hep-th0402111.tex
\count100=1 
\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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        \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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\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\rmh{{\rm h}}  \def\rmi{{\rm i}}  \def\rmj{{\rm j}} 
\def\rmk{{\rm k}}  \def\rml{\ell}  \def\rmm{{\rm m}} 
\def\rmn{{\rm n}} 
\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\plpl{{+\!\!\!\!\!{\hskip 0.009in}%
{\raise-1.0pt\hbox{$_+$}}  {\hskip 0.0008in}}} 
\def\mimi{{-\!\!\!\!\!{\hskip 0.009in}%
{\raise-1.0pt\hbox{$_-$}}  {\hskip 0.0008in}}} 

\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\pln#1#2#3{Phys.~Lett.~{\bf {#1}B} (20{#2}) #3}
\def\npn#1#2#3{Nucl.~Phys.~{\bf B{#1}} (20{#2}) #3}
\def\prln#1#2#3{Phys.~Rev.~Lett.~{\bf #1} (20{#2}) #3}
\def\prn#1#2#3{Phys.~Rev.~{\bf D{#1}} (20{#2}) #3}
\def\cqgn#1#2#3{Class.~and Quant.~Gr.~{\bf {#1}} (20{#2}) #3} 
\def\cmpn#1#2#3{Comm.~Math.~Phys.~{\bf {#1}} (20{#2}) #3} 
\def\jmpn#1#2#3{Jour.~Math.~Phys.~{\bf {#1}} (20{#2}) #3} 
\def\apn#1#2#3{Ann.~of Phys.~{\bf {#1}} (20{#2}) #3} 
\def\prepn#1#2#3{Phys.~Rep.~{\bf {#1}C} (20{#2}) #3}
\def\jhepn#1#2#3{Jour.~High Energy Phys.~{\bf {#1}} (20{#2}) #3}
\def\ptpn#1#2#3{Prog.~Theor.~Phys.~{\bf {#1}} (20{#2}) #3}
\def\ijmpn#1#2#3{Int.~Jour.~Mod.~Phys.~{\bf A{#1}} (20{#2}) #3}
\def\ncn#1#2#3{Nuovo Cim.~{\bf {#1}} (20{#2}) #3}
\def\ibidn#1#2#3{{\it ibid.}~{\bf {#1}} (20{#2}) #3}
\def\grgn#1#2#3{Gen.~Rel.~Grav.~{\bf{#1}} (20{#2}) {#3} }
\def\plan#1#2#3{Phys.~Lett.~{\bf A{#1}} (20{#2}) {#3}}
\def\mpln#1#2#3{Mod.~Phys.~Lett.~{\bf A{#1}} (20{#2}) #3} 
\def\zpn#1#2#3{Zeit.~f\"ur Phys.~{\bf{#1}C} (20{#2}) {#3}} 
\def\jgtpn#1#2#3{Jour.~of Group Theory for Physicists, %
{\bf{#1}} (20{#2}) {#3}}
\def\rmpn#1#2#3{Rev.~Mod.~Phys.~{\bf {#1}} (20{#2}) {#3}}
\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\Check#1{\hbox{\LARGE\v{}}\!\!\!\!\!{#1}} 
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%%%%%%%%% Worksheet %%%%%%%%%%%%%
% \end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \thispagestyle{empty}

\doit0{
{\bf Preliminary Version (FOR YOUR EYES
ONLY!)\hfill\today
} \\[-0.25in] 
%{\hfill\today} 
\\[-0.3in]  
}

{\hbox to\hsize{\hfill
hep-th/0402111}} 
\vskip -0.06in 
{\hbox to\hsize{\hfill CSULB--PA--04--1}} 
\vskip -0.14in 
\hfill 
%{(Revised Version)} 
\\ 

\begin{center} 

\vskip 0.03in 

{\Large\bf $\alephnull\,$-$\,$Extended Supersymmetric} 
\\ 
{\Large\bf Chern-Simons Theory}
\\{\Large\bf for Arbitrary Gauge Groups} %
{\hskip 0.5pt}%
\footnote{Work supported 
in part by NSF Grant \# 0308246}
\\    [.1in] 

\baselineskip 9pt 

\vskip 0.36in 

Hitoshi ~N{\smallcmr ISHINO}%
\footnote{E-Mail: hnishino@csulb.edu}
~and 
~Subhash ~R{\smallcmr AJPOOT}%
\footnote{E-Mail: rajpoot@csulb.edu} 
\\[.16in]  {\it Department of Physics \& Astronomy}
\\ [.015in] 
{\it California State University} \\ [.015in]  
{\it 1250 Bellflower Boulevard} \\ [.015in]  
{\it Long Beach, CA 90840} \\ [0.02in] 


\vskip 2.1 in 

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

\vskip 0.1in 

\baselineskip 14pt 


~~~We present a model of supersymmetric non-Abelian
Chern-Simons theories  in three-dimensions with
arbitrarily many  supersymmetries, called
$~\alephnull\-$extended supersymmetry.  The number of 
supersymmetry $~N$~ equals the dimensionality of  any
non-Abelian gauge group $~G$~ as $~N  = \hbox{dim}\, G$. 
Due to the  supersymmetry parameter in the adjoint
representation of a local gauge group $~G$, 
supersymmetry has to be local.  The minimal  coupling
constant is to be quantized, when   the homotopy mapping
is nontrivial:  $~\pi_3(G) = \ZZ$.  Our results indicate that
there is still a lot of freedom to be explored for
Chern-Simons type theories in three dimensions, 
possibly related to M-theory.  



\vskip 0.32in

\leftline{\small PACS: 11.30.Pb, 11.15.Bt, 
11.15.�q, 04.65.+e, 11.25.Tq} 
\vskip -0.05in 
\leftline{\small Key Words:  Chern-Simons, Alephnull 
Extended Supersymmetry, Supergravity, Topology}   
\vskip -0.05in 
\vfill\eject 

\baselineskip 18.0pt 

\oddsidemargin=0.03in 
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%\bottomfraction=.25

\pageno=2 


\leftline{\bf 1.~~Introduction}  

Three-dimensional space-time (3D) is peculiar in the
sense that Chern-Simons (CS) theories  
%%% 
\ref\djt{S.~Deser, R.~Jackiw and S.~Templeton, 
\prl{48}{82}{975}; \ap{140}{82}{372};  
C.R.~Hagen, \ap{157}{84}{342}; \pr{31}{85}{331}.}%
%%% 
\ref\witten{E.~Witten, \cmp{121}{89}{351};
K.~Koehler, F.~Mansouri, C.~Vaz and L.~Witten,
\mpl{5}{90}{935}; \jmp{32}{91}{239}.}%
%%% 
\ref\carlip{S.~Carlip, J.~Korean Phys.~Soc.~{\bf 28} (1995)
S447, gr-qc/9503024, {\it and references therein};  
{\it `Quantum Gravity in 2+1 Dimensions'}, Cambridge
University Press (1998).}  
%%%  
can accommodate arbitrarily many supersymmetries 
%%% 
\ref\at{A.~Achucarro and P.K.~Townsend,
\pl{180}{86}{89}; \pl{229}{89}{383}.}%
%%% 
\ref\ngscs{H.~Nishino and S.J.~Gates, Jr., 
\ijmp{8}{93}{3371}.}%
%%% 
\ref\ngaleph{H.~Nishino and S.J.~Gates, Jr.,  
hep-th/9606090, \np{480}{96}{573}.}% 
%%%
\ref\nps{W.G.~Ney, O.~Piguet and W.~Spalenza, 
{\it `Gauge Fixing of Chern-Simons N-Extended 
Supergravity'}, hep-th/0312193.}.  
%%%. 
Some typical  examples were given in \at\ with 
the gauge groups $~OSp(p|2;\IR) 
\otimes OSp(q|2;\IR)$, or with $~N=8M$~ and $~N=8M-2$~
supersymmetries in \ngscs, or $~SO(N)$~ symmetries in
\ngaleph.  Another example is conformal supergravity CS
theory with $~^\forall N\-$extended supersymmetries 
with local $~SO(N)$~ gauge symmetries
%%% 
\ref\rpvn{M.~Ro\v cek and P.~van
Nieuwenhuizen, \cqg{3}{86}{43}.}%
\ngscs.  Since arbitrarily many supersymmetries are 
allowed in these models \at\ngscs\ngaleph\nps\rpvn,
we sometimes call them ~$\alephnull\-$extended 
supersymmetric models.  

However, all of these known theories so far have rather 
limited gauge groups, such as $~OSp$ \at\ or $~SO(N)$
\ngscs.  In this brief report, we generalize these results,
constructing supersymmetric CS (SCS) theory with
non-Abelian gauge field strengths for arbitrary gauge
group $~G$.  In our formulation, $~G$~ can be any
arbitrary classical gauge group $~A_n,~B_n, ~ C_n, ~D_n$,
as well as any exceptional gauge group $~F_2, ~G_4,~E_6, 
~E_7$~ and $~E_8$.  We can include nontrivial trilinear
interactions for non-Abelian gauge group $~G$.  In our
system, the number of supersymmetries equals the
dimensionality of the gauge group as ~$N = \hbox{dim}\,
G$.  



\bigskip\bigskip\medskip 

% \newpage 



\leftline{\bf 2.~$~\alephnull\-$Extended 
Local Supersymmetries with Arbitrary Gauge Group}   

Our model is based on spinor charges transforming 
as the adjoint representation of an arbitrary gauge
group $~^\forall G$.  It has been well-known that local
supersymmetry is needed, when spinor charges
are non-singlet under a local gauge group
$~G$.  In our model, the gauge group $~G$~ can be
completely arbitrary with the relationship $~N =
\hbox{dim}\,G$.    

Our field content is similar to the models given in 
section 2 in \ngaleph, namely, $~(e\du\m m, 
\psi\du\m I, A\du\m I, 
\newline  B\du\m I,  C\du\m I, \l^I)$~ 
where $~\psi\du\m I$~ is the gravitino in the adjoint 
representation of $~^\forall G$, while both $~B\du\m I$~ 
and $~A\du\m I$~ play a role of the gauge field for $~G$, 
while $~C\du\m I$~ is a vector field, transforming as 
the adjoint representation of $~G$.  The gaugino field 
$~\l^I$~ is a Majorana spinor.  Our total action
$~I\equiv \int d^3 x\, \Lag$~ has the lagrangian composed
of the three main parts:  a supergravity lagrangian,
a $~B F\-$type term and SCS terms, as   
$$ \li{ \Lag \equiv \, & 
      - \frac 14 e R(\o) 
      + \frac 12 \e^{\m\n\r} 
       (\Bar\psi_\m{}^I D_\n\psi_\r{}^I) 
   + \frac 12 g \e^{\m\n\r} C\du\m I G\du{\n\r} I \cr 
& + \frac 12 g h \e^{\m\n\r} 
      (F\du{\m\n} I A\du\r I - \frac 13 g f^{I J K} 
    A\du\m I A\du\n J A\du\r K ) 
    + \frac 12  g h e (\Bar\l{}^I \l^I)  ~~, 
&(2.1) \cr } $$ 
where we adopt the signature $~(\eta\low{m n}) 
=\hbox{diag.}\,(-,+,+)$~ and 
$$ \li{ & D_{\[ \m} \psi\du{\n\]} I 
      \equiv \partial_{\[ \m} \psi\du{\n\]} I 
     + \frac 14 \o\du{\[\m | }{r s}\g_{r s} \psi\du{ |\n\]} I 
     + g f^{I J K} B\du{\[\m}J \psi\du{\n\]} K  ~~. 
&(2.2\rma) \cr
 & F\du{\m\n} I 
    \equiv \partial_\m A\du\n I - \partial_\n A\du\m I 
    + g f^{I J K} A\du\m J A\du\n K ~~, 
&(2.2\rmb)  \cr 
& G\du{\m\n}I   
       \equiv \partial_\m B\du\n I - \partial_\n B\du\m I 
      + g f^{I J K} B\du\m J B\du\n K ~~, 
&(2.2\rmc)  \cr 
& H\du{\m\n} I 
    \equiv ( \partial_\m C\du\n I 
           + g f^{I J K} B\du\m J C\du\n K ) 
           - ({\scst \m \leftrightarrow \n }) 
      \equiv D_\m C\du\n I - D_\n C\du\m I ~~.   
&(2.2\rmd)  \cr } $$ 
The structure constant $~f^{I J K}$~ of the gauge group 
$~^\forall G$~ plays a crucial role in our formulation.  
The covariant derivative $~D_\m$~ 
has both the Lorentz connection and the minimal 
coupling to $~B\du\m I$~ which is the gauge field of $~G$.
The constants $~g$~ and $~h$~ are {\it a priori} 
nonzero and arbitrary.  As usual in supergravity 
%%% 
\ref\pvn{See {\it e.g.}, P.~van Nieuwenhuizen,
\prep{68}{81}{189}.},    
%%%
the Lorentz connection
$~\o\du\m{r s}$~  is an independent variable with 
an algebraic field equation  
%%% 
\ref\ms{N.~Marcus and J.H.~Schwarz, 
\np{228}{83}{}145.}:   
%%%
$$ \li{ & \o_{m r s} \eqdot \Hat\o_{m r s} 
     \equiv \frac 12 (\Hat C_{m r s}  
     - \Hat C_{m s r} + \Hat C_{s r m} ) ~~,  
&(2.3\rma) \cr 
&\Hat C\du{\m\n} r 
       \equiv\partial_\m e\du\n r 
       - \partial_\n e\du\m r 
       - (\Bar\psi_\m{}^I \g^r \psi_\n{}^I) ~~.  
&(2.3\rmb) \cr } $$ 
Note that the $~C G\-$term is nothing but a 
$~B F\-$term.  

The structure of the lagrangian (2.1) is very similar 
to that in section 2 in \ngaleph, but there are also 
differences.  The most important one is that the 
gauge group $~G$~ in the present case is completely 
arbitrary, and the gravitino is in the adjoint 
representation of $~G$.  There is {\it no}  
restriction on the gauge group $~G$.  

Our total action $~I$~ is invariant under
local supersymmetry
$$ \li{ \d_Q e\du\m m 
     = \, & + (\Bar\e{}^I \g^m\psi\du\m I) ~~,  
&(2.4\rma) \cr 
\d_Q \psi\du\m I = \, & + \partial_\m \e^I 
      + \frac 14 \Hat \o\du\m{r s} \g_{r s} \e^I 
      + g f^{I J K} B\du\m J \e^K 
      + g f^{I J K}\g^\n \e^J \Hat H\du{\m\n} K  \cr 
\equiv \, & + D_\m \e^I 
     + g f^{I J K}\g^\n \e^J \Hat H\du{\m\n} K ~~, 
&(2.4\rmb) \cr 
\d_Q A\du\m I = \, & + f^{I J K} (\Bar\e{}^J \g_\m\l^K ) ~~, 
&(2.4\rmc)\cr 
\d_Q B\du\m I = \, & + f^{I J K }
      (\Bar\e{}^J \g^\n \calR\du{\m\n} K ) 
    + h f^{I J K} (\Bar\e{}^J \g_\m \l^K) ~~, 
&(2.4\rmd)\cr 
\d_Q C\du\m I = \, & - f^{I J K} (\Bar\e{}^J \psi\du\m K) 
      + h f^{I J K} (\Bar\e{}^J \g_\m \l^K) ~~, 
&(2.4\rme)\cr
\d_Q \l^I = \, & - \frac 12 f^{I J K} \g^{\m\n} 
     \e^J ( 2 F\du{\m\n} K +  G\du{\m\n} K  
      +  \Hat H\du{\m\n} K  ) 
     - \frac 12 \l^I (\Bar\e{}^J \g^\m \psi\du\m J) \cr 
& - 2 f^{I J K} f^{K L M} (\g_{\[\m} \psi\du{\n\]} J ) 
     (\Bar\e{}^L \g_\m \Tilde\calR\du\n M ) ~~, 
&(2.4\rmf) \cr } $$ 
As usual in supergravity \pvn, $~\Hat
H\du{\m\n} I$~ is the supercovariantization of 
$~H\du{\m\n} I$:  
$$ \li{ & \Hat H\du{\m\n} I \equiv 
      H\du{\m\n} I + f^{I J K} (\Bar\psi\du\m J\psi\du\n K ) ~~,
&(2.5) \cr } $$ 
and $~\calR\du{\m\n} I$~ is the gravitino field strength:
$$ \li{ & \calR\du{\m\n} I 
     \equiv ( \partial_\m \psi\du \n I 
     + \frac 14 \Hat\o\du\m{r s}\g_{r s} \psi\du\n I 
    + g f^{I J K} B\du\m J \psi\du\n K ) 
     - ({\scst \m \leftrightarrow \n }) 
     = D_\m\psi\du\n I - D_\n \psi\du\m I {~~, ~~~~~ ~~}
&(2.6) \cr } $$ 
while $~\Tilde \calR\du\m I$~ is its Hodge
dual: $~\Tilde\calR\du m I \equiv (1/2) \e\du m{r s} 
\calR\du{rs} I$.  

The on-shell closure of our system is rather easy to see, 
because of the field equations 
$$\li{ & F\du{\m\n} I \eqdot 0 ~~, ~~~~
     G\du{\m\n} I \eqdot 0 ~~, ~~~~
     \Hat H\du{\m\n} I \eqdot 0 ~~, ~~~~
     \calR\du{\m\n} I \eqdot 0~~, ~~~~ \l^I \eqdot 0~~, 
&(2.7) \cr } $$ 
where the symbol $~\eqdot$~ is for a field equation.  
Among the field strengths $~F,~G,~H$, only $~H$~ has the 
field equation with the supercovariantized field
strength, due to the minimal $~g B\psi\-$coupling.   
To be more specific, the closure of gauge algebra is 
$$ \li{ & \[ \d_Q(\e_1) , \d_Q(\e_2) \] 
     = \d_P(\xi^m) + \d_G (\xi^m) + \d_Q (\e_3^I) 
      + \d_L (\l^{r s}) + \d_\L + \d_{\Tilde\L} ~~, 
&(2.8) \cr } $$
where $~\d_P, ~\d_G$~ and $~ \d_L$~ are
respectively the translation, general coordinate
and local Lorentz transformations, while $~\d_\L$~ is 
the gauge transformation of the group $~G$, and 
$~\d_{\Tilde\L}$~ is an extra symmetry of 
$~C\du\m I$~ for our action, acting like 
$$ \li{ & \d_{\Tilde \L} C\du\m I 
       = \partial_\m\Tilde \L^I + g f^{I J K} B\du\m J \Tilde\L^K 
    \equiv D_\m \Tilde \L^I ~~,  
&(2.9) \cr } $$     
leaving other fields intact.  The parameters in (2.8) are 
$$ \li{ & \xi^m \equiv + (\Bar\e{}^I_2 \g^m\e_1^I) ~~, ~~~~
     \l^{r s} \equiv +  \xi^\m \Hat\o\du\m{r s} 
      + 2 g f^{I J K} (\Bar\e{}^I_1 \e_2^J) \Hat H^{r s\, K} ~~, 
&(2.10\rma) \cr
& \e_3^I \equiv - \xi^\m\psi\du\m I ~~, ~~~~ 
     \L^I \equiv - \xi^\m A\du\m I ~~, ~~~~
     \Tilde \L^I \equiv - \xi^\m B\du\m I ~~.
&(2.10\rmb) \cr } $$ 
Due to the field equation (2.7), the existence of the last 
term with $~\Hat H$~ in (2.10a) does not matter for 
on-shell closure.  

Since the parameters $~g$~ and $~h$~ have been 
arbitrary, we can think of interesting cases.  First, 
if $~h=0$, then we have no SCS terms, but with the 
$~C G\-$term which is a kind of $~B F\-$term.    
Second, if $~g h\neq 0$, then we have generally 
some quantization for the coefficients for the CS term,
when the gauge group ~$G$~ has nontrivial
$~\pi_3\-$homotopy mapping.  To be more specific, 
$$ ~\pi_3(G) 
= \cases{  \ZZ  & (for  ~$G = A_n, ~B_n,  ~
     C_n, ~D_n ~~(n\ge 2, ~G\neq D_2), 
      ~G_2, ~F_4, ~E_6, ~E_7, ~ E_8) { ~,~~~~ ~~~~~} $   \cr  
\ZZ \oplus \ZZ & (for $~G = SO(4))$~, \cr 
0 & (for ~$G = U(1))~. $ \cr } 
\eqno(2.11) $$  
For a gauge group with $~\pi_3(G) = \ZZ$, 
the quantization condition is \djt 
$$ \li{ & g h = \fracmm \ell {8\pi} ~~~~(\ell= \pm 1, \pm 2, 
\cdots) ~~. 
&(2.12) \cr } $$ 
Therefore, as long as $~h\neq 0$, the minimal 
coupling constant $~g$~ should be generally quantized in 
this model.  
Third, when we  keep $~g h \neq 0$~ restricted as in 
(2.12), and take a special limit $~g \rightarrow 0$,
we still have the effect of the CS term leaving the action
topologically nontrivial, even though we lose the minimal
coupling.   
 


% \bigskip\bigskip\medskip 

\newpage


\leftline{\bf 3.~~Comments} 

In this brief report, we have presented a  model of
$~\alephnull\-$extended supersymmetric non-Abelian CS 
theories.  The total action is invariant 
under $~N =\hbox{dim}~G\-$extended and local
non-Abelian gauge symmetry, closing gauge algebra. 
Interestingly, we have two different  
constants $~g$~ and $~h$~ which are {\it a priori} 
arbitrary.  Depending on the gauge group $~G$~ 
with nontrivial $~\pi_3\-$homotopy mapping,
the  combination $~g h$~ is to be quantized as 
$~gh = \ell /(8\pi)~~(\ell\in\ZZ)$.  

The generalization of supersymmetric
CS theories to certain special non-Abelian
gauge group is not so surprising, like some examples 
for $~SO(N)$~ for $~^\forall N=1,2,\cdots$~ shown in 
\ngaleph.  However, the important new aspect of our
present results is that non-Abelian CS theory with
arbitrarily many extended supersymmetries $~^\forall 
N$~ for an arbitrary gauge group $~^\forall G$, to our
knowledge, has been presented in this  paper for the first
time.  Note also that it is due to the special topological 
feature of CS theories in 3D that makes it possible to 
generalize the number of supersymmetries up to 
infinity, consistent also with local supersymmetry.  

As is usual with non-Abelian CS theories
\djt\witten\at\ngscs\ngaleph, our action is nontrivial,
even if $~F\du{\m\n} I \eqdot 0$~ on-shell.  This is due to
the presence of  the non-vanishing $~g A\wedge A\wedge
A\-$term, even for a pure-gauge solution 
$~F\du{\m\n}I\eqdot 0$.  It is also important that we have 
nontrivial trilinear interactions for the topological
vector field $~A_\m$~ with arbitrarily many
supersymmetries.  For example, even though a prototype
{\it free} system with $~^\forall N~$ supersymmetries was 
given in \ms, no generalization to trilinear interactions
had been successful, {\it except} those in 
\at\ngscs\ngaleph\nps\rpvn.  

It has been commonly believed that there
is a limit for $~N$~ in 3D for interacting models with 
physical fields, like the limit $~N\le 8$~ in 4D
%%% 
\ref\hls{R.~Haag, J.T.~Lopuszanski and 
M.~Sohnius, \np{88}{75}{257};  
W.~Nahm, \np{135}{78}{149};
J.~Strathdee, \ijmp{2}{87}{273}.}.  
%%% 
In this sense, our model establishes a
counter-example of such wisdom.  
However, our result does not seem to contradict with 
the general analysis on supersymmetry algebra in 3D 
\hls.  We understand that our result is a consequence of 
the special feature of 3D that has not been well
emphasized  in the past, even though some
non-interacting models  in \ms\ had certain indication of 
$~\alephnull\-$supersymmetry.  
As a matter of fact, from a certain viewpoint, 
the existence of $~\alephnull\-$supersymmetric
interacting theories in dimensions $~D\le 3$~ is 
not unusual.  For example, in 1D there are 
analogous $~\alephnull\-$supersymmetries 
%%% 
\ref\gr{S.J.~Gates, Jr.~and L.~Rana, 
\pl{342}{95}{132}, hep-th/9410150;
\pl{352}{95}{50}, hep-th/9504025;
\pl{369}{96}{262}, hep-th/9510151; 
\pl{369}{96}{269}, hep-th/9510152; 
S.J.~Gates Jr, W.D.~Linch and J.~Phillips, 
hep-th/0211034.}.

Physics in 3D is supposed to be closely
related to M-theory, in terms of supermembrane theory
%%% 
\ref\bst{E.~Bergshoeff, E.~Sezgin and P.~Townsend, 
\pl{189}{87}{75}, \ap{185}{88}{330}.}.  
%%%  
Our result here seems to indicate there is still a lot
of freedom to be explored for Chern-Simons theories in
3D.  In fact, we have in our recent paper 
%%%
\ref\nr{H.~Nishino and S.~Rajpoot, 
{\it `Supermembrane with Non-Abelian Gauging
and Chern-Simons Quantization'}, 
hep-th/0309100.}  
%%%
that Chern-Simons terms with quantization arise in
supermembrane action \bst\ upon compactifications with
Killing vectors.  

\newpage

With these encouraging results at hand, we expect that 
there would be more un-explored models with 
$~\alephnull\-$supersymmetry in 3D.    
We are grateful to S.J.~Gates, Jr.~and C.~Vafa for
stimulating  discussions.  


% \newpage

 
\bigskip\bigskip\bigskip\bigskip


\listrefsr

% \listrefsr

\vfill\eject


\end{document} 



%%%%%% Worksheet %%%%%%%%

[1] E. Cremmer, B. Julia and J. Scherk Phys.Lett. B76, 409
(1978)
[2] M. J. Duff, Bristol, UK: IOP (1999) 513 p.
[3] P. K. Townsend, hep-th/9507048; hep-th/9612121
[4] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys.
Proc. Suppl. 62, 348 (1998)
[5] W. Nahm, Nucl. Phys. B135, 149 (1978).
[6] L. Castellani, P. Fr\' e, F. Giani, K. Pilch and P. van
Nieuwenhuizen, Annals Phys. 146, 35 (1983).
[7] H. Nicolai, P. K. Townsend and P. van Nieuwenhuizen,
Lett. Nuovo Cim. 30, 315 (1981).
[8] R. D'Auria and P. Fr\' e, Nucl. Phys. B 201 (1982) 101
[Erratum-ibid. B 206 (1982) 496].
[9] P. Horava, Phys. Rev. D59, 046004 (1999) [hepth/
9712130].
[10] I. Bandos, N. Berkovits and D. Sorokin, Nucl. Phys. B
522, 214 (1998) [hep-th/9711055].
[11] H. Nishino, Mod. Phys. Lett. A 14, 977 (1999) [hepth/
9802009].
[12] Y. Ling and L. Smolin, Nucl. Phys. B 601, 191 (2001)
[hep-th/0003285].
[13] R. Troncoso and J. Zanelli, Phys.Rev. D58, R101703
(1998); R. Troncoso and J. Zanelli, hep-th/9902003.
[14] M. Ba~nados, R. Troncoso and J. Zanelli, Phys. Rev. D54,
2605 (1996).
[15] A. Achucarro and P.K. Townsend, Phys. Lett. B180, 89
(1986). E. Witten, Nucl. Phys. B 311, 4 (1988).
[16] M. Ba~nados, C. Teitelboim and J.Zanelli, Phys. Rev.
D49, 975 (1994).
[17] B. Zumino, �Chiral Anomalies And Dierential Geometry",
UCB-PTH-83/16 Lectures given at Les Houches
Summer School on Theoretical Physics, Les Houches,
France, Aug 8 - Sep 2, 1983.
[18] J. Zanelli, Phys. Rev. D 51, 490 (1995) [hep-th/9406202].
[19] D. Lovelock, J. Math. Phys. 12, 498 (1971).
[20] A.H. Chamseddine, Nucl. Phys. B346, 213 (1990).
[21] M. Ba~nados, L.J. Garay and M. Henneaux, Phys. Rev.
D53, R593 (1996); Nucl. Phys. B476, 611 (1996).
[22] D.G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656
(1985).
[23] J.T. Wheeler, Nucl.Phys. B268, 737 (1986); B273, 732
(1986).
[24] M. Ba~nados, submitted to Physical Review D. See also
M. Ba~nados, Nucl. Phys. Proc. Suppl. 88, 17 (2000) [hepth/
9911150].
[25] K. Bautier, S. Deser, M. Henneaux and D. Seminara,
Phys. Lett. B 406, 49 (1997) [hep-th/9704131].
[26] See U. Gran, hep-th/0105086 for a useful Mathematica
package to do calculations with Dirac matrices.
[27] T. L. Curtright and P. G. Freund, Nucl. Phys. B 172,
413 (1980).
[28] H. Casini, R. Montemayor and L. F. Urrutia, Phys. Lett.
B 507, 336 (2001) [hep-th/0102104].
[29] C. M. Hull, hep-th/0107149.
[30] R. D'Auria, E. Maina, T. Regge and P. Fr\'e, Annals Phys.
135 (1981) 237.
[31] A.H. Chamseddine and H. Nicolai, Phys. Lett. B96, 89
(1980).
[32] E. Witten, Nucl. Phys. B 443, 85 (1995) [hepth/
9503124].
[33] E. Bergshoe, C. Hull and T. Ortin, Nucl. Phys. B 451,
547 (1995) [hep-th/9504081].

%%%%%%%%%%%%%


$~F\du{\m_1\m_2} I_1~, ~\cdots, ~
F\du{\m\low{2M-3}\m\low{2M-2}}{I\low{M-1}} ~, ~
\newline F\du{y\m\low{2M-1}} {I\low M}$, except for the
last  factor 
$~ F\du{y\m\low{2M-1}}{I\low M} = \partial_y 
A\du{\m\low{2M-1}}{I\low M}$, like  



\ngaleph{H.~Nishino and S.J.~Gates, Jr.,
hep-th/9606090, \np{480}{96}{573-588}.  


%%% 
\ref\aaortin{{\it See, e.g.,} N.~Alonso-Alberca and T.~Ortin,
\npn{651}{03}{263}, \hepth{0210011}.}. 
%%% 

In other words, we have given
the supersymmetric non-Abelian CS 
theory for an arbitrary gauge group $~G$~ with 
arbitrarily many extended supersymmetries, which is
the most general compared with similar theories in the
past \ngscs\ngaleph.  

%%%
\ref\siegel{W.~Siegel, \pl{128}{83}{397}.}%
%%%

%%% 
\ref\bns{E.~Bergshoeff, H.~Nishino and
E.~Sezgin, \pl{166}{86}{141};  G.~Atkinson,
U.~Chattopadhyay and S.J.~Gates, Jr.,
\ap{168}{86}{387}.}.   
%%% 



