hep-th0203102.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
        \setlength{\unitlength}{.4mm}
        \thinlines
        \par
        \begin{picture}(349,16)
        \put(325,16){\line(1,0){4}}\put(330,16){\line(1,0){4}}
 \put(340,16){\line(1,0){4}}\put(335,0){\line(1,0){4}}
        \put(340,0){\line(1,0){4}}\put(345,0){\line(1,0){4}}
        \put(329,0){\line(0,1){16}}\put(330,0){\line(0,1){16}}
        \put(339,0){\line(0,1){16}}\put(340,0){\line(0,1){16}}
        \put(344,0){\line(0,1){16}}\put(345,0){\line(0,1){16}}
        \put(329,16){\oval(8,32)[bl]}\put(330,16){\oval(8,32)[br]}
        \put(339,0){\oval(8,32)[tl]}\put(345,0){\oval(8,32)[tr]}
        \end{picture}
        \par
        \thicklines
        \vskip.2in}
\def\border{                                            % border
        \setlength{\unitlength}{1mm}
        \newcount\xco
        \newcount\yco
        \xco=-24
        \yco=12
        \begin{picture}(140,0)
        \put(\xco,\yco){$\ktl$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcl$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbl$}}
        \xco=158
        \yco=12
        \put(\xco,\yco){$\ktr$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcr$}
        \advance\yco by-2
        \ifnum\yco>-240
        \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}
        \par\vskip-8mm}
\def\bordero{               % alternate border
        \setlength{\unitlength}{1mm}
        \newcount\xco
        \newcount\yco
        \xco=-24
        \yco=12
        \begin{picture}(140,0)
        \put(\xco,\yco){$\ktl$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcl$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbl$}}
        \xco=158
        \yco=12
        \put(\xco,\yco){$\ktr$}
        \advance\yco by-1
        {\loop
        \put(\xco,\yco){$\kcr$}
        \advance\yco by-2
        \ifnum\yco>-240
        \repeat
        \put(\xco,\yco){$\kbr$}}
        \put(-20,12){\ooo bacdefghidfghghdhededbihdgdfdfhhdheidhd%
hebaaahjhhdahbahgdedgehgfdiehhgdigicba}
        \put(-20,-241.5){\ooo ababaighefdbfghgeahgdfgafagihdidihiid%
hiagfedhadbfdecdcdfagdcbhaddhbgfchbgfdacfediacbabab}
        \end{picture}
        \par\vskip-8mm}
\def\headpic{                                           % UM heading
        \indent
        \setlength{\unitlength}{.4mm}
        \thinlines
        \par
        \begin{picture}(29,16)
        \put(165,16){\line(1,0){4}}
        \put(170,16){\line(1,0){4}}
        \put(180,16){\line(1,0){4}}
        \put(175,0){\line(1,0){4}}
        \put(180,0){\line(1,0){4}}
        \put(185,0){\line(1,0){4}}
        \put(169,0){\line(0,1){16}}
        \put(170,0){\line(0,1){16}}
        \put(179,0){\line(0,1){16}}
        \put(180,0){\line(0,1){16}}
        \put(184,0){\line(0,1){16}}
        \put(185,0){\line(0,1){16}}
        \put(169,16){\oval(8,32)[bl]}
        \put(170,16){\oval(8,32)[br]}
        \put(179,0){\oval(8,32)[tl]}
        \put(185,0){\oval(8,32)[tr]}
        \end{picture}
        \par\vskip-6.5mm
        \thicklines}

\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}

\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
                              \scriptscriptfont0\fivebf
            \textfont1\tenmib \scriptfont1\sevenmib
                               \scriptscriptfont1\fivemib
            \textfont2\tenbsy \scriptfont2\sevenbsy
                               \scriptscriptfont2\fivebsy}
 %
\def\rlx{\relax\leavevmode}                  
 % Guess what this is for...
\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
                      {\hbox{$\scriptscriptstyle\BMfont#1$}}
                 \else{$\BMfont#1$}\fi}

\font\tenmib=cmmib10
\font\sevenmib=cmmib10 at 7pt % =cmmib7 % if you have it
\font\fivemib=cmmib10 at 5pt  % =cmmib5 % if you have it
\font\tenbsy=cmbsy10
\font\sevenbsy=cmbsy10 at 7pt % =cmbsy7 % if you have it
\font\fivebsy=cmbsy10 at 5pt  % =cmbsy5 % if you have it
 %
\def\BMfont{\textfont0\tenbf \scriptfont0\sevenbf
                              \scriptscriptfont0\fivebf
            \textfont1\tenmib \scriptfont1\sevenmib
                               \scriptscriptfont1\fivemib
            \textfont2\tenbsy \scriptfont2\sevenbsy
                               \scriptscriptfont2\fivebsy}
 %
\def\BM#1{\rlx\ifmmode\mathchoice
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\BMfont#1$}}
                      {\hbox{$\scriptstyle\BMfont#1$}}
                      {\hbox{$\scriptscriptstyle\BMfont#1$}}
                 \else{$\BMfont#1$}\fi}

\def\inbar{\vrule height1.5ex width.4pt depth0pt}
\def\sinbar{\vrule height1ex width.35pt depth0pt}
\def\ssinbar{\vrule height.7ex width.3pt depth0pt}
\font\cmss=cmss10
\font\cmsss=cmss10 at 7pt
\def\ZZ{{}Z {\hskip -6.7pt} Z{}} 
\def\Ik{\rlx{\rm I\kern-.18em k}}  % Yes, I know. This ain't capital.
\def\IC{\rlx\leavevmode
             \ifmmode\mathchoice
                    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
                    {\hbox{\kern.33em\inbar\kern-.3em{\rm C}}}
                    {\hbox{\kern.28em\sinbar\kern-.25em{\rm C}}}
                    {\hbox{\kern.25em\ssinbar\kern-.22em{\rm C}}}
             \else{\hbox{\kern.3em\inbar\kern-.3em{\rm C}}}\fi}
\def\IP{\rlx{\rm I\kern-.18em P}}
\def\IR{\rlx{\rm I\kern-.18em R}}
\def\IN{\rlx{\rm I\kern-.20em N}}
\def\Ione{\rlx{\rm 1\kern-2.7pt l}}

%
%%% apple lw
\def\unredoffs{} \def\redoffs{\voffset=-.31truein\hoffset=-.59truein}
\def\speclscape{\special{ps: landscape}}

\newbox\leftpage \newdimen\fullhsize \newdimen\hstitle\newdimen\hsbody
\tolerance=1000\hfuzz=2pt\def\fontflag{cm}
%
\catcode`\@=11 % This allows us to modify PLAIN macros.
\hsbody=\hsize \hstitle=\hsize %take default values for 

% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
\def\nolabels{\def\wrlabeL##1{}\def\eqlabeL##1{}\def\reflabeL##1{}}
\def\writelabels{\def\wrlabeL##1{\leavevmode\vadjust{\rlap{\smash%
{\line{{\escapechar=` \hfill\rlap{\sevenrm\hskip.03in\string##1}}}}}}}%
\def\eqlabeL##1{{\escapechar-1\rlap{\sevenrm\hskip.05in\string##1}}}%
\def\reflabeL##1{\noexpand\llap{\noexpand\sevenrm\string\string%
\string##1}}}
\nolabels
%
% tagged sec numbers
\global\newcount\secno \global\secno=0
\global\newcount\meqno \global\meqno=1
%
\def\newsec#1{\global\advance\secno by1\message{(\the\secno. #1)}
%\ifx\answ\bigans \vfill\eject \else \bigbreak\bigskip \fi %if desired
\global\subsecno=0\eqnres@t\noindent{\bf\the\secno. #1}
\writetoca{{\secsym} {#1}}\par\nobreak\medskip\nobreak}
\def\eqnres@t{\xdef\secsym{\the\secno.}\global\meqno=1
\bigbreak\bigskip}
\def\sequentialequations{\def\eqnres@t{\bigbreak}}\xdef\secsym{}
%
\global\newcount\subsecno \global\subsecno=0
\def\subsec#1{\global\advance\subsecno by1%
\message{(\secsym\the\subsecno.%
 #1)}
\ifnum\lastpenalty>9000\else\bigbreak\fi
\noindent{\it\secsym\the\subsecno. #1}\writetoca{\string\quad
{\secsym\the\subsecno.} {#1}}\par\nobreak\medskip\nobreak}
%
\def\appendix#1#2{\global\meqno=1\global\subsecno=0%
\xdef\secsym{\hbox{#1.}}
\bigbreak\bigskip\noindent{\bf Appendix #1. #2}\message{(#1. #2)}
\writetoca{Appendix {#1.} {#2}}\par\nobreak\medskip\nobreak}
\def\eqnn#1{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1\wrlabeL#1}
\def\eqna#1{\xdef #1##1{\hbox{$(\secsym\the\meqno##1)$}}
\writedef{#1\numbersign1\leftbracket#1{\numbersign1}}%
\global\advance\meqno by1\wrlabeL{#1$\{\}$}}
\def\eqn#1#2{\xdef #1{(\secsym\the\meqno)}\writedef{#1\leftbracket#1}%
\global\advance\meqno by1$$#2\eqno#1\eqlabeL#1$$}
%
%                        footnotes
\newskip\footskip\footskip8pt plus 1pt minus 1pt 
% \footskip sets footnote baselineskip 
\font\smallcmr=cmr5 
\def\footnotefont{\smallcmr}
\def\f@t#1{\footnotefont #1\@foot}
\def\f@@t{\baselineskip\footskip\bgroup\footnotefont\aftergroup%
\@foot\let\next}
\setbox\strutbox=\hbox{\vrule height9.5pt depth4.5pt width0pt} %
\global\newcount\ftno \global\ftno=0
\def\foot{\global\advance\ftno by1\footnote{$^{\the\ftno}$}}
%
%say \footend to put footnotes at end
%will cause problems if \ref used inside \foot, %
%instead use \nref before
\newwrite\ftfile
\def\footend{\def\foot{\global\advance\ftno by1\chardef\wfile=\ftfile
$^{\the\ftno}$\ifnum\ftno=1\immediate\openout\ftfile=foots.tmp\fi%
\immediate\write\ftfile{\noexpand\smallskip%
\noexpand\item{f\the\ftno:\ }\pctsign}\findarg}%
\def\footatend{\vfill\eject\immediate\closeout\ftfile{\parindent=20pt
\centerline{\bf Footnotes}\nobreak\bigskip\input foots.tmp }}}
\def\footatend{}
%
\global\newcount\refno \global\refno=1
\newwrite\rfile
%% We have tampered after #1 in \items which was originally %
% \item and also 
%% the argument of \xdef without [ ].  Also \\ after \items{#1}.
%
% We have to be careful about \ref, when using \label and \eq commands.
\def\ref{[\the\refno]\nref}%
\def\nref#1{\xdef#1{[\the\refno]}\writedef{#1\leftbracket#1}%
\ifnum\refno=1\immediate\openout\rfile=refs.tmp\fi%
\global\advance\refno by1\chardef\wfile=\rfile\immediate%
\write\rfile{\noexpand\Item{#1}\reflabeL{#1\hskip.31in}\pctsign}%
\findarg\hskip10.0pt}%  
%       horrible hack to sidestep tex \write limitation
\def\findarg#1#{\begingroup\obeylines\newlinechar=`\^^M\pass@rg}
{\obeylines\gdef\pass@rg#1{\writ@line\relax #1^^M\hbox{}^^M}%
\gdef\writ@line#1^^M{\expandafter\toks0\expandafter{\striprel@x #1}%
\edef\next{\the\toks0}\ifx\next\em@rk\let\next=\endgroup%
\else\ifx\next\empty%
\else\immediate\write\wfile{\the\toks0}%
\fi\let\next=\writ@line\fi\next\relax}}
\def\striprel@x#1{} \def\em@rk{\hbox{}}
%
\def\lref{\begingroup\obeylines\lr@f}
\def\lr@f#1#2{\gdef#1{\ref#1{#2}}\endgroup\unskip}
%
\def\semi{;\hfil\break}
\def\addref#1{\immediate\write\rfile{\noexpand\item{}#1}} %now 
% unnecessary
%
\def\listrefs{\footatend\vfill\supereject\immediate\closeout%
\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp%
\vfill\eject}\nonfrenchspacing}
%
% The following is the revision of \listrefs to put the list in 
% the same page.
\def\listrefsr{\immediate\closeout\rfile\writestoppt
\baselineskip=14pt\centerline{{\bf References}}%
\bigskip{\frenchspacing%
\parindent=20pt\escapechar=` \input refs.tmp\vfill\eject}%
\nonfrenchspacing}
% The following is the revision of \listrefs to put the list %
% in the same page with the smaller fonts.
\def\listrefsrsmall{\immediate\closeout\rfile\writestoppt
\baselineskip=11pt\centerline{{\bf References}}
\font\smallreffonts=cmr9 \font\it=cmti9 \font\bf=cmbx9%
\bigskip{ {\smallreffonts% 
\parindent=15pt\escapechar=` \input refs.tmp\vfill\eject}}}
%
\def\startrefs#1{\immediate\openout\rfile=refs.tmp\refno=#1}
%
\def\xref{\expandafter\xr@f}\def\xr@f[#1]{#1}
\def\refs#1{\count255=1[\r@fs #1{\hbox{}}]}
\def\r@fs#1{\ifx\und@fined#1\message{reflabel %
\string#1 is undefined.}%
\nref#1{need to supply reference \string#1.}\fi%
\vphantom{\hphantom{#1}}\edef\next{#1}\ifx\next\em@rk\def\next{}%
\else\ifx\next#1\ifodd\count255\relax\xref#1\count255=0\fi%
\else#1\count255=1\fi\let\next=\r@fs\fi\next}
%
\def\figures{\centerline{{\bf Figure Captions}}%
\medskip\parindent=40pt%
\def\fig##1##2{\medskip\item{Fig.~##1.  }##2}}
%
% this is ugly, but moore insists
% The following is skipped on 09/14/01 

\newwrite\ffile\global\newcount\figno \global\figno=1
% The following is skipped on 09/14/01.  
\doit0{
\def\fig{fig.~\the\figno\nfig}
\def\nfig#1{\xdef#1{fig.~\the\figno}%
\writedef{#1\leftbracket fig.\noexpand~\the\figno}%
\ifnum\figno=1\immediate\openout\ffile=figs.tmp%
\fi\chardef\wfile=\ffile%
\immediate\write\ffile{\noexpand\medskip\noexpand%
\item{Fig.\ \the\figno. }
\reflabeL{#1\hskip.55in}\pctsign}\global\advance\figno by1\findarg}
%
\def\listfigs{\vfill\eject\immediate\closeout\ffile{\parindent40pt
\baselineskip14pt\centerline{{\bf Figure Captions}}\nobreak\medskip
\escapechar=` \input figs.tmp\vfill\eject}}
%
\def\xfig{\expandafter\xf@g}\def\xf@g fig.\penalty\@M\ {}
\def\figs#1{figs.~\f@gs #1{\hbox{}}}
\def\f@gs#1{\edef\next{#1}\ifx\next\em@rk\def\next{}\else
\ifx\next#1\xfig #1\else#1\fi\let\next=\f@gs\fi\next}
%
}

\newwrite\lfile
{\escapechar-1\xdef\pctsign{\string\%}\xdef\leftbracket{\string\{}
\xdef\rightbracket{\string\}}\xdef\numbersign{\string\#}}
\def\writedefs{\immediate\openout\lfile=labeldefs.tmp %
\def\writedef##1{%
\immediate\write\lfile{\string\def\string##1\rightbracket}}}
%
\def\writestop{\def\writestoppt%
{\immediate\write\lfile{\string\pageno%
\the\pageno\string\startrefs\leftbracket\the\refno\rightbracket%
\string\def\string\secsym\leftbracket\secsym\rightbracket%
\string\secno\the\secno\string\meqno\the\meqno}% 
\immediate\closeout\lfile}}
%
\def\writestoppt{}\def\writedef#1{}
%
\def\seclab#1{\xdef #1{\the\secno}\writedef{#1\leftbracket#1}%
\wrlabeL{#1=#1}}
\def\subseclab#1{\xdef #1{\secsym\the\subsecno}%
\writedef{#1\leftbracket#1}\wrlabeL{#1=#1}}
%
\newwrite\tfile \def\writetoca#1{}
\def\leaderfill{\leaders\hbox to 1em{\hss.\hss}\hfill}
%       use this to write file with table of contents
\def\writetoc{\immediate\openout\tfile=toc.tmp
   \def\writetoca##1{{\edef\next{\write\tfile{\noindent ##1
   \string\leaderfill {\noexpand\number\pageno} \par}}\next}}}
%       and this lists table of contents on second pass
\def\listtoc{\centerline{\bf Contents}\nobreak%
 \medskip{\baselineskip=12pt
 \parskip=0pt\catcode`\@=11 \input toc.tex \catcode`\@=12 %
 \bigbreak\bigskip}} 
%
\catcode`\@=12 % at signs are no longer letters
%

% The following is to lift the bottom of the body from page number:
\countdef\pageno=0 \pageno=1
\newtoks\headline \headline={\hfil} 
\newtoks\footline 
 \footline={\bigskip\hss\tenrm\folio\hss}
 %\footline={\hss\tenrm\folio\hss}
\def\folio{\ifnum\pageno<0 \romannumeral-\pageno \else\number\pageno 
 \fi} 

\def\nopagenumbers{\footline={\hfil}} 
\def\advancepageno{\ifnum\pageno<0 \global\advance\pageno by -1 
 \else\global\advance\pageno by 1 \fi} 
\newif\ifraggedbottom

\def\raggedbottom{\topskip10pt plus60pt \raggedbottomtrue}
\def\normalbottom{\topskip10pt \raggedbottomfalse} 

\def\on#1#2{{\buildrel{\mkern2.5mu#1\mkern-2.5mu}\over{#2}}}
\def\dt#1{\on{\hbox{\bf .}}{#1}}                % (big) dot over
\def\Dot#1{\dt{#1}}

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

\font\smallcmr=cmr6 scaled \magstep2 
\font\largetitle=cmr17 scaled \magstep1 
\font\LargeLarge=cmr17 scaled \magstep5 

\def\alephnull{~$\large{\aleph_0}\,$~} 
\def\alephnulllarge{${\displaystyle\aleph_0}$~} 
\def\Bo{\bo{\hskip 0.03in}} 
\def\lrad#1{ \left( A {\buildrel\leftrightarrow\over D}_{#1} B\right) } 
\def\.{.$\,$} 
\def\alpr{\a{\hskip 1.2pt}'} 
\def\dim#1{\hbox{dim}\,{#1}} 
% \font\goth = eufm7 scaled \magstep3 
% \font\gothsmall = eufm5 scaled \magstep3 
\def\Real{\raise-1pt\hbox{\goth R}\!\raise-1pt\hbox{\goth e}\,} 
\def\Imaginary{\raise-1pt\hbox{\goth I}\!\raise-1pt\hbox{\goth m}\,} 
\def\Realsmall{\raise-3pt\hbox{\goth R}\!\raise-3pt\hbox{\gothsmall e}\,} 
\def\Imaginarysmall{\raise-3pt\hbox{\goth I}\!\raise-3pt% 
\hbox{\gothsmall m}\,} 
\def\leftarrowoverdel{{\buildrel\leftarrow\over\partial}} 
\def\rightarrowoverdel{{\buildrel\rightarrow\over\partial}} 
\def\Dsl{{}D \!\!\!\! /{}} 
\def\ee{{\hskip 0.6pt}e{\hskip 0.6pt}} 
 
\def\eppp#1{\big( \Bar\e \star \g_{#1} \psi_1 \big) 
     \star \big(\Bar\psi_2 \star \g^{#1} \psi_3 \big)} 
\def\ppep#1{\big( \Bar\psi_1 \star \g_{#1} \psi_2 \big) 
     \star \big(\Bar\e \star \g^{#1} \psi_3 \big)}  
\def\epppo{\big( \Bar\e \star \psi_1 \big) 
     \star \big(\Bar\psi_2 \star \psi_3 \big)} 
\def\ppepo{\big( \Bar\psi_1 \star \psi_2 \big) 
     \star \big(\Bar\e \star \psi_3 \big)}  
\def\rpppp#1{\big( \Bar\psi_1 \star \g_{#1} \psi_4 \big) 
     \star \big(\Bar\psi_3 \star \g^{#1} \psi_2 \big)} 
\def\lppppo{\big( \Bar\psi_1 \star \psi_2 \big) 
     \star \big(\Bar\psi_3 \star \psi_4 \big)} 
\def\rppppo{\big( \Bar\psi_1 \star \psi_4 \big) 
     \star \big(\Bar\psi_3 \star \psi_2 \big)}  
\def\starproduct#1#2#3#4{{#1} \star {#2} \star {#3} \star {#4}}
\def\grass#1{(-1)^{#1}}

\def\cosete{E_{8(+8)} / SO(16)} 
\def\cosetf{F_{4(-20)} / SO(9)} 
\def\cosets{SO(8,n) / SO(n) \times SO(8)} 

\def\ZZ{Z\!\!\! Z} 

\def\atmp#1#2#3{Adv.~Theor.~Math.~Phys.~{\bf{#1}} (19{#2}) {#3}} 

\def\bea{\begin{eqnarray}} 
\def\eea{\end{eqnarray}} 

\def\Christoffel#1#2#3{\left\{ {\buildrel #1 
\over{\scst #2 #3}} \right\} } 

\def\Christoffelsmall#1#2#3{\big\{ {\buildrel #1 
\over{\scst #2 #3}} \big\} } 

\def\hepth#1{hep-th/{#1}} 

\def\binomial#1#2{\left(\,{\buildrel 
{\raise4pt\hbox{$\displaystyle{#1}$}}\over 
{\raise-6pt\hbox{$\displaystyle{#2}$}}}\,\right)} 

\font\smallcmr=cmr6 scaled \magstep2 

\def\plpl{{+\!\!\!\!\!{\hskip 0.009in}{\raise -1.0pt\hbox{$_+$}} 
{\hskip 0.0008in}}} 
\def\mimi{{-\!\!\!\!\!{\hskip 0.009in}{\raise -1.0pt\hbox{$_-$}} 
{\hskip 0.0008in}}} 

\def\ul{\underline} 
\def\un{\underline} 
\def\-{{\hskip 1.5pt}\hbox{-}} 
\def\fracmm#1#2{{{#1}\over{#2}}} 
\def\fracms#1#2{{{\small{#1}}\over{\small{#2}}}} 
\def\low#1{{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}} 

\def\ip{{=\!\!\! \mid}} 
\def\Lslash{${\rm L}{\!\!\!\! /}\, $} 
\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~} 
\def\eqstar{~{\buildrel * \over =}~} 

\def\framing#1{\doit{#1} 
{\framingfonts{#1} 
\border\headpic 
}} 

\def\eqdot{~{\buildrel{\normalsize\cdot} \over =}~} 
\def\Check#1{\raise0pt\hbox{$\buildrel{\raise0pt\hbox{$_{_{_\smile}}$}}% 
\over{\raise1pt\hbox{${#1}$}}$}} 

% If we need the framing in the cover page, put 1 after the following 
% \framing-command, and put 0 otherwise: 
\framing{0} 

% \thispagestyle{empty}

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


\doit0{\vskip -0.1in} 
{\hbox to\hsize{\hfill hep-th/0203102}}\par 
{\hbox to\hsize{\hfill CSULB--PA--02--1}}\par 
\vskip -0.1in 
\hfill 
%{(Revised Version)} 
\\ 

\begin{center} 
%\vglue .25in 

%\vskip -0.6in 
\vskip 0.06in 

{\large\bf Supersymmetric $~E_{8(+8)} / SO(16)$~ Sigma-Model
Coupled ~to} 
\\[0.026in] 
{\large\bf $~N=1$~ Supergravity ~in~ 
Three$\,$-$\,$Dimensions}
\\[.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 2.4in 

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

\baselineskip 14.5pt 

~~~A three-dimensional simple $~N=1$~ supergravity theory with a
supersymmetric sigma-model on the coset $~\cosete$~ is constructed. 
Both bosons and fermions in the matter multiplets are in the spinorial
$~{\bf  128}\-$representation of $~SO(16)$~ with the same chirality. 
Due to their common chirality, this model can not be obtained from the
maximal $~N=16$~ supergravity.  By introducing an independent vector 
multiplet, we can also gauge an arbitrary subgroup of $~SO(16)$~ 
together with a Chern-Simons term.  Similar $~N=1$~ supersymmetric 
$~\s\-$models coupled to supergravity are also constructed for the
cosets $~\cosetf$~ and $~SO(8,n) / SO(8) \times SO(n)$.  

\vskip 0.7in

\leftline{\small PACS: ~04.65, ~02.40, ~02.20.T, ~11.15, ~11.15.B, 
11.15.K} 
\vskip -0.05in
\leftline{\small Key Words: Supergravity, Supersymmetry, 
Three-Dimensions, Exceptional Groups,}
\vskip -0.05in
\leftline{Sigma-Models, M-Theory}
\vfill\eject 

\baselineskip 15.3pt

\oddsidemargin=0.03in
\evensidemargin=0.01in
\hsize=6.5in
%\vsize=7.8in
\textwidth=6.5in 
\textheight=9in 
\flushbottom
\footnotesep=1.0em
\footskip=0.26in 
\def\baselinestretch{0.8} 
%\footheight=1in 
%\bottomfraction=.25

\pageno=2



\leftline{\bf 1.~~Introduction} 

There have been recently considerable developments in three-dimensional 
(3D) extended supergravity theories $~2\le N \le 16$~ 
%%%  
\ref\ms{N.~Marcus and J.H.~Schwarz, \np{228}{83}{145}.}% 
%%% 
\ref\ns{H.~Nicolai and H.~Samtleben, Phys.~Rev.~Lett.~{\bf 86}
(2001) 1686, hep-th/0010076; JHEP{\bf 04} (2001) 022.}. 
%%% 
The $~N=16$~ maximal supergravity theory has a non-trivial $~\s\-$model 
on the coset $~E_{8(+8)} / SO(16)$~ originally explored in
\ms, and its most general gaugings have been intensively studied in
\ns.  A more unified treatment of general extended supergravities
in 3D has been also given in 
%%% 
\ref\dwtn{B.~de Wit, A.K.~Tollst\'en and H.~Nicolai,
\np{392}{93}{3}, \hepth{9208074}.}.       
%%% 

It has been well understood that other non-maximally extended 
supergravities with lower  $~N$~ can be rather easily obtained
by suitable truncations of the $~N=16$~ maximal supergravity
\ms\dwtn\ns.  A typical example is $~N=12$~ supergravity with the coset
$~E_{7(-5)}/SO(12)\times Sp(1)$~ \ms\dwtn, which is a quaternionic
K\"ahler manifold 
%%% 
\ref\bw{J.~Bagger and E.~Witten, Phys.~Lett.~{\bf B115} (1982) 202;
Nucl.~Phys.~{\bf B222} (1983) 1.}.  

The maximal $~N=16$~ supergravity \ms\dwtn\ns\ corresponds to 
11D supergravity, and therefore it is supposed to be large 
enough to accommodate most of the possible target spaces coupled to
lower $~N\-$extended as well as simple supergravity in 3D.  Moreover,
the spinorial $~{\bf 128}\-$representation of $~SO(16)$~ in $~\cosete$~
is so peculiar that it seems difficult to couple such a coset to any
lower extended supergravity theories.  On the other hand,  
studies in 4D reveal that any K\"ahler manifold $~\s\-$model as  
the consistent target space coupled to $~N=1$~ supergravity is further
restricted to quaternionic K\"ahler manifold, when coupled to
$~N=2$~ supergravity \bw.  Therefore, in 3D we expect an analogous
restriction on $~\s\-$model target spaces.  
As a matter of fact, any Riemannian manifold can be the
consistent target space coupled to $~N=1$~ 
supergravity, while any K\"ahler manifold can be consistent with 
$~N=2$~ supergravity \dwtn.  The general possible
target spaces for $~\s\-$models coupled to extended supergravities are
categorized in \dwtn\ in an exhaustive and unified fashion, starting
with $~N=1$~  simple supergravity.  

However, the formulation of $~N=1$~ supersymmetric
$~\s\-$models in \dwtn\ treats the matter fermions $~\chi$~ 
as the world-vector representation of the coset $~G/H$, like
the $~\s\-$model coordinate scalars.  There is a subtlety about this
treatment, because there is a difference between 
the chiral $~{\bf 128}$~ and anti-chiral $~{\bf
\Bar{128}}\-$representation of $~SO(16)$ for the fermions $~\chi$, where the
former was not covered as a special case in \dwtn.  The method
in \dwtn\ applied to the $~N=1$~ as a special case is equivalent to the
truncation of the $~N=16$~ supergravity, in which the
fermions will always come out to be in the anti-chiral $~{\bf
\Bar{128}}\-$representation.       

There is another important motivation of studying the coset 
$~\cosete$~ with $~N=1$~ supersymmetry.  In 3D, it seems true that
$~N=1$~ supersymmetric $~\s\-$model can be formulated without
supergravity.  If  both local $~N=16$~ supersymmetry and global global
$~N=1$~  supersymmetry can realize the supersymmetrization of the same
coset $~\cosete$~ with exactly the same  physical degrees of freedom,
then this provides a good motivation of studying such a globally
$~N=1$~ supersymmetric models.  This link between global and local
supersymmetries  is associated with one important aspect of M-theory   
%%% 
\ref\mtheory{C.~Hull and P.K.~Townsend, \np{438}{95}{109};
E.~Witten, \np{443}{95}{85}; 
P.K.~Townsend, {\it `Four Lectures on
M-Theory'}, in {\it `Proceedings of ICTP Summer School on High Energy
Physics and Cosmology'}, Trieste (June 1996), hep-th/9612121; 
{\it `M-theory from its Superalgebra'}, hep-th/9712004.}, 
%%%    
namely, globally supersymmetric 1D matrix theory formulation
%%% 
\ref\bfss{T.~Banks, W.~Fischler, S.H.~Shenker and L.~Susskind, 
\pr{55}{97}{5112}.}%  
%%% 
\ref\mtheoryrev{{\it For reviews of M(atrix)-theory, see, e.g.,}  
A.~Bilal, Fort.~f\"ur Phys.~{\bf 47} (1999) 5; 
T.~Banks, {\it `TASI Lecture Note on Matrix Theory'},
\hepth{9911068};  W.~Taylor IV, {\it The M(atrix) 
Model of M-Theory'}, Lectures for NATO school {\it `Quantum Geometry'}
(Iceland 1999), \hepth{0002016}; {\it and references
therein}}            
%%% 
is supposed to reproduce 11D supergravity 
%%% 
\ref\cjs{E.~Cremmer, B.~Julia and N.~Scherk, \pl{76}{78}{409};
E.~Cremmer and B.~Julia, \np{159}{79}{141}.}   
%%% 
with local supersymmetry, and this aspect may well be associated 
with the maximal coset $~\cosete$~ in 3D.   
Additionally, a similar relation between local and global 
supersymmetries is found in the AdS/CFT correspondence, {\it i.e.}, 
the relationship between Type IIB superstring in 10D compactified into
5D and a globally $~N=4$~ supersymmetric Yang-Mills
theory in 4D 
%%% 
\ref\adscft{I.~Klebanov, Nucl.~Phys.~{\bf B496} (1997) 231, 
hep-th/9702076; S.~Gubser, I.~Klebanov and A.A.~Tseytlin,
Nucl.~Phys.~{\bf B499} (1997) 217, hep-th/9703040;    
S.~Gubser and I.~Klebanov, Phys.~Lett.~{\bf B 413} (1997) 41, 
hep-th/9708005;   
J.~Maldacena, Adv.~Theor.~Math.~Phys.~{\bf 2} (1998)
231, hep-th/9711200;   
S.~Gubser, I.~Klebanov and A.~Polyakov,
Phys.~Lett.~{\bf B428} (1998) 105, hep-th/9802109;      
E. Witten,
Adv.~Theor.~Math.~Phys.~{\bf 2} (1998) 253, hep-th/9802150.}%   
%%%  
\ref\revadscft{{\it For reviews, see, e.g.,} O.~Aharony, S.~Gubser,
J.~Maldacena, H.~Ooguri and  Y.~Oz, Phys.~Rept.~323 (2000) 183;
hep-th/9905111;  E.~D'Hoker and D.Z.~Freedman, {\it `Supersymmetric 
Gauge Theories and the AdS/CFT Correspondence'}, \hepth{0201253}.},  
%%% 
or analogous relationship between AdS(3) and CFT in 2D 
%%% 
\ref\gst{M.~G\"unaydin, G.~Sierra and P.K.~Townsend, 
\np{274}{86}{429}.}.     
%%% 

Motivated by these viewpoints, we will in this paper perform the coupling
of supersymmetric $~\s\-$model on the coset $~\cosete$~ to $~N=1$~ 
supergravity, with the matter fermion in the spinorial $~{\bf
128}\-$representation.  As has been mentioned, our methodology different
from \dwtn\ is that both the $~\s\-$model scalars
$~\varphi\low A$~ and the fermions $~\chi\low A$~ are in the $~{\bf
128}\-$representation of $~SO(16)$.  

We also conjecture that we can apply the same technique to other 
series of cosets, such as all the cosets given in Table 1 below.  
\vspace{0.10cm} 
\begin{center} 
\footnotesize 
\begin{tabular}{|c|c|c|c|} \hline 
~~~ & ~~~ & ~~~ & \\ 
\noalign {\vskip -0.20cm} 
Supersymmetries & Cosets $~G/H$ & $~\varphi~$ & $~\chi ~$  
\\  
~~~ & ~~~ & ~~~ & \\ 
\noalign {\vskip -0.20cm} 
\hline\hline
~~~ & ~~~ & ~~~ & \\ 
\noalign {\vskip -0.26cm} 
$N=16$ & $E_{8(+8)} / SO(16)$ & {\bf 128} & ${\bf\Bar{128}}$ \\ 
\hline
~~~ & ~~~ & ~~~ & \\ 
\noalign {\vskip -0.26cm} 
$N=12$ & ~$E_{7(-5)} / SO(12) \times Sp(1)$~ & {\bf 64} &
     ${\bf\Bar{64}}$ 
\\ \hline 
~~~ & ~~~ & ~~~ & \\ 
\noalign {\vskip -0.26cm} 
$N=10$ & ~$E_{6(-14)} / SO(10) \times U(1)$~ & {\bf 32} &
     ${\bf\Bar{32}}$ 
\\ \hline 
{\hskip 0.1in} & {\hskip 0.1in} & {\hskip 0.1in} & \\ 
\noalign {\vskip -0.2cm} 
$N=9$ & $F_{4(-20)} / SO(9)$ & {\bf 16} & ${\bf 16}$ \\ 
%{\hskip 0.1in} & {\hskip 0.1in} & {\hskip 0.1in} & \\ 
%\noalign {\vskip -0.6cm} 
\hline 
% \noalign {\vskip -0.6cm} 
% ~~~ & ~~~ & ~~~ & \\ 
% \noalign {\vskip -0.32cm} 
% $N=4$ & $G_{2(+2)} / SO(4)$ & {\bf 8} & ${\bf\Bar 8}$ \\ 
% \hline 
\end{tabular} 
\end{center} 
\begin{center} 
\vskip -0.02in
{Table 1:  ~Cosets and Representations Coupled to 
Extended Supergravities in 3D} 
\end{center} 
\vskip -0.12in

\noindent The common feature of these cosets is that both the bosons
$~\varphi$~ and fermions $~\chi$~ belong to the {\it spinorial}
representation of $~SO(n)$~ in $~H$.  This conjecture is based on two
main reasons:  First, the coset $~\cosete$~ is large enough to generate
all of the cosets in Table 1 by its `truncations'.  Second, we can
utilize the peculiar feature in 3D that both bosons and fermions can be
in the same representations.  In fact, we can overcome the difference
in chirality, by choosing $~\chi\low A$~ to be chiral (undotted) spinors 
instead of anti-chiral (dotted) spinors, as will be done for the cosets
$~\cosete$~ and $~\cosetf$.  As a by-product, another $~N=1$~
supersymmetric $~\s\-$model on $~SO(8,n) / SO(8) \times SO(n)$, which
has been known in $~N=8$~ supergravity \ms, is
coupled to supergravity.  We will also add  kinetic terms and a
Chern-Simons term for a vector multiplet for  gauging as a possible
generalization.   


\bigskip\bigskip\medskip 

% \newpage 


\leftline{\bf 2.~~$~N=1$~ Supergravity with $~\s\-$Model 
on $~\cosete$}  

The construction of $~N=1$~ supergravity shares many aspects with the 
$~N=16$~ case \ms\ns\ that we can take advantage of.  However, we have
also to pay attention to the difference of our formulation from the 
$~N=16$~ case \ms\ns.  The most important difference is that the
representation of our fermions $~\chi\low A$~ belong to the
chiral $~{\bf 128}~$ of $~SO(16)$.  In ref.~\dwtn,
$~N=1$~ supersymmetric
$~\s\-$models are understood as a special case of general $~N\-$extended 
supergravities.  However, in that formulation, the fermions 
$~\chi_{\Dot A}$~ always belong to the anti-chiral 
$~{\bf\Bar{128}}\-$representation, after appropriate truncations from
$~N\ge 2$~ into $~N=1$.

To see how a different assignment is possible, we look into the
linearized transformation of the latter for the fields 
$~(\varphi\low A, \chi_{\Dot A})$:\footnotew{We are  using the same
notations as in \ns\ {\it e.g.}, the metric $~\big(\eta_{m n}
\big) = \hbox{diag.}~(+,-,-)$, except for minor differences, such as
putting all the spinorial (or vectorial) representation indices as
subscripts (or superscripts), {\it etc.}.}  
$$ \li{ & \d_Q \varphi\low A = + \big(\G^I \big)_{A\Dot B} 
     \big(\Bar\e{}^I \chi_{\Dot B} \big) ~~, ~~~~ 
     \d_Q \chi\low{\Dot A}
     = + \frac i 2 \big(\G^I \big)_{B \Dot A} \g^\m \e^I 
      \, \partial_\m \varphi\low B~~.  
&(2.1) \cr } $$ 
The $~{\scst A,~B,~\cdots~=~1,~2,~\cdots, ~128}$~ and 
$~{\scst \Dot A,~\Dot B,~\cdots~=~\Dot 1,~\Dot 2,~\cdots, ~\Bar{128}}$~
are respectively the chiral and anti-chiral spinorial representations of
$~SO(16)$.  Due to the index $~{\scst I~=~1,~2,~\cdots,~16}$~ for  
the vectorial representation $~{\bf 16}$~ of $~SO(16)$~ on 
$~\e^I$, we need to supply $~\G\-$matrices in the r.h.s.,
in order to match the indices on the l.h.s.  
This structure is no longer imperative, when we need only $~N=1$~
supersymmetry.  The simplest assignment is 
$~(\varphi\low A, \chi\low A)$, with bosons and fermions in the
same $~{\bf 128}$'s.  Now the linear transformation rule is 
much simplified as 
$$ \li{ & \d_Q \varphi\low A = + \big(\Bar\e\chi\low A \big) ~~,~~~~ 
    \d_Q \chi\low A 
    = + \frac i 2 \g^\m \e \, \partial_\m \varphi\low A~~.  
&(2.2) \cr } $$ 
Note that this special trick is possible, thanks to the peculiar feature
of 3D supersymmetry, where the same numbers of bosonic and fermionic
degrees of freedom can occupy exactly the same representation 
%%% 
\ref\strathdee{J.~Strathdee, \ijmp{2}{87}{273}.}.      
%%% 
Notice also that within $~N=16$, we can not put both $~\varphi$~ and 
$~\chi$~ in the same representation, due to the presence of the 
$~\G\-$matrix saturating the $~{\scst I}\-$index as in
(2.1).  Therefore, our multiplet $~(\varphi\low A, \chi\low A)$~ 
does not come from the maximal $~N=16$~ supergravity \ms\ns.  

In ref.~\dwtn, a unified treatment of all the extended supergravities 
in 3D is presented including $~N=1$~ with the
$~\s\-$model fermions $~\chi_{\Dot A}$~ for $~N=1$~ in \dwtn\ 
in the anti-chiral  $~{\bf\Bar{128}}$~ of $~H$~ in
$~G/H$.  The difference here is that the particular first   
Gamma matrix $~\big(\G^1\big)_{A \Dot B}$~ among the 
$~\G^I\-$matrices is used to compensate the dottedness of each sides in
\dwtn, while our (2.2) does not need it.  This is the reason our
formulation is not covered as a special case in the unified formulation
in \dwtn.  

Our next step is to couple such a $~\s\-$model to an $~N=1$~
supergravity multiplet.  This is a routine construction, with
Noether and quartic terms as in \ms\ns.  Our
field content is $~(e\du\m m , \psi_\m, 
\varphi\low A, \chi\low A)$, where $~{\scst A, ~B,~\cdots~=~1,~2,
~\cdots,~128}$~ are for the $~{\bf 128}~$ 
of $~SO(16)$.  The Maurer-Cartan form for coset
representative $~\calV$~ for the coset $~\cosete$~ is \ms\ns  
$$ \li{ & \calV^{-1} \partial_\m \calV 
   = + P_{\m A} Y_A + \frac 12 Q\du\m{I J} X^{I J} ~~, 
&(2.3) \cr } $$ 
where $~X^{I J}$~ and $~Y_A$~ are respectively the generators in 
$~E_8$~ in the directions of $~SO(16)$~ and the remaining 
coset $~\cosete$.  

Our lagrangian for $~N=1$~ supergravity with the coset 
$~\cosete$~ with a cosmological constant is rather simple:  
$$ \li{ \Lag\,\low{E8} = & - \frac 14 e R (e,\o) 
     + \frac 12 \e^{\m\n\r} 
     \big( \Bar\psi_\m D_\n (\omega) \psi_\r\big) 
      + \frac 14 e g^{\m\n}  P_{\m A} P_{\n A} 
     - \frac i 2 e \big(\Bar\chi\low A \g^\m 
     D_\m (\o, Q) \chi\low A \big) \cr 
& - \frac 12 e \big( \Bar\psi_\m  
    \g^\n \g^\m \chi\low A \big) P_{\n A} \
     -  \frac 18 e \big( \Bar\psi_\m \g^\n\g^\m \psi_\n \big) 
     \big(\Bar\chi \chi \big) 
   + \frac18 e \big(\Bar\chi \chi \big)^2 
     - \frac1{96} e \big(\Bar\chi\G^{I J}\g_\m \chi \big)^2 \cr
& + 2 m^2 e + \frac 12 m e \big( \Bar\psi_\m \g^{\m\n} \psi_\n \big) 
      - \frac12 m e \big(\Bar\chi \chi\big) ~~,  
&(2.4) \cr } $$ 
whose action $~I\low{E8} \equiv \int d^3 x\, \Lag\,\low{E8}$~ is 
invariant under supersymmetry 
$$\li{ \d_Q e\du\m m = \, & + i \big( \Bar\e\g^m\psi_\m \big)~~, ~~~~ 
    \d_Q \psi_\m = + D_\m (\omega) \e + i m \g_\m \e ~~, \cr 
\d_Q \chi\low A  
 = \, & +\frac i 2  \g^\m \e \Hat P_{\m A} 
     - \frac 14 \big( \G^{I J} \chi\big)_A 
     \S^{I J} ~~~~ \Big( \S \equiv \tanh\frac L 2 S \Big) ~~, \cr 
\d_Q \varphi\low A 
= \, & + \fracm L {\sinh L} 
    \big( \Bar\e \chi\low A \big) 
         \equiv \Big( \fracm L {\sinh L} S \Big)_A  ~~.   
&(2.5) \cr } $$ 
As in \ms, $~L$~ acts on a function $~f$~ of $~\varphi\low A$~ like $~ L
f \equiv \[\, \calV, f \, \]\equiv \[\, \exp\big( \varphi\low A Y_A
\big), f \, \]$.  The $~\o\du\m{r s} $~ is treated as an independent
spinor connection as in the first-order formalism  
%%% 
\ref\pvn{P.~van Nieuwenhuizen, \prep{68}{81}{189}.},   
%%% 
and $~R(e,\o) \equiv e\du m\m e\du n\n R\du{\m\n}{m n}(\o)$.  
Relevantly, we have the covariant derivative 
$$\li{ & D_\m (\o, Q) \chi\low A \equiv 
     \partial_\m \chi\low A + \frac 14 \o\du\m{r s} \g_{r s} \chi\low A 
     + \frac 14 Q \du\m{I J} \big( \G^{I J} \chi\big)_A~~. 
&(2.6) \cr } $$ 
Other notations, such as 
$~\big(\Bar\chi\chi)\ \equiv \big(\Bar\chi\low A\chi\low A\big)$,  
are the same as in \ms.  The positive coefficient for the cosmological
constant in (2.4)  implies anti-de Sitter background for our 3D
supergravity, as desired.  

Note also that our $~N=1$~ supergravity here can {\it not} be obtained
from the maximal $~N=16$~ \ms\ns\ by truncations covered in
\dwtn.  This is because both bosons and fermions of the $~\s\-$model are
in the $~{\bf 128}$~ of $~SO(16)$, while those in 
\ms\dwtn\ns\ are in the chiral $~{\bf 128}$~ and anti-chiral
$~{\bf\Bar{128}}$.  To put it differently, even though 
$~N=1$~ is much `smaller' than $~N=16$, it still maintains the  same
degrees of freedom $~128 + 128$~ as the $~N=16$~ theory \ms\ns, while
the chirality assignment differentiates our theory from any descendants
from the $~N=16$~ maximal supergravity \ns.  In other words, there are 
two versions of $~N=1$~ supersymmetric $~\cosete$~ $~\s\-$models coupled
to  supergravity, one with $~\chi\low A$~ in the $~{\bf 128}$~ as above,
and  another with $~\chi_{\Dot A}$~ as in \dwtn.  We can apply 
a similar method to the coset $~\cosetf$~ which will be performed 
in a more unified fashion in the next section.    



\bigskip\bigskip\bigskip

% \newpage 


\leftline{\bf 3.~~Possibility of Gaugings}     

As for the gauging of subgroups of $~E_8$~ similar to that in $~N=16$~
\ns, there seems to be some obstruction, due to the simple nature of
$~N=1$~ supergravity.  To be more specific, following \ns, we set up 
the possible form of an additional lagrangian in terms of a vector
$~B\du\m m$, and three functions $~A_1, ~A_{2 A}$~ and $~A_{3 A}$~ 
of the scalars $~\varphi\low A$:
$$\li{ \Lag\,\low{\rm E8} \, \big|_{g,\,B} 
     \equiv \, & - \frac14 \e^{\m\n\r} 
     B\du\m m \big( \partial_\n B_{\r m} 
    + \frac 13 g f_{m n p} B\du\n n B\du\r p \big) \cr 
& + \frac 12 g e A_1 
     \big( \Bar\psi_\m \g^{\m\n} \psi_\n \big)
      + i g e A_{2 A} \big( \Bar\chi\low A \g^\m \psi_\m \big) 
      + \frac 12 g e A_{3 A B} \big( \Bar\chi\low A \chi\low B \big)  \cr 
& + a_1 g^2  e \big( A_1 \big)^2 
    + a_2 g^2 e \big( A_{2 A} \big)^2 ~~,   
&(3.1) \cr } $$ 
with the minimal coupling constant $~g$, and the transformation
rule modifications 
$$ \li{ & \d_Q B\du\m\calM = + i \calV\du A\calM 
     \big( \Bar\e \g_\m \chi\low A\big) ~~, \cr 
& \d_Q \psi_\m \big|_g = + i g A_1 \g_\m \, \e~~, ~~~~
     \d_Q \chi\low A \big|_g = + g A_{2 A} \, \e ~~.    
&(3.2) \cr } $$ 
Note that due to the neutral gravitino $~\psi_\m$~ under 
$~SO(16)$, there is no $~\psi\-$linear term in $~\d_Q B_\m$, 
in contrast to \ns.    
Accordingly, the Maurer-Cartan form (2.3) is modified to 
$~\calV^{-1} \calD_\m \calV \equiv \calV^{-1} \partial_\m \calV 
+ \frac 12 g B\du\m{I J} X^{I J} \calV 
\equiv \calP_{\m A} Y_A + \frac12 \calQ\du\m{I J} X^{I J}$~ \ns.     

The first obstruction shows up in the commutator on $~B\du\m\calM$, as
the $~g\-$dependent term like $~\[ \d_1 , 
\d_2 \] B\du\m\calM \big|_g = + 2 i g \calV\du A\calM \big(\Bar\e_2 
\g_\m \e_1\big) A_{2 A}$.  The problem is that this term does not 
seem to be absorbed into the leading term that gives the desirable 
translation term $~\approx \big( \Bar\e_2 \g^\r \e_1 \big) 
G_{\r\m}{}^\calM$, by the use of the duality field equation relating 
$~P_\m$~ to the field strength $~G_{\m\n}{}^\calM$~ of 
$~B_\m{}^\calM$.  This forces us to impose the condition $~A_{2A} =0$, 
which in turn leads to the constancy of both $~A_1$~ and $~A_{3 A B}$, 
seen as follows:  
As in \ns, we see that certain conditions to be satisfied by 
the $~A's$, categorized as the 
vanishing of the following terms in $~\d_Q \Lag_{\rm total}$:  
(i) The $~g \psi P$~ or $~g \psi D A_1\-$terms.  (ii) The  
$~g \chi P\-$terms.  (iii) The $~g^2\psi\-$terms.  (iv) 
The $~g^2\chi\-$terms.  (v) The $~g\psi\chi^2\-$terms.  These
respectively  yield the conditions 
$$ \li{ & \calD_\m A_1 = A_{2 A} \calP_{\m A} ~~, 
&(3.3\rma) \cr 
&  \calD_\m A_{2 A}  - \frac12 A_1 \calP_{\m A} 
        - \frac 12 A_{3 A B} \calP_{\m B} 
     + \frac 12 T_{A|B} \calP_{\m B} = 0 ~~, 
&(3.3\rmb) \cr 
& (a_1 - 2) \big( A_1 \big)^2  
      + (a_2 + 1) \big( A_{2 A} \big)^2 = 0 ~~,
&(3.3\rmc) \cr
& ( 2a_1 + a_2 - 3) A_1 A_{2 A} 
    + (a_2 + 1) A_{2 A} A_{3 A B} 
     + a_2 T_{A | B} A_{2 B} = 0 ~~, 
&(3.3\rmd) \cr 
& A_1 \d_{A B} + A_{3 A B} = 0 ~~,  
&(3.3\rme) \cr } $$ 
where $~T_{\calA | \calB} \equiv \calV\du\calA m \Theta_{m n} 
\calV\du\calB n$~ in the notation in \ns.   Once $~A_{2A} = 0$~ is
accepted, then from (3.3a) the only non-trivial solutions seem to be 
$~A_1 = \hbox{const.}, ~A_{2 A} = 0$, which with (3.3c) and (3.3e)
imply that $~a_1 = +2 $~ and $~A_{3A} = - \d_{A B} A_1$.  At this
stage we have   
$$ \li{ & A_1 = \hbox{const.}\not= 0.~~, ~~~~
     A_{2 A} = 0 ~~, ~~~~A_{3A B} = - \d_{A B} A_1~~, ~~~~
      a_1 = + 2~~.    
&(3.4) \cr } $$ 
The only remaining condition is (3.3b) with the last term left over,  
which comes originally from the minimal coupling 
between $~B\du\m\calM$~ and $~\varphi\low A\-$scalars, implying 
that $~T_{A | B} =0$.  Therefore, if we
do not require the minimal couplings, all of the conditions in (3.3)
are satisfied.  For an obvious reason, we no longer  need $~a_2$, and
this is nothing other than our result (2.4) with the  cosmological
constant with
$~m\equiv g A_1$.   

This result indicates that there is no allowed gauge group in this
formulation in contrast to $~N=16$~ \ns.  In  a certain
sense, this is reasonable, because $~N=1$~ supergravity does not have so
much freedom as 
$~N=16$~ supergravity, and this may be the price to be paid.  However,
as will be mentioned, our method for $~\cosete$~ can be
easily applied to other cosets, such as those in Table 1.  In that
sense, even this $~N=1$~ simple supergravity still has rich enough
structures, when coupled to non-trivial $~\s\-$models.  To put it
differently, in $~N=1$~ supergravity, we have more freedom to choose
the coset out of certain series such  as those in Table 1 all of which
can consistently couple to
$~N=1$~  supergravity.  In this sense, we do not need the freedom of 
`gauging' that plays an important role in the case of $~N=16$~ maximal 
supergravity \ns.  Since the supergravity multiplet does not have  any
physical degrees of freedom in 3D, the actual difference in  the number
of supersymmetries $~N$~ may not be so crucial in 3D as 
analogous supergravities in dimensions 4D or higher.  

The gauging we have failed is an `internal' gauging, where  
$~B\du\m{I J}$~ transformed into $~\chi$, with no independent
gaugino $~\l$.  We can show, however, that certain `external' gauging
with an additional multiplet $~\big( A\du\m{(r)}, 
\l^{(r)} \big)$~ like the $~N=1$~ $~\s\-$model in 2D 
%%% 
\ref\bns{E.~Bergshoeff, H.~Nishino, and E.~Sezgin, 
\pl{166}{86}{141}.}   
%%% 
is indeed possible.  For simplicity, we consider only the cosets $~G/H =
\cosete$~ or $~\cosetf$~ in a more unified notation until the end of
this section.   For such a purpose, it is more convenient to switch to 
a notation using the metric 
$~g\low{\a\b}$~ on $~G/H$, instead of using coset representatives
$~\calV$~ as we did above.  The metric $~g\low{\a\b}\equiv 
V_{\a A} V_{\b A}$~ is defined by the vielbein
$~V_{\a A}$~ in the Maurer-Cartan form for the gauged case:   
$$\li{ & \calV^{-1} \calD_\m \calV 
    \equiv \calV^{-1} \partial_\m \calV 
    + g A\du\m{(r)} \calV^{-1} T^{(r)} \calV 
    = \big(\calD_\m \phi^\a \big) V \du\a A Y_A 
     + \frac 12 \big(\calD_\m \phi^\a \big)  Q\du\a{I J}
     X^{I J}~~, \cr  
& \calD_\m \phi^\a \equiv  
     \partial_\m \phi^\a - g A\du\m{(r)} \xi^{\a(r)} ~~, 
&(3.5) \cr } $$ 
with the curved indices $~{\scst \a,~\b,~\cdots~=~1,~2,~\cdots,~ {\rm
dim}\,(G/H)}$, and spinorial indices $~{\scst A,~B,~\cdots~=~
1,~2,~\cdots,~d_{\rm S}}$~ where $~d_{\rm S}$~ is the  dimensionality
of the spinorial representation of $~H$.   The indices $~{\scst (r),
~(s),~\cdots~=~ (1),~(2),~\cdots,}$
\newline ${\scst (d_0)}$~ on the generators
$~T^{(r)}$~ are for the adjoint representation of an arbitrary
gauged subgroup $~H_0\subset H$~ with $~d_0~\equiv~ \hbox{dim}\, H_0$. 
Accordingly, we use $~\phi^\a$~ for the coordinates for the coset
$~\cosete$~ or $~\cosetf$, which are the same as $~\varphi\low A$~ at the
linear order, but different at higher orders.   

Our total field content is now $~(e\du\m m, \psi_\m, \phi^\a,
\chi\low A, A\du\m{(r)}, \l^{(r)} \big)$~ for 
$~N=1$~ supergravity coupled to $~\s\-$model on $~\cosete$~ or
$~\cosetf$~ with an `external' gauging.  As an additional 
generalization, we can also add a Chern-Simons term for the field 
strength $~F\du{\m\n}{(r)}$ 
%%%
\ref\cs{S.J.~Gates Jr., M.T.~Grisaru, M.~Ro\v cek and W.~Siegel, {\it
Superspace}, Benjamin/Cummings, Reading, MA, 1983; 
A.~Ac\'ucarro and P.K.~Townsend, \pl{180}{86}{89}; 
H.~Nishino and S.J.~Gates, Jr., \ijmp{8}{93}{3371}.}.  
%%% 
The corresponding lagrangian is  
$$\li{\Lag\low{G/H} = & - \frac 14 e R (e,\o)  
     + \frac12 \e^{\m\n\r} \big( \Bar\psi_\m D_\n(\o) \psi_\r \big) \cr 
& + \frac 14\ee g\low{\a\b} \, g^{\m\n} 
      \big( \partial_\m\phi^\a \big) 
       \big( \partial_\n\phi^\b \big)   
   - \frac i 2 \ee \big( \Bar\chi\low A \g^\m\calD_\m (\o,Q,A) 
      \chi\low A \big) \cr 
& - \frac 14\ee g^{\m\n} g^{\r\s} F\du{\m\r}{(r)}  F\du{\n\s}{(r)} 
     - i \ee \big( \Bar\l{}^{(r)} \g^\m D_\m (\o, A) 
      \l^{(r)} \big) \cr 
& - \frac 12\ee \big( \Bar\psi_\m \g^\n\g^\m \chi\low A \big) 
      V_{\a A} \calD_\n \phi^\a 
   - \frac i 2\ee \big( \Bar\psi_\m \g^{\r\s} \g^\m \l^{(r)} \big)
    F\du{\r\s}{(r)}        
     - g \ee \big( \Bar\chi\low A \l^{(r)} \big) 
     V_{\a A} \xi^{\a (r)}  \cr 
& - \frac 18\ee \big(\Bar\psi_\m \g^\n\g^\m \psi_\n\big) 
      \big( \Bar\chi\low A \chi\low A\big) 
    + \frac 18 \ee \big( \Bar\chi\low A \chi\low A \big)^2 
    + \frac1{12} \ee R_{A B C D} 
     \big(\Bar\chi\low A\g_\m \chi\low B\big) 
      \big(\Bar\chi\low C\g^\m \chi\low D\big) \cr 
&      + \frac 12 \ee \big( \Bar\psi_\m\psi^\m \big) 
       \big(\Bar\l{}^{(r)} \l^{(r)}\big)  
       + \frac 12 \ee \big( \Bar\l{}^{(r)} \l^{(r)} \big)^2     
       + \frac 12 \ee \big( \Bar\chi\low A \chi\low A \big) 
          \big( \Bar\l{}^{(r)} \l^{(r)} \big)  \cr   
& + \frac12 c \, m \, \e^{\m\n\r} \big( F_{\m\n}{}^{(r)} A_\r{}^{(r)} 
     - \frac 13 g f^{(r)(s)(t)} A \du\m{(r)} 
          A \du\n{(s)} A \du\r{(t)} \big) \cr 
& + 2 m^2\ee + \frac 12 m \ee \big( \Bar\psi_\m \g^{\m\n} \psi_\n \big) 
     - \frac 12 m \ee \big( \Bar\chi\low A \chi\low A \big) 
     + ( 2c - 1) m \ee \big( \Bar\l{}^{(r)} \l^{(r)} \big) ~~, 
&(3.6) \cr } $$ 
whose action is invariant under supersymmetry 
$$ \li{ & \d_Q e\du\m m = + i \big( \Bar\e\g^m\psi_\m \big)~~, ~~~~ 
     \d_Q \psi_\m = 
      + D_\m (\o) \e + i m \g_\m \e ~~, \cr 
& \d_Q \phi^\a = + V\du A\a \big( \Bar\e\chi\low A\big) ~~, ~~~~ 
    \d_Q \chi\low A =  
     + \frac i 2  \g^\m \e V_{\a A} \Hat\calD_\m \phi^\a  
     - \frac 14 \big( \d_Q \phi^\a \big) 
     Q\du\a{I J} \big( \G^{I J} \chi\big)_A ~~, \cr 
& \d_Q A\du\m {(r)} = 
     + i \big( \Bar\e \g_\m \l^{(r)} \big) ~~, ~~~~
     \d_Q \l^{(r)} 
    = - \frac 14 \g^{\m\n} \e \Hat F_{\m\n}{}^{(r)} ~~.  
&(3.7) \cr } $$
The penultimate line of (3.6) has a coefficient
with an arbitrary constant $~c$~ that might be determined by the
quantization of the Chern-Simons term.  This constant enters also
the gaugino mass term.  
The covariant derivative $~\calD_\m$~ contains both the  
composite and minimal couplings: 
$$ \li{ \calD_\m(\o, Q,A) \chi\low A 
\equiv \, &  + \partial_\m \chi\low A 
     + \frac 14 \o\du\m{r s} \g_{r s} \chi\low A \cr 
& + \frac 14 \big( \calD_\m\phi^\a \big) Q\du\a{I J} 
    \big( \G^{I J} \chi\big)_A 
    + 2 g A\du\m{(r)} \big( D_{A}(Q) \xi\low{B}{}^{(r)} \big)  
     \chi\low B {~~, ~~~~~ ~~~~~}  
&(3.8) \cr } $$  
The covariant derivative $~D_A(Q)$~ 
acts on the $~\xi$'s like $~D_A (Q) \xi\du B{(r)} 
\equiv V\du A\a \big[\, \partial_\a \xi\du B{(r)} + 
(1/4) Q\du\a{I J} \big( \G^{I J} \big)_{A B} \xi\du B{(r)} \,\big]$.   
Relevantly, the curvature 
tensor $~R_{A B C D}$~ is defined by 
$$ \li{ & R_{A B C D} 
    \equiv V\du A\a V\du B \b R_{\a\b C D}
    \equiv \frac 14 V\du A\a V\du B \b R\du{\a\b}{I J} 
      \big( \G^{I J}\big)_{C D} ~~, 
&(3.9) \cr } $$ 
with the curvature tensor $~R\du{\a\b}{I J}$~ for the composite 
connection $~Q\du\a {I J}$~ for the isotropy group $~H$~ \dwtn.      
More explicitly, both for the cosets $~\cosete$~ and $~\cosetf$, 
we have \dwtn~\footnotew{The no sign-flip in our definition (3.9)
compared with
\dwtn\ costed an extra minus sign in (3.10).}
$$ \li{ & R_{A B C D} = - \frac 18 \big(\G^{I J} \big)_{A B} 
      \big(\G^{I J} \big)_{C D} ~~, 
&(3.10) \cr } $$ 
independent of the value $~n$~ in $~SO(n)$~ in $~H$.    
This notation thus unifies the previous cases of  
$~\cosete$~ and $~\cosetf$.    
A crucial relationship in the invariance confirmation of our 
action is 
$$ \li{ & D_{A}(Q) \xi\low{B}{}^{(r)} = 
     + \frac 14 \big(\G^{I J} \big)_{A B} 
     Q\du\a{I J} \xi^{\a (r)} ~~,  
&(3.11) \cr } $$
which is proven by other relationships, such as 
$~\xi\low A{}^{(r)} = + (1/60) \tr\big( \calV^{-1}\, T^{(r)} \,\calV 
\, Y_A\big)$.  

The $~g\-$linear $~(\Bar\chi\l)\-$term in (3.6) is 
much like similar terms in 2D heterotic $~\s\-$model \bns\ 
or $~N=2$~ gauged hypermultiplet couplings in 6D 
%%% 
\ref\nishinosezgin{H.~Nishino and E.~Sezgin,
\pl{144}{84}{187}; \np{278}{86}{353}.}.  
%%% 
Compared with \nishinosezgin, due to the neutral gravitino under $~H_0$,
we have no mixture term between $~\psi_\m$~ and $~\l$~ linear in $~g$.  
The Chern-Simons term with the kinetic term for the vector field 
leads to the $~A_\m\-$field equation 
$$ \li{& D_\n F^{\m\n\, (r)} 
     \eqdot c \, m \, e^{-1} \e^{\m\n\r} 
     F\du{\n\r}{(r)} + J^{\m\,(r)}~~,   
&(3.12) \cr } $$ 
which is called `generalized self-duality' condition 
in odd dimensions
%%%
\ref\sd{P.K.~Townsend, K.~Pilch and P.~van Nieuwenhuizen, 
\pl{136}{84}{38}; Addendum: \ibid{137B}{84}{443}.}.  
%%%

Since we no longer have strong restriction by the `internal'  gauging,
there is no condition on the allowed gauge  group $~H_0$~ as a subgroup
of $~SO(16)$~ for $~\cosete$~ or of $~SO(9)$~ for $~\cosetf$. 
Additionally, the $~N=1$~  case has more freedom than $~N=16$, for
choosing the  gravitino mass $~m$~ independently of the minimal gauge
coupling constant $~g$.  In this sense, this $~N=1$~ system is much
closer to the $~N=1$~ case in 2D \bns\ than $~N=16$~ in 3D, except that
the former forbids the cosmological constant due to chirality.
We can apply similar methods to other cosets in Table 1, which are
to be skipped in this paper.    



% \bigskip\bigskip\bigskip

\newpage 


\leftline{\bf 4.~~Application to Coset $~SO(8,n) / SO(8) \times
SO(n)$} 

We have so far realized a supersymmetric $~\s\-$model with the
non-trivial representations $~{\bf 128} + {\bf 128}$~ on such a huge
coset $~\cosete$~ with simple $~N=1$~ supergravity, which is different
from the formulation in \dwtn.  A next natural question is whether 
this formulation is possible only with the cosets
$~\cosete$~ and $~\cosetf$, or are there others?  In fact, this question
can be answered in the affirmative, shown by a similar construction for 
$~\cosets$~ which is not contained in Table 1, but has been known as a
consistent coset for $~N=8$~ supergravity \ms.  The common feature here
is that the fermions on $~\cosets$~ are also in the spinorial $~{\bf
8}_{\rm S}$~ of $~SO(8)$.   

There is also a slight difference between $~\cosets$~ and $~G/H = 
\cosete$~ or $~\cosetf$, due to the additional isotropy group
$~SO(n)$~ that needs additional care.  We use a notation 
similar to \ms, and assign the $~({\bf 8}_{\rm S}, {\bf
n})\-$representation of
$~SO(8)\times SO(n)$~ to the fermions $~\chi\low{A \, a}$. 
Accordingly, our field content is $~\big( e\du\m m, \psi_\m,
\phi^\a, \chi\low{A a}, A\du\m{(r)}, \newline \l^{(r)} \big)$, where
$~{\scst A,~B,~\cdots~=~1,~2,~\cdots,~8}$~ are for the chiral 
$~{\bf 8}_{\rm S}$~ of $~SO(8)$, and $~{\scst a,~b,~\cdots~=~
1,~2,~\cdots,~n}$~ for the vectorial $~{\bf n}$~ of $~SO(n)$. 
The difference here from refs.~\ms\dwtn\ is that our  
$~\chi\low{A a}$~ is in the $~{\bf 8}_{\rm S}$~ but not the $~{\bf
8}_{\rm C}\-$representation of $~SO(8)$.  

Our result for $~N=1$~ supergravity with the $~\cosets$~ 
is summarized by the lagrangian  
$$\li{\Lag\low{SO(8,n)} = & - \frac 14 e R (e,\o) 
     + \frac12 \e^{\m\n\r} \big( \Bar\psi_\m D_\n(\o) \psi_\r \big) \cr 
& + \frac 14\ee g\low{\a\b} \, g^{\m\n} 
      \big( \partial_\m\phi^\a \big) 
       \big( \partial_\n\phi^\b \big)   
   - \frac i 2 \ee \big( \Bar\chi\low{A a} \g^\m\calD_\m (\o,Q,A) 
      \chi\low{A a} \big) \cr 
& - \frac 14\, e\, g^{\m\n} g^{\r\s} F\du{\m\r}{(r)}  F\du{\n\s}{(r)} 
     - i \, e \big( \Bar\l{}^{(r)} \g^\m D_\m (\o, A) 
      \l^{(r)} \big) \cr 
& - \frac 12\, e\, \big( \Bar\psi_\m \g^\n\g^\m \chi\low{A a} \big) 
      V_{\a A a} \calD_\n \phi^\a 
   - \frac i 2\ee \big( \Bar\psi_\m \g^{\r\s} \g^\m \l^{(r)} \big)
    F\du{\r\s}{(r)}        
     - g\ee \big( \Bar\chi\low{A a} \l^{(r)} \big) 
     V_{\a A a} \xi^{\a (r)}  \cr 
& - \frac 18\, e\, \big(\Bar\psi_\m \g^\n\g^\m \psi_\n\big) 
      \big( \Bar\chi\low{A a} \chi\low{A a} \big) 
      + \frac 3{32}\ee \big( \Bar\chi\low{A a} \chi\low{A a} \big)^2 
     - \frac1{64}\ee \big( \Bar\chi \g_\m \G^{I J} \chi\big)^2 
     - \frac 1{1536}\ee \big( \Bar\chi\G^{I J K L} \chi\big)^2 \cr 
&      + \frac 12\ee \big( \Bar\psi_\m\psi^\m \big) 
       \big(\Bar\l{}^{(r)} \l^{(r)}\big)  
       + \frac 12\ee \big( \Bar\l{}^{(r)} \l^{(r)} \big)^2     
       + \frac 12\ee \big( \Bar\chi\low{A a} \chi\low{A a} \big) 
       \big( \Bar\l{}^{(r)} \l^{(r)} \big)  \cr   
& + \frac12 c \, m \, \e^{\m\n\r} \big( F_{\m\n}{}^{(r)} A_\r{}^{(r)} 
     - \frac 13 g f^{(r)(s)(t)} A \du\m{(r)} 
          A \du\n{(s)} A \du\r{(t)} \big) \cr 
& + 2 m^2\ee + \frac 12 m\, e\, 
     \big( \Bar\psi_\m \g^{\m\n} \psi_\n \big) 
     - \frac 12 m\ee \big( \Bar\chi\low{A a} \chi\low{A a} \big) 
     + ( 2c - 1) m\ee \big( \Bar\l{}^{(r)} \l^{(r)} \big) 
    {~~, ~~~~~ ~~~~~}  
&(4.1) \cr } $$ 
and the supersymmetry transformation rule 
$$ \li{ & \d_Q e\du\m m =  
     + i \big( \Bar\e\g^m\psi_\m \big)~\, , ~~~ 
     \d_Q \psi_\m =  + D_\m (\o) \e + i m \g_\m \e ~\, , \cr 
& \d_Q \phi^\a = + V\du{A a}\a \big( \Bar\e\chi\low{A a}\big) ~\, , 
     ~~~ \d_Q \chi\low{A a} =  
     + \frac i 2  \g^\m \e V_{\a A a} \Hat\calD_\m \phi^\a
     - \big( \d_Q \phi^\a \big) \Big[\, \frac 14 
     Q\du\a{I J} \big( \G^{I J} \chi_a \big)_A 
     + Q_{\a a b} \chi_{A b} \, \Big] ~\, , \cr 
& \d_Q A\du\m{(r)} = + i \big( \Bar\e \g_\m \l^{(r)} \big) ~\, , 
     ~~~ \d_Q \l^{(r)} 
     = - \frac 14 \g^{\m\n} \e \Hat F_{\m\n}{}^{(r)} ~\, .  
&(4.2) \cr } $$ 
Our notation is much like that in \ms, and other relevant 
relationships are also similar to the previous section, such as the
covariant  derivative 
$$ \li{ D_\m(\o , Q) \chi\low{A a} 
      \equiv \, & + \partial_\m\chi\low{A a} 
      + \frac 1 4 \o\du\m{r s} \g_{r s} \chi\low{A a} \cr 
& + \frac 14 Q\du\m{I J} \big(\G^{I J} \big)_{A B} \chi\low{B a} 
     + Q_{\m a b} \chi\low{A b} 
+ 2 g A\du\m{(r)} V\du{A a}\a \big( D_\a (Q) \xi\low{B b}{}^{(r)} \big) 
     \chi\low{B b} {~~.~~~~~ ~~~~~}      
&(4.3) \cr } $$ 
As before, we use notations like $~Q\du\m{I J} \equiv 
\big( \calD_\m\phi^\a \big) Q\du\a{I J}$.  The Killing vectors 
$~\xi^{\a \, (r)}$~ are in the directions of an arbitrary gauged group 
$~H_0\subset SO(8,n) \times SO(n)$~ with $~{\scst (r),~(s), ~\cdots 
~=~(1), ~(2), ~\cdots,~(d)}$~ for $~d\equiv \hbox{dim}\, H_0$.  
The relationship corresponding to (3.11) is now 
$$ \li{ & V\du{A a} \a D_\a(Q) \,\xi\low{B b}{}^{(r)} 
        = + \frac 14 \big(\G^{I J} \big)_{A B} \d_{a b} 
        Q\du\a{I J} \xi^{\a (r)} 
        + \d_{A B} Q_{\a a b} \xi^{\a (r)} ~~.    
&(4.4) \cr } $$

As is easily seen, even though there is an additional factor 
group in $~\cosets$, it exhibits a parallel structure between (4.1)
and  (3.6).  Thus we can treat both of these cosets in 
a more unified fashion like that in \dwtn, the details of which we skip
in this paper due to space limitation.  

Note that the case with $~\cosets$~ can not come from the $~N=8$~
supergravity \ms\ns\ {\it via} any truncations.  The reason is that in
our theory $~\chi\low{A\a}$~ belongs to the
$~{\bf 8}_{\rm S}$~ of $~SO(8)$, while 
$~\phi^\a$~ is equivalent to $~\varphi\low{A a}$~ in the $~{\bf 8}_{\rm
S}$~ of $~SO(8)$.  In other words, we have the $~(\varphi, \chi)$~ 
in the $~({\bf 8}_{\rm S}, {\bf 8}_{\rm S})$~ of $~SO(8)$,  
while those in \ms\dwtn\ are either $~({\bf  
8}_{\rm V}, {\bf 8}_{\rm C})$~ or $~({\bf 8}_{\rm S}, {\bf 8}_{\rm
C})$~ because of the `triality' of $~SO(8)$, but not 
$~({\bf 8}_{\rm S}, {\bf 8}_{\rm S})$~ due to the chirality-flipping by
the $~\G\-$matrix
$~\big(\G^I\big)_{A \Dot A}$~ \dwtn.  In particular, for the
above-mentioned reason, both fermions and bosons in $~N=8$~ in \ms\ can
not be put into the same representations like ours.  Notice also that
this situation is different from the case of $~\cosetf$, in which
$~N=9$~ supergravity with $~\cosetf$~ has the same Majorana 
$~{\bf 16}\-$representation for the whole $~\s\-$model multiplet, due to
the oddness of $~9$~ in $~SO(9)$.  (Cf.~Table 1.)  In other words,  
$~N=9$~ supergravity with $~\cosetf$~ can reproduce  
$~N=1$~ supergravity with the same coset by some truncations.  




%\newpage


\bigskip\bigskip\bigskip



\leftline{\bf 5.~~Globally $N=1$~ Supersymmetric $~\s\-$Model on
$~\cosete$}    

As we promised, we next clarify the realization of these
$~\s\-$models only with {\it global} supersymmetry without
supergravity.  Even though this looks rather straightforward, after we
have constructed the local case, the consequences seem non-trivial for
two reasons.  The first reason is that for general supergravity
theories, there has been another wisdom about the couplings to
supergravity restricting the algebraic structure of
$~\s\-$model cosets.  A typical example is 
$~N=2$~ supersymmetric $~\s\-$model in 4D.  It has been well-known \bw\
that {\it global} $~N=2$~ supersymmetry in 4D requires the coset to be
hyper-K\"ahler manifold, which is further restricted to quaternionic
K\"ahler manifold, when coupled to $~N=2$~ supergravity.  From this
example, we expect that the same coset may not be consistent, if 
couplings to supergravity are turned off, when going to global 
supersymmetry.  In 3D, however, we see that the coset structure of
$~\cosete$~ is `independent' of the couplings to supergravity, {\it
i.e.,} the same coset with exactly the same field representations can be
supersymmetrized even with {\it global}
$~N=1$~ supersymmetry.  The second reason is related  to the possible
link with M-theory \mtheory\mtheoryrev, or the question between global
and local supersymmetrization of the maximal coset coming from 11D
supergravity \cjs, as mentioned in the Introduction.   

The corresponding lagrangian is easily constructed by truncating all
the supergravity fields in section 2.  However, one caveat is
that when we study the $~\chi^4\-$quartic terms, there arises 
certain subtlety related to the Fierz rearrangements.     
Our lagrangian for $~N=1$~ globally supersymmetric $~\s\-$model 
on $~G/H = \cosete$~ or $~\cosetf$~ thus-obtained is 
$$ \li{ \Lag\,\low{\rm global~G/H} = 
& + \frac 14 g_{\a\b} \big(\partial_\m\phi^\a \big) 
     \big(\partial^\m\phi^\b \big)  
     - \frac i 2 \big(\Bar\chi\low A \g^\m 
     \calD_\m (Q) \chi\low A \big) 
     + \frac1{12} R_{A B C D} 
     \big(\Bar\chi\low A \g_\m \chi\low B \big) 
     \big(\Bar\chi\low C \g^\m \chi\low D \big) \cr 
& - \frac 14 \big( F\du{\m\n}{(r)} \big)^2 
     - i \big( \Bar\l{}^{(r)} \g^\m D_\m (A) \l^{(r)} \big) \cr 
& + \frac 12 \, c \, m \, \e^{\m\n\r} \big( F\du{\m\n}{(r)} 
     A\du\r{(r)} - \frac 13 g f^{(r)(s)(t)}  
     A\du\m{(r)} A\du\n{(s)} A\du\r{(t)} \big) 
			 	+ 2 c \, m \, \big(\Bar\l{}^{(r)} \l^{(r)} \big) 
    {~~,~~~~~ ~~~~~} 
&(5.1) \cr } $$ 
with global $~N=1$~ supersymmetry 
$$\li{ & \d_Q \phi^\a = V\du A\a \big( \Bar\e \chi\low A \big) ~~, 
~~~~ \d_Q \chi\low A 
     = + \frac i 2 \g^\m \e V_{\a A} \partial_\m \phi^\a  
     - \frac 14 \big( \d_Q \phi^\a \big) Q\du\a{I J} 
     \big( \G^{I J} \chi\big)_A ~~, \cr 
& \d_Q A\du\m{(r)} = + i \big( \Bar\e \g_\m\l^{(r)}\big) ~~, ~~~~
      \d_Q \l^{(r)} = -\frac14 \g^{\m\n} \e F\du{\m\n}{(r)} ~~. 
&(5.2) \cr } $$ 
It is interesting to see the absence of $~\big(\Bar\chi
\chi\big)^2\-$term that was present in the local case (2.4).  
This can be traced back to the absence of the $~\psi\-$torsion
in the variation
of the $~\chi\-$field kinetic term.  Needless to say, we do not
have the cosmological constant, when supersymmetry is realized 
globally.  

As for `external' gaugings, there seems to be a problem related to the
absence of the Noether term.  In the local case (3.6), the $~g \chi
F\-$terms out of the variations of the Noether term and $~g \big(\chi
\l\big)\-$term cancelled each other.  However, in the global case, the
absence of the former requires the absence of the latter, which in turn
implies that the $~g \l \partial \phi$~ terms in
$~\d_Q\Lag$~ have no counter-term, indicating the failure of  the
minimal coupling.        

This result leads to the natural conjecture that a similar
procedure can be performed to get globally $~N=1$~ supersymmetric
$~\s\-$models for other cosets in Table 1, as well as $~SO(8,n) / SO(8)
\times SO(n)$, whose confirmation we skip in this paper.  





\bigskip\bigskip\bigskip




% \newpage 


\leftline{\bf 6.~~Concluding Remarks} 

In this paper, we have carried out an explicit construction of $~N=1$~ 
supergravity in 3D coupled to a supersymmetric $~\s\-$model on the
`maximal' coset $~\cosete~$ with both bosons and fermions in the same
chiral $~{\bf 128}\-$representation of $~SO(16)$, which is not covered
as a special case in the unified formulation in \dwtn.   We have also
seen that such a supersymmetric $~\s\-$model is possible even without
coupling to supergravity.  

The results in our present paper elucidate several important new aspects
of supergravity in 3D:  First, we can actually couple supergravity to
$~\s\-$model on huge cosets like $~\cosete$~ with the 
$~\s\-$model multiplet in the $~({\bf 128}, {\bf 128})$~ of $~SO(16)$,
which can not be obtained from $~N=16$~ supergravity
\ms\dwtn\ns\ by truncations.  This is due to   the difference of the
field representations $~({\bf 128}, {\bf\Bar{128}})$~ in the latter. 
Second, we can apply this method to other cosets like those in Table
1.  As an  explicit example, we have dealt with the metric notation
compatible with $~\cosetf$.  Third, we can also add some cosmological 
constant and gravitino mass term, as in $~N=16$~ supergravity \ns.  
Fourth, we have seen that the `internal' gauging is not possible for 
any subgroup in the same fashion as in \ns.  However, since we can
choose such a wide variety of cosets within $~N=1$~ supergravity as in
Table 1, we do not regard  this as a drawback.  Fifth, we have seen
that we can introduce an additional vector multiplet for `external'
gauging, with no restriction on allowed gauge groups for the subgroups
$~H_0\subset H$~ of $~G/H$~ for $~\cosete$~ or $~\cosetf$.  We can also
add a Chern-Simons term for the vector multiplet, leading to the
`generalized self-duality' field equation.  Sixth, we have further seen
that the $~N=1$~ globally supersymmetric $~\s\-$model on the coset
$~\cosete$~ is realized without supergravity, and the same seems also
true for other cosets in Table 1, and $~SO(8,n) / SO(8) \times SO(n)$.  

Our result in this paper also constitutes a good counter-example against
the conventional wisdom that since supergravity in 11D \cjs\ with
$~128+128$~ degrees of freedom naturally yields the maximal supergravity
in  dimensions $~D\le 10$, and that any other `lower' $~N$~ supergravity
within that dimension is most likely obtained by truncations from the
maximal supergravity.  Hence, any lower $~N$~ supergravity has
necessarily `fewer' degrees of freedom.  Therefore, some degrees of
freedom in $~128 + 128$~ in the maximal $~N=16$~ supergravity in 3D are
supposed to be lost in the truncation to reach our $~N=1$~
supergravity.  However, as we have seen, the total degrees of freedom of
our $~N=1$~ supergravity are still $~128 + 128$~ in the irreducible
representations of $~SO(16)$, with the total number maintained to be the
same as $~N=16$~ supergravity \ms\dwtn\ns.       

Note also that our $~N=1$~ supergravity with $~\cosete$~ is {\it not}
obtained by a truncation from the maximal $~N=16$~ supergravity
in 3D \ms\dwtn\ns.  The reason is that our $~\s\-$model fields 
$~(\varphi\low A, \chi\low A)$~ are both in the same chiral $~{\bf 
128}$~ of $~SO(16)$, while those in \ms\dwtn\ns\ have 
$~(\varphi\low A, \chi_{\Dot A})$~ in the chiral $~{\bf 128}$~
and anti-chiral $~{\bf \Bar{128}}\-$representations.  
Even though our $~N=1$~ theory has obviously fewer 
supersymmetries compared with the maximal $~N=16$~ \ms\ns, our
system has different representations, and yet the same 
$~128 + 128$~ physical degrees of freedom.  The fact that our $~N=1$~
system does not come from $~N=16$, but has the same degrees of
freedom $~128 + 128$~ is peculiar to 3D, where supergravity has no
physical degree of freedom.  From these considerations, we may regard
our $~N=1$~ supersymmetric $~\s\-$model on $~\cosete$~ as the `chiral'
version of $~\cosete$, while that from $~N=16$~ \ms\dwtn\ns\ as a
`non-chiral' version.  We can further regard this new feature as
`duality' or even `triality' for all the cosets 
in Table 1.\footnotew{This is with the exception of 
$~\cosetf$~ due to the same chirality shared by bosons and fermions.}   

It has been known that any Riemannian manifold can be the consistent
target space for locally $~N=1$~ supersymmetric $~\s\-$models in 3D
\dwtn.  Since the coset $~\cosete$~ is also a Riemannian manifold, it is
no wonder that $~N=1$~ supergravity can be coupled to $~\cosete$~
itself.  However, the important point is that our fermions $~\chi$~ are
in the $~{\bf 128}$~ of $~SO(16)$~ instead of the  
$~{\bf\Bar{128}}$~ considered in the construction in
\dwtn.  In other words, our result has opened a new direction for
possible $~\s\-$model representations in $~N=1$~ supergravity.  

Our result makes sense also from the following  viewpoint:  Since there
is no physical degree of freedom for the supergravity multiplets in 3D,
`extended supersymmetries' do not have such strong significance as in 4D
or higher.  Once we recognize this crucial point, it is not a 
far-fetched wishful-thinking to expect that our $~N=1$~ supergravity has
an equally good chance of being directly related to M-theory
\mtheory\mtheoryrev\ in 11D \cjs\ as the maximal $~N=16$~ supergravity
\ms\dwtn\ns.    

\doit0{
Our formulation also elucidated an interesting aspect that 
the coset $~\cosete$~ which is `larger' than quaternionic K\"ahler 
manifolds, such as $~E_{7(-5)} / SO(12) \times SO(3)$, can be
supersymmetrized without supergravity.  This is 
an interesting result, because in 4D local $~N=2$~ supersymmetry 
necessitates the quaternionic K\"ahler instead of 
hyper K\"ahler structure for consistent supersymmetric 
$~\s\-$models.  It is usually expected that 
quaternionic K\"ahler manifolds are associated with local 
and extended supergravity.  Our results showed that this is not 
necessarily the case, because $~\cosete$~ even larger then a 
quaternionic K\"ahler manifold $~E_{7(-5)} / SO(12) \times SO(3)$~ can 
be $~N=1$~ supersymmetrized without local supersymmetry in 3D.  
}

It seems also true that our $~N=1$~ supergravity with $~\cosete$~ has no
higher-dimensional `ancestor' theory among conventional ones, such as
11D supergravity \cjs.  This is inferred from the fact that our
multiplet $~(\varphi\low A, \chi\low A)$~ forms the `irreducible' 
$~{\bf 128} + {\bf 128}~$ of $~SO(16)$~ with the coset $~\cosete$, whose
higher-dimensional origin is not clear.  This is in a sharp contrast to
the multiplet $~(\varphi\low A, \chi_{\Dot A})$~ in the maximal $~N=16$~
supergravity \ms\ns\ with its direct origin in 11D supergravity \cjs. 
Thus our $~N=1$~ $~\cosete$~ supergravity seems to be disconnected from
the conventional higher-dimensional theories, as well as from other
higher $~N$~ within 3D \dwtn.  Note also that the `minimal'
supergravity  by simple dimensional reduction from $~N=1$~ supergravity
in 4D is $~N=2$~ supergravity in 3D.  In this sense, our $~N=1$~ 
supergravity in 3D is more `chiral', like $~N=1$~ supergravity in 10D.  

We have also emphasized the connection between the global and
local supersymmetries, such as the same coset $~\cosete$~ realized
both with $~N=16$~ local and $~N=1$~ global supersymmetries.  A close
link between global and local supersymmetries for the same coset may be
the first manifestation of their important relationship 
with M-theory \mtheory\mtheoryrev, which is yet to be uncovered.  
The AdS/CFT correspondence between Type IIB supergravity and a globally
$~N=4$~ supersymmetric Yang-Mills theory in 4D \adscft\revadscft\gst\ is
another suggestive example in this direction.  The redundancy of local
supersymmetry for supersymmetric $~\s\-$models suggests a more direct
link between local supergravity with AdS(3) and global supersymmetric
theories already within 3D before going down to 2D.  

It is a peculiar nature in 3D that there is certain freedom for
supersymmetric theories.  This can be  observed roughly in two different
directions.  The first one is, as we have seen, so much freedom of
constructing $~\s\-$models within $~N=1$~ supergravity, where we can
accommodate even the `largest' coset
$~\cosete$.  The second direction is to go to higher values of $~N$~ up
to $~N\rightarrow \infty$~ with no limit, in terms of Chern-Simons
formulations  
%%% 
\ref\rpvnat{M.~Ro{\v c}ek and P.~van Nieuwenhuizen, \cqg{3}{86}{43}; 
A.~Achucaro and P.K.~Townsend, \pl{180}{86}{89}; 
\pl{229}{89}{383}.},  
%%% 
called \alephnull supergravity theories 
%%% 
\ref\ng{H.~Nishino and S.J.~Gates, Jr., hep-th/9606090,
\np{480}{96}{573}.},        
%%% 
in the absence of $~\s\-$models.  
Even though these two directions seem complementary to each other, they
may well be the manifestation of different phases of a more fundamental 
theory such as M-theory \mtheory\mtheoryrev\bfss.  Or, turning the
table around, we can use 3D supergravity as a `working ground' for
the better understanding of M-theory itself \mtheory\mtheoryrev\bfss.     

We believe that our result in this paper has the potential to generate 
other supergravity models with $~N=1$~ supergravity in 3D that have 
not been explored in the past.  We emphasize that the crucial 
ingredient in our work is the fact that in 3D the supergravity 
multiplets have no physical degree of freedom.  This fact is 
in sharp contrast with what we know in higher dimensions.  

We are grateful to P.K.~Townsend for informing us about 
supersymmetric $~\s\-$models in 3D.  

\bigskip\bigskip\bigskip\bigskip



% \newpage


\listrefsrsmall

% \listrefsr

\vfill\eject


\end{document} 

