hep-th0207246.tex
\count100=1 
% \input defrrr.tex 

\documentstyle[12pt]{article}

\def\doit#1#2{\ifcase#1\or#2\fi}

% Underline for text or math
\catcode`@=11
\catcode`@=12

% Accents and foreign (in text):
\let\under=\b                   % bar-under (but see \un above)
\let\ced=\c                     % cedilla
\let\du=\d                      % dot-under
\let\um=\H                      % Hungarian umlaut
\let\sll=\l                     % slashed (suppressed) l (Polish)
\let\Sll=\L                     % " L
\let\slo=\o                     % slashed o (Scandinavian)
\let\Slo=\O                     % " O
\let\tie=\t                     % tie-after (semicircle 
%   connecting two letters)
\let\br=\u                      % breve

% Abbreviations for Greek letters
\def\a{\alpha} \def\b{\beta} \def\c{\chi} \def\d{\delta}
\def\e{\epsilon} \def\f{\phi} \def\g{\gamma}
\def\h{\eta} \def\i{\iota} \def\j{\psi} \def\k{\kappa}
\def\l{\lambda} \def\m{\mu} \def\n{\nu} \def\o{\omega}
\def\p{\pi} \def\q{\theta} \def\r{\rho} \def\s{\sigma}
\def\t{\tau} \def\u{\upsilon} \def\x{\xi} \def\z{\zeta}
\def\D{\Delta} \def\F{\Phi} \def\G{\Gamma} \def\J{\Psi}
\def\L{\Lambda} \def\O{\Omega} \def\P{\Pi} \def\Q{\Theta}
\def\S{\Sigma} \def\U{\Upsilon} \def\X{\Xi}

% Use the following definitions of \pmb, \bfalpha, etc, 
% for `boldfaced' Greek characters. (03/05/99) 
\def\pmb#1{\setbox0=\hbox{${#1}$}%
   \kern-.025em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern.05em\copy0\kern-\wd0
   \kern-.035em\copy0\kern-\wd0
   \kern-.025em\box0 }

\def\bfalpha{\pmb{\alpha}} \def\bfbeta{\pmb{\beta}} 
\def\bfgamma{\pmb{\gamma}}
\def\bfdelta{\pmb{\delta}} \def\bfepsilon{\pmb{\epsilon}}
\def\bfzeta{\pmb{\zeta}}  \def\bfeta{\pmb{\eta}} 
\def\bftheta{\pmb{\theta}}
\def\bfiota{\pmb{\iota}}  \def\bfkappa{\pmb{\kappa}}
\def\bflambda{\pmb{\lambda}} \def\bfmu{\pmb{\mu}}  
\def\bfnu{\pmb{\nu}}
\def\bfxi{\pmb{\xi}} \def\bfomicron{\pmb{o}}  \def\bfpi{\pmb{\pi}}
\def\bfroh{\pmb{\rho}} \def\bfsigma{\pmb{\sigma}}  
\def\bftau{\pmb{\tau}}
\def\bfupsilon{\pmb{\upsilon}} \def\bfphi{\pmb{\phi}}  
\def\bfchi{\pmb{\chi}}
\def\bfpsi{\pmb{\psi}} \def\bfomega{\pmb{\omega}}

\def\bfAlpha{\pmb{A}} \def\bfBeta{\pmb{B}} \def\bfGamma{\pmb{\Gamma}}
\def\bfDelta{\pmb{\Delta}} \def\bfEpsilon{\pmb{E}} \def\bfZeta{\pmb{Z}} 
\def\bfEta{\pmb{H}} \def\bfTheta{\pmb{\Theta}} \def\bfIota{\pmb{I}} 
\def\bfKappa{\pmb{K}} \def\bfLambda{\pmb{\Lambda}} \def\bfMu{\pmb{M}} 
\def\bfNu{\pmb{N}} \def\bfXi{\pmb{\Xi}} \def\bfOmicron{\pmb{O}} 
\def\bfPi{\pmb{\Pi}} \def\bfRoh{\pmb{P}} \def\bfSigma{\pmb{\Sigma}} 
\def\bfTau{\pmb{T}} \def\bfUpsilon{\pmb{\Upsilon}} 
\def\bfPhi{\pmb{\Phi}} 
\def\bfChi{\pmb{X}} \def\bfPsi{\pmb{\Psi}} \def\bfOmega{\pmb{\Omega}}

% Math symbols
\def\dvp{\raisebox{-.45ex}{\rlap{$=$}} 
\raisebox{-.45ex}{$\hskip .48ex { |}$}} 
\def\dvm{\raisebox{-.45ex}{\rlap{$=$}} }
\def\uvp{\raisebox{.45ex}{\rlap{$=$}} 
\raisebox{.45ex}{$\hskip .48ex { 
|}$}}   
\def\uvm{\raisebox{.45ex}{\rlap{$=$}} }
\def\DP{{\scriptsize{\dvp}}~~}\def\DM{{\scriptsize{\dvm}}~~}
\def\UP{{\scriptsize{\uvp}}~~}
\def\UM{{\scriptsize{\uvm}}~~} % 2-d vector indices
%\def\bo{{\raise.15ex\hbox{\large$\Box$}}} % D'Alembertian
\def\bo{{\raise-.46ex\hbox{\large$\Box$}}} % D'Alembertian
\def\cbo{\Sc [}                            % curly "
\def\pa{\partial}                          % curly d
\def\dell{\bigtriangledown}               % hi ho the dairy-o

\def\pr{\prod}                            % product
\def\iff{\leftrightarrow}                % <-->
\def\conj{{\hbox{\large *}}}             % complex conjugate

\def\ltap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$<$}} %< or ~
\def\gtap{\raisebox{-.4ex}{\rlap{$\sim$}} \raisebox{.4ex}{$>$}} %> or ~
\def\TH{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-4.7mu %
\llap H \;}}
\def\face{{\raise.2ex\hbox{$\displaystyle \bigodot$}\mskip-2.2mu %
\llap {$\ddot
        \smile$}}}                           % happy face
\def\dg{\sp\dagger}                         % hermitian 
% conjugate
\def\ddg{\sp\ddagger}                      % double dagger
\font\tenex=cmex10 scaled 1200

% Math stuff with one argument
\def\sp#1{{}^{#1}}                 % superscript (unaligned)
\def\sb#1{{}_{#1}}                              % sub"
\def\oldsl#1{\rlap/#1}                          % poor slash
\def\sl#1{\rlap{\hbox{$\mskip 1 mu /$}}#1} % good slash for lower case

\def\Sl#1{\rlap{\hbox{$\mskip 3 mu /$}}#1}      % " upper
\def\SL#1{\rlap{\hbox{$\mskip 4.5 mu /$}}#1}    % " fat stuff (e.g., M)
\def\PMMM#1{\rlap{\hbox{$\mskip 2 mu | $}}#1}   % 
\def\PMM#1{\rlap{\hbox{$\mskip 4 mu ~ \mid $}}#1}       % 

%\def\Tilde#1{{\widetilde{#1}}\hskip 0.03in}                    
\def\Tilde#1{{\widetilde{#1}}\hskip 0.015in}     % big tilde
\def\Hat#1{\widehat{#1}}                        % big hat
\def\Bar#1{\overline{#1}}                       % big bar
\def\bra#1{\left\langle #1\right|}              % < |
\def\ket#1{\left| #1\right\rangle}              % | >
\def\VEV#1{\left\langle #1\right\rangle}        % < >
\def\abs#1{\left| #1\right|}                    % | |
\def\leftrightarrowfill{$\mathsurround=0pt \mathord\leftarrow 
 \mkern-6mu
        \cleaders\hbox{$\mkern-2mu \mathord- \mkern-2mu$}\hfill
        \mkern-6mu \mathord\rightarrow$}
\def\dvec#1{\vbox{\ialign{##\crcr
        \leftrightarrowfill\crcr\noalign{\kern-1pt\nointerlineskip}
        $\hfil\displaystyle{#1}\hfil$\crcr}}}           % <--> accent
\def\dt#1{{\buildrel {\hbox{\LARGE .}} \over {#1}}}% dot-over 
% for sp/sb
\def\dtt#1{{\buildrel \bullet \over {#1}}}              % alternate "
\def\der#1{{\pa \over \pa {#1}}}              % partial derivative
\def\fder#1{{\d \over \d {#1}}}               % functional derivative

% Math stuff with more than one argument
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise.20ex
        \hbox{$\scriptstyle{#2}$}}}}}   % fraction
\def\sfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{\small$#1$}}\over
        \vphantom1\smash{\raise.4ex\hbox{\small$#2$}}}}
       %alternate fraction
\def\bfrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over
        \vphantom1\smash{\raise.3ex\hbox{$#2$}}}}       % "
\def\afrac#1#2{{\vphantom1\smash{\lower.5ex\hbox{$#1$}}\over#2}} % "
\def\partder#1#2{{\partial #1\over\partial #2}} % partial derivative 
 % of
\def\parvar#1#2{{\d #1\over \d #2}}               % variation of
\def\secder#1#2#3{{\partial^2 #1\over\partial #2 \partial #3}}  %second
 %  "
\def\on#1#2{\mathop{\null#2}\limits^{#1}}       % arbitrary accent
\def\bvec#1{\on\leftarrow{#1}}                % backward vector accent
\def\oover#1{\on\circ{#1}}                      % circle accent

% Aligned equations
\newskip\humongous \humongous=0pt plus 1000pt minus 1000pt
\def\caja{\mathsurround=0pt}
\def\eqalign#1{\,\vcenter{\openup2\jot \caja
        \ialign{\strut \hfil$\displaystyle{##}$&$
        \displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}
\newif\ifdtup
\def\panorama{\global\dtuptrue \openup2\jot \caja
        \everycr{\noalign{\ifdtup \global\dtupfalse
        \vskip-\lineskiplimit \vskip\normallineskiplimit
        \else \penalty\interdisplaylinepenalty \fi}}}
\def\li#1{\panorama \tabskip=\humongous      % eqalignno
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}
\def\eqalignnotwo#1{\panorama \tabskip=\humongous
        \halign to\displaywidth{\hfil$\displaystyle{##}$
        \tabskip=0pt&$\displaystyle{{}##}$
        \tabskip=0pt&$\displaystyle{{}##}$\hfil
        \tabskip=\humongous&\llap{$##$}\tabskip=0pt
        \crcr#1\crcr}}

% The following is to be skipped when using harvmac.tex
\doit0{
\def\ref#1{$\sp{#1)}$}
}

% Text style parameters
\topmargin=0in                          % top margin (less 1") (LaTeX)
\headheight=0in                         % height of heading (LaTeX)
\headsep=0in            % separation of heading from body (LaTeX)
\textheight=9in                         % height of body (LaTeX)
%\footheight=3ex                           % height of foot (LaTeX)
\footheight=15ex                           % height of foot (LaTeX)
\footskip=4ex    % distance between bottoms of body & foot (LaTeX)
\textwidth=6in                          % width of body (LaTeX)
\hsize=6in                              % " (TeX)
\parskip=\medskipamount          % space between paragraphs (LaTeX)
\lineskip=0pt                           % minimum box separation (TeX)
\abovedisplayskip=1em plus.3em minus.5em        %space above equation 
%   (either)
\belowdisplayskip=1em plus.3em minus.5em        % " below
\abovedisplayshortskip=.5em plus.2em minus.4em  %"above 
%   when no overlap
\belowdisplayshortskip=.5em plus.2em minus.4em  %"below
\def\baselinestretch{1.2}       % magnification for line spacing 
%   (LaTeX)
\thicklines                         % thick straight lines for pictures 
%   (LaTeX)

% Title page
% \thispagestyle{empty}               % no heading or foot on title page 
%   (LaTeX)
\def\oldheadpic{                                % old UM heading
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        \vskip.2in}
\def\border{                                            % border
        \setlength{\unitlength}{1mm}
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        \begin{picture}(140,0)
        \put(\xco,\yco){$\ktl$}
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        \yco=12
        \put(\xco,\yco){$\ktr$}
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        \repeat
        \put(\xco,\yco){$\kbr$}}
        \put(-20,11){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics % 
University of Maryland Elementary Particle Physics}
        \put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
        \end{picture}
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        \repeat
        \put(\xco,\yco){$\kbl$}}
        \xco=158
        \yco=12
        \put(\xco,\yco){$\ktr$}
        \advance\yco by-1
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        \put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhd%
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\def\headpic{                                           % UM heading
        \indent
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        \par
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        \put(165,16){\line(1,0){4}}
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\def\endtitle{\end{quotation}\newpage}  % end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} %
\nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
\def\delsl{{{\partial\!\!\! /}}}
\def\nablasl{{{\nabla\!\!\!\!\!{\hskip 1.0pt}/}}}
\def\caldsl{{\calD\!\!\! /}}    
\def\calO{{\cal O}}
\def\asym{({\scriptstyle 1\leftrightarrow \scriptstyle 2})}
\def\Lag{{\cal L}}
\def\du#1#2{_{#1}{}^{#2}}
\def\ud#1#2{^{#1}{}_{#2}}
\def\dud#1#2#3{_{#1}{}^{#2}{}_{#3}}
\def\udu#1#2#3{^{#1}{}_{#2}{}^{#3}}
\def\calA{{\cal A}}\def\calB{{\cal B}}\def\calC{{\cal C}}
\def\calD{{\cal D}}
\def\calE{{\cal E}}\def\calF{{\cal F}}\def\calG{{\cal G}}
\def\calH{{\cal H}}
\def\calI{{\cal I}}\def\calJ{{\cal J}}\def\calK{{\cal K}}
\def\calL{{\cal L}} 
\def\calM{{\cal M}}\def\calN{{\cal N}}\def\calO{{\cal O}}
\def\calP{{\cal P}}
\def\calQ{{\cal Q}}\def\calR{{\cal R}}\def\calS{{\cal S}}
\def\calT{{\cal T}}
\def\calU{{\cal U}}\def\calV{{\cal V}}\def\calW{{\cal W}}
\def\calX{{\cal X}}
\def\calY{{\cal Y}}\def\calZ{{\cal Z}}
\def\rma{{\rm a}} \def\rmb{{\rm b}} \def\rmc{{\rm c}} 
\def\rmd{{\rm d}} 
\def\rme{{\rm e}} \def\rmf{{\rm f}} \def\rmg{{\rm g}} 
\def\Mpl{M_{\rm Pl}}
\def\mw{m\low{\rm W}}
\def\Mgut{M\low{\rm GUT}}
%\def\Re{{\rm Re}\,\,}  \def\Im{{\rm Im}\,\,} 
\def\Re{{\cal R}e\,}  \def\Im{{\cal I}m\,}
\def\tildef{{\tilde f}}
\def\calDsl{{\calD\!\!\!\! /}}
\def\plpl{{{\hskip0.03em}{}{+{\hskip -0.563em}
{\raise-0.03em\hbox{$_+$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}} 
\def\mimi{{{\hskip0.03em}{}{-{\hskip -0.563em}
{\raise-0.05em\hbox{$_-$}}
{\hskip 0.05pt}}{}{\hskip0.03em}}}   
\def\E{{\cal E}}
\def\order#1#2{{\cal O}({#1}^{#2})}
\def\alp{\alpha{\hskip 0.007in}'}
\def\oalp#1{\alp^{\hskip 0.007in {#1}}}
\def\naive{{{na${\scriptstyle 1}\!{\dot{}}\!{\dot{}}\,\,$ve}}}
\def\pl#1#2#3{Phys.~Lett.~{\bf {#1}B} (19{#2}) #3}
\def\np#1#2#3{Nucl.~Phys.~{\bf B{#1}} (19{#2}) #3}
\def\prl#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (19{#2}) #3}
\def\pr#1#2#3{Phys.~Rev.~{\bf D{#1}} (19{#2}) #3}
\def\cqg#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (19{#2}) #3} 
\def\cmp#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\jmp#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (19{#2}) #3} 
\def\ap#1#2#3{Ann.~of Phys.~{\bf {#1}} (19{#2}) #3} 
\def\prep#1#2#3{Phys.~Rep.~{\bf {#1}C} (19{#2}) #3}
\def\jhep#1#2#3{Jour.~High Energy Phys.~{\bf {#1}} (19{#2}) #3}
\def\ptp#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (19{#2}) #3}
\def\nc#1#2#3{Nuovo Cim.~{\bf {#1}} (19{#2}) #3}
\def\ibid#1#2#3{{\it ibid.}~{\bf {#1}} (19{#2}) #3}
\def\grg#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (19{#2}) {#3} }
\def\pla#1#2#3{Phys.~Lett.~{\bf A{#1}} (19{#2}) {#3}}
\def\mpl#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (19{#2}) #3} 
\def\zp#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (19{#2}) {#3}} 
\def\jgtp#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (19{#2}) {#3}}
\def\rmp#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (19{#2}) {#3}}
\def\hepth#1{hep-th/{#1}} \def\heplat#1{hep-lat/{#1}} 
\def\hepph#1{hep-ph/{#1}} 
\def\szet{{${\scriptstyle \b}$}}
\def\ula{{\underline a}} \def\ulb{{\underline b}} 
\def\ulc{{\underline c}} \def\uld{{\underline d}} 
\def\ule{{\underline e}} \def\ulf{{\underline f}} 
\def\ulg{{\underline g}} \def\ulm{{\underline m}} 
\def\ulp{{\underline p}} \def\uln{{\underline n}}
\def\ulq{{\underline q}} \def\ulr{{\underline r}}
\def\ul{\underline} 
\def\un{\underline} 
%\def\uln{{\underline n}} 
\def\uln#1{\underline{#1}}
\def\ulp{{\underline p}} \def\ulq{{\underline q}} 
\def\ulr{{\underline r}}
\def\ulA{{\un A}} \def\ulM{{\underline M}} 
\def\hatm{\hat m}\def\hatn{\hat n}\def\hatr{\hat r}\def\hats{\hat s}
\def\hatt{\hat t}
\def\<<{<\!\!<} \def\>>{>\!\!>} 
\def\Check#1{{\raise0.6pt\hbox{\Large\v{}}{\hskip -10pt}{#1}}}
\def\cdm{{\Sc D}_{--}} \def\cdp{{\Sc D}_{++}}
\def\Pisl{{\Pi\!\!\!\! /}\,}
\def\eqques{{~\,={\hskip -11.5pt}\raise -1.8pt\hbox{\large ?}
{\hskip 4.5pt}\,}}
%\def\fracmm#1#2{{{#1}\over{#2}}}
%\def\fracm#1#2{\hbox{\large{${\frac{{#1}}{{#2}}}$}}}
\def\fracm#1#2{\,\hbox{\large{${\frac{{#1}}{{#2}}}$}}\,}
\def\fracmm#1#2{\,{{#1}\over{#2}}\,}
\def\half{{\fracm12}}
\def\ha{\half}
\def\frac#1#2{{\textstyle{#1\over\vphantom2\smash{\raise -.20ex
        \hbox{$\scriptstyle{#2}$}}}}}   % fraction
%\def\dt#1{\Dot{#1}}
\def\sqrttwo{{\sqrt2}}
\def\scst{\scriptstyle}
\def\itrema{$\ddot{\scriptstyle 1}$}
\def\Bo{\bo{\hskip 0.03in}}
\def\lrad#1{ \left( A 
{\buildrel\leftrightarrow\over D}_{#1} B\right)} 
\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
\def\Vec#1{{\overrightarrow{#1}}}
\def\.{.$\,$}
\def\-{{\hskip 1.5pt}\hbox{-}}
\def\kd#1#2{\d\du{#1}{#2}}

\def\footnotew#1{\footnote{\hsize=6.5in {#1}}} 

\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
\def\ip{{=\!\!\! \mid}}
\def\gg{{\hbox{\sc g}}}
\def\tr{\,\,{\rm tr}\,} \def\Tr{\,\,{\rm Tr}\,}
\def\det{{\rm det}\,} 
\def\Norm{|\hskip -2.0pt|}
\def\Sgn{\hbox{\rm Sgn}\,}
\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} 
\def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

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%%%%%%%%%%%% Worksheet %%%%%%%%%%%%%%%%
% \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \thispagestyle{empty}

%{\bf Preliminary Version (FOR YOUR EYES ONLY!)\hfill\today}
% \\[-0.2in] 
% {\hfill\today} \\[-0.3in] 

\doit0{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0207246}}\par 
{\hbox to\hsize{\hfill CSULB--PA--02--3}}\par 
\vskip -0.1in 
\hfill 
%{(Revised Version)} 
\\ 

\begin{center} 
%\vglue .25in 

%\vskip -0.6in 
\vskip 0.01in 

{\large\bf Gauged~ $N=2$~ Supergravity~ in~ 
Nine$\,$-$\,$Dimensions} 
\\  
{\large\bf and ~Domain ~Wall ~Solutions} \\[.1in]

\baselineskip 9pt 

\vskip 0.26in 

\doit0{ 
Hitoshi ~N{\small ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} 
\\[.16in] 
{\it Department of Physics} \\ [.015in] 
{\it University of Maryland} \\ [.015in] 
{\it College Park, MD 20742-4111} \\[.20in]   
and \\[0.2in] 
} 

Hitoshi ~N{\small ISHINO}\footnotew{E-Mail: 
hnishino@csulb.edu} 
~and~ Subhash ~R{\small AJPOOT}\footnotew{E-Mail: 
rajpoot@csulb.edu} 
\\[.16in] 
{\it Department of Physics \& Astronomy} \\ [.015in] 
{\it California State University} \\ [.015in] 
{\it Long Beach, CA 90840} \\[0.02in] 


\vskip 2.3in 

{\bf Abstract}\\[.1in]  
\end{center} 

\baselineskip 14.5pt 


~~~We present massive $~N=2$~ supergravity with
$~SO(2)\-$gauging in nine-dimensions by direct construction.  A
full lagrangian and transformation rules are fixed, respectively up
to quartic and quadratic fermion terms.  Corresponding to the
generalized Scherk-Schwarz dimensional reduction utilizing
$~SL(2,\IR)$~ symmetry, this theory allows three arbitrary mass
parameters $~m_0,~m_1$~ and $~m_2$~ in addition to the
minimal gauge coupling $~g$, so that our system has the most
general form compared with other results in the past.  Unlike
ordinary gauged maximal supergravity theories in other
dimensions, the scalar potential is positive definite for arbitrary
values of the mass parameters.  As an application, we also analyze
the stability and supersymmetry for 7-brane domain wall solutions
for this gauged maximal supergravity, keeping the three mass
parameters.   


\vskip 0.35in

\leftline{\small PACS: ~04.20.F, 04.50, 04.65, 11.15, 
11.27, 04.70} 
\vskip -0.05in
\leftline{\small Key Words: ~Supergravity, Gauging, 
Nine-Dimensions, D-Brane, M-Theory} 
\vskip -0.05in 
\vfill\eject 

\baselineskip 15.8pt

\oddsidemargin=0.03in
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\footskip=0.26in 
\def\baselinestretch{0.8} 
%\footheight=1in 
%\bottomfraction=.25

\pageno=2



\leftline{\bf 1.~~Introduction} 


There has been increasing recognition of the importance of
maximal supergravity theories in dimensions between four and
eleven, due to their relationship with M-theory or superstring
theory.  Gauged or massive maximal
supergravity, such as type IIA theory in 10D 
%%% 
\ref\romans{L.~Romans, \pl{169}{86}{374}.}
%%%  
have more importance, due to the possible duality with
non-massive theories 
%%% 
\ref\singh{H.~Singh, {\it `Duality Symmetric Massive Type II
Theories  in D=8 and D=6'}, \hepth{0109147}, JHEP {\bf 0204}
(2002) 017.}.  
%%%  
Furthermore, these massive maximal supergravity theories
are dual to other non-massive theories, such as massive type IIA
theory is related to type IIB theory under T-duality  
%%% 
\ref\bho{E.~Bergshoeff, C.M.~Hull and T.~Ort\'in, 
\hepth{9504081}, \np{451}{95}{547};  E.~Bergshoeff, M.~de
Roo, M.B.~Green, G.~Papadopoulos, P.K.~Townsend,
\hepth{9601150}, 
\np{470}{96}{113}.}.  
%%%  
Additionally, the importance of massive supergravity
theories with  cosmological constants is associated with the AdS
background which in turn is related to brane-domain
wall/AdS/CFT correspondence 
%%% 
\ref\adscft{L.~Randall and R.~Sundrum, \hepth{9905221},
\prl{83}{99}{3370}; \ibid{83}{99}{4690};  
J.~Maldacena, \hepth{9711200}, 
Adv.~Theor.~Math.~Phys.~{\bf 2} (1998) 231;   
H.J.~Boonstra, K.~Skenderis and P.K.~Townsend, 
\hepth{9807137}, JHEP {\bf 9901} (1999) 003.}.    
%%% 

Massive supergravity theories are generated by new 
generalized dimensional reduction scheme utilizing certain 
$~\s\-$model symmetry 
%%% 
\ref\llp{I.V.~Lavrinenko, H.~L\"u and C.N.~Pope,
hep-th/9710243,  Class.~Quant.~Grav.~{\bf 15} (1998)
2239.}%
%%% 
\ref\mo{P.~Meessen, T.~Ort\' in, hep-th/9806120, 
Nucl.~Phys.~{\bf B541} (1999) 195.},  
%%%  
as a generalization of Scherk-Schwarz type dimensional 
reduction
%%% 
\ref\scherk{J.~Scherk and J.H.~Schwarz, Nucl.~Phys.~{\bf
B153} (1979) 61.}.   
%%%  
Such massive supergravity is possible in space-time
dimensions up to $~D\le 10$~ 
%%% 
\ref\ssdiverse{{\it `Supergravity in Diverse Dimensions'},
Vols.~{\bf 1} \& {\bf 2}, A.~Salam and E.~Sezgin, {\it eds}.,
North-Holland, World Scientific (1989); {\it  and references
therein}.},  
%%%  
while gauged supergravity is possible only in $~D\le 9$~ 
\ssdiverse.  
Therefore, it seems that 9D is the unique
maximal space-time dimensions for massive maximal supergravity
with gauging \ssdiverse.  

Motivated by these developments, there have been some works
on maximal supergravity in 9D.  The first work was in 
%%% 
\ref\kk{N.~Khviengia and Z.~Khviengia,  {\it `D=9 Supergravity
and p-Brane Solitons'}, \hepth{9703063}.}, 
%%%  
in which the non-gauged $~N=2$~ supergravity lagrangian and
transformation rule were given.  However, the results in \kk\ seem
to suffer from certain flaws caused by technical but fundamental
mistakes.  One example is the mistreatment of the barred spinors,
in the dimensional reduction from 11D into 9D.  For example, the
barred spinor parameter of supersymmetry $~\Bar\e$~ in 11D
does not stay just the same 
$~\Bar\e$~ in 9D any longer, after the simple dimensional
reduction \scherk.  The reason is that the charge  conjugation
matrix $~\Hat C$~ in 11D is antisymmetric, while that
$~C$~ in 9D is symmetric.  Therefore, there must be an extra
antisymmetric matrix, such as the second Pauli matrix $~\s_2$~
should be present:  $~\Hat C = \s_2 \otimes C$.  This
affects many terms given in \kk, such as the missing
$~\s_2\-$matrix for the 
$~F_{\m\n\r\s}\-$terms in the gravitino transformation rule
$~\d_{\rm Q}\psi_\m$, that in turn leads to the non-closure of
supersymmetry on the neunbein $~e\du\m m$, caused by the
unwanted $~e\du\m m\big(\Bar\e_2
\g^{\n\r\s\t}\e_1\big) F_{\n\r\s\t}\-$term.  This is due to the
flipping property of gamma-matrices in 9D that leaves this
combination non-vanishing, thus violating the fundamental closure
of supersymmetry.  

The first version of $~SO(2)\-$gauging of maximal supergravity
in 9D was given in 
%%%
\ref\cowdall{P.M.~Cowdall, {\it `Novel Domain Wall and 
Minkowski Vacua of D=9 Maximal SO(2) Gauged Supergravity'},
\hepth{0009016}.},    
%%% 
based on a generalized dimensional reduction  combining
the $~SL(2,\IR)$~ symmetry with the extra coordinate 
dependence \llp\mo\ from $~N=2$~ supergravity 10D into 9D, 
as a generalization of Scherk-Schwarz dimensional reduction  
\scherk.  
Recently, it has been claimed  
%%% 
\ref\gm{J.~Gheerardyn and P.~Meessen, \hepth{0111130}, 
Phys.~Lett.~{\bf B525} (2002) 322.}  
%%%  
that the Minkowski background in 9D \Minknine\  is
realized as the vacuum solution for gauged $~N=2$~
supergravity,  without preserving supersymmetry.  However, this 
is unusual for supergravity, because in ordinary supergravity
a scalar potential is minimized with supersymmetry, unless
there is a nonzero cosmological constant.  On the other hand, it
has been already  well-known that the scalar potential for gauged
$~N=2$~ supergravity  in 9D is positive definite \cowdall% 
%%% 
\ref\hull{C.M.~Hull, {\it private communications}.}.  
%%%  
To deal with such a subtle issue, it seems also imperative to derive
both the total lagrangian and the transformation rule in a
consistent manner.  For example, even though supersymmetry
transformation rules for gauged maximal supergravity in 9D were
given in \gm, it is still important to see the mutual  consistency
between the total lagrangian and transformation rules.  

Considering these recent developments, it seems imperative to
establish a more complete system of massive gauged $~N=2$~
supergravity  in 9D with a consistent lagrangian and
supersymmetry transformation rule.  In this Letter, we give a more
complete result for $~N=2$~ gauged maximal supergravity in 9D,
based on direct construction within 9D, instead of dimensional
reduction from 11D or 10D.  We perform the
$~SO(2)\-$gauging of the
$~\s\-$model coset $~SL(2,\IR) / SO(2)$, by a vector field
available in the multiplet.   We keep all the three possible mass
parameters $~m_1,~m_2$~ and $~m_3$~ in addition to the
minimal gauge coupling $~g$.  These  mass parameters
corresponds to the generalization \llp\ of the dimensional
reduction by \scherk, so that our system is the most general
compared with other systems \kk\cowdall\gm\ in the past.  




\bigskip\bigskip\medskip 

% \newpage 


\leftline{\bf 2.~~Lagrangian and Transformation Rule}   


The field content of our gauged $~N=2$~ supergravity in 9D is 
$~\big( e\du\m m, \psi_{\m\, A}, A_{\[3\]}, B_{\[2\] \, \a}, 
A_\m , 
\newline B_{\m
\, \a} , \chi\low{\,i\, A} , \l_A , L\du\a i , \varphi)$.  Here the
gravitino 
$~\psi_{\m\, A}$~ is a pair of Majorana spinors forming 
$~{\bf 2}_{\rm s}$~ of $~SO(2)$~ with the spinorial $~{\bf
2}_{\rm s}\-$indices $~{\scst A,~B,~\cdots}$~ usually
suppressed.  The
$~A_{\m\n\r}$~ and $~A_\m$~ are both real, while
$~B_{\m\n\, \a}$~ and
$~B_{\m\n\r\, \a}$~ both carry the curved two-dimensional
index $~{\scst \a~=~1,~2}$~ for the coset $~SL(2,\IR) /
SO(2)$.  The fermion 
$~\chi\low{\,i\,A}$~ carries both the $~{\bf 2}_{\rm
v}\-$index $~{\scst i}$~ of $~SO(2)$~ and $~{\bf 2}_{\rm
s}\-$index $~{\scst A}$~ of
$~SO(2)$, the latter of which is usually suppressed.  This
$~\chi\low {\, i}$~ is also subject to the extra constraint $~\t_i
\chi\low{\, i} =0$~  in order to have the right degrees of
freedom.  The fermion
$~\l_A$~ is in the $~{\bf 2}_{\rm s}$~ of $~SO(2)$.  The real
scalars
$~L\du\a i$~ are the coset representatives for
$~SL(2,\IR)/SO(2)$, while
$~\varphi$~ is a real scalar dilaton.  As for notations, we use
$~\big(\eta\low{m n}\big) \equiv \hbox{diag.}\,\big( +--\cdots
-), ~\{ \g_m, \g_n\} = + 2 \eta\low{m n} I$, and
$~\e^{\[9-n][n]} \g_{\[n\]} = (-1)^{n(n-1)/2} \, (n!) \,
\g^{\[9-n\]}$.  Here we use the symbol such as 
$~{\scst \[ n \]}$~ for a normalized antisymmetrized 
indices, {\it e.g.,} $~\g^{\[n\]}$~ is equivalent to
$~\g^{m_1\cdots m_n}$.  We use also the $~\t\-$matrices
defined by 
$$ \li{ & \t_1 \equiv \pmatrix{ 0 & 1 \cr 1 & 0 \cr } ~~, ~~~~
          \t_2 \equiv \pmatrix{ 1 & 0 \cr 0 & -1 \cr } ~~, ~~~~
          \t_3 \equiv \pmatrix{ 0 & -1 \cr 1 & 0 \cr } ~~,  
&(2.1) \cr } $$  
so that $~\t_i \t_j = \d_{i j} + \t_{i j} 
= \d_{i j} + \e\low{i j} \t_3$, and $~\big(\t_3 \big)^2 = -
I$.  We need to assign the indices $~{\scst i~=~1,~2}$~ for
$~{\bf 2}_{\rm v}$~ on the {\it symmetric} $~2\times 2$~
matrices, and this is why  our $~\t\-$matrices have the
numbering different from the standard Pauli matrices.  The
multiplication of fermions by these matrices are  such as
$~\big(\Bar\psi_\m
\t_3 \l\big) \equiv 
\Bar\psi_m{}_A \big(\t_3\big)_{A B} \l_B$, where due to the 
positive definite metric for $~SO(2)$, there is no distinction for 
super/subscripts of $~{\scst A,~B}$.   For each value
of  the $~{\scst A}\-$index, the flipping property for the
spinorial  product is such that $~\big( \Bar\psi \g^{\[n\]}
\chi\big)   = -(-1)^{n(n-1)/2} \big( \Bar\chi \g^{\[n\]}
\psi\big)$, as in the 
\hbox{$~N=1$~} supergravity in 9D 
%%%
\ref\gns{S.J.~Gates, Jr., H.~Nishino and E.~Sezgin, 
\cqg{3}{86}{21}.}.%
%%%
\footnotew{There is a crucial sign error with a 
flipping property equation in \gns.  The equation 
$~\big(\Bar\psi \g^{\m_1\cdots\m_n} \chi\big) = (-1)^n 
\big(\Bar \chi\g^{\m_n\cdots\m_1}\psi\big)$~ given below
eq.~(22b) in
\gns\ should be replaced by $~\big(
\Bar\psi\g^{\m_1\cdots\m_n}
\chi\big) = -\big(\Bar \chi\g^{\m_n\cdots\m_1} \psi\big) = - 
(-1)^{n(n-1)/2} \, \big( \Bar\chi
\g^{\m_1\cdots\m_n} \psi\big)$.  This can be reconfirmed by
the  aid of 
%%%
\ref\kt{T.~Kugo and P.K.~Townsend, \np{211}{83}{157}.}.}  
%%%
Relevantly, we have $~\big(i\Bar\psi\g^{\[n\]}
\chi\big)^\dagger = + \big(i \Bar\psi\g^{\[n\]} \chi\big)$. 

The coset representative form the $~\s\-$model quantities
$~\calP$~ and $~\calQ$~  defined by \ssdiverse
$$ \li{ & X_{\m \, i j} \equiv L\du i \a \big( \partial_\m 
      L_{\a j} 
       + g \eta\low{\a\b} \e^{\b\g} A_\m L_{\g j} \big) 
     \equiv L\du i \a \calD_\m L_{\a j} 
     = \calP_{\m\, i j} + \calQ_{\m\, i j} ~~, \cr  
& \calP_{\m i j} \equiv X_{\m (i j)}~~, ~~~~
      \calQ_{\m i j} \equiv X_{\m \[i j\]} 
       \equiv \calQ_\m  \, \e_{i j} ~~, 
        ~~~~\calP_{\m i i } \equiv 0 ~~, \cr   
& \eta\low{\a\b} \equiv m_0 \d_{\a\b} 
     + m_1 \big( \t_1\big)_{\a\b} 
     + m_2 \big( \t_2\big)_{\a\b} = \eta\low{\b\a} ~~.    
&(2.2) \cr } $$  
The condition $~\calP_{\m i i } \equiv 0$~ is from the 
unimodular nature of $~L\du\a i$~ for $~SL(2,\IR) / SO(2)$.   
The $~\eta\low{\a\b}\-$matrix can be also regarded as a
`metric'  tensor, in the sense that it lowers the index
$~{\scst\b}$~ on
$~\e^{\b\g}$~ in $~X_{\m\, i j}$.   Note that the three
constants $~m_0, ~m_1$~ and 
$~m_2$~ correspond to the mass parameters in \cowdall\ 
created by a generalized dimensional reduction \llp\ from 10D
into 9D using the $~SL(2,\IR)\-$symmetry, as a generalization
of the Scherk-Schwarz dimensional reduction \scherk.
The explicit representation of the coset representatives 
$~L$'s and the metric on the coset $~SL(2,\IR) / SO(2)$~
\cowdall\gm\ are 
$$ \li{ & \big( L_{\a i} \big) = 
     e^{\phi/2} \pmatrix{ e^{-\phi} & - \r \cr 
       0 & -1 \cr } ~~, ~~~~
     \big( L\du i \a \big) = 
     e^{- \phi/2} \pmatrix{ e^\phi & - e^\phi \r \cr 
       0 & -1 \cr } ~~, \cr  
& \big( g\low{\a\b} \big) \equiv \big( L_{\a i} L_{\b i} \big)   
    = e^\phi \pmatrix{ \r^2 + e^{-2\phi} & \r \cr 
     \r & 1 \cr } ~, ~~~
    \big( g^{\a\b} \big) \equiv \big( L\du i\a L\du i \b \big)    
    = e^\phi \pmatrix{ 1 & - \r \cr 
     - \r & \r^ 2 + e^{-2\phi}  \cr } {\, , ~~~~~ ~~~} 
&(2.3) \cr } $$ 
where $~g_{\a\b}$~ corresponds to the matrix
$~\calM$~ in \cowdall.  

Another important quantity for gauged supergravity is  the
matrix $~S_{i j}$~ defined by 
$$ \li{ S_{i j} \equiv & \eta\low{\a\b} L\du i \a L\du j \b 
     = S_{j i} \cr 
= & \pmatrix { (m_0+m_2) e^{\phi} 
     - 2 m_1 e^{\phi} \rho + (m_0-m_2) e^{\phi} \rho^2   
     & ~~ & - m_1 +(m_0-m_2) \rho \cr 
     - m_1 +(m_0-m_2) \rho
     & ~~ & (m_0 -m_2) e^{-\phi}  \cr}   
    \equiv \pmatrix{ A & B \cr B & C \cr } \, , \cr    
S \equiv & S_{i i} = \eta\low{\a\b} g^{\a\b} = A + C ~~,     
&(2.4) \cr } $$  
which is very similar to other maximally extended supergravity
models \ssdiverse, such as $~N=2$~ in 8D 
%%% 
\ref\sseight{A.~Salam and E.~Sezgin, \np{258}{85}{284}.}   
%%% 
or $~N=4$~ in 7D   
%%% 
\ref\ppvn{M.~Pernici, K.~Pilch and P.~van Nieuwenhuizen, 
\pl{143}{84}{103}.}.  
%%%

% \newpage

With these preliminaries, we are ready to give our lagrangian  
$$\li{ e^{-1} \Lag = & -\frac 1 4 R 
     - \frac i 2 \big( \Bar\psi_\m\g^{\m\n\r} 
      \calD_\n\psi_\r \big)
     - \frac 1{48} \expfour F_{\[4\]}^2 
     + \frac 1{12} \expmtwo \big( G_{\[3\] \, i} \big)^2  
     - \frac 14 \expmeight F_{\m\n}^2 \cr  
& - \frac 14 \expsix \big( G_{\m\n\, i} \big)^2
    + \frac 12 \big( \partial_\m\varphi\big)^2  
    + \frac i 2 \big( \Bar\chi\low{\,i} \g^\m\calD_\m
     \chi\low{\,i} \big) 
    + \frac i 2 \big( \Bar\l\g^\m \calD_\m\l \big) 
     + \frac 14 \big( \calP_{\m\, i j} \big)^2 \cr  
& + \exptwo F_{\[4\]} \Big[ 
     - \frac i {96} \big( \Bar\psi{}^\r \t_3 
     \g\low{\[\r|} \g^{\m\n\[4]} \g\low{|\s\]} \psi^\s \big)  
     -  \frac i{24{\sqrt{14}}} 
      \big(\Bar\psi_\m \t_3 \g^{\[4\]} \g^\m
      \l\big) \cr  
& ~~~~~ ~~~~~ ~~~~~ ~~~ - \frac i{96}  
      \big(\Bar\chi\low{\,i} \t_3 \g^{\[4\]} \chi\low{\,i}  \big)    
      + \frac i {224}  
     \big( \Bar\l \t_3 \g^{\[4\]} \l \big) \, \Big] \cr  
& + \expmone G_{\[ 3 \] \, i}  \Big[ 
     - \frac i {24} 
      \big( \Bar\psi{}^\r \t_i\g\low{\[\r|} \g^{\[3\]}
      \g\low{|\s\]} \psi^\s \big)
      + \frac i {12} \big(\Bar\psi_\m \g^{\[3\]} 
       \g^\m\chi\low{\,i}  \big) \cr  
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ + \frac i {12\sqrt 4} 
       \big( \Bar\psi_\m \t_i \g^{\[3\]} \g^\m\l \big) 
         G_{\[3\]\, i}  
       + \frac i {6\sqrt{14}} \big( \Bar\chi\low {\,i} 
       \g^{\[3\]} \l \big)   
        - \frac i {28} \big( \Bar\l \t_i \g^{\[3\]} \l \big)  
       \,\Big]  \cr  
& + \expmfour F_{\m\n} \Big[ - \frac i 8  
     \big( \Bar\psi{}^\r \g\low{\[\r|} \g^{\m\n} 
      \g\low{|\s\]} \psi^\s \big)  
    + \frac i {\sqrt{14}}\big( \Bar\psi_\m\g^{\r\s} 
    \g^\m\l\big) \cr   
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ + \frac i 8 \big(
   \Bar\chi\low{\,i} \g^{\m\n} \chi\low{\,i} \big) 
    - \frac {9i}{56} \expmfour \big(\Bar\l\g^{\m\n}\l\big) 
    \, \Big] {~~~~~ ~~~~~ } \cr  
& + \expthree G_{\m\n\, i}\Big[ 
      - \frac i 8 \big( \Bar\psi{}^\r \t_i \g\low{\[\r|} 
       \g^{\m\n} \g\low{|\s\]} \psi^\s \big)
     + \frac i 4 \big( \Bar\psi_\m \g^{\r\s} 
      \g^\m \chi\low{\,i} \big)
      -  \frac{3i}{4\sqrt{14}} 
       \big( \Bar\psi_\m \t_i \g^{\r\s} \g^\m\l\big)  \cr  
& ~~~~~ ~~~~~ ~~~~~ ~~~~~ + \frac{3i}{2\sqrt{14}}  
      \big( \Bar\chi\low{\,i} \g^{\m\n} \l \big)  
      - \frac i {28} \expthree 
     \big( \Bar\l \t_i \g^{\m\n} \l \big) \,\Big] \cr  
& + \frac i 2 
     \big( \Bar\psi_\m \t_i \g^\n \g^\m 
      \chi\low{\,j} \big) \calP_{\n\,i j} 
     + \frac i {\sqrt 2} \big( \Bar\psi_\m\g^\n\g^\m\l\big) 
     \partial_\n\varphi \cr  
& + \frac 1{576} e^{-1} \e^{\[4\]\[4\]'\m}
      F_{\[4\]} F_{\[4\]'} A_\m 
       - \frac1{216} e^{-1} \e^{\[3\]\[3\]'\[3\]''} \, 
     \e^{\a\b} \, 
    \Tilde G_{\[3\]\,\a} \Tilde G_{\[3\]' \,\b} A_{\[3\]''} \cr  
& - \frac1{36} e^{-1} \e^{\[3\]\[3\]' \[2\]\m} 
     \e^{\a\b} \e^{\g\d} 
      \Tilde G_{\[3\]\, \a} \Tilde G_{\[3\]'\, \b} 
      B_{\[2\]\, \g} B_{\m\, \d} \cr  
& - \frac i {16} g \expfour S 
     \big(\Bar\psi_\m\t_3 \g^{\m\n}\psi_\n \big) 
       - \frac i {2\sqrt{14}} g \expfour S 
     \big(\Bar\psi_\m \t_3 \g^\m \l \big) 
      - \frac i {16}  g \expfour S 
     \big( \Bar\chi\low{\,i} \t_3 \chi\low{\, i} \big) \cr  
& - \frac{9i}{116} g \expfour S \big( \Bar\l \t_3 \l \big) 
     + \frac i 2 g \expfour S_{i j} 
     \big( \Bar \chi\low{\,i} \t_3 \chi\low{\, j} \big) 
     + \frac i4 g \expfour S_{i j} 
     \big( \Bar\psi_\m \t_3 \t_i\g^\m \chi\low{\, j} \big) \cr  
& - \frac {2i}{\sqrt{14}}S_{i j} \big( \Bar\chi\low{\,i} 
      \t_3 \t_j\l\big) 
     - \frac 1{32} g^2 \expeight \Big[ 2 \big(S_{i j} \big)^2 
     - S^2 \Big] {~~, ~~~~~ ~~~}  
&(2.5) \cr } $$
up to quartic fermion terms.  
Our action $~I = \int d^9 x\, \Lag$~
is invariant under supersymmetry
$$ \li{ \d_Q e\du\m m = & - i  \big(\Bar\e\g^m \psi_\m \big)
~~, \cr 
\noalign{\vskip 0.05in} 
\d_Q \psi_\m = & + \calD_\m \e 
    + \frac1{112} \exptwo \t_3 
    \big( \g\du\m{\n\[3\]} 
     - \frac{16} 3 \d\du\m\n\g^{\[3\]} \big) \e 
      \Hat F_{\n\[3\]} \cr  
&  +  \frac1{42} \expmone \t_i 
     \big( \g\du\m{\n\[2\]} 
      - \frac{15} 2 \d\du\m\n \g^{\[2\]} \big) 
        \Hat G_{\n\[2\]\, i}   
      + \frac1{28} \expmfour 
     \big( \d\du\m{\n\r} 
      - 12\d\du\m\n\g^\r \big) \e \Hat F_{\n\r} \cr  
&  + \frac 1{28} \expthree \t_i       
       \big( \g\du\m{\n\r} -12 \d\du\m\n \g^\r \big)\e
       \Hat G_{\n\r\, i} 
     - \frac1{56} g \expfour \t_3 \g_\m \e S 
     \equiv \Hat\calD_\m \e ~~, \cr 
\noalign{\vskip 0.05in} 
\d_Q A_{\m\n\r} = & - \frac{3 i} 2 \expmtwo
     \big( \Bar\e \t_3 \g_{\[\m\n} \psi_{\r\]} \big) 
     + \frac i{\sqrt{14}} \expmtwo
     \big( \Bar\e \t_3\g_{\m\n} \l \big) \cr 
&  + 6 \e^{\a\b} B_{\[\m\n|\, \a} 
     \big( \d_Q B_{|\n\] \, \b} \big) 
     + 24 \e^{\a\b} A_{\[\m|} B_{|\n| \, \a} \big( \d_Q
         B_{|\r\] \, \b} \big)      \cr 
\noalign{\vskip 0.05in} 
\d_Q B_{\m\n\, \a} = & - i \expone L\du\a i 
    \big( \Bar\e \t_i \g_{\[\m} \psi_{\n\]} \big) 
     - \frac i2 \expone L\du\a i
     \big(\Bar\e\g_{\m\n}\chi\low{\,i} \big) \cr  
& - \frac i {2\sqrt{14}} L\du\a i 
    \big( \Bar\e \t_i \g_{\m\n} \l \big) 
     - 4 B_{\[ \m | \,\a } \big( \d_Q A_{|\n\]} \big) ~~, \cr 
\noalign{\vskip 0.05in} 
\d_Q A_\m  = & - \frac i 2 \expfour\big( \Bar\e\psi_\m\big) 
       - \frac {2i}{\sqrt{14}}\expfour 
       \big( \Bar\e\g_\m\l\big) ~~, \cr
\noalign{\vskip 0.05in} 
\d_Q B_{\m \, \a} = & - \frac i 2 \expmthree L_{\a \, i } 
      \big(\Bar\e\t_i \psi_\m\big) 
     - \frac i 2 \expmthree L_{\a\, i} 
     \big(\Bar\e\g_\m\chi\low i \big) 
      + \frac{3 i}{2{\sqrt{14}}}L_{\a\, i} 
     \big(\Bar\e\t_i \g_\m\l\big) ~~, \cr 
\noalign{\vskip 0.05in} 
\d_Q \chi\low{\,i} = & + \frac12 \t_j \g^\m \e 
      \Hat\calP_{\m \, i j}
    - \frac 14 \expthree \big(\d_{i j}- \frac12 \t_i \t_j \big) 
     \g^{\m\n} \e \Hat G_{\m\n\, j} \cr  
&  - \frac 18 \expmone 
     \big( \d_{i j} - \frac 12 \t_i\t_j \big) \g^{\[3\]} \e 
     \Hat G_{\[3\]\, j}  + \frac 1 8 g \expfour 
     \big( \d_{i j} - \e_{i j} \t_3\big) \t_3 \t_k \e S_{j k} ~~, \cr 
\noalign{\vskip 0.05in} 
\d_Q \l = &  - \frac1{\sqrt{14}}\expmfour \g^{\m\n} \e
     \Hat F_{\m\n} 
     + \frac3{4\sqrt{14}} \expthree \t_i \g^{\m\n} \e
      \Hat G_{\m\n\, i} \cr  
& - \frac1{12\sqrt{14}}\expmone \t_i
     \g^{\[3\]} \e \Hat G_{\[3\]\, i} 
     + \frac 1{24\sqrt{14}}\exptwo \t_3 \g^{\[4\]} \e
     \Hat F_{\[4\]} 
    + \frac 1{\sqrt2} \g^\m\e \Hat D_\m\varphi \cr  
& + \frac1{2\sqrt{14}} g \expfour \t_3 \e S ~~, \cr 
   \noalign{\vskip 0.05in} 
L\du i \a \d_Q L_{\a \, j} =  & - i \big( \Bar\e \t_i
      \chi\low{\,j} \big) ~~, 
       ~~~~ \d_Q \varphi = - \frac i {\sqrt2} \big( \Bar\e\l\big)
      ~~,   
&(2.6) \cr }  $$  
up to quadratic fermion terms.  As usual, the {\it hatted} field 
strengths, {\it e.g.,} $~\Hat G_{\m\n\m\,\a}$~, {\it etc.}~are 
supercovariantized ones \ssdiverse.  Each term in
$~\d_Q\chi\low i$~ has the right projection, such  that the extra
constraint $~\t_i \chi\low i =0$~ is desirably satisfied.  Note that
all the three mass parameters $~m_0,~m_1$~ and 
$~m_2$~ are arbitrary in our formulation with the
$~SO(2)\-$gauging, so that our result here is the most general 
compared with other past references \kk\cowdall\gm.  

The field strengths are defined by 
$$ \li{ & F_{\m\n\r\s} \equiv 4 \calD_{\[\m} A_{\n\r\s\]} 
    + 12 \e^{\a\b} B_{\[\m\n|\,\a} G_{| \r\s\]\,\b} 
    - 48 \e^{\a\b} A_{\[\m|} B_{|\n|\, \a} 
     G_{| \r\s\]\, \b} \cr 
& ~~~~~ ~~~~~ ~~ -24 g \e^{\a\b} \eta\low{\b\g} \e^{\g\d} 
     A_{\[\m| } B_{|\n\r| \, \a} B_{|\s\] \, \d} 
      + 3 g \e^{a\b} \eta\low{\b\g}\e^{\g\d} 
        B_{\[\m\n| \,\a} B_{|\r\s\] \, \d} ~~,  
&(2.7\rma)  \cr  
& G_{\m\n\r\,\a} \equiv 3\calD_{\[\m} B_{\n\r\]\,\a} 
     - 6 F_{\[\m \n} B_{\r\]\,\a} ~, 
     ~~~ \Tilde G _{\m\n\r\,\a} \equiv 
     G _{\m\n\r\,\a} + 6 A_{\[\m } G_{\n\r\] \,\a} ~,
&(2.7\rmb)  \cr  
& G_{\m\n\, \a} \equiv 2\calD_{\[\m} B_{\n\]\,\a} 
     - \frac12 g \eta_{\a\b}\e^{\b\g} B_{\m\n\,\g} ~~, ~~~~
    F_{\m\n} \equiv 2 \partial_{\[\m} A_{\n\]} {~~. ~~~~~ ~~~}
&(2.7\rmc) \cr } $$  
The $~\Tilde G _{\m\n\r\,\a} $~ is 
used in the Chern-Simons terms in the lagrangian.  
All the field strengths with the index $~{\scst i}$~ 
are  defined by the multiplication of the above field
strengths  by $~L\du i\a$~ {\it e.g.,} $~G_{\m\n\r\, i}\equiv
L\du i\a  G_{\m\n\r\,\a}$.  
As usual in supergravity theories \ssdiverse, the extra
transformations 
$~B_{\[1\]}\wedge\d_Q A_{\[1\]}$~ in $~\d_Q B_{\[2\]}$~
or $~B_{\[2\]}\wedge\d_Q B_{\[1\]}$~ or
$~A_{\[1\]} \wedge B_{\[1\]}\wedge \d_Q B_{\[1\]} $~ in
$~\d_Q A_{\[3\]}$~ are determined by the requirement that
the supersymmetry transformations of their field strengths will
have only field strengths \ssdiverse.     
The covariant derivative $~\calD_\m$~ acts on fermions and
potentials, as 
$$ \li{ & \calD_{\[ \m |} \psi_{|\n\]} 
       \equiv D_{\[ \m |} (\Hat\o) \psi_{|\n\]} 
      + \frac 1 4 \calQ_{\[\m| \, i j} \t_{i j} \psi_{|\n\]} 
       = D_{\[ \m |} (\Hat\o) \psi_{|\n\]} 
      + \frac 1 2 \calQ_{\[\m|} \t_3 \psi_{|\n\]} ~~, \cr  
& \calD_\m\chi\low{\, i} \equiv D_\m (\Hat\o) \chi\low{\, i} 
     + \frac14 \calQ_{\m \, j k} \t_{j k} \chi\low{\, i} 
     + \calQ_{\m i j} \chi\low{\, j}~~, ~~~~
      \calD_\m \l\equiv D_\m (\Hat\o) \l 
     + \frac 14 \calQ_{\m \, i j }\t_{i j} \l ~~, \cr 
& \calD_\m B_{\n\r\,\a} \equiv 
     \partial_\m B_{\n\r\,\a} 
       + g \eta\low{\a\b} \e^{\b\g} A_{\m} 
      B_{\n\r\,\g} ~,  ~~~
     \calD_\m B_{\n\,\a} \equiv 
     \partial_\m B_{\n\,\a} 
       + g \eta\low{\a\b} \e^{\b\g} A_{\m} 
      B_{\n \,\g} ~,  {~~~~~ ~~~~~}     
&(2.8) \cr } $$ 
where the $~\calQ$'s has also the implicit $~SO(2)$~ minimal
coupling {\it via} (2.2). 
The derivatives on the $~S$'s give 
$$ \li{ & \partial_\m S_{i j} = - 2 S_{(i | k} \, X_{\m\, |j) k} ~~, 
     ~~~~ \partial_\m S = - 2 S_{i j}\, X_{\m \, i j}
          = - 2 S_{i j} \calP_{\m \, i j}  ~~.  
&(2.9) \cr } $$

As in the massive type IIA \romans\ or the gauged $~N=2$~ in
8D \sseight, we need to put a linear term in the tensor field
$~B_{\m\n\, \a}$~ in the field strength $~G_{\m\n\, \a}$~ for
the gauged case with $~g\neq0$.  Relevantly, some useful
relationships for invariance confirmation are the Bianchi
identities 
$$ \li{ & \calD_{\[ \m} G_{\n\r\] \, \a} 
     \equiv - \frac16 \, g\, \eta\low{\a\b} 
     \, \e^{\b\g} \, G_{\m\n\r\,\g} 
%
% = - \frac16 g L_{\a i} \, S_{i j} \, \e_{j k} \, G_{\m\n\r\, k} 
     ~~,  ~~~~\calD_{\[ \m} G_{\n\r\s] \, \a} 
     \equiv - 3 F_{\[\m\n} G_{\r\s\] \,\a} 
     { ~~, ~~~~~ } 
&(2.10\rma) \cr 
& \calD_{\[\m} F_{\n\r\s\t\]} 
     = + 4 \e^{\a\b} G_{\[\m\n\r| \,\a} G_{|\s\t\]\,\b} ~~.  
&(2.10\rmb) \cr } $$
Eq.~(2.10b) confirms the validity of 
the explicitly $~g\-$dependent terms in (2.7a).  
     
Even though there are some $~A_\m\-$explicit terms in 
(2.7a), we can show that $~F_{\m\n\r\s}$~ is invariant 
under the local $~SO(2)$~ transformation with the parameter
$~\a$:  
$$\li{ & \d_\a A_\m = \partial_\m \a~~, ~~~~
     \d_\a B_{\m\,\a} = - g \a \eta\low{\a\b} 
     \e^{\b\g} B_{\m\,\g} ~~, ~~~~
     \d_\a B_{\m\n\,\a} = - g \a \eta\low{\a\b} 
     \e^{\b\g} B_{\m\n\,\g}~~.  
&(2.11) \cr } $$  
We can show also the covariance (or invariance) of all the 
$~G\-$field (or $~F\-$field) strengths, {\it except} the 
{\it tilded} one $~\Tilde G_{\m\n\r\,\a}$, as will be
mentioned in the paragraph below (2.12).  

As has been well-known for massive maximal supergravity theories
\romans, the $~SO(2)\-$gauging breaks the covariance of the
field strength $~G_{\m\n\r\,\a}$~ under  the proper
antisymmetric tensor gauge transformation of
$~B_{\m\n\,\a}$, upsets the right propagating degrees of 
freedom.  However, this is compensated by the absorption of
$~B_{\m\n\,\a}$~ into $~G_{\m\n\,\a}$.  After this,
$~B_{\m\n\,\a}$~ loses its original tensor gauge covariance, but
this does not pose any problem, because it becomes massive with
the right propagating degrees of freedom.  A similar situation can
be also found for the gauged maximal supergravity in 8D
\sseight.  

Note that there are two ways to recover the non-gauged case of
$~N=2$~  supergravity in 9D \kk,\footnotew{We cite \kk\ with
the caveat mentioned in the Introduction in mind.} either  by
putting $~g=0$~ or by requiring  all the mass parameters to zero:
$~m_0 = m_1 = m_2=0$.  Even though
$~\eta\low{\a\b}$~ becomes non-invertible in the latter case, it
does not matter, because $~\eta\low{\a\b}$~ is always with the
minimal couplings with $~g A_\m$~ which are not needed in the
non-gauged case anyway.  

The validity of the Chern-Simons terms with the $~\e\-$tensors
in  (2.5) can be reconfirmed by the supersymmetric variation:  In
terms of differential forms, we get 
$$ \li{ \d_Q & \Big[ \,\frac1{576} F_4 F_4 A_1 
     - \frac1{216} \e^{\a\b} \Tilde G_{3\a} \Tilde G_{3\b} A_3 
     - \frac1{36} \e^{\a\b} \e^{\g\d} 
      \Tilde G_{3\a} \Tilde G_{3\b} 
     B_{2\g} B_{1\d} \, \Big] \cr 
=\, & \frac 1{144} \big(\d_Q^{(0)} \! A_3\big) 
     F_4 F_2 
     + \frac 1{572}F_4 F_4 \big( \d_Q^{(0)} \! A_1 \big)  
     + \frac1{144} \e^{\a\b} \big( \d_Q^{(0)} \!  B_{2\a} \big) 
     G_{3\b} F_4 
     - \frac1{216} \e^{\a\b} G_{3\a} G_{3\b} 
     \big(\d_Q^{(0)} \! A_3\big) {~,~~~~~ ~~~} 
&(2.12) \cr } $$ 
where $~\d_Q^{(0)}$~ is for the fermion-linear terms in
$~\d_Q$~ without the extra terms.  Thus, all the higher-order
terms cancel themselves desirably, leaving only field 
strengths.\footnotew{We  have confirmed this fact explicitly up
to quintic order terms.  Our result also agrees with \cowdall\ up to
notation-dependent numerical coefficients.}  There are many other
intrinsic consistency  checks, such as the $~SO(2)\-$invariance of
the  Chern-Simons terms in the lagrangian, in particular with 
$~\Tilde G_3$, whose details will be reported elsewhere   
%%%
\ref\next{H.~Nishino and S.~Rajpoot, {\it in preparation.}}.    

It is noteworthy that the scalar potential is positive definite:    
$$ \li{ & V = + \frac1{32} \, \expeight \, 
     \Big[ 2 \big( S_{i j}\big)^2 
              - S^2 \Big]  = + \frac1{32} \, \expeight \, 
     \Big[ \big(A - C)^2 + 4 B^2 \Big] ~~,      
&(2.13) \cr } $$  
agreeing with \cowdall.  Note that this positive definite potential
for gauged maximal supergravity is very peculiar to 9D.   The main
technical reason is that the index range is
$~{\scst i~=~1.~2}$, so that even if all the $~\s\-$model scalar
fields have zero backgrounds, the potential (2.13) is minimized to
zero by the balance between $~2 \big( S_{i j}\big)^2$~ and
$~S^2$.  This feature is rather uncommon to   other gauged
maximal supergravity theories in other dimensions \ssdiverse.   For
example, even though the gauged maximal supergravity in 8D
\sseight\ has formally the same combination $~2\big( T_{i
j}\big)^2 -  T^2$, this scalar potential is not positive definite in
8D,  due to the different index range $~{\scst i~=~1,~2,~3}$~
for the coset 
$~SL(3,\IR) / SO(3)$.  This situation is similar to other lower
dimensions, such as gauged $~N=4$~ supergravity in 7D \ppvn,
or $~N=8$~ gauged supergravity in 4D 
%%% 
\ref\dwn{B.~De Wit and H.~Nicolai, \pl{108}{82}{285}.}    
%%%  
all of which have scalar potentials with regions with 
negative values.      



% \bigskip\bigskip\bigskip 

\newpage 


\leftline{\bf 3.~~7-Brane Domain Solutions and Supersymmetry} 

We next analyze the stability analysis of  7-brane domain wall
solutions.   Even though there have been general formulae
developed, based on Bogomol'nyi equations 
%%%
\ref\bogomolnyi{E.B.~Bogomol'nyi, Sov.~Jour.~Nucl.~Phys.~{\bf
24} (1976) 449;  E.~Witten and D.~Olive, Phys.~Lett.~{\bf 78B}
(1978) 77;  K.~Shiraishi and S.~Hirenzaki, Int.~Jour.~of
Mod.~Phys.~{\bf A6}  (1991) 2635;  E.~Moreno, C.~Nu\~ nez
and F.A.~Schaposnik, hep-th/9802175,  Phys.~Rev.~{\bf D58}
(1998) 025015;  H.~de Vega and F.A.~Schaposnik,
Phys.~Rev.~Lett.~{\bf 56} (1986) 2564, Phys.~Rev.~{\bf D34}
(1986) 3206.}  
%%%  
for domain wall solutions in arbitrary space-time
dimensions 
%%%
\ref\gl{G.W.~Gibbons and N.D.~Lambert, hep-th/0003197,
Phys.~Lett.~{\bf B488} (2000) 90.}%  
%%%
\ref\cg{A.~Chamblin and G.W.~Gibbons, \hepth{9909130}, 
Phys.~Rev.~Lett.~{\bf 84} (2000) 1090.}, 
%%% 
we still need to look into the explicit forms of Killing spinor
conditions for surviving supersymmetry.  
There are a few reasons for this.  

First, for example, it was claimed in \gm\ that when all the
$~\s\-$model scalar fields have zero backgrounds, minimizing the
scalar potential, this vacuum has no preserved supersymmetry. 
However, this conclusion disagrees with \cowdall.  It is also rather
unusual that the background solution minimizing the scalar 
potential lacks supersymmetry.  This is because in an ordinary
supergravity theory, it is more common and natural that the
minimal point of the scalar potential maintains supersymmetry. 
We analyze this  problem of supersymmetry for \Minknine, by
keeping all the three mass parameters
$~m_1,~m_2$~ and $~m_3$.  

Second, general prescriptions based on Bogomol'nyi equations in
\gl\cg\  seem to overlook certain subtleties related to 
the internal indices and $~\g\-$matrix properties 
depending on different space-time dimensions.  For
example, eq.~(12) in \gl\ with the $~\g\-$matrices as a
general form common to any space-time dimensions, needs
some care.  In fact, since ref.~\gl\ uses the signature
$~(-++\cdots+)$, all the $~\G_m\-$matrix in
eq.~(12) in \gl\ should be replaced by 
$~i \G_m$~ in the case of 9D, as is clear from our explicit
transformation rule (2.6) based on the signature
$~(+--\cdots-)$.  The existence of such an
imaginary unit causes a crucial difference for the Killing spinor
equations.  Due to these subtleties, we look into the Killing spinor
equations directly instead of using the general prescriptions
\gl\cg.  

We start with the space-time metric of 7-brane domain wall
solutions 
$$ \li{ & d s^2 = e^{2 \a(r)} \, \eta\low{\un\m\un\n} \, 
     d x^{\un\m} \,  d x^{\un\n} - d r^2 ~~,  
&(3.1) \cr } $$  
\vskip-0.3in 
\noindent where $~r \equiv x^8 $, and $~{\scst \un\m, ~\un\n,
~\cdots~=~1,~2,~\cdots,~7}$, so that
$~\big(\eta\low{\un\m\un\n}\big)
\equiv \hbox{diag.}~(+\overbrace{--\cdots-}^7)$, and the last
term is negative due to
$~g\low{88} = -1$.  Now the three Killing spinor equations are
$~\d_Q
\psi_\m = 0 , ~
\d_Q \chi\low{\,i} = 0 , ~\d_Q \l = 0$, where all the scalars 
$~\a,~\phi,~\r$~ and $~\varphi$~ depend only on $~r$. 
Under these   conditions, and other bosonic fields with zero
backgrounds, we get the Killing spinor conditions
$$ \li { & \phi ' = \pm \frac12 g e^{4\varphi/{\sqrt7}} 
      \big( A- C\big) ~~,   
&(3.2\rma)  \cr  
& \rho ' = \pm g e^{4\varphi/{\sqrt7} -\phi} B ~~,  
&(3.2\rmb)  \cr  
& \varphi' = \pm \frac1{2\sqrt7} g \expfour\big( A + C \big) 
    = - 2{\sqrt7} \a' ~~,  
&(3.2\rmc)  \cr  
& \a ' = \mp \frac1{28} g 
     \expfour \big( A + C\big) ~~,    
&(3.2\rmd) \cr } $$  
where the prime $~'$~ denotes the derivative with respect to
$~r$, and
$~A, ~B,~C$~ are given in (2.4).  As has been mentioned, these
forms are `roughly' the same as those derived from general
Bogomol'nyi equations in \gl\cg.  The important difference is that 
the Dirac matrix $~\g^{(8)}$~ in our signature
$~\big(+ - \cdots -)$~ has no real eigenvalues due to
$~\big(\g^{\[8\]}\big)^2 = -I$, so that a condition like
$~\g^{\[8\]} \e = \pm\e $~ does not make sense.  
The key point is the necessity of $~\t_3$~ in front, and the
right chirality on 8D for the Killing spinor
$~\e$~ should be like $~\t_3 
\g^{(8)} \e = \pm \e$.\footnotew{This is because $~\t_3
\g^{\[8\]}$~ has real eigenvalues.  When in the opposite
signature $~\big(-++\cdots+)$, 
$~\g^{(8)}$~ should have an imaginary unit in front, but   
the presence of $~\t_3$~ is still essential for Killing 
spinor equations.}  Accordingly,
the right condition for a general Killing spinor equation  
$$ \li{ & \big( a + b \t_3 \g^{(8)} \big) \e = 0 ~~,  
&(3.3) \cr } $$  
with generally field-dependent coefficients $~a$~ and 
$~b$~ is 
$$ \li{ & a \pm b = 0 ~~,    
&(3.4) \cr } $$ 
respectively for $~\t_3 \g^{(8)} \e = \pm \e$~   in our
signature $~\big(+--\cdots-)$.  In 
fact, all the $~\g^{(8)}$~ in all the Killing spinor equations in
(2.6) for the domain wall metric appear always with $~\t_3$~  in
front.  Note also that we do not use the arbitrary constant
$~c$~ \cowdall\ in so-called `superpotential' $~W$~ \cg\gl,
because our computation  based on Killing spinor equations
necessitate {\it no} such an arbitrary constant
$~c$.\footnotew{In our notation, so-called `superpotential'
\cowdall\cg\gl\ is proportional to 
$~W\approx S = \eta\low{\a\b} g^{\a\b}$.}  

Our field equations for the bosonic fields $~\phi,~\r,~\varphi,
~\a$~ and $~A_\m$~ are obtained as
$$\li{ & \phi'' + 8 \a ' \phi '  
					- e^{2\phi} \big(\r')^2
     - \frac14 g^2 \expeight\big( A^2 - C^2 \big) \eqdot 0 ~~,
&(3.5\rma)  \cr  
& \r '' + 8\a ' \r' + 2\phi' \r ' 
     - \frac 12 g^2 e^{8\varphi/{\sqrt7} - \phi} 
     B \big( A + C \big)  \eqdot 0 ~~,  
&(3.5\rmb)  \cr  
& \varphi'' + 8\a' \varphi' - \frac1{4{\sqrt7}} 
     g^2 \expeight 
      \Big[ \, \big(A - C \big)^2 + 4 B^2 \,\Big] \eqdot 0~~, 
&(3.5\rmc)  \cr  
& \a'' + \frac1{14} \big( \phi' \big)^2 
      + \frac1{14} e^{2\phi} \big( \r' \big)^2 
      + \frac2 7 \big( \varphi' \big)^2 \eqdot 0 ~~,  
&(3.5\rmd)  \cr  
& \big( \a'\big)^2 - \frac1{112} \big( \phi' \big)^2 
      - \frac1{112} e^{2\phi} \big( \r' \big)^2 
      - \frac1{28} \big( \varphi' \big)^2 
      + \frac1{14} V\eqdot 0 ~~,  
&(3.5\rme) \cr 
& 2 B \phi ' - e^\phi \big( A - C\big) \rho ' \eqdot 0 ~~,     
&(3.5\rmf)  \cr } $$   
where $~\eqdot$~ denotes a field equation.  
When $~m_1=m_2 =0$, (3.5) and (3.2)
agree with the corresponding equations in ref.~\cowdall.   

It is not too difficult to show that the Killing spinor condition 
(Bogomol'nyi equation) (3.2) is the sufficient condition of all the
bosonic field equations in (3.5).  This provides us with a good
consistency check of the total system.  Relevantly, since
the condition (3.2) is stronger than the field equations (3.5),
there are certain solutions that satisfy field equations (3.5), but
not the former.  This is nothing but the well-known fact that
certain background solutions do not maintain supersymmetry.  

As the most basic case, consider first the background solutions
$~\phi = \rho = \varphi = \a =0$.  Obviously, this can not
satisfy  (3.2), because $~A = C = B = 0$~ leads to
$~m_0 = m_1 = m_2 = 0$~ which means no gauging.  However,
it is true that this set of solutions trivially satisfy all the bosonic
field equations (3.5) including the $~A_\m\-$field equation.  To 
avoid misunderstanding, we emphasize that the vacuum solution
is  {\it not} realized, unless $~m_0 = m_1 = m_2 = 0$~ or
$~g=0$.  In  other words, if $~g=0$~ with no gauging from the
outset,  there {\it is} a supersymmetric \Minknine\ as the trivial
background,    in agreement with \gm.  

As for the puzzle that the minimizing point of the scalar potential
does not preserve supersymmetry, we take the flowing
standpoint.  Namely, this \Minknine\ solution represents only the
vacuum but not domain walls, so that the usual argument based
on the Nester tensor 
%%% 
\ref\nester{J.M.~Nester, Phys.~Lett.~{\bf A83} (1981) 241; 
E.~Witten, \cmp{80}{81}{241};  P.K.~Townsend,
\pl{148}{84}{55}.}% 
%%% 
\ref\bcr{K.~Behrndt and M.~Cveti\v c, \hepth{9909058}, 
Phys.~Lett.~{\bf B475} (2000) 253;   M.~Cveti\v c, S.~Griffies
and S.J.~Rey, \hepth{9201007}, 
\np{381}{92}{301}; W.~Boucher, \np{242}{84}{282}.}  
%%%  
does not apply.  To be more specific, the Nester tensor
\nester\ in  our system  
$$ \li{ & N^{\m\n} \equiv 
     \big( \Bar\eta\g^{\m\n\r} \Hat\calD_\r \eta\big)  ~~
&(3.6) \cr } $$  
satisfies the off-shell identity  
$$ \li{ & D_\n N^{\m\n} 
     = \big( \Hat\calD_\n\Bar\eta \big) \g^{\m\n\r} 
         \big( \Hat\calD_\r \eta \big) 
     + \big( \d_\eta \Bar\chi\low{\, i} \big) \g^\m 
         \big( \d_\eta\chi\low{\, i} \big) 
     + \big( \d_\eta \Bar \l \big) \g^\m 
          \big( \d_\eta\l \big) 
      - 2 e^{-1} \big( \Bar\eta\g^\n \eta\big) 
         \, \fracmm{\d\Lag}{\d g^{\m\n}} {~~, ~~~~~ ~~} 
&(3.7) \cr } $$  
where $~\eta$~ is a commuting spinor for
supersymmetry, distinguished from the anti-commuting one
$~\e$, while $~\d_\eta$~ is
$~\d_Q$~ with $~\e$~ replaced by $~\eta$.  Accordingly, 
$~\Hat\calD_\m\eta$~ is the same as $~\d_Q\psi_\m$~ with 
~$\e$~ replaced by
$~\eta$.  The last term in (3.7) disappears on-shell.  The 
$~D_\m N^{\m 0}$~ is negative definite under the Witten
condition 
$~\g^\m\d_\eta\psi_\m =0$~ \nester\bcr, so that the
integration of
$~N^{\m\n}$~ over a space-like boundary that encloses the
domain wall is positive definite \nester\bcr
$$ \li{ & \frac 12\int_{\partial\S} \, d \S_{\m\n} \, N^{\m\n} 
    = \int_{\partial\S} \, d\S_{0 \n} \, N^{0\n} 
     = - \int_\S \,  d\S_0 \, D_\m N^{\m 0} \ge 0~~.    
&(3.8) \cr } $$   
On the other hand, the surface integral 
$~\int d \S_{0\m} N^{0 \m}~$ can be separately evaluated with
two terms coming from $~\d_Q\psi_\m$~ (2.6b):  The tension
(energy density) $~\s$~ of the domain wall and the central
charge term with $~g S$~ \bcr.  Now the main obstruction 
against \Minknine\ to preserve supersymmetry in the gauged
case is eq.~(3.2c) or (3.2d) from $~\d_Q\psi_\m=0$~ not
satisfied, while (3.2a) and (3.2b) are just the same as the
minimization of the scalar potential $~V$.  This is equivalent to
state that $~\Hat\calD_\m\eta$~ is {\it not} zero, when the
potential is minimized, unless trivially $~g=0$~ or $~m_0 = m_1 =
m_2=0$.  Moreover, in the case of \Minknine, there is no domain
wall from the outset, so that there is no domain wall tension
$~\s$~ in (3.7).  Therefore the usual argument based on the 
domain wall and AdS background \bcr\ does not apply here,
either.   In any case, using Nester tensor argument turns out to be
equivalent to analyzing the Killing spinor conditions.  

The second basic case is when $~\varphi\neq 0,~\r=\phi=0$. 
First, the  Killing spinor equations (3.2a) and (3.2b) imply that 
$~A = C, ~B=0$, which in turn leads to
$~m_1=m_2=0,~A=C=m_0\equiv m$.   Now (3.2c), with  the
upper sign without loss of generality, implies
that 
$$ \li{ & \varphi' = \fracm1{\sqrt7} g m \expfour ~~,  
&(3.9) \cr } $$  
which can be easily integrated to give 
$$ \li{ & \expmfour = - \frac 4 7 g m r + \tilde b_0 ~~, ~~~~ 
     e^{2\a} = \m e^{-\varphi/{\sqrt7}} 
         = \big( a_0 g m r + b_0\big)^{1/4} ~~,    
&(3.10) \cr } $$ 
with certain constants $~a_0~,b_0,~\tilde b_0,~\m$.   The
satisfaction of the Killing spinor conditions guarantees the 
stability as well as the satisfaction of all the bosonic field
equations.  This is similar to \cowdall, except that we do not 
have such an ambiguity as the constant $~c$~ in \cowdall. 

A more interesting case is when $~\rho = 0,~ \phi\neq 0~,
\varphi\neq 0$~ similar to \cowdall.  In this case, (3.2) becomes 
$$ \li{ \varphi ' & = + \frac1{\sqrt7} g \expfour 
       \big( m_0 \cosh \phi + m_2 \sinh\phi \big) 
       = - 2{\sqrt 7} \a ' ~~, \cr 
\phi '  & =  +g \expfour \big( m_0\sinh\phi 
     + m_2\cosh\phi \big) ~~.    
&(3.11) \cr } $$  
Dividing these two equations, we can integrate over
$~r$~ to get 
$$ \li{ & m_0 \sinh\phi + m_2 \cosh\phi = \tilde\m
      e^{{\sqrt7}\varphi} ~~,   
&(3.12) \cr } $$ 
where $~\tilde\m$~ is a real constant.  This can be used in
(3.11) to get the integration 
$$ \li { & \int \fracmm{d\phi} {(m_0\sinh\phi + m_2
      \cosh\phi)^{11/7}} = \tilde a g r + \tilde b ~~, 
&(3.13) \cr } $$  
with some real constants $~\tilde a $~ and $~\tilde b $.  This
integral can be performed as 
$$ \li{ \int d\phi ~ \big( p e^\phi + q e^{-\phi} \big)^\n  
& = \sum_{n=0}^\infty \, 
       \fracmm{\G(\n+1) \, p^n \, q^{\n -n}} 
       { n! \, \G(\n-n+1) \, (2n-\n) } \, e^{(2n-\n)\phi} 
     + \hbox{const.}  \cr  
& = - \fracm 1 \n q^\n \, e^{-\n\phi} 
      F\left( -\n, -\fracm\n 2, 
     \fracm{2-\n} 2; -\fracm p q \, e^{2\phi} \right) 
     + \hbox{const.}  ~~ 
&(3.14)
\cr } $$  
The first equality is from the expansion: $~\big( P + Q
\big)^\n = \sum_{n=0}^\infty 
\G(\n+1)/\big[\,n!\,\G(\n-n +1)\big] \, P^n \, Q^{\n -n}$, which
is  further simplified by the standard hypergeometric function 
%%% 
\ref\integral{I.S.~Gradshteyn and I.M.~Ryzhik,  {\it `Table of
Integrals Series and Products'}, Academic Press,  New York
(1980).}, 
%%% 
$$ \li{ & F\big( \a,\b,\g; z \big) 
     \equiv \fracmm{\G(\g)} {\G(\a)\, \G(\b)} 
      \, \sum_{n=0}^\infty \, 
      \fracmm{\G(\a+n) \, \G(\b+n)} {n!\, \G(\g+n) } z^n ~~.
&(3.15) \cr} $$    
Therefore, the integral in (3.13) is (3.14)  
with $~p \equiv (m_2+m_0)/2,~q \equiv (m_2 -m_0)/2$~ and 
$~\n=-11/7$.  This solution is a generalization of the domain
wall solution in \cowdall\ to the case $~m_2\neq 0$.  The only
caveat here is that  in \cowdall\ the special value $~c=-1$~ was 
taken for the arbitrary constant in so-called `superpotential'
$~W$, while we have {\it no} such an ambiguity within our direct
construction, as has been demonstrated.

To summarize, our the 7-brane domain wall solution is 
$$\li{ & e^{11\phi/7} \, 
     F\left( \fracm{11}7 , \fracm{11}{14} , 
    \fracm{25}{14} ; \fracm{m_0+m_2}{m_0-m_2} 
     e^{2\phi} \right) = a g r + b ~~,  \cr  
& e^{2\a} = \hat\m e^{-\varphi/\sqrt7} 
       = \m \big( m_0 \sinh \phi 
      + m_2 \cosh\phi \big)^{-1/7} ~~,     
&(3.16) \cr } $$  
where $~a, ~b, ~\m, ~\hat\m$~ are appropriate real constants. 
This solution  preserves a half of the original
$~N=2$~ supersymmetry, guaranteeing its stability.  This is also
consistent with the  general expectation based on Nester tensor
\nester\bcr.  Our solution is a generalization of  a similar solution
in \cowdall\ to the case of
$~m_1\neq0,~m_2\neq 0$~ with the caveat that we do not  use
the constant $~c$~ in \cowdall.  

Even though we do not exhaust all the cases of other non-trivial
scalar fields in order to save space in this Letter, our results for
the field equations (3.5) and Killing spinor equations (3.2) with
the total lagrangian with the general mass parameters
$~m_0,~m_1$~ and $~m_2$~ will be of great help for possible
future studies.    



\bigskip\bigskip\bigskip 

% \newpage 



\leftline{\bf 4.~~Concluding Remarks} 

In this paper we have performed a direct construction of gauged
$~N=2$~ supergravity in 9D, with the lagrangian (2.5) and the 
supersymmetry transformation rule (2.6).  We did not use  the
dimensional reductions \scherk\llp\ from 11D or 10D,  due to
several advantages.  First, we can directly confirm the consistency
of the whole system, performing the variation under
supersymmetry.  Since such a confirmation is indispensable even
for dimensional reductions \scherk\llp, it is more economical to
commence with the construction directly within 9D.  In fact, 
it is by such a direct computation that we can exclude practical
mistakes like sign-errors or overlooked matrix $~\s_2$~ which
are fatal for supersymmetry.  Second, when dealing only with
bosonic fields in dimensional reduction \scherk\llp, there is always
some ambiguity for field redefinitions.  However, it is the fermionic
sector that plays a decisive role for the total consistency {\it via}
supersymmetric invariance.  Moreover, since the fermionic
transformation rules are crucial for Killing spinor equations, we can
not discard fermionic contributions, as well as the total lagrangian
at the same time; They should be treated simultaneously with
mutual consistency.  In fact, in our direct construction, we have no
ambiguity such as the constant $~c$~ given in \cowdall\ in
so-called `superpotential' $~W$~ \gl\cg.  Third, there may be
some new degrees of freedom for the direct construction 
that are not easily obtained by dimensional reductions, 
{\it e.g.,} similarly to gauged $~N=8$~ supergravity within 4D 
%%% 
\ref\dewitnicolai{B.~de Wit and H.~Nicolai, \pl{108}{82}{285}; 
C.~Hull, \pl{142}{84}{39}.}. 
%%%   
Fourth, since there have been already certain results by 
dimensional reductions from 10D or 11D \kk\cowdall\gm, the
direct construction provides a new methodology, as well as the
consistency check for the past results.  Our results with the three
mass parameters also provide more general couplings than those
studied in the past.   

There seem to be a few reasons for the `delay' in the past for
such a direct  construction of gauged $~N=2$~ supergravity
within 9D.  First, even though it sounds in principle very
straightforward to obtain this system by a dimensional reduction
\llp\ from 10D or 11D \ssdiverse, a practical computation turns
out to be unexpectedly complicated.  While the purely bosonic
sector has been studied with relatively detailed computations
\mo, it is the remaining fermionic terms with  Noether/Pauli
couplings that play a crucial role for Killing spinor equations as well
as the consistency of the total action.  Even though the number of
lagrangian terms by dimensional reduction \scherk\llp\ looks
rather `small' as in (2.5), the invariance confirmation of the total
action has a considerable number of sectors.  Actually,  there can
be 63 different sectors of the structure 
$~(\hbox{fermion}) \times (\hbox{boson}) \times
(\hbox{boson}')$~ at the cubic order in the variation of the
lagrangian under supersymmetry even {\it before} gauging, aside
from quartic fermion terms.  Such an invariance confirmation does
not seem to have been accomplished in \kk, not to mention  more
basic linear-order closures on all the bosonic fields.    Second, the
$~SO(2)\-$gauging process needs additional 23 different sectors
to be confirmed for the invariance, up to cubic fermionic terms in
the variation.  Third, as we have seen, the fermion $~\chi\low{\,i}
$~ has an extra $~SO(2)$~ index, that necessitates an additional
irreducibility condition $~\t_i\chi\low{\,i} =0$.  This simplifies
some parts of the computation, but it also increases the number
of different sectors in the variations.    

We have accomplished in this Letter such complicated
computations by a direct construction within 9D, and we have the
total control of this system.  We have also confirmed the 
non-trivial structures of field strengths, such as 
$~F_{\m\n\r\s}$~ in (2.7).  In addition to the lagrangian and
transformation rule, we also clarified the Killing spinor
(Bogomol'nyi) equations (3.2) and bosonic field equations (3.5)
for the 7-brane domain wall configuration (3.1) with the mass
parameters $~m_0,~m_1,~m_2$.  As an application of our most
general result, we have given a new domain wall solution (3.16) as
a generalization of a similar solution in \cowdall.  Even though we
have not exhausted all the other possible solutions, these
equations constitute the working ground for possible future
studies of domain wall solutions for the gauged $~N=2$~
supergravity in 9D.  

By analyzing gauged maximal supergravity in 9D, we have
encountered some new aspects of supergravity.  For example, the
most fundamental \Minknine\ background does not maintain
supersymmetry in the gauged case.  This is peculiar to 
the 9D case, because ordinary gauged maximal supergravity
\ssdiverse\ has negative cosmological constant, leading to AdS
background, such as in 10D massive type IIA 
\romans, gauged $~N=4$~ supergravity in 7D \ppvn, or gauged
$~N=8$~ supergravity in 5D 
%%% 
\ref\grw{M.~G\"unaydin, L.J.~Romans and N.P.~Warner,
\pl{154}{85}{268}; \np{272}{86}{598}.}. 
%%%  
Therefore, it seems unusual to have the scalar potential (2.13)
manifestly positive definite with no cosmological constant at the
minimum, without preserving supersymmetry in \Minknine. 
However, we have understood this situation from the standpoint
that the usual argument of stability \bcr\ based on the Nester
tensor \nester\ does not apply to such a vacuum solution, which
does not have domain walls with boundaries.  

Even though we have performed only the gauging of the
$~SO(2)\-$subgroup of $~SL(2,\IR)$, it will be straightforward
to generalize our result to the non-compact
$~SO(1,1)\-$gauging, following the prescription in 
%%% 
\ref\hull{C.M.~Hull, {\it `Gauged D=9 Sueprgravities and 
Scherk-Schwarz Reduction'}, \hepth{0203146}.}, 
%%%  
with the lagrangian and the transformation rule well
established at hand.  

We could also repeat similar direct constructions of maximal
supergravity theories with more general mass parameter in 
dimensions $~D\le 8$, such as $~N=4$~ supergravity in 7D 
\ppvn\ or $~N=8$~ supergravity in 5D \grw.  Even though these
new mass parameters are naturally understood as the
generalizations \llp\ of Scherk-Schwarz dimensional reductions
\scherk, direct constructions have certain advantages from a
practical viewpoint.  

We are grateful for C.M.~Hull, P.~Meessen, and E.~Sezgin for
helpful discussions.  Special acknowledgment is addressed 
to E.~Bergshoeff for informing us about ref.~%
%%% 
\ref\bergshoeff{E.~Bergshoeff, U.~Gran and D.~Roest, 
hep-th/0203202, Class.~Quant.~Grav.~{\bf 19} (2002)
4207.},  
%%% 
where domain wall solutions in $~N=2$~ 
supergravity in 9D are systematically given.  


\newpage

\listrefsrsmall

% \listrefsr

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\end{document} 

