considering the tradeoff between cost and performance.

\subsection{Sensitivity}

\begin{figure}
\epsfig{file=rmiller_fig9.eps,width=14.0cm}
\caption{\label{sourcebin}
(a) Significance for a one day observation of a
Crab-like source for three trigger conditions
(40, 120, 400 pixels) as a function of the source bin size.
(b) Energy distribution of the
detected $\gamma$-showers for the three trigger conditions.
The source location is
$\left|\delta-\lambda\right|\simeq\,5^{\mathrm{o}}$,
where $\delta$ is the source declination and $\lambda$
is the latitude of the detector site.}
\end{figure}

\begin{figure}
\epsfig{file=rmiller_fig10.eps,width=14.0cm}
\caption{\label{quality}
(a) Significance/day for a Crab-like source as a function
of the trigger condition with and without
$\gamma$/hadron-separation for the $150\times 150\,\mathrm{m}^{2}$
prototype.  (b) Quality factor as a function of the
trigger condition. (c) Sensitivity as a function of the source
position $\left|\delta-\lambda\right|$.}
\end{figure}

The ultimate characteristic of a detector is its sensitivity
to a known standard candle.
In this section, the methods described so far are
combined to estimate the overall
point source sensitivity of a pixellated scintillation detector.
As indicated in Equation\,\ref{equation1}, the sensitivity
of an air shower array depends on its
angular resolution $\sigma_{\theta}$, its effective area
$A_{eff}$, the trigger probabilities
for source and background showers, and the quality factor of the
$\gamma$/hadron-separation. However, as most of the parameters
are functions of the primary energy, the sensitivity depends on the
spectrum of the cosmic ray background and the spectrum of the
source itself. The significance $S$ therefore has to be
calculated using
\begin{equation}
S=\frac{\int A^{eff}_{\gamma}(E)~\epsilon_{\gamma}(E)
~J_{\gamma}(E)~{\mathrm d}E~~f_{\gamma}~T}
       {\sqrt{\int A^{eff}_{p}(E)~(1\,-\,\epsilon_{p}(E))
~J_{p}(E)~{\mathrm d}E~~\Delta\Omega~T}}
\label{equation4}
\end{equation}
where $J_{\gamma}$ and $J_{p}$ are the photon and proton
energy spectrum, and
$f_{\gamma}$ is the fraction of $\gamma$-showers
fitted within the solid angle bin
$\Delta\Omega=2\,\pi\,(1-\mathrm{cos}\,\theta)$.
Other parameters have their standard meaning.

The Crab Nebula is commonly treated as a standard candle
in $\gamma$-ray astronomy; this allows
the sensitivity of different telescopes to be compared.
The differential spectrum of the Crab
at TeV energies has been measured by the
Whipple collaboration \cite{whipple_crab}:
\begin{equation}
J_{\gamma}(E) = (3.20\,\pm0.17\,\pm0.6)\times10^{-7}\,
E_{\mathrm{TeV}}^{-2.49\,\pm0.06\,\pm0.04}\,
\mathrm{m}^{-2}\,\mathrm{s}^{-1}\,\mathrm{TeV}^{-1}.
\label{crab_rate}
\end{equation}
The sensitivity of the detector to a Crab-like source
can be estimated using Equation\,\ref{equation4} and
the differential proton background flux measured by the
JACEE balloon experiment \cite{jacee}:
\begin{equation}
\frac{dJ_{p}(E)}{d\Omega}=(1.11^{+0.08}_{-0.06})\times10^{-1}\,
E_{\mathrm{TeV}}^{-2.80\,\pm0.04}\,
\mathrm{m}^{-2}\,\mathrm{sr}^{-1}\,\mathrm{s}^{-1}\,
\mathrm{TeV}^{-1}.
\label{bg_rate}
\end{equation}

Calculating $S$ using Equation\,\ref{equation4} is not
straightforward since $\epsilon_{\gamma}$
is not a constant, but rather a function of the event size
and core position and, therefore, angular reconstruction accuracy.
This equation can be solved, however, by a Monte Carlo
approach: Using the Crab and cosmic-ray proton spectral
indices, a pool of simulated $\gamma$- and proton showers
($\cal{O}$($10^{6}$) events of each particle type at
each altitude) is generated with energies from 50\,GeV
to 30\,TeV and with zenith angles
$0^{\mathrm o}\ge\theta\ge45^{\mathrm o}$.
A full Julian day source transit can be simulated for a given
source bin size and declination
by randomly choosing $\gamma$-ray and proton-induced showers
from the simulated shower pools at rates
given by Equations\,\ref{crab_rate} and~\ref{bg_rate}. The only
constraint imposed on the events is that
they have the same zenith angle as the source bin at the
given time. The showers are then fully reconstructed,
trigger and core location veto cuts, as well as
$\gamma$/hadron-separation cuts, are applied; $\gamma$-showers
are required to fall into the source bin. This procedure
produces distributions of pixel multiplicity, core
position, etc. reflecting instrumental resolutions and responses.

Because the angular resolution varies with the number of
pixels, the optimal source bin size also varies.
Figure\,\ref{sourcebin}\,(a) shows the significance for a full-day
observation of a Crab-like source with
$\left|\delta-\lambda\right|\simeq\,5^{\mathrm{o}}$,
where $\delta$ is the source declination and $\lambda$
is the latitude of the detector site,
as a function of the source bin size for three trigger conditions.
As the number of pixels increases, the optimal source bin size
decreases from $0.6^{\mathrm{o}}$ (40 pixels) to
$0.2^{\mathrm{o}}$ (400 pixels).
Figure\,\ref{sourcebin}\,(b) shows
how the energy distribution of detected $\gamma$-showers
changes with trigger condition. The
median energy for a 40 pixel trigger is 600\,GeV,
with a substantial fraction of showers having energies
below 200\,GeV. For a 120 pixel trigger, $E_{med}$ is 1\,TeV.

For a 1 day Crab-like source transit, Figure\,\ref{quality}\,(a)
shows how the significance varies as a
function of the trigger condition with and without
$\gamma$/hadron-separation. If no $\gamma$/hadron-separation
is applied the sensitivity increases and then falls above
500 pixels because of the finite size of the detector.
However, as shown in Section\,\ref{sec_bg}, above 500 pixels
the quality factor of the
$\gamma$/hadron-separation counterbalances the loss of area.
Figure\,\ref{quality}\,(b) shows the quality factor
derived solely from the ratio of source significance with
and without separation. As expected, $Q$ increases
dramatically with pixel number, leading to significances
well above $3\,\sigma$ per day for energies above
several TeV.
The sensitivity also depends on the declination of the source.
Results quoted so far refer to sources
$\left|\delta-\lambda\right|\simeq\,5^{\mathrm{o}}$.
Figure\,\ref{quality}\,(c) shows how the
expected significance per source day transit changes with
the source declination $\delta$.

\begin{table}[ht]
\caption{\label{table2} Significances $S$ for a 1 day
observation of a Crab-like source with
$\left|\delta-\lambda\right|\simeq\,5^{\mathrm{o}}$
for different altitudes and trigger conditions. $E_{med}$ is the
median energy of the detected source particles, values
in parentheses denote significances after
$\gamma$/hadron separation.}
\vskip 0.5cm
\small
\centerline{
\begin{tabular}{|c|c|| c c | c || c c | c |}
\hline
   &    & \multicolumn{3}{|c||}{$150\times 150\,\mathrm{m}^{2}$}
        & \multicolumn{3}{|c|}{$200\times 200\,\mathrm{m}^{2}$} \\
\hline
altitude & trigger & \multicolumn{2}{|c|}{$S \left[\sigma\right]$}
& log($E_{med}^{\mathrm{GeV}})$
         & \multicolumn{2}{|c|}{$S \left[\sigma\right]$}
& log($E_{med}^{\mathrm{GeV}})$ \\
\hline
\hline
4500\,m & 40   & 1.3 & (1.3) & 2.8 & 1.8 & (1.8) & 2.8 \\
        & 1000 & 1.6 & (2.5) & 3.8 & 1.9 & (2.7) & 3.8 \\
\hline
3500\,m & 40   & 0.9 & (0.9) & 3.1 & 1.2 & (1.2) & 3.0 \\
        & 1000 & 0.8 & (1.2) & 4.1 & 1.3 & (1.7) & 4.0 \\
\hline
2500\,m & 40   & 0.4 & (0.4) & 3.3 & 0.6 & (0.6) & 3.2 \\
        & 1000 & 0.5 & (0.6) & 4.3 & 0.8 & (1.1) & 4.2 \\
\hline
\end{tabular}}
\end{table}
\normalsize

\begin{table}[ht]
\caption{\label{table3} Expected rates [kHz] for a
$150\times 150\,\mathrm{m}^2$ detector at
different altitudes. Cores are randomly distributed
over $300\times 300\,\mathrm{m}^{2}$ and no core
veto cut is applied.}
\vskip 0.5cm
\small
\centerline{
\begin{tabular}{|l||c|c|c|}
\hline
trigger & 4500\,m & 3500\,m & 2500\,m \\
\hline
\hline
10   & 34.5  & 18.9  & 10.5  \\
40   &  6.7  &  3.6  &  2.1  \\
100  &  1.8  &  1.0  &  0.6  \\
400  &  0.2  &  0.2  &  0.1  \\
1000 &  0.05 &  0.04 &  0.02 \\
\hline
\end{tabular}}
\end{table}
\normalsize

Table\,\ref{table2} summarizes the dependence of the
detector performance on the size and the altitude
of the detector. Significance scales with $\sqrt{A_{eff}}$
as expected from Equation\,\ref{equation1}. Detector
altitude, however, is more critical. Although at 2500\,m
$10\,\sigma$ detections of a steady Crab-like source
per year are possible, it is only at 4000\,m altitudes where
the sensitivity is sufficient to detect
statistically significant daily variations of source emissions.

It is noteworthy that for the canonical design at 4500\,m,
a trigger condition as low as 10 pixels still produces
$1.2\,\sigma$ per day at median energies of about 280\,GeV
corresponding to an event rate of 34.5\,kHz.
Predicted event rates for different trigger conditions at
various altitudes are summarized in Table\,\ref{table3}.
Sustained event rates below $\sim$\,10\,kHz are achievable
with off-the-shelf data acquisition electronics; higher
rates may also be possible. The event rates estimated here
are relatively low compared to the rate of
single cosmic-ray muons; because of the optical isolation and
low-cross talk of individual detector elements
single muons are unlikely to trigger the detector even with
a low pixel multiplicity trigger condition.

\section{Conclusion}

To fully develop the field of VHE $\gamma$-ray astronomy
a new instrument is required - one capable of
continuously monitoring the sky for VHE transient emission
and performing a sensitive systematic survey for
steady sources. To achieve these goals such an instrument
must have a wide field of view, $\sim\,100\,\%$ duty
cycle, a low energy threshold, and background rejection
capabilities. Combining these features we have shown that a
detector composed of individual scintillator-based pixels
and 100$\%$ active area provides high sensitivity
at energies from 100\,GeV to beyond 10\,TeV. Detailed
simulations indicate that a source with the intensity
of the Crab Nebula would be observed with an energy dependent
significance exceeding $\sim\,3\,\sigma$/day.
AGN flares, or other transient phenomena, could be detected
on timescales $\ll$1 day depending on their
intensity - providing a true VHE transient all-sky monitor.
A conservative estimate of the
sensitivity of a detector like the one described here
(the PIXIE telescope) is shown in Figure\,\ref{sensi_comp}
compared to current and future ground- and space-based
experiments. The plot shows the sensitivities for both 50
hour and 1 year source exposures, relevant for transient and
quiescent sources, respectively. A detector based on
the PIXIE design improves upon first-generation detector
concepts, such as Milagro, in two principal ways: fast
timing and spatial mapping of air shower particles.
The sensitivity of a sky map produced by this detector in
1 year reaches the flux sensitivity of current air-Cherenkov
telescopes (for a 50 hour exposure), making the detection of
AGN in their quiescent states possible.

\begin{figure}
\epsfig{file=rmiller_fig11.eps,width=14.0cm}
\caption{\label{sensi_comp} Predicted sensitivity of some
proposed and operational
ground-based telescopes. The dashed and dotted lines show
the predicted sensitivity of the telescope described
here (PIXIE) at an altitude $>4000$\,m. The numbers are based
on a $5\,\sigma$ detection for the given exposure
on a single source. EGRET and GLAST sensitivities are for
1 month of all-sky survey. The ARGO sensitivity is
taken from \cite{argo}, all others from \cite{glast_proposal}.
Information required to extrapolate
the ARGO sensitivity to higher energies is not given
in \cite{argo}.}
\end{figure}

The cost for a detector based on the conceptual design
outlined is estimated conservatively at between
\$\,500 and \$\,1000 per pixel; the cost of scintillator
is the dominant factor. A proposal to perform a detailed
detector design study (evaluation of detector
materials, data acquisition prototyping, and investigation
of construction techniques) leading to a final
design is currently pending.

Although unlikely to surpass the sensitivity of
air-Cherenkov telescopes for detailed single source
observations, a sensitive all-sky VHE telescope could
continuously monitor the observable sky at VHE energies.
In summary, non-optical ground-based VHE astronomy {\em is}
viable, and the development of an all-sky VHE telescope with
sensitivity approaching that of the existing narrow field of
view air-Cherenkov telescopes, will contribute to
the continuing evolution of VHE astronomy.

\begin{ack}
We thank the authors of CORSIKA for providing us with the
simulation code; we also acknowledge
D.G. Coyne, C.M. Hoffman, J.M. Ryan, and D.A. Williams
for their useful comments. This research is
supported in part by the U.S. Department of Energy Office
of High Energy Physics, the U.S. Department of Energy
Office of Nuclear Physics, the University of California (RSM),
and the National Science Foundation (SW).
\end{ack}


\section{Introduction}

The eleven-dimensional interpretation of the massive Type IIA
supergravity of Romans \cite{Romans} has devoted a lot of
attention recently.
It was shown in \cite{BDHS} that it is not possible
to construct a covariant
supergravity theory in eleven dimensions with cosmological
constant.
One might assume that the mass arises in the Type IIA theory from
the dimensional reduction
procedure, \`a la Scherk-Schwarz, however a theory was obtained
through this kind of construction in \cite{HLW} with the result
that it was not
Romans' supergravity but a different massive supergravity for which
there is no action.

A massive eleven-dimensional supergravity was proposed
in \cite{BLO} with the peculiarity that it
can only be formulated
when the eleventh direction is compact, with
Lorentz invariance taking place in the other ten dimensional
coordinates. This feature
circumvents the no-go theorem of \cite{BDHS}. The explicit
eleven-dimensional supergravity action depends on the Killing vector
associated to translations along the eleventh coordinate, and
gives the massive Type IIA supergravity action after a direct
dimensional reduction along this direction.

More recently \cite{Hull3}
a proposal for massive M-theory has been given
based on the connection between M-theory and Type IIB and the
relation between the Scherk-Schwarz reduction of Type IIB and
the reduction of massive Type IIA. It remains an interesting open
problem to make contact with the description of massive
eleven-dimensional supergravity with a Killing vector.

M-branes propagating in a massive background were constructed in
\cite{LO,BLO} and referred to generically as massive M-branes.
Their effective actions are described by gauged
sigma-models in which the Killing isometry is gauged. They contain
as well new couplings proportional to the mass which can be
interpreted in terms of ``massive'' solitons.

Furthermore, two massive branes in the Type IIA theory
can be obtained from a single massive M-brane \cite{BLO}.
They are obtained as either direct or double
dimensional reduction of the massive M-brane effective
action along the space-time coordinate
where the isometry is realized.

In this paper we will construct the effective action of the massive
Type IIA KK-monopole.
We will follow two approaches.
First of all, we will obtain the action of the IIA KK-monopole
by performing a double dimensional reduction in the effective action
of the massive eleven-dimensional KK-monopole.
As explained in \cite{BEL} the construction of
a massive KK-monopole in eleven dimensions
is more subtle than that of an ordinary
brane, since
the monopole is already described by a gauged sigma-model
in the massless case.

The massive M-KK-monopole giving
rise to the massive D6-brane after a direct dimensional reduction
was constructed in \cite{BEL}. There
it was shown that in order to assure
invariance under massive gauge transformations the gauge field
associated to the Taub-NUT isometry
had to transform proportionally to the
mass. Also, new couplings to the Born-Infeld
field had to be introduced.

In this eleven-dimensional massive monopole the ``mass
isometry direction''
coincides with the Taub-NUT direction,
and the dimensionally reduced action gives the massive D6-brane.
However, in order to obtain a ten dimensional KK-monopole we are not
interested in eliminating the isometry in the Taub-NUT direction
once a double dimensional reduction is performed.
Therefore we first have to find the worldvolume action of a more
general M-KK-monopole in which the invariance under massive
transformations is achieved by gauging
an isometry other than the Taub-NUT isometry.
Double dimensional reduction along this
new isometry direction
will give rise to the action of the massive IIA KK-monopole (see Figure 1).
We present
these actions in sections \ref{massive-MKK}, \ref{MKK-->IIAKK} and
\ref{mIIAKK}.

\vskip 12pt
\begin{figure}[!ht]
\begin{center}
\leavevmode
\epsfxsize=13.5cm
\epsfysize=4cm
\epsffile{cuadro3.eps}
\caption{\small {\bf Dimensional reductions of the (massive) M-KK-monopole.}
In this figure we display the reductions of the
massless M-KK-monopole
and its two possible massive extensions.
The massless M-KK-monopole is described by a gauged sigma-model
with Killing vector ${\hat k}$.
Reducing along a worldvolume
coordinate gives rise to the Type IIA KK-monopole, the reduction along
${\hat k}$ gives the D6-brane and the reduction along a transversal
coordinate different than ${\hat k}$ gives a 6-brane (KK-6A)
\cite{U1,U2,U3,MO}, described by a gauged sigma-model. The KK-6A brane
is not associated to a central
charge in the Type IIA supersymmetry algebra
\cite{Hull}.
There are two possible massive
extensions depending on whether the {\it massive} isometry, ${\hat h}$,
is chosen to be ${\hat h}={\hat k}$ or ${\hat h} \neq {\hat k}$, such that
reducing along ${\hat h}$ gives a massive Type IIA brane.
When ${\hat h}={\hat k}$ we obtain a massive D6-brane.
When ${\hat h}\neq {\hat k}$ we obtain a massive IIA
KK-monopole (KK-5A in the Figure),
if ${\hat h}$ lies in a worldvolume direction of the M-KK-monopole;
or a massive KK-6A brane, if ${\hat h}$ lies in a transversal
direction.}
\end{center}
\end{figure}

An alternative way to derive the action of the massive IIA KK-monopole
is to start with the action of the IIB NS-5-brane
constructed in \cite{EJL} and perform a (massive) T-duality
transformation. The massive T-duality rules for the
target space (dual) potentials that couple to the 5-brane are given in
Appendix A. We present the details of the calculation of the massive IIA
KK-monopole action by this procedure in section
\ref{IIBNS5-->IIAKK}. The result coincides with that given
in section \ref{mIIAKK}
following the double dimensional reduction approach
(see Figure 2).

\vskip 12pt
\begin{figure}[!ht]
\begin{center}
\leavevmode
\epsfxsize= 9cm
\epsfysize= 4cm
\epsffile{cuadro1.eps}
\caption{\small {\bf Derivation of the massive IIA KK-monopole.}
In this Figure we show the two procedures that we have followed to obtain
the massive IIA KK-monopole. In the first one
we perform a double dimensional reduction
along the ${\hat h}$ direction of the massive M-KK-monopole
with two gauged isometries, the Taub-NUT ${\hat k}$ and ${\hat h}$.
In the second case, we obtain the same massive
IIA KK-monopole via a massive T-duality transformation on the
IIB NS-5-brane effective action.}
\label{fig:cuadro1}
\end{center}
\end{figure}

\section{The Massive M-KK-monopole}
\label{massive-MKK}

Let us start by recalling the action of the massless M-KK-monopole constructed
in \cite{BJO} \footnote{For some recent work on KK-monopoles
see \cite{varios}.}.
The KK-monopole in eleven dimensions behaves like a
6-brane, and its field content is that of the 7-dimensional vector
multiplet, involving 3 scalars and 1 vector. Since the embedding
coordinates describe $11-7=4$ degrees of freedom one scalar has to be
eliminated by gauging an isometry of the background\footnote{For
constructions of gauged sigma models with WZ term in arbitrary
dimensions see \cite{HS}.}.

The Taub-NUT
space of the monopole is isometric in its Taub-NUT direction.
Let us denote ${\hat k}$ the Killing vector associated to this
isometry:

\begin{equation}
\delta {\hat X}^{\hat \mu}=-{\hat \sigma}^{(0)}{\hat k}^{\hat \mu}\, ,
\end{equation}

\noindent such that the Lie derivatives of all target space
fields and gauge
parameters with respect to ${\hat k}$ vanish.
The effective action of the monopole is constructed by replacing
ordinary derivatives by covariant derivatives:

\begin{equation}
\partial_{\hat \imath}{\hat X}^{\hat \mu}\rightarrow D_{\hat \imath}
{\hat X}^{\hat \mu}=\partial_{\hat \imath}{\hat X}^{\hat \mu}+
{\hat A}_{\hat \imath}{\hat k}^{\hat \mu}\, ,
\end{equation}

\noindent with ${\hat A}_{\hat \imath}$ a dependent gauge field
given by:

\begin{equation}
{\hat A}_{\hat \imath}=|{\hat k}|^{-2}\partial_{\hat \imath}
{\hat X}^{\hat \mu}{\hat k}_{\hat \mu}\, ,
\end{equation}

\noindent where $|{\hat k}|^2=-{\hat k}^{\hat \mu}
{\hat k}^{\hat \nu}{\hat g}_{{\hat \mu}{\hat \nu}}$.
Accordingly, the gauge transformation of ${\hat A}_{\hat \imath}$ is:

\begin{equation}
\delta {\hat A}_{\hat \imath}=\partial_{\hat \imath}
{\hat \sigma}^{(0)}\, .
\end{equation}

\noindent In this way the effective metric becomes:

\begin{equation}
{\hat g}_{{\hat \mu}{\hat \nu}}D_{\hat \imath}
{\hat X}^{\hat \mu}D_{\hat \jmath}{\hat X}^{\hat \nu}
={\hat \Pi}_{{\hat \mu}{\hat \nu}}\partial_{\hat \imath}
{\hat X}^{\hat \mu}\partial_{\hat \jmath}
{\hat X}^{\hat \nu}\, ,
\end{equation}

\noindent with

\begin{equation}
{\hat \Pi}_{{\hat \mu}{\hat \nu}}=
{\hat g}_{{\hat \mu}{\hat \nu}}+|{\hat k}|^{-2}
{\hat k}_{\hat \mu}{\hat k}_{\hat \nu}\, .
\end{equation}

\noindent Since ${\hat \Pi}_{{\hat \mu}{\hat \nu}}
{\hat k}^{\hat \nu}=0$ the isometry direction is effectively
eliminated from the action.

The construction of the ${\hat \sigma}^{(0)}$--invariant WZ term has
been given in \cite{BEL}.
We summarize the target space and worldvolume field content in
Tables \ref{table1} and \ref{table2}.

\begin{table}[!ht]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Target space & Gauge     \\
Field        & Parameter \\
\hline\hline
${\hat g}_{{\hat \mu}{\hat \nu}}$
& \\
\hline
${\hat C}_{{\hat \mu}{\hat \nu}{\hat \rho}}$ &
 ${\hat \chi}_{{\hat \mu}{\hat \nu}}$ \\
\hline
$(i_{\hat k} {\hat C})_{{\hat \mu}{\hat \nu}}$&
$(i_{\hat k} {\hat \chi})_{{\hat \mu}}$\\
\hline
${\hat {\tilde C}}_{{\hat \mu}_1 \dots {\hat \mu}_6}$
& ${\hat {\tilde \chi}}_{{\hat \mu}_1 \dots {\hat \mu}_5}$ \\
\hline
$({i}_{\hat k} {\hat {\tilde C}})_{{\hat \mu}_1 \dots {\hat \mu}_5}$
& $({i}_{\hat k} {\hat {\tilde \chi}})_{{\hat \mu}_1 \dots {\hat \mu}_4}$ \\
\hline
${\hat N}_{{\hat \mu}_1 \dots {\hat \mu}_8}$
& ${\hat \Omega}_{{\hat \mu}_1 \dots {\hat \mu}_7}$ \\
\hline
$({i}_{\hat k} {\hat {\tilde N}})_{{\hat \mu}_1 \dots {\hat \mu}_7}$ &
 $(i_{\hat k} {\hat \Omega})_{{\hat \mu}_1 \dots {\hat \mu}_6}$\\
\hline
\end{tabular}
\end{center}
\caption{\label{table1} \small {\bf Target space fields in the
M-KK-monopole.}
This table shows the 11-dimensional target space
fields that couple to the M-KK-monopole,
together with their gauge parameters. We also include
the contractions with the Killing vector ${\hat k}^{\hat \mu}$.
The field ${\hat {\tilde C}}$ is the Poincar\'e dual of ${\hat C}$,
the 3-form of eleven-dimensional supergravity,
and ${\hat N}$ is the Poincar\'e dual of the Killing vector,
considered
as a 1-form ${\hat k}_{\hat \mu}$.
}
\renewcommand{\arraystretch}{1}
\end{table}

\begin{table}[h]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Worldvolume     & $\sharp$ of   \\
Field           & d.o.f         \\
\hline\hline
${\hat X}^{{\hat \mu}}$ & $11 - 7 - (1) = 3$ \\
\hline
$ {\hat \omega}^{(1)}_{{\hat \imath}}$ &
 $7 - 2 = 5$\\
\hline
${\hat \omega}^{(6)}_{{\hat \imath}_1 \dots {\hat \imath}_6}$
& $-$ \\
\hline
\end{tabular}
\end{center}
\caption{\small \label{table2} {\bf Worldvolume fields.}
In this table we summarize the
worldvolume fields, together with their
degrees of freedom, that occur in the worldvolume
action of the M--theory KK--monopole.
The worldvolume scalars ${\hat X}^{\hat \mu}$ are the embedding coordinates,
${\hat \omega}^{(1)}$ is a 1-form and
${\hat \omega}^{(6)}$ is a non propagating 6-form that describes the
tension of the monopole. Due to the gauging the embedding scalars
describe 3 and not 4 degrees of freedom as indicated in the table.
}
\renewcommand{\arraystretch}{1}
\end{table}

The resulting effective action gives the D6-brane of the Type
IIA theory after a direct dimensional reduction along the Taub-NUT
direction \cite{BJO}.

In \cite{BEL} the effective action of the M-KK-monopole giving
rise to the D6-brane of the massive Type IIA theory was also
constructed. As discussed in \cite{BLO} eleven-dimensional
massive branes are described by gauged sigma models with gauge
coupling constant proportional to $m$. The massive D6-brane is
obtained by reducing the massive KK-monopole in which the two
Killing vectors associated to the mass and the Taub-NUT
isometries coincide. The gauge field ${\hat A}$ must be attributed
mass transformation rules and extra terms need to be added to the
WZ part. As we mentioned in the Introduction this massive
M-KK-monopole cannot give rise to the massive Type IIA KK-monopole
after a double dimensional reduction, since the gauged isometry
disappears in the reduction. However it is possible to construct
a massive M-KK-monopole in which the two isometry directions
associated to the mass and the Taub-NUT space are different.
Double dimensionally reducing along the direction associated to the
mass will give rise to the effective action of the massive IIA
KK-monopole.

It was shown in \cite{BLO} that a massive brane in eleven dimensions is
obtained by gauging an isometry generated by a Killing vector ${\hat h}$:

\begin{equation}
\delta_{{\hat \rho}^{(0)}}{\hat X}^{\hat \mu}=\frac{m}{2}
l_p^2 \, {\hat \rho}^{(0)}{\hat h}^{\hat \mu}\, ,
\end{equation}

\noindent through the introduction of a gauge field
${\hat b}_{\hat \imath}$ transforming as\footnote{Here
$l_p$ is the eleven-dimensional Planck length.}:

\begin{equation}
\delta {\hat b}_{\hat \imath}=\partial_{\hat \imath}
{\hat \rho}^{(0)}- l_p^{-2} \,
(i_{\hat h}{\hat \chi})_{\hat \imath}\, ,
\end{equation}

\noindent where $(i_{\hat h}{\hat \chi})$ is (the pull-back of)
the interior product of the gauge parameter of the
eleven-dimensional 3-form ${\hat C}$ with the Killing vector
${\hat h}$. Substituting ordinary derivatives by covariant
derivatives

\begin{equation}
{\cal D}_{\hat \imath}{\hat X}^{\hat \mu}=\partial_{\hat \imath}
{\hat X}^{\hat \mu}-\frac{m}{2} l_p^2 \,
{\hat b}_{\hat \imath}{\hat h}^{\hat \mu}
\end{equation}

\noindent one assures invariance under massive transformations:

\begin{equation}
\label{massivetr}
\delta_{\hat \chi}{\hat L}_{{\hat \mu}_1\dots {\hat \mu}_r}=
r\frac{m}{2} (-1)^r (i_{\hat h}{\hat \chi})_{[{\hat \mu}_1}
(i_{\hat h}{\hat L})_{{\hat \mu}_2\dots {\hat \mu}_r]}\, ,
\end{equation}

\noindent for a rank $r$ 11 dim form ${\hat L}$, and

\begin{equation}
\delta_{\hat \chi}{\hat g}_{{\hat \mu}{\hat \nu}}=-m
(i_{\hat h}{\hat \chi})_{({\hat \mu}}
(i_{\hat h} {\hat g})_{{\hat \nu})}\, ,
\end{equation}

\noindent for the 11 dim metric.

These transformations give rise to the known
massive transformations of the Type IIA background fields
after dimensional reduction.
Together with the gauging it is
necessary to include additional ${\hat b}$--dependent terms
(proportional to the mass) to achieve invariance under massive
transformations.

This suggests that we should construct the massive M-KK-monopole
by substituting:

\begin{equation}
{\hat \Pi}_{{\hat \mu}{\hat \nu}}\partial_{\hat \imath}
{\hat X}^{\hat \mu}\partial_{\hat \jmath}{\hat X}^{\hat \nu}
\rightarrow {\hat \Pi}_{{\hat \mu}{\hat \nu}}
{\cal D}_{\hat \imath}{\hat X}^{\hat \mu}{\cal D}_{\hat \jmath}
{\hat X}^{\hat \nu}\, ,
\end{equation}

\noindent with ${\cal D}{\hat X}$ as defined above. This can
also be written as:

\begin{equation}
{\hat \Pi}_{{\hat \mu}{\hat \nu}}{\cal D}_{\hat \imath}
{\hat X}^{\hat \mu}{\cal D}_{\hat \jmath}{\hat X}^{\hat \nu}
={\hat g}_{{\hat \mu}{\hat \nu}}D_{\hat \imath}
{\hat X}^{\hat \mu}D_{\hat \jmath}{\hat X}^{\hat \nu}\, ,
\end{equation}

\noindent with

\begin{equation}
\label{covdev}
D_{\hat \imath}{\hat X}^{\hat \mu}\equiv
\partial_{\hat \imath}{\hat X}^{\hat \mu}
+{\hat A}_{\hat \imath}{\hat k}^{\hat \mu}
-\frac{m}{2}l_p^2 \, {\hat b}_{\hat \imath}
{\hat h}^{\hat \mu}\, ,
\end{equation}

\noindent and:

\begin{equation}
\label{Atrans}
{\hat A}_{\hat \imath}=|{\hat k}|^{-2}{\hat k}_{\hat \mu}
\left( \partial_{\hat \imath}{\hat X}^{\hat \mu}-\frac{m}{2} \right.
\left. l_p^2 \, {\hat b}_{\hat \imath}{\hat h}^{\hat \mu}\right)\, .
\end{equation}

\noindent ${\hat \Pi}_{{\hat \mu}{\hat \nu}}$ transforms as a
metric under massive transformations iff

\begin{equation}
\label{vectortr}
\delta {\hat k}_{\hat \mu}=-\frac{m}{2}
(i_{\hat h}{\hat \chi})_{\hat \mu}(i_{\hat h}{\hat k})\, ,
\end{equation}

\noindent which implies that the Killing vector associated to the
Taub-NUT isometry must be attributed a massive transformation:

\begin{equation}
\label{ktrans}
\delta {\hat k}^{\hat \mu}=\frac{m}{2}(i_{\hat k}i_{\hat h}
{\hat \chi}){\hat h}^{\hat \mu}\, .
\end{equation}

\noindent The transformation rule (\ref{vectortr}) is that of a
vector under massive transformations (see (\ref{massivetr})).
We showed in \cite{BEL} that ${\hat k}_{\hat \mu}$ had to be
considered as a target space 1-form in order to construct the
action of the 11 dim KK-monopole. In particular, the monopole is
charged with respect to its dual 8-form\footnote{To be precise,
with respect to its interior product with the Killing vector
${\hat k}$.}. We see here that this is also the case regarding
massive transformations. With this transformation rule the interior
product of ${\hat k}$ with any eleven-dimensional $r$-form
transforms according to (\ref{massivetr}) (see Appendix B).

The fact that ${\hat k}^{\hat \mu}$ transforms under massive
transformations implies that the Killing condition and the
massive transformations do not commute\footnote{They commute
if $(i_{\hat k}i_{\hat h}{\hat \chi})=0$, which is not the
most general case. Note, however, that if in ten dimensions we
set the component of the RR 1-form along the Taub-NUT
direction to zero, which can always be done since $C^{(1)}$
is a non-physical Stueckelberg field \cite{Romans,BRGPT}, then
this condition is satisfied.}.

In the system of adapted coordinates to the isometry generated
by ${\hat h}$: ${\hat h}^{\hat \mu}=\delta^{\hat \mu}{}_y$, we
can define:

\begin{equation}
{\hat k}^y=\frac{m}{2} l_p^2 \, {\hat \omega}^{(0)}\, ,
\end{equation}

\noindent with

\begin{equation}
\delta {\hat \omega}^{(0)}=\frac{1}{2\pi\alpha^\prime}
(i_{\hat k}i_{\hat h}{\hat \chi})\, .
\end{equation}

\noindent In the reduction to ten dimensions ${\hat k}^{\mu}$
(with ${\mu}$ a ten dimensional index) will be the Killing
vector generating translations along the Taub-NUT direction,
and ${\hat \omega}^{(0)}$ an extra worldvolume scalar
needed to compensate for certain massive transformations.

It is easy to check that the invariance under ${\hat \sigma}^{(0)}$
is preserved\footnote{In the kinetic term. We will comment
later on the WZ term.} if the ${\hat b}$ field transforms as:

\begin{equation}
\label{btrans}
\delta {\hat b}_{\hat \imath}=
\partial_{\hat \imath} {\hat \rho}^{(0)}
-{\hat \sigma}^{(0)}\partial_{\hat \imath}
{\hat \omega}^{(0)}
- l_p^{-2} \, (i_{\hat h}{\hat \chi})_{\hat \imath}\, .
\end{equation}

The gauge transformation rule of ${\hat A}_{\hat \imath}$ (given
by (\ref{Atrans})) is still $\delta {\hat A}_{\hat \imath}=
\partial_{\hat \imath}{\hat \sigma}^{(0)}$, as in the massless case.

The action that we propose for the massive M-theory KK-monopole
is the following:

\begin{eqnarray}
\label{MKKaction}
{\hat S}&=&-T_{{\rm mMKK}}\int d^7 {\hat \xi} {\hat k}^2\sqrt{|{\rm det}
(D_{\hat \imath}{\hat X}^{\hat \mu}D_{\hat \jmath}
{\hat X}^{\hat \nu}{\hat g}_{{\hat \mu}{\hat \nu}}+
 l_p^2 |{\hat k}|^{-1}{\hat {\cal K}}^{(2)}_{{\hat \imath}
{\hat \jmath}})|}
\nonumber \\
&&+\,\, \frac{1}{7!} l_p^2 \, T_{{\rm mMKK}}\int d^7 {\hat \xi}
\,\, \varepsilon^{{\hat \imath}_1\dots{\hat \imath}_7}
{\hat {\cal K}}^{(7)}_{{\hat \imath}_1\dots {\hat \imath}_7}\, .
\end{eqnarray}

\noindent The covariant derivatives are defined by (\ref{covdev})
and ${\hat {\cal K}}^{(2)}$
is the massive field strength of the worldvolume field
${\hat \omega}^{(1)}$:

\begin{equation}
{\hat {\cal K}}^{(2)}=2\partial {\hat \omega}^{(1)}+
l_p^{-2} \, D{\hat X}^{\hat \mu}D
{\hat X}^{\hat \nu}(i_{\hat k}{\hat C})_{{\hat \mu}{\hat \nu}}
-m l_p^2 \, \partial{\hat \omega}^{(0)}{\hat b}  \, .
\end{equation}

\noindent Finally, ${\hat {\cal K}}^{(7)}$ is given by:

\begin{equation}
\begin{array}{rcl}
{\hat {\cal K}}^{(7)} &=&
7 \left\{ \partial {\hat \omega}^{(6)}
+m {\hat \omega}^{(7)}
-\frac32  l_p^2 \, m{\hat d}^{(5)}
\left(2\partial{\hat\omega}^{(1)}
-ml_p^2 \,\partial {\hat \omega}^{(0)}{\hat b} \right) \right. \\
& & \\
& &
\left.  - {1 \over 7} l_p^{-2} \,
D {\hat X}^{{\hat \mu}_1} \dots D
 {\hat X}^{{\hat \mu}_7}
(i_{\hat k} {\hat N})_{{\hat \mu}_1 \dots {\hat \mu}_7}
+3 D  {\hat X}^{{\hat \mu}_1} \dots
D  {\hat X}^{{\hat \mu}_5}
 (i_{\hat k}{\hat {\tilde C}})_{{\hat \mu}_1 \dots {\hat \mu}_5}
{\hat {\cal K}}^{(2)}\right.
\\ & & \\
& &
\left. - {5} l_p^{-2} \, D  {\hat X}^{{\hat \mu}_1}
\dots D  {\hat X}^{{\hat \mu}_7}
{\hat C}_{{\hat \mu}_1 \dots {\hat \mu}_3}
({i}_{\hat k} {\hat C})_{{\hat \mu}_4 {\hat \mu}_5}
 ({i}_{\hat k} {\hat C})_{{\hat \mu}_6 {\hat \mu}_7} \right.\\
& &\\
& &
\left. -15 D {\hat X}^{{\hat \mu}_1} \dots
D {\hat X}^{{\hat \mu}_5}
{\hat C}_{{\hat \mu}_1 \dots {\hat \mu}_3} ({i}_{\hat k} {\hat C})_{{\hat \mu}_4 {\hat \mu}_5}
 \left( 2\partial {\hat \omega}^{(1)}
-m l_p^2 \, \partial{\hat \omega}^{(0)}{\hat b}\right)\right.
\\ & & \\
& &
\left. -60 l_p^2 \,  D {\hat X}^{{\hat \mu}_1} \dots
D {\hat X}^{{\hat \mu}_3} {\hat C}_{{\hat \mu}_1 \dots
{\hat \mu}_3}\partial {\hat \omega}^{(1)}
\left( \partial {\hat \omega}^{(1)}-
m l_p^2 \, \partial {\hat \omega}^{(0)}{\hat b} \right)\right.\\
& &\\
& &
\left. -60 l_p^4 \,  {\hat A} \left( 2\partial {\hat \omega}^{(1)}
-3m l_p^2 \, \partial{\hat \omega}^{(0)}{\hat b} \right)
\partial {\hat \omega}^{(1)}\partial {\hat \omega}^{(1)}
 \right\} \, . \\
\end{array}
\end{equation}

\noindent This action is invariant under the gauge transformations given in
Appendices B and \ref{massiveMKK}.
Invariance under ${\hat \rho}^{(0)}$ transformations is
assured by the presence of covariant derivatives. It can also be seen that
with the transformation rule (\ref{btrans}) for the ${\hat b}$ field
the action is
invariant under ${\hat \sigma}^{(0)}$ transformations.

Notice that we have introduced a new auxiliary field,
${\hat d}^{(5)}$, associated to the dual massive transformations
with parameter $(i_{\hat h}{\hat \Sigma})$ (see \cite{BLO}).
This field transforms proportionally to
$(i_{\hat k}i_{\hat h} {\hat \Sigma})$ (see Appendix C.1), since
only contractions with the Taub-NUT Killing vector need to be
cancelled in the KK-monopole action.

Moreover, ${\hat b}$ is a non-propagating field in this action, playing
the role of gauge field for the ${\hat h}$ isometry\footnote{Additional
$m{\hat b}$ couplings are necessary as well in the WZ term to cancel
certain gauge variations.}.
We will see in the next section that it disappears in the double
dimensionally reduced action. This means that there are no fundamental
strings ending on the monopole.

When ${\hat k}$ and ${\hat h}$ are parallel, as in \cite{BEL},
${\hat \omega}^{(1)}$ transforms
like ${\hat b}$ (in (\ref{btrans})) \footnote{In this case
${\hat \omega}^{(0)}=0$.}
and it is not necessary to
introduce this new field.
Furthermore, an
extra term $-15m l_p^6 \, {\hat b}\partial {\hat b}\partial {\hat b}
\partial {\hat b}$ in the
WZ part of the action accounts for the variation of the ${\hat
  \omega}^{(7)}$ field\footnote{With the
  modification of the transformation law of ${\hat \omega}^{(6)}$
  found in \cite{BEL}.}, which is also not needed.
This is also the case for ${\hat \omega}^{(0)}$ and
${\hat d}^{(5)}$.

In Table \ref{table-mMKK} we summarize the worldvolume fields
present in (\ref{MKKaction}). All these fields can be given an
interpretation in terms of solitons on the KK-monopole.

\begin{table}[h]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Worldvolume     & Field  \\
Field           & Strength  \\
\hline\hline
${\hat \omega}^{(1)}_{{\hat \imath}}$ &
 ${\hat {\cal K}}^{(2)}_{{\hat \imath}{\hat \jmath}}$ \\
\hline
${\hat d}^{(5)}_{{\hat \imath}_1{\hat \imath}_2}$ &
$ $  \\
\hline
${\hat \omega}^{(6)}_{{\hat \imath}_1 \dots {\hat \imath}_6}$
& ${\hat {\cal K}}^{(7)}_{{\hat \imath}_1 \dots {\hat \imath}_7}$ \\
\hline
${\hat \omega}^{(7)}_{{\hat \imath}_1 \dots {\hat \imath}_7}$
 & \\
\hline
\end{tabular}
\end{center}
\caption{\label{table-mMKK} \small {\bf Worldvolume field content of the
massive M-theory KK-monopole.} In this table we give the
worldvolume fields, together with their field strengths,
present in the worldvolume action of the massive M--theory KK--monopole.}
\renewcommand{\arraystretch}{1}
\end{table}

We find, as in the massless case, a 1-form ${\hat \omega}^{(1)}$,
describing a 0-brane soliton in the worldvolume of the KK-monopole.
Its dual 4-form describes a 3-brane soliton.
They correspond to the intersections: $(0|{\rm M}2,{\rm MKK})$
and $(3|{\rm M}5,{\rm MKK})$,
respectively.
The monopole contains as well a
4-brane soliton which couples to the 5-form dual to one of the embedding
scalars: $(4|{\rm MKK},{\rm MKK})_{1,2}$.
All these intersections have been discussed in \cite{BREJS,BGT}.

In the massive case the field ${\hat d}^{(5)}$ couples to the 5-brane
soliton represented by the configuration\footnote{We use here a
notation where $\times (-)$ indicates
  a worldvolume (transverse) direction. The first $\times$ in a
  row indicates the time direction.} \cite{deRoo,BGT}:

\begin{displaymath}
 (5|{\rm M}9,{\rm MKK})=
\mbox{
$\left\{ \begin{array}{c|cccccccccc}
         \times & \times & \times  & \times & \times & z & - & \times & \times  & \times  & \times \\
         \times & \times & \times   & \times  & \times & \times & \times & z & -  & - & -
                                    \end{array} \right.$}
\end{displaymath}

\noindent Here the $z$-direction in the monopole corresponds to the
isometry direction of the Taub-NUT space. A single M9-brane
contains as well a Killing isometry in its worldvolume, as has been
discussed in \cite{BvdS,BEHHLS,proci}. This
isometric direction has been depicted as well as a $z$-direction.
The 5-brane soliton predicted by the M-KK and M9-brane worldvolume
supersymmetry algebras \cite{BGT} is realized as a 4-brane
soliton given that it cannot develop a worldvolume direction along the
isometry of the M9-brane. This is in agreement with the worldvolume
field content that we have found for the massive M-KK-monopole, since
the only worldvolume field to which this soliton can couple
is the 5-form ${\hat d}^{(5)}$.

The 6-form ${\hat \omega}^{(6)}$ is interpreted as the tension of
the monopole and couples to the 5-brane soliton
realized as the embedding of an M5-brane on the
KK-monopole \cite{Tsey,BREJS}.
In the massive case it also
plays the role of Stueckelberg field for the auxiliary field
${\hat \omega}^{(7)}$ (see their ${\hat \rho}^{(6)}$
transformation rules in Appendix C.1).

Finally, ${\hat \omega}^{(7)}$ couples to the 6-brane soliton describing
the embedding of the monopole in an M9-brane: $(6|{\rm M}9,{\rm MKK})$.
The M9-brane contains a 1-form vector field in its worldvolume
\cite{proci}. The dual of this massive 1-form in the nine
dimensional worldvolume is a 7-form field which from the point of view
of the M9-brane is the worldvolume field that couples to the
6-brane soliton.

\section{Double Dimensional Reduction:\\
Massive MKK $\rightarrow$ Massive KK-5A}
\label{MKK-->IIAKK}

We can now proceed and perform the double dimensional reduction
of the action constructed in the previous section.
We will see that the Killing isometry associated to translations of the
Taub-NUT coordinate is restored in this process.

In adapted coordinates
${\hat h}^{\hat \mu}=\delta^{\hat \mu}{}_y$, we take the ansatz
for the worldvolume reduction:
\begin{equation}
{\hat X}^y=Y={\hat \xi}^6
\end{equation}

\noindent with all other worldvolume fields and gauge parameters
independent of ${\hat \xi}^6$.

The ${\hat k}$ transformation rule (\ref{ktrans}) implies that
this vector gets a $y$-component under massive gauge transformations.
Therefore we reduce it as:

\begin{equation}
{\hat k}^\mu=k^\mu \, ,\qquad
{\hat k}^y = m 2\pi\alpha^\prime \omega^{(0)} \, .
\end{equation}

\noindent $k^\mu$ will be the Killing vector associated to the isometry
of the Taub-NUT space of the IIA KK-monopole.

In order to keep track of all the
gauge transformations we have to introduce a compensating gauge
transformation:

\begin{equation}
\delta {\hat \xi}^{{\hat \imath}}=\delta^{{\hat \imath}6}
[-\Lambda^{(0)}+\frac{m}{2}(2\pi\alpha^\prime)
{\rho}^{(0)}-\frac{m}{2}(2\pi\alpha^\prime)\sigma^{(0)}
\omega^{(0)}]\, ,
\end{equation}
\noindent where $\Lambda^{(0)}$ is a g.c.t. in the direction $Y$.

The reduction rules for the background fields and gauge parameters
can be found for instance in \cite{BEL}.
The worldvolume fields reduce as:

\begin{equation}
\begin{array}{rclrcl}
l_p^2 \, {\hat b}_i&=& 2 \pi \alpha^\prime \, b_i \, ,&
 l_p^2 \, {\hat b}_6 &=& 2 \pi \alpha^\prime \, v^{(0)} \, ,\\
& & \\
l_p^2 \, {\hat \omega}^{(1)}_i &=& 2 \pi \alpha^\prime \, {\omega}^{(1)}_i\, ,& \,\,\,\,\,\,
l_p^2 \, {\hat \omega}^{(1)}_6 &=& 2 \pi \alpha^\prime \, \omega^{(0)} \, ,\\
& & \\
l_p^2 \, {\hat d}^{(5)}_{i_1\dots i_5}&=& 2 \pi \alpha^\prime \, d^{(5)}_{i_1\dots i_5}
\, , & \,\,\,\,\,\,
l_p^2 \, {\hat \omega}^{(6)}_{i_1\dots i_5 6}&=&
2 \pi \alpha^\prime \, \omega^{(5)}_{i_1\dots i_5} \, ,\\
\end{array}
\end{equation}

\begin{displaymath}
l_p^2 \, {\hat \omega}^{(7)}_{i_1\dots i_6 6}=
-\frac37 (2\pi\alpha^\prime)^2 \, v^{(0)}
\partial_{[i_1}\omega^{(5)}_{i_2\dots i_6]}
+\frac67 (2 \pi \alpha^\prime) \left(1-\frac{m}{2}(2\pi\alpha^\prime) v^{(0)}\right)
 \, \omega^{(6)}_{i_1\dots i_6} \, .
\end{displaymath}

The (modified) gauge transformations of the new, reduced, worldvolume
fields can be found in Appendix C.2.

The reduction of ${\hat \omega}^{(6)}$ holds up to a total
derivative (see Appendix C).
The reduction of ${\hat d}^{(5)}_{i_1\dots i_4 6}$ gives a
worldvolume 4-form which is gauge invariant and contributes to the reduced
action with a decoupled term, therefore we have fixed it to zero. The scalar
field $v^{(0)}$ also has vanishing transformation law.
However this field contributes to the double dimensionally reduced
action as:

\begin{equation}
\int d^6\xi (1-\frac{m}{2}(2\pi\alpha^\prime)v^{(0)})
{\cal L}_{{\rm mAKK}}\, ,
\end{equation}

\noindent and is constrained to a constant by
the equation of motion of the worldvolume field $\omega^{(5)}$
playing the role of tension of the IIA monopole. In general it modifies
the tension of the new massive p-brane as \cite{BLO}:

\begin{equation}
\label{modifi}
T_{\rm mIIA} = \left( 1 - {m \over 2} (2 \pi \alpha^\prime)v^{(0)} \right)
T_{\rm mM} \, .
\end{equation}

This has the implication that for the particular value
$v^{(0)}=2/m(2\pi\alpha^\prime)$ the brane tension vanishes.
The physical mechanism behind this phenomenon is unclear and
it would be interesting to investigate (see the Conclusions for a
further discussion).

The dependent gauge field ${\hat A}$ reduces as:

\begin{eqnarray}
{\hat A}_i&=&\left( 1+e^{2\phi}|k|^{-2}(i_k C^{(1)}+
m\pi\alpha^\prime\omega^{(0)})^2 \right)^{-1}\times\nonumber\\
&&\times\left(A_i-e^{2\phi}
|k|^{-2}(C^{(1)}_i-m\pi\alpha^\prime b_i)(i_kC^{(1)}+
m\pi\alpha^\prime\omega^{(0)}) \right)\, ,
\end{eqnarray}

\noindent where $A_i\equiv |k|^{-2}k_{\mu}\partial_i X^\mu$, and

\begin{eqnarray}
{\hat A}_6&=&- \left( 1+e^{2\phi}|k|^{-2}(i_k C^{(1)}+m\pi\alpha^\prime
\omega^{(0)})^2 \right)^{-1}\times\nonumber\\
&&\times \,\, e^{2\phi}
|k|^{-2}(1-m\pi\alpha^\prime v^{(0)})(i_k C^{(1)}+
m\pi\alpha^\prime\omega^{(0)}) \, .
\end{eqnarray}

\section{The Action of the Massive IIA KK-monopole}
\label{mIIAKK}

The double dimensional reduction of the action of the massive M-KK-monopole
gives:

\begin{eqnarray}
\label{accionmasiva}
S&=&
-T_{\rm mAKK} \int d^6\xi \,\, k^2
e^{-2\phi}
\sqrt{1+e^{2\phi}k^{-2}(i_k C^{(1)}+m\pi\alpha^\prime\omega^{(0)})^2}
\times
\nonumber \\
&&\hspace{-1.5cm}
\times \sqrt{\Biggl|{\rm det} \left( \Pi_{ij}
-(2\pi\alpha^\prime)^2
k^{-2}{\cal K}^{(1)}_i {\cal K}^{(1)}_j
+\frac{(2\pi\alpha^\prime)k^{-1}e^{\phi}}{\sqrt{1+e^{2\phi}
k^{-2}(i_k C^{(1)}+m\pi\alpha^\prime\omega^{(0)})^2}}
{\cal K}^{(2)}_{ij} \right) \Biggr|}\nonumber \\
&&+\,\, \frac{1}{6!}(2\pi\alpha^\prime)T_{\rm mAKK} \int d^6\xi
\epsilon^{i_1\dots i_6}
{\cal K}^{(6)}_{i_1\dots i_6}\, .
\end{eqnarray}

\noindent The covariant derivative is defined as:
$D_i X^\mu=\partial_i X^\mu+A_i k^\mu$ and
\begin{equation}
\Pi_{ij}  =D_i X^{\mu}D_j X^{\nu}g_{\mu\nu} \, .
\end{equation}

\noindent The tension $T_{\rm mAKK}$ is the modified tension
given by (\ref{modifi}).

The gauge invariant forms ${\cal K}^{(2)}$ and ${\cal K}^{(1)}$
are the field strengths of
$\omega^{(1)}$ and $\omega^{(0)}$, respectively:

\begin{equation}
\label{curvas}
\begin{array}{rcl}
{\cal K}^{(2)} &=& 2\partial \omega^{(1)}+
\frac{1}{2\pi\alpha^\prime}(i_k C^{(3)})
- 2 {\cal K}^{(1)} (DX C^{(1)})
\\& &\\& &
+{m \over 2} \omega^{(0)} (DXDX B) - m (2 \pi \alpha^\prime) \omega^{(0)}
\partial \omega^{(0)} A \, ,\\ & &\\
{\cal K}^{(1)} &=& \partial {\omega}^{(0)}-\frac{1}{2\pi\alpha^\prime}
(i_k B) \, ,\\
\end{array}
\end{equation}

\noindent and the WZ term is the field strength of the worldvolume field
$\omega^{(5)}$, playing the role of tension of the IIA KK-monopole:

\begin{equation}
\begin{array}{rcl}
\label{WZKK}
{\cal K}^{(6)} &=& 6 (\partial \omega^{(5)}+m\omega^{(6)})
-3m(2\pi\alpha^\prime)d^{(5)}\partial\omega^{(0)}+
\frac{1}{2\pi\alpha^\prime}(i_k N)
\\
& & \\ & &
-15 (i_k C^{(5)})(2\partial\omega^{(1)}
+\frac{1}{2\pi\alpha^\prime}(i_k C^{(3)})+\frac{m}{2}\omega^{(0)}B)
\\
& & \\ & &
-6((i_k {\tilde B})
-\frac{m}{2}(2\pi\alpha^\prime)\omega^{(0)}C^{(5)})
{\cal K}^{(1)} \\
& & \\ & &
 -60 (2 \pi \alpha^\prime) DX^{{\mu}} DX^{{\nu}} DX^{{\rho}}
C^{(3)}_{{\mu} {\nu} {\rho}}
\, {\cal K}^{(1)} {\cal K}^{(2)}
\\ & & \\ & &
+{30 \over 2\pi\alpha^{\prime}}B
 (i_k C^{(3)})^2
+30DX^{\mu} DX^{\nu} DX^{\rho} C^{(3)}_{\mu \nu \rho}
( i_k C^{(3)}) {\cal K}^{(1)}
\\ & & \\ & &
-180 (2 \pi \alpha^\prime) DX^{\mu} DX^{{\nu}} B_{{\mu} {\nu}}
(\partial\omega^{(1)})^2
\\ & & \\ & &
+\frac{20}{2\pi\alpha^\prime}C^{(3)}(i_k B)(i_k C^{(3)}+
m\pi\alpha^\prime\omega^{(0)} B)\\ & & \\ & &
-\frac{15}{4}m^2(2\pi\alpha^\prime)
(\omega^{(0)})^2 DX^{\mu_1}\dots DX^{\mu_6}B^3_{\mu_1\dots\mu_6}
\\ & & \\ & &
-45m(2\pi\alpha^\prime)\omega^{(0)}DX^{\mu_1}\dots DX^{\mu_4}
B^2_{\mu_1\dots\mu_4}\partial\omega^{(1)}\\
& & \\ & &
+\frac{15}{2}m\omega^{(0)}B^2(i_k C^{(3)})
-360 (2 \pi \alpha^\prime)^2 A (\partial \omega^{(1)}+
\frac{m}{4}\omega^{(0)}B)^2
\partial \omega^{(0)}
\\ & & \\ & &
+ 15 (2 \pi \alpha^\prime)^2 { e^{2 \phi} |k|^{-2} \left(i_k C^{(1)}+
m\pi\alpha^\prime\omega^{(0)}\right) \over
1 + e^{2 \phi} |k|^{-2}
\left(i_k C^{(1)}+m\pi\alpha^\prime\omega^{(0)}\right)^2}
{\cal K}^{(2)}{\cal K}^{(2)}{\cal K}^{(2)}
 \, .\\
\end{array}
\end{equation}

In Table \ref{table-mIIAKK} we summarize the worldvolume field content.
The gauge transformation rules of background and worldvolume fields
can be found in Appendices B, C.2 and reference \cite{BLO}.
We summarize our notation for the Type IIA background fields in
Table \ref{IIAback}.

\begin{table}[h]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Worldvolume        & Field  \\
Field              & Strength  \\
\hline\hline
$ {\omega}^{(0)} $ &
${\cal K}^{(1)}_i$ \\
\hline
$ {\omega}^{(1)}_i$ &
 ${\cal K}^{(2)}_{ij}$ \\
\hline
$d^{(5)}$ & $ $  \\
\hline
${\omega}^{(5)}_{i_1\dots i_5}$
& ${\cal K}^{(6)}_{i_1\dots i_6}$ \\
\hline
${\omega}^{(6)}_{i_1\dots i_6}$
& \\
\hline
\end{tabular}
\end{center}
\caption{\label{table-mIIAKK} \small {\bf Worldvolume fields of the
massive IIA KK-monopole.}
In this table we give the
worldvolume fields, together
with their field strengths,
that occur in the effective
action of the massive IIA KK-monopole.}
\renewcommand{\arraystretch}{1}
\end{table}

\begin{table}[!ht]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Target space & Gauge     & Dual  & Gauge    \\
Field        & Parameter & Field & Parameter \\
\hline\hline
${g}_{{\mu}{\nu}}$, $\phi$
& $-$ & $-$ & $-$ \\
\hline
$B_{\mu \nu}$ & $\Lambda_\mu$ & ${\tilde B}_{\mu_1 \dots \mu_6}$
& ${\tilde \Lambda}_{\mu_1 \dots \mu_5}$ \\
\hline
$C^{(1)}_\mu$ & $\Lambda^{(0)}$ & $C^{(7)}_{\mu_1 \dots \mu_7}$
& $\Lambda^{(6)}_{\mu_1 \dots \mu_6}$ \\
\hline
${C}^{(3)}_{{\mu}{\nu}{\rho}}$ & $\Lambda^{(2)}_{\mu \nu}$ &
$C^{(5)}_{\mu_1 \dots \mu_5}$
& $\Lambda^{(4)}_{\mu_1 \dots \mu_4}$ \\
\hline
$k_{\mu}$ & $-$ & ${N}_{\mu_1 \dots \mu_7}$ &
${\Omega}^{(6)}_{\mu_1\dots\mu_6}$ \\
\hline
\end{tabular}
\end{center}
\caption{\label{IIAback} \small {\bf Target space fields of the type IIA
superstring.}
The type IIA background contains the NS-NS sector:
$( g_{\mu \nu}, \phi, B_{\mu \nu})$,
the RR sector:
$( C^{(1)}, C^{(3)})$,
and the Poincar{\'e} duals of the RR fields and the NS-NS 2-form $B$:
$( C^{(5)}, C^{(7)}, {\tilde B})$. The Kaluza-Klein monopole couples to a
new field $N$, dual to the Killing vector associated to the Taub-NUT
isometry, considered as a 1-form $k_\mu$.}
\renewcommand{\arraystretch}{1.5}
\end{table}

The action of the ten dimensional IIA KK-monopole is manifestly invariant
under translations of the Taub-NUT coordinate, since all the reduced fields
and gauge parameters have vanishing Lie derivative with respect to $k$. This
symmetry has been restored in the reduction by a mechanism in which the
$y$ component of the Killing vector in eleven dimensions gives rise to an
auxiliary worldvolume field that is needed to compensate the massive
variation of the Stueckelberg field $i_k C^{(1)}$.
A further check of this action is that for $m=0$ reduces to
the action of the massless IIA KK-monopole \cite{BEL}.

It is also worth noting that the field $b$ has
disappeared in the reduced action. This reflects the fact  that there
are no fundamental strings
ending on the monopole. Nevertheless there is a string-like object,
described by $\omega^{(1)}$, ending on the monopole.
In fact, the worldvolume fields that couple
to the soliton solutions of a KK-monopole are those necessary to
construct invariant field strengths for the fields
$i_k C^{(p+1)}$ (see \cite{EJL}).
These field strengths have the form:

\begin{equation}
{\cal K}^{(p)} = p\partial\omega^{(p-1)}+\frac{1}{2\pi\alpha^\prime}
(i_k C^{(p+1)})+\dots  \, ,
\end{equation}

\noindent so that $\omega^{(p-1)}$ couples to a
$(p-2)$-brane soliton which describes the boundary of a $p$-brane
ending on the monopole, with one of its worldvolume directions
wrapped around the Taub-NUT direction of the monopole. Since the
target space field associated to $\omega^{(1)}$ is $(i_k C^{(3)})$
it describes a wrapped D2-brane ending on the monopole.

We also find the soliton configurations:
$(2|{\rm D}4,{\rm KK})$, $(3|{\rm NS}5,KK)$ and $(3|{\rm KK},{\rm KK})_{1,2}$
(see \cite{Papa,EJL}). The only modifications due to the mass occur
in the explicit expressions of the field strengths, where
new terms proportional to the mass appear, which involve
worldvolume fields that already propagated in the massless case.
There is however an exception, and that is the presence of the
$d^{(5)}$ worldvolume field associated to the dual massive
transformations, which couples to a 4-brane soliton.
This soliton is a domain wall in the six dimensional
worldvolume and is described by the configuration:

\begin{displaymath}
 (4|{\rm D}8,{\rm KK})=
\mbox{
$\left\{ \begin{array}{c|ccccccccc}
         \times & \times & \times  & \times & \times & - & \times & \times & \times  & \times \\
         \times & \times & \times   & \times  & \times & \times & z & - & -  & -
                                    \end{array} \right.$}
\end{displaymath}

\noindent which is obtained by reducing the $(4|{\rm M}9,{\rm KK})$ soliton
configuration of the M-theory KK-monopole along the isometric
direction of the M9-brane.
This intersection is related to the Hanany-Witten
configuration \cite{HW}:

\begin{equation}
\label{conf1}
 \begin{array}{c|c}
{\rm D5}:\ \ \          \times & \times    \times   -   -   -  \times  \times   \times   - \\
{\rm NS5}:\          \times & \times   \times  \times  \times  \times  -  -   -   -    \\
{\rm D3}: \ \ \       \times & \times \times - - - - - - \times
                         \end{array}
\end{equation}

\noindent by T-duality along the 3,4,9 directions.
It is also T-dual to the intersection $(4|{\rm D}7,{\rm NS}5)$ in Type IIB,
corresponding to a 4-brane soliton in the IIB NS-5-brane
\cite{EGJP}.

There is as well another 4-brane soliton in the KK-monopole
worldvolume \cite{BREJS} which is already present in the
massless case. This is the embedding of the D4-brane on
the monopole: $(4|{\rm D}4,{\rm KK})$, and it couples to the worldvolume
field $\omega^{(5)}$, describing the tension of the monopole.
This is the reduction of the 5-brane soliton $(5|{\rm M}5,{\rm KK})$, which
couples to the tension ${\hat \omega}^{(6)}$ of the M-KK-monopole.
Finally, the reduction of the 6-brane soliton $(6|{\rm M}9,{\rm KK})$
gives a 5-brane
soliton realized as the embedding of the KK-monopole on a
KK-7A-brane\footnote{This brane is obtained by reducing the M9-brane
along a worldvolume direction other than the $z$-direction,
but it is not predicted by the Type IIA supersymmetry algebra.
As discussed in \cite{proci}, this is also the case for the
KK-6A brane, obtained by reducing the M-KK-monopole along a
transverse direction different from the Taub-NUT direction.
These branes are required by U-duality of M-theory on a
d-torus, as it has been shown in \cite{U1,U2,U3}.}.

\section{Massive $T$-duality:\\
 IIB NS-$5$ $\rightarrow$ Massive IIAKK}
\label{IIBNS5-->IIAKK}

In this section we obtain the massive IIA KK-monopole through
a ``massive'' T-duality transformation in the action of the
IIB NS-5-brane.

The action of the Type IIB NS-5-brane was constructed in
\cite{EJL} and is given by:

\begin{eqnarray}
\label{5brane-action}
\hspace{-.5cm} S &=&
-T_{{\rm NS-5B}} \int d^6 \xi \,\,
e^{-2\varphi} \sqrt{1 + e^{2 \varphi} (C^{(0)})^2} \, \times
\nonumber \\
&&\hspace{+1cm}
\times \, \sqrt{|{\rm det}
( {\j} - (2 \pi \alpha^\prime) {e^{\varphi} \over \sqrt{1 + e^{2 \varphi}
(C^{(0)})^2}} {\tilde {\cal F}} )|} \, \nonumber \\
&&\hspace{-.5cm}+ \,\, {1 \over 6!}(2 \pi \alpha^\prime) T_{{\rm NS-5B}} \int d^6\xi
\,\,  \epsilon^{i_1 \dots i_6}
{\tilde {\cal G}}^{(6)}_{i_1 \dots i_6} \, .
\end{eqnarray}
Here
${\tilde {\cal F}}=2\partial c^{(1)}+\frac{1}{2\pi\alpha^\prime} C^{(2)}$
and:
\begin{equation}
\begin{array}{rcl}
{\tilde {\cal G}}^{(6)} &=& \left\{
6\partial {\tilde c}^{(5)} - {1 \over 2 \pi \alpha^\prime}{\tilde {\cal B}}
- {45 \over 2(2 \pi \alpha^\prime)} {\cal B}C^{(2)}C^{(2)} -15 C^{(4)} {\tilde {\cal F}}
\right. \\ & &\\
& & - 180 (2 \pi \alpha^\prime) {\cal B} \partial c^{(1)}\partial c^{(1)}
-90 {\cal B} C^{(2)} \partial c^{(1)}
\\& &\\& &
\left.
+15 (2 \pi \alpha^\prime)^2 {C^{(0)} \over e^{-2 \varphi} + (C^{(0)})^2}
{\tilde {\cal F}}{\tilde {\cal F}}{\tilde {\cal F}}
\right\} \, .\\
\end{array}
\end{equation}

In Table \ref{IIBback} we have summarized our notation for the
Type IIB background fields.

\begin{table}[!ht]
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Target space & Gauge     & Dual  & Gauge    \\
Field        & Parameter & Field & Parameter \\
\hline\hline
${\j}_{{\mu}{\nu}}$, $\varphi$
& $-$ & $-$ & $-$ \\
\hline
${\cal B}_{\mu \nu}$ & $\Lambda_\mu$ & ${\tilde {\cal B}}_{\mu_1 \dots \mu_6}$
 & ${\tilde \Lambda}_{\mu_1 \dots \mu_5}$ \\
\hline
$C^{(0)}$ & $-$ & $-$ & $-$ \\
\hline
${C}^{(2)}_{{\mu}{\nu}}$ & $\Lambda^{(1)}_\mu$
& $C^{(6)}_{\mu_1 \dots \mu_6}$
& $\Lambda^{(5)}_{\mu_1 \dots \mu_5}$ \\
\hline
\end{tabular}
\end{center}
\caption{\label{IIBback} \small
{\bf Target space fields of the type IIB superstring.}
The Type IIB background contains
the common sector:
$( {\j}_{\mu \nu}, \varphi, {\cal B}_{\mu \nu} )$,
the RR sector: $(C^{(0)}, C^{(2)}, C^{(4)})$,
and the Poincar{\'e} duals of the 2-forms $C^{(2)}$ and ${\cal B}$:
$( C^{(6)}, {\tilde {\cal B}})$.
}
\renewcommand{\arraystretch}{1}
\end{table}

We apply now a T-duality transformation along a transverse
direction. The worldvolume fields do not change rank after
T-duality since the original and dual branes have the same
number of worldvolume dimensions.
Moreover, we have a new scalar field $Z'$, which is the T-dual
of the coordinate along which we perform the duality transformation.
The T-duality rules for the worldvolume fields are given by:

\begin{equation}
\begin{array}{rcl}
Z' &=& (2 \pi \alpha^\prime) \omega^{(0)} \, ,\\
& &\\
c^{(1) \prime} &=& - \omega^{(1)}
- {m \over 4} (2 \pi \alpha^\prime) (\omega^{(0)})^2 \partial Z \, ,\\
& &\\
\partial{\tilde c}^{(5) \prime} &=& \partial\omega^{(5)} +
60 (2 \pi \alpha^\prime)^2 \partial Z \partial \omega^{(1)} \partial \omega^{(1)}
\partial \omega^{(0)}
\\& &\\
& &+m\omega^{(6)}-\frac12 m(2\pi\alpha^\prime)
d^{(5)}\partial\omega^{(0)}\, .
\end{array}
\end{equation}

\noindent Here $Z$ is the Taub-NUT coordinate of the
KK-monopole. Its occurrence on the right hand side of the two
expressions above is required by gauge invariance, and assures
the gauged sigma-model structure necessary to describe the
KK-monopole.

Using the massive T-duality rules given in (\ref{IIB-IIA-massive}) and
(\ref{atres}) for the background fields,
we find the following transformations for the worldvolume
curvatures:

\begin{equation}
{\tilde {\cal F}}^{\prime} = - {\cal K}^{(2)} \, ,\qquad
{\tilde {\cal G}}^{(6) \prime}= {\cal K}^{(6)} \, ,
\end{equation}

\noindent where ${\cal K}^{(2)}$ is given by
(\ref{curvas}) and ${\cal K}^{(6)}$ by (\ref{WZKK}).
Substituting in the IIB NS-5-brane effective action we
recover the expression (\ref{accionmasiva})
for the effective action of the massive IIA KK-monopole that we
obtained from eleven dimensions.
This provides a check of that action as well as of the massive
T-duality rule (\ref{atres}) given in the Appendix.

One remark is in order at this point. In general the double
dimensional reduction of a massive M-brane gives a tension
\cite{BLO}:

\begin{equation}
T=\left( 1- {m \over 2} (2 \pi \alpha^\prime) v^{(0)} \right) {\hat T} \, \int d {\hat \xi}
\end{equation}

\noindent for the reduced brane, where ${\hat \xi}$ is the
compact worldvolume direction. A particular example is the massive
IIA KK-monopole obtained in the previous section.
This suggests that in the massive case the tensions should transform
under T-duality as:

\begin{equation}
T^\prime_{{\rm B5}}=T_{{\rm mAKK}}=
\left( 1- {m \over 2} (2 \pi \alpha^\prime) v^{(0)} \right) T_{{\rm mMKK}}\, ,
\end{equation}

\noindent which seems, however, an arbitrary choice from the
T-duality point of view. We leave some especulations about this point for the
Conclusions.

\section{Conclusions}

We have constructed the worldvolume effective action of the
Type IIA KK-monopole propagating in a background with
non-vanishing cosmological constant. The worldvolume field
content of this brane consists on a scalar, a 1-form, two
5-forms and a 6-form. One of the two 5-forms is interpreted as
the tension of the monopole, whereas the other one is
associated to dual massive transformations in the worldvolume
of the monopole. The 6-form and the tension of the monopole
are interpreted as a Stueckelberg pair with respect to massive
transformations.

These worldvolume fields have been interpreted
in terms of soliton solutions propagating in the worldvolume of
the monopole. The new feature with respect to the massless case
is that there is a 4-brane soliton realized as the domain wall
intersection of the KK-monopole with a D8-brane, and a 6-brane
soliton corresponding to the embedding of the monopole on
a KK7-brane.

We have already mentioned that the KK7-brane belongs
to a particular class of Type IIA brane solutions which are not
predicted by the spacetime supersymmetry algebra. It is unclear
why this happens. These branes have been encountered already
in the literature in a different context, namely they are required
in order to fill up multiplets of BPS states in representations
of the U-duality symmetry group of M-theory on a d-torus
\cite{U1,U2,U3,MO}. As we discuss in \cite{proci}, where we
calculate the kinetic part of their worldvolume effective actions,
these branes do not have an obvious interpretation in weakly coupled
string theory  since they scale with the coupling constant more singularly
than $1/{g_s^2}$.

The derivation of the KK-monopole effective action from eleven
dimensions gives  a tension depending on the mass and on a Wilson line
of the gauge field associated to the massive isometry
(which is set to a constant value
by the integration of the worldvolume 5-form playing the role
of tension of the monopole) (see (\ref{modifi})).
In particular, the tension can vanish for a certain value
of the Wilson line. In fact, this is a general feature for the Type IIA
massive branes that are obtained from eleven dimensions by a
double dimensional reduction \cite{BLO}.
The presence of covariant derivatives with respect to massive
transformations in the eleven-dimensional massive branes implies
that in the double dimensional reduction the brane can be
wrapped along the eleventh direction with unusual
winding number, which depends on the mass and on a Wilson line
of the gauge field.
Therefore, consistency implies that $(1-\frac{m}{2}(2\pi\alpha^\prime)
v^{(0)})$ has to be an integer, i.e. the winding number, with
the tension of the reduced brane proportional to it.
The Wilson line may be interpreted as a distance
$2 \pi \alpha^\prime v^{(0)}$ corresponding to a D8-brane,
since a massive IIA brane is a brane in the presence
of D8-branes.
This is similar to the interpretation of the masses
of the string states in terms of the D8-brane positions
in a Type ${\rm I}^\prime$ construction.
Since the D8-branes are T-dual to the
D9-branes, the arbitrariness of $v^{(0)}$ from the T-duality point
of view might be originated by Wilson lines in Type IIB with
D9-branes.

Finally it would be of interest to analyze the six dimensional
interacting gauge theory that arises from considering a system
of parallel massive KK-monopoles in the limit in which gravity is
decoupled \cite{sixdi}.

\section*{Acknowledgements}

We would like to thank E.~Bergshoeff for useful discussions.
The work of E.E.~ is part of the research program of the
Dutch Foundation FOM.