hep-th0002077.tex
\documentstyle[12pt]{article}

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

\doit0{
% Horizontal alignment of accents: bars, hats, tildes, etc.
\skewchar\fivmi='177 \skewchar\sixmi='177 \skewchar\sevmi='177
\skewchar\egtmi='177 \skewchar\ninmi='177 \skewchar\tenmi='177
\skewchar\elvmi='177 \skewchar\twlmi='177 \skewchar\frtnmi='177
\skewchar\svtnmi='177 \skewchar\twtymi='177
\def\@magscale#1{ scaled \magstep #1}
}

% For framing, put 1 in \doit below as well as in \framing{1}.     
\doit{0}{
% Framing 
\def\framingfonts#1{
\doit{#1}{\font\twfvmi  = ammi10   \@magscale5 % math italic
\skewchar\twfvmi='177 \skewchar\fivsy='60 \skewchar\sixsy='60
\skewchar\sevsy='60 \skewchar\egtsy='60 \skewchar\ninsy='60
\skewchar\tensy='60 \skewchar\elvsy='60 \skewchar\twlsy='60
\skewchar\frtnsy='60 \skewchar\svtnsy='60 \skewchar\twtysy='60
\font\twfvsy  = amsy10   \@magscale5 % math symbols
\skewchar\twfvsy='60
% Fonts
\font\go=font018			% Gothic
\font\sc=font005			% script
\def\Go#1{{\hbox{\go #1}}}	% Gothic for single characters in equations
\def\Sc#1{{\hbox{\sc #1}}}	% script for single characters in equations
\def\Sf#1{{\hbox{\sf #1}}}	% sans serif for single characters in equations
\font\oo=circlew10	      % thick circles (hollow: ` and a-n , solid:  p-z)
\font\ooo=circle10			% thin circles   
\font\ro=manfnt				% font with rope
\def\kcl{{\hbox{\ro 6}}}		% left-handed rope
\def\kcr{{\hbox{\ro 7}}}		% right-handed rope
\def\ktl{{\hbox{\ro \char'134}}}	% top end for left-handed rope
\def\ktr{{\hbox{\ro \char'135}}}	% " right
\def\kbl{{\hbox{\ro \char'136}}}	% " bottom left
\def\kbr{{\hbox{\ro \char'137}}}	% " right
}}  
}

% Underline for text or math
\catcode`@=11
%\def\un#1{\relax\ifmmode\@@underline#1\else
%	$\@@underline{\hbox{#1}}$\relax\fi}
\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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	\par
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	\vskip.2in}
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	\setlength{\unitlength}{1mm}
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	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
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	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
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	{\loop
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	\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}
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	\begin{picture}(140,0)
	\put(\xco,\yco){$\ktl$}
	\advance\yco by-1
	{\loop
	\put(\xco,\yco){$\kcl$}
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	\repeat
	\put(\xco,\yco){$\kbl$}}
	\xco=158
	\yco=12
	\put(\xco,\yco){$\ktr$}
	\advance\yco by-1
	{\loop
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\def\headpic{						% UM heading
	\indent
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	\thinlines
	\par
	\begin{picture}(29,16)
	\put(165,16){\line(1,0){4}}
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\def\endtitle{\end{quotation}\newpage}			% end title page

% Section heading and reference stuff
\def\sect#1{\bigskip\medskip \goodbreak \noindent{\bf {#1}} \nobreak \medskip}
\def\refs{\sect{References} \footnotesize \frenchspacing \parskip=0pt}
\def\Item{\par\hang\textindent}
\def\Itemitem{\par\indent \hangindent2\parindent \textindent}
\def\makelabel#1{\hfil #1}
\def\topic{\par\noindent \hangafter1 \hangindent20pt}
\def\Topic{\par\noindent \hangafter1 \hangindent60pt}
\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil}
\def\]{\rfloor{\hskip 0.35pt}\!\!\!\rceil}
%\def\[{\lfloor{\hskip 0.35pt}\!\!\!\lceil\,}
%\def\]{\,\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\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\Re{{\cal R}e\,} 
\def\Im{{\cal I}m\,}
\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}
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\def\derx{\partial_x} \def\dery{\partial_y} \def\dert{\partial_t} 
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% There are more than one line for the definition of \footnotew below:
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% \noindent #1}}}
% \baselineskip 16pt\oddsidemargin=0.03in 
% \evensidemargin=0.01in\hsize=6.5in\textwidth=6.5in} 

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

\def\low#1{\hskip0.01in{\raise -3pt\hbox{${\hskip 1.0pt}\!_{#1}$}}}
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\def\dt#1{\Dot{#1}}
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\def\Norm{|\hskip -2.0pt|}
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\def\hati{{\hat i}} \def\hatj{{\hat j}} \def\hatk{{\hat k}} 
\def\hatl{{\hat l}} \def\hatm{\hat m} \def\hatn{\hat n} \def\hatr{\hat r}
\def\hats{\hat s} \def\hatt{\hat t}

\begin{document}

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%%%%%%%%%%%%%%%%%%%%%%%% harvmactest.tex %%%%%%%%%%%%%%%%%%%%%%%%
%
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\def\bigans{b }
\message{ big or little (b/l)? }\read-1 to\answ
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\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% use \nolabels to get rid of eqn, ref, and fig labels in draft mode
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%%%%%%%%%%%%%%%%%%%%%%%%% End of harvmac.tex %%%%%%%%%%%%%%%%%%%%%%%%
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\def\f@t#1{#1\@foot}
\def\@foot{\strut\egroup}
\def\footstrut{\vbox to\splittopskip{}}
\skip\footins=\bigskipamount 
\count\footins=1000
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This is a test for footnote.\footnote*{This is a footnote.}
\end{document} 
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\def\nopagenumbers{\footline={\hfil}} 
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\def\ul{\underline} \def\un{\underline} 
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\framing{0}

\footskip=0.26in 


\vskip 0.07in

{\hbox to\hsize{\hfill hep-th/0002077}} \par
{\hbox to\hsize{~~~~~ ~~~~~ \hfill UMDEPP 00--052}} \par\vskip -0.1in
\hfill 
{(Revised Version)}
\\
\begin{center}

{\bf\largecmrseventeen ~Fundamental ~Extended ~Objects} \\ 
{\bf\largecmrseventeen ~for ~Chern-Simons~
Supergravity}$\,$\footnote{This  work is supported in part by NSF grant
\# PHY-93-41926.} \\[.1in] 


\baselineskip 9pt 

\vskip 0.32in

Pablo M{\smallcmr ORA}\footnote{Also at 
Instituto de F\'isica, Facultad de Ciencias, 
Igu\'a 4225, Montevideo, Uruguay.  \newline 
E-Mail: pmora@physics.umd.edu} ~and ~ 
Hitoshi N{\smallcmr ISHINO}\footnote{E-Mail: 
nishino@nscpmail.physics.umd.edu} \\[.2in]
{\it Department of Physics} \\ [.015in]
{\it University of Maryland at College Park} \\ [.015in]
{\it College Park, MD 20742-4111, USA} \\[.020in]   

\vskip 1.5in 

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

\begin{quotation}

We propose a class of models in which extended objects are introduced in
Chern-Simons supergravity in such a way that those objects appear on the same
footing as the target space.  This is motivated by the idea that branes are
already first quantized object, so that it is desirable to have a formalism
that treats branes and their target space in a similar fashion.  Accordingly,
our models describe interacting branes, as gauge systems for supergroups.  We
also consider the case in which those objects have boundaries, and discuss
possible links to superstring theory and/or M-theory, by studying the fermionic
$~\k\-$symmetry of the action.  


\endtitle


\oddsidemargin=0.03in
\evensidemargin=0.01in
\hsize=6.5in
\textwidth=6.5in 
\textheight=8.7in 
\baselineskip 14.8pt


\centerline{\bf 1.~~Introduction}

     
Gauge supergravity theories, based on lagrangians of the Chern-Simons 
form involving gauge fields for a gauge supergroup containing the super
Poincar\'e group of the appropriate dimension, have recently attracted a
great deal of attention.  Since the construction of (2+1)-dimensional
theories 
%%%
\ref\achu{A.~Achucarro and P.K.~Townsend, \pl{180}{86}{89}.}%
\ref\witone{E.~Witten, \np{311}{88}{46}; \ibid{323}{89}{113}.}%
\ref\ng{H.~Nishino and S.J.~Gates, Jr., \ijmp{8}{93}{3371}.}
and its quantization \witone\ these
theories have been extended to higher dimensions by Chamseddine 
%%%
\ref\chamseddine{A.H.~Chamseddine, 
\pl{233}{89}{291}; \np{346}{90}{213}.}
%%%
and further developed by Ba\~nados, Troncoso and Zanelli 
%%%
\ref\zanone{M.~Ba\~nados, R.~Troncoso and J.~Zanelli, 
\pr{54}{96}{2605}.}%
\ref\zantwo{R.~Troncoso and J.~Zanelli, \pr{58}{98}{101703}, 
{\it `Higher Dimensional Gravity and Local Anti-de Sitter Symmetry'}, 
hep-th/9907109.}.  Also Green
%%%
\ref\mile{M.B.~Green, \pl{223}{89}{157}.}, 
%%%  
or Moore and Seiberg 
%%%
\ref\moor{G.~Moore and N.~Seiberg, \pl{220}{89}{422}.}
gave possible links between 
Chern-Simons theory for a supergroup and superstring theory,  
while Ho\v rava 
%%%
\ref\hora{P.~Ho\v rava, \pr{59}{99}{046004}.}
%%%
suggested that M-theory 
%%%
\ref\hulltownsend{C.~Hull and P.K.~Townsend, \np{348}{95}{109}.}%
\ref\witten{E.~Witten, \np{443}{95}{85}.}%
\ref\townsend{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.}
%%%
may actually be a field theory of this class.  

In this Letter we will suggest a natural way to introduce fundamental
extended objects in Chern-Simons supergravities by embedding lower
dimensional lagrangians for Chern-Simons supergravity in higher
dimensions.  We may also think of this model as a set of Chern-Simons  
supersymmetric branes\footnotew{Chern-Simons bosonic branes in a somewhat
different framework have been considered in 
%%%
ref.~\ref\zaikov{R.P.~Zaikov,
\pl{211}{88}{281}; \pl{263}{91}{209}; \pl{266}{91}{303}.}.}
%%%
embedded into a background
%vjr change
%old: that is itself a Chern-Simons brane, 
%new: 
described also by a Chern-Simons action (and can be considered
itself a  Chern-Simons brane, even though it is not embedded in a larger
manifold) providing in effect a `brane-target space democracy'. 
It is important to remark, 
however, that supersymmetry is introduced here as a gauge symmetry, in
the sense that  the super Poincar\'e group (or any of it extensions, like
the super conformal or super (anti) de Sitter groups) are realized as
parts of a gauge group.  That is to  be contrasted with the standard
approach of getting spacetime supersymmetry  by defining the extended
objects as embeddings in flat or curved superspace.   

A heuristic motivation for the kind of models presented here is that if
there is some underlying pregeometric theory in which the de Rham
complex get physical content through some kind of antighost field
variables, we must expect the effective action to be a sum of pieces of
diverse dimensions of the form considered here.  We believe that this
simple observation is the key to understand the existence of extended
objects in a fundamental theory of nature.
 
We also consider the case in which the embedded supersymmetric branes
have boundaries.  In such a case, we need to add boundary terms in order 
to preserve gauge invariance in a way resembling the action of Dixon 
%%%
{\it et al.}~\ref\duffone{J.A.~Dixon, M.J. Duff and E. Sezgin, 
\pl{279}{92}{265}.}  
%%%
for extended objects coupled to gauge
fields except in two very important respects:  First, we implement
supersymmetry as a part of the gauge (super)group.  Second, we have a
bulk(s) with nontrivial degrees of freedom, which can be ignored only in
the case that the gauge potential for the supergroup is pure-gauge.  We
will show that in this case and for a particular choice of the gauge
supergroup one can establish a link with ten-dimensional (10D)
superstring theory 
%%%
\ref\superstring{M.B.~Green, J.H.~Schwarz and E.~Witten, 
{\it `Superstring Theory'}, Vols. 1 \& 2,
(Cambridge Univ. Press, 1987).}.  
%%%

\bigskip\bigskip\bigskip

 
\centerline{\bf 2.~~Action for Extended Objects}

We consider a (super)group $~G$~ with generators $~T^I$, 
the gauge potential 1-form $~A=A_m{}^I T_I \, dx^m $~ and 
the curvature 2-form 
$~F=dA+A^2$, defined on a manifold $~M^{D+1}$~ of even dimension $~D+1$. 
If we have an invariant polynomial $~P(F)$~ defined as the formal sum
\begin{equation}
P(F)=\sum_{n=1}^{(D+1)/2} \alpha _n \STr \big( F^n \big) ~~,   
\end{equation} 
here $~\STr \big( T^{I_1}\dots T^{I_n} \big) 
= g^{I_1\cdots I_n}$~ stands for an
invariant  symmetric trace on the algebra  of $~G$, we define the action
for our system by  
\be
S=\sum_{d~{\rm even}}^{D-1}\alpha_d\int_{\Omega^{d+1}}I^0_{d+1}~~.  
\ee 
Here the $~I^0_{d+1}(F,A)$~ are the Chern-Simons forms associated with  
$~P(F)$, so locally
\be
P(F)= \sum_{d~{\rm even}}^{D-1} \a_d \STr\Big( F^{\frac{d+2}2} \Big) 
= \sum_{d~{\rm even}}^{D-1} \alpha_d \, dI^0_{d+1}(F,A) ~~,   
~~~ (n = \frac d 2 + 1) ~~, 
\ee 
and $~M^{D+1}$~ has $~\Omega^D$~ as its boundary, while the $~\Omega
^{d+1}$'s are submanifolds of $~\Omega^D$~ of odd dimension. The
dynamical variables are the gauge potentials $~A$~ and the embedding
coordinates of the submanifolds $~X^m_{(d)}(\chi _{(d)}^i)$, where
the $~\chi_{(d)}^i$~ with $~{\scst i~=~1,~...,~d+1}$~ 
are local coordinates in $~\Omega^{d+1}$.  
Notice that all these manifolds are supposed to be non compact
at least  in what would be the `time direction'.  As observed in
ref.~\zantwo, each of the dimensionless coefficients $~\alpha_d$~ can
consistently take only a  discrete set of values  as a result of the
requirement that the action must  be independent of the way in which the
manifolds $~\Omega ^{d+1}$~ are  extended into manifolds 
$~M^{d+2}$~ included into $~M^{D+1}$~ such that $~\partial
\Omega^{d+1}\equiv M^{d+2}$.$\,$\footnote{For instance  
$P(F)=\STr[e^{i F/(2\pi)}]$ would work.}

It seem to us that even though any symmetric trace is allowed at this
level, if we are looking for a effective theory corresponding to some
underlying  pregeometric theory then Chern characters may be favored on
the grounds of their  additivity  under a Stieffel (direct) sum of
bundles, which would correspond to  overlapping `brane elements' at a
point. However we will keep our considerations  as general as possible 
in what follows.

Suppose now that we allow for the possibility that the extended objects 
with world-volume $~\Omega^{d+1}$~ have boundaries $~S^d\equiv
\partial  \Omega^{d+1}$. In that case the action is not gauge invariant
anymore,  because of a boundary term in the gauge variation $~\d_{\rm G}$~ 
of the Chern-Simons actions 
%%%
\ref\zumino{B.~Zumino, in {\it `Relativity, Groups and
Topology II'} (B.S.~De Witt and R.~Stora eds.),
North Holland, Amsterdam, 1984.}%
\ref\alvarez{L.~Alvarez-Gaum\' e and P.~Ginsparg, 
\ap{161}{85}{423}.}.  
Locally 
\be
\delta_{\rm G}I^0_{d+1}(F,A) =d I^1_d (F,A,\lambda ) ~~, \nn 
\ee
where we are using the notation conventions of ref.~\duffone.
One way to cancel this variation (actually also the global variation) 
is to add to the action boundary terms of a form 
analogous to the one in \duffone% 
\footnote{It is important however to point out that our 
action is not just the action of \duffone\ for a supergroup, as for our
action there is no background metric or Ramond-Ramond (RR) 
p-form fields, and furthermore our branes do have a bulk with non trivial 
degrees of freedom and a nontrivial dynamics while the kinetic term
`lives' at the boundary.  The branes of \duffone\ would correspond to the
boundary of ours in the pure gauge case (when bulk degrees of freedom go
away),  and if we use the WZNW term from $A$ to mimic the RR field.}
(see also
%%% 
ref.~\ref\delduc{E.~Bergshoeff, F.~Delduc and E.~Sokatchev,
\pl{262}{91}{444}.})
%%% 
so that the complete action now reads
\bea && 
S = \sum_{d~{\rm even}}^{D-1}\alpha_d \Bigg\{ \int_{\Omega^{d+1}}
I^0_{d+1} +
\bigg[ \frac1 2 \int_{S^d} d^d \zeta_{_{(d)}} {\sqrt{-\gamma_{_{(d)}}}}
\bigg( \gamma_{_{(d)}}^{i j} \STr (J_i J_j) - (d-2) \bigg)  \nn \\
&& ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ 
-\int_{S^d} (C_d(A,K) - b_d) \bigg] \Bigg\} ~~,  
\la{sone} 
\eea
where the $~\zeta$'s are local coordinates in $~S^d$~ and 
\be
J_i{}^I =A_m{}^I \fracm{\partial X^m }{\partial \zeta ^i}-
K_\iota{}^I \fracm{\partial y^\iota} {\partial \zeta ^i} ~~, 
~~~(\hbox{or}~~ J= A - K~~ \hbox{as 1-form})
\ee
where $~K$~ is the left-invariant Maurer-Cartan form in the (super)group 
manifold G, the $~y^\iota$'s are coordinates on that (super)group 
manifold, while 
$~K_\iota(y)=g^{-1}(y)\partial g(y)/\partial y^\iota $~ satisfies $~dK+K^2=0$~ 
(the differentials are taken in $~S^d$~ for ~$y\-$dependent quantities), and
$~b_d$~ is a $~d\-$form such that $~d b_d=-I^0_{d+1}(0,K)$, while
$~C_d(A,K)$~ is chosen in order to preserve gauge invariance in a way 
described below. The auxiliary metrics $~\gamma_{_{(d)}}$~ have purely
algebraic equation of motion 
that can be used to eliminate it from the
action.  The $~b_d\-$term is a generalization of Wess-Zumino-Novikov-Witten
(WZNW)-term.  For example, the $~d=2$~ case corresponds to the usual 
(super)string formulation \superstring\ on $~d=2$~ world-sheet 
with an additional 3D bulk term with $~I_3^0$.   
We must add to the list of  our dynamical variables the functions
$~X_{_{(d)}}^m (\zeta_{_{(d)}}^i)$~ (or equivalently
$~\chi_{_{(d)}}^k (\zeta_{_{(d)}}^i)$), $~y_{_{(d)}}^\iota
(\zeta_{_{(d)}}^i)$~ and $~\gamma_{_{(d)}}^{ij}(\zeta_{_{(d)}}^k)$. 

Under (super)gauge transformations, each term of our action transforms
as \duffone
\be
\delta_{\rm G} A=d\lambda +\[ A,\lambda \]~~, ~~~~
          \delta K=d\lambda + \[ K,\lambda \]~~, 
\end{equation}
and accordingly $~J$~ transforms covariantly
\begin{equation}
\delta_{\rm G} J = \[ J,\lambda \]~~.
\ee
Also 
\be
\delta_{\rm G} I^0_{d+1}(F,A)=d I^1_d(F,A,\lambda )~~,~~~~
        \delta_{\rm G} b_d =-I_d^1(0,K,\lambda )~~, 
\ee
and we require 
\be
\delta_{\rm G} C_d =I^1_d (F,A,\lambda )-I_d^1(0,K,\lambda ) 
     + (\hbox{exact form}) ~~, 
\ee
which is satisfied for \duffone
\be
C_d = k_{01}I_{d+1}^0 (F_t,A_t) = \int_0^1 d t \, l_t~I^0_{d+1}(F_t,A_t)~~,
\la{cp} \\
\ee
where we identify 
\be
A \equiv A_1~~,~~K\equiv A_0~~, 
\ee 
for 
\be
A_t=t A_1+(1-t)A_0~~,~~F_t=d A_t+A_t^2=t F+t(t-1)(A_1-A_0)^2 ~~.  
\ee
For $~d=2 $~ for instance, $~C_2=\STr (A K)$.   
The ~$l_t$~ in eq.~(11) is defined to act on arbitrary polynomials 
\be
l_t A_t=0~~,~~~~ 
l_t F_t = A_1 - A_0 ~~, 
\ee
with the convention that ~$l_t$~ is defined to act as an antiderivation.
Then the Cartan homotopy
operator is defined by \alvarez
\be
k_{01}{\cal P}(F_t,A_t) =\int_0^1 \d t~l_t{\cal P}(F_t,A_t) ~~,  
\ee
where $~\d t$~ is for the $~t\-$integration, to be distinguished from 
other one-forms such as $~d A_t$ ~\alvarez.  By integrating 
\be
\big( l_td+dl_t\big) {\cal P}(F_t,A_t)
=\frac{\partial}{\partial t}{\cal P}(F_t,A_t)
\ee
over $~t$~ from $~0$~ to $~1$, we get the Cartan homotopy formula 
\alvarez 
\be
\big( k_{01}d+dk_{01}\big) {\cal P}(F_t,A_t)
	={\cal P}(F_1,A_1)-{\cal P}(F_0,A_0) ~~. 
\ee
Putting all this together we see that the action (5) 
is indeed gauge invariant.  

A more compact form of the action (5) can be given by considering the
Cartan homotopy formula for ${\cal P}(F_t,A_t)=I^0_{d+1}(F_t,A_t)$, then
the action (5) is rewritten as 
\be
S = \sum_{d~{\rm even}}^{D-1}\alpha_d \Bigg\{ \int_{\Omega^{d+1}} 
     k_{01} \STr \Big( F{\low t}^{\frac{d+2}2} \Big)  
     + \frac1 2 \int_{S^d} d^d \zeta_{_{(d)}} 
     {\sqrt{-\gamma_{_{(d)}}}}
     \bigg[ \gamma_{_{(d)}}^{ij} \STr \big( J_i J_j \big) 
	- (d-2)  \bigg] \Bigg\} ~. {~~} 
\la{stwo} 
\ee
Here we used also the relationship $~\int_{\Omega^{d+1}} I_{d+1}^0 (0,K) = 
- \int_{S^d} b_d$.  This form of the action makes its gauge
invariance more manifest, because $k_{01}$ is a gauge invariant operator.
We can generalize our action one step further, and treat ~$A_0 = K$~ and
~$A_1 = A$~ in a symmetric fashion by letting ~$A_0$~ be an arbitrary
gauge field and not just pure-gauge, then the action (5) is just a
particular case of the action (18).  The price we have paid is doubling
the number of bulk degrees of freedom.\footnotew{The Cauchy problem (as
well as causality issues, considering the absence of a background metric)
for the action (18) seems formidable, though probably not much worse than
the same problems for standard Chern-Simons supergravities. In the case
that the manifold in which all the branes are embedded has boundaries, we
would need boundary conditions for the gauge fields, as well as specifying
whether the extended objects have boundaries on that boundary or not, in
order to have a well posed Cauchy problem.}

%change: I change some main text into the footnote for the previous paragraph

It is worthwhile to note that the $~\STr(J^2)$~ in the `kinetic' term in
the action (5) is gauge invariant by itself, while the
bulk and the other boundary terms transform into each other under gauge 
transformations.  That would in principle mean that the coefficient of
that term can be chosen independently.  However, as will be seen in the
next section, the $~\k\-$symmetric Green-Schwarz
string action obtained as a certain limit fixes that coefficient,
because the $~\k\-$symmetry mixes the kinetic and WZNW-term of the
action.  In the standard formulation of the heterotic string, we have the
choice between a `fermionic' 
%%%
\ref\hull{C.~Hull and E.~Witten, \pl{160}{85}{398}.}
%%%
and a `bosonic'
%%%
\ref\nilsson{M.~Duff, B.~Nilsson and C.~Pope, \pl{163}{85}{343}.}
%%%  
formulation, depending on whether the gauge group acts on
world-sheet fermions or on bosonic group manifold coordinates. Our
action is analogous to the latter, 
but we might think of an alternative fermionic formulation in which the
quantum anomaly of a quadratic kinetic boundary term cancels out the
boundary gauge variation of the Chern-Simons bulk term, and the WZNW-term
is a quantum effect, so that all relative coefficients are determined.  

%change:next paragraph added
The $\kappa$-simmetry should in principle rule out the infinite 
number of generally covariant and gauge invariant terms that 
we can build out of the gauge covariant objects $F$ and $J$ 
(and their pull-backs) and the gauge invariant $\gamma$'s. 
However, some alternative forms of the kinetic term may also 
yield $\kappa$-symmetric actions, for instance Born-Infeld-like 
kinetic terms as
$$\int_{S^d} d^d \zeta_{_{(d)}} 
 \STr \left[\sqrt{-\sdet\big\{ J_iJ_j+(F_0)_{ij}+(F_1)_{ij}\big\} }\right]
\eqno(19\rma) $$
or
$$\int_{S^d} d^d \zeta_{_{(d)}} \STr \left[\sqrt{-\sdet\big\{ \STr (J_i
J_j) +(F_0)_{ij}+(F_1)_{ij}\big\} }\right]
\eqno(19\rmb) $$
where the superdeterminant is taken in the curved indices $~{\scst i,j}~$
of the  pull-backs on ~$S^d$~ while the supertraces are taken on the 
group indices. Notice that because supersymmetry is 
realized as a part of the
gauge group for our models the kinetic terms given before are 
at once a gauge non-abelian and 
fermionic supersymmetric (as well as generally covariant) generalization 
of the bosonic abelian Born-Infeld action. Out of these possible alternative 
kinetic terms, the second one is the one that corresponds to a more
straightforward generalization of the metric-eliminated version of our original
one. The above actions with Born-Infeld- like kinetic terms would provide us a
framework where fundamental strings and Dp-branes can be treated in an
equivalent fashion. Duality arguments would likely help in fixing 
the precise form of the kinetic term.
%vjr change: references added
For a review of the literature on D-branes and Born-Infeld
terms see \ref\tseytlin{A.A.~Tseytlin,  
{\it `Born-Infeld action, supersymmetry and string theory`}, 
hep-th/9908105.} and references therein.  (Cf.~A related recent work in
%%% 
\ref\nishinobi{H.~Nishino,  
{\it `Non-Linear Realization of {\small\alephnull}$\!$-Extended Supersymmetry'}, 
hep-th/0002029.})  
%%%

%vjr change: paragraph added next about referee coment
Notice that our actions of eqs.~(5) and (18) describe 
a system of any number of interacting branes with or 
without boundaries which are charged with respect to the gauge fields 
of the model. The currents carried by the branes can be computed by taking 
the functional derivatives of different pieces of the action with respect to
the gauge potentials. 
 
\bigskip\bigskip



\centerline{\bf 3.~~ From $~OSp(32|1)$~ to Type IIA Green-Schwarz
Superstring} 
 
We next give some intermediate steps to obtain the action for 10D Type
IIA Green-Schwarz superstring from a Maurer-Cartan form for the M-theory
group $~OSp(32|1)$~ in 11D \townsend%
%%%
\ref\bars{I.~Bars, \pr{55}{97}{2373}.}%
%%%
\hora.  Subsequently, we consider how we can understand the  usual
fermionic $~\k\-$symmetry 
%%%
\ref\kappasym{M.~Green and J.H.~Schwarz, \pl{136}{84}{367}; 
W.~Siegel, \pl{128}{83}{397}.}
%%%
for the Green-Schwarz
superstring \superstring\ in our formulation.
We hope this
provides us with a clearer picture about the link between our extended
objects descrbed by our action (5) or (18) on Chern-Simons supergravity
\chamseddine\zanone\zantwo\hora\ with 10D Green-Schwarz superstring
\superstring.  

%vjr change:footnote pass to main text and new discussion
An earlier work towards establishing a link between 
Chern-Simons theory for a supergroup and superstring theory has been done by
Green in \mile\ (Cf.~also \moor).   
However, several differences with our model should be clarified here:\\
(i) In the last section, a Chern-Simons action for a supergroup extension 
of the super Poincar\'e group is discussed for a three dimensional base 
manifold, however the action does not contain boundary terms 
which would render it gauge invariant without gauge-fixing in 
the boundary.  Also a gauge-invariant kinetic term is absent in 
\mile.  \\ 
(ii) In this section, the WZNW action corresponding to the Chern-Simons action
of Section 2 will be considered with a kinetic term now included, but in 
such a way that the model is again not gauge invariant without 
gauge-fixing. 
\\
(iii)  Our branes are embedded in a larger spacetime and interacting 
with other branes, nemely, the pure gauge condition is not 
required for us in general, as it is for ref.~\mile.  That means 
that we can consider generic backgrounds, and not only flat superspace.

Consider the group $~G$~ as the M-theory group $~OSp(32|1)$~ 
with generators $~P_a$, $~Q_{\alpha}$, 
$~M_{ab}$~ and $~Z_{a_1...a_5}$, and the symmetric trace
is just the standard symmetricized supertrace in the adjoint
representation of $~G$.  If we consider the case of `elementary'
fundamental objects of size one in proper coordinates (as opposite to
`cosmic' extended objects), living in a large background\footnotew{Notice
that  this is {\it a priori} a somewhat unnatural limit in our
approach.}  then dimensional arguments show that the $~n\-$dimensional
integrals in  our action scale as $~{\cal O}(1/L^n)$~ where $~L$~ is the 
characteristic scale of the background. That means that the dominant
terms  in the action will be those of lower dimensions that are
non-vanishing.  In the case of the group $~OSp(32|1)$~ 
the generators are traceless in the adjoint representation, 
so that the 0-brane (particle) 
terms are absent and the dominant contribution comes from the 2-brane. We
notice that if $~A=0$, and the pure-gauge $~K$~ is of the form
$~K = g^{-1} d g$, then we compute essentially up to 
$~{\cal O}(R^{-1/2})\-$terms.  This is equivalent to keeping only 
$~\big(X^a, \theta^\a\big)$~ among the coordinates $~\big(y^\iota\big)$.  
Plugging this ansatz back into the action, restricting the boundary of
the 2-brane to a 10D boundary of the 11D base
manifold, identifying the gauge parameters $~X^a$~ with the $~D=11$~
coordinates, we get the kinetic term from $~\STr\big( J^2 \big)$~ 
in the original action, and the WZNW-term from the $~b_2\-$term. 

The $~\hbox{\it OSp}(32|1)$~ algebra is dictated by
%%%
\ref\preitschopf{R.~D'Auria, P.~Fr\'e, \np{201}{82}{101}; 
Erratum-ibid.~{\bf B206} (1982) 496;
T.~Hurth, P.~van Nieuwenhuizen, A.~Waldron, C.~Preitschopf, 
\pr{55}{97}{7593}.}%  
%%%
\ref\holtenproyen{{\it See, e.g.,} J.W.~Holten and A.V.~Proyen, 
\jmp{15}{82}{3763}.}
%%% 
$$ \li{ & \{ Q_\a , Q_\b \} = a_2 \frac 1 R 
	\big(\Hat\g^{\hat a\hat b} \big)_{\a\b} \Hat M_{\hat a\hat b} 
     + i a_5 \frac1{\sqrt R} 
	\big( \Hat\g^{\hat a_1\cdots \hat a_5} \big)_{\a\b}
     \Hat Z_{\hata_1\cdots\hata_5} ~~, 
&(20\rma) \cr 
& \[ \Hat M_{\hata\hatb} , Q_\a \] 
     = - \frac 1 2 \big( \Hat\g_{\hata\hatb} \big)\du\a\b Q_\b ~~,~~~~
     \[ \Hat M_{\hata\hatb} , \Hat M^{\hatc\hatd} \]  
     = - 4 \d\du{\[\hata}{\[\hatc} \Hat M\du{\hatb\]}{\hatd\]} ~~, 
&(20\rmb) \cr 
& \[ \Hat Z_{\hata_1\cdots \hata_5} , Q_\a \] 
    =  i b_3 \frac 1 {\sqrt R} 
	\big(\Hat\g_{\hata_1\cdots \hata_5} \big)\du\a\b Q_\b ~~, ~~~~ 
    \[ \Hat M_{\hata\hatb} , \Hat Z^{\hatc_1\cdots\hatc_5} \] 
      = - 10 \d\du{\[ \hata}{\[ \hatc_1} 
    \Hat Z\du{\hatb\]}{\hatc_1\cdots\hatc_5\]}  ~~, 
&(20\rmc) \cr 
& \[ \Hat Z_{\hata_1\cdots\hata_5}, \Hat Z_{\hatb_1\cdots\hatb_5} \] 
    = \frac1{\sqrt R} f\du{\hata_1\cdots\hata_5~\hatb_1\cdots\hatb_5} 
      {\hatc_1\cdots\hatc_5} \Hat Z_{\hatc_1\cdots \hatc_5} ~~,
&(20\rmd) \cr } $$
in the $~SO(1,10)$~ manifest notation.
The $~a$'s and $~b$'s are appropriate constants, and 
$~f$'s in (20d) is a structure constant for the $~\[ \Hat Z,\Hat Z
\]\-$commutator, whose detail is not important here.   
Here all the indices and operators with {\it hats} refer to the 
11D,\footnotew{For $~Q$'s we do not need this distinction, because 
all of its Majorana components are kept in 10D upon the 
compactification.  Relevantly, the indices $~{\scst \a,~\b,~\cdots}$~ 
are Majorana indices in 10D.} to be distinguished from those in 10D into
which we are now  going to perform the compactification on a torus from
the former.  The constant $~R$~ denotes the radius of such a torus.  We
assigned appropriate powers of $~R$~ in such a way that in the limit
$~R\rightarrow \infty$~ recovers the usual super Poincar\'e algebra in
10D, with the consistent mass dimensions, namely $~\[\Hat
P_{\hata}\]_{\rm M} = +1,~\[ \Hat M_{\hata\hatb} \]_{\rm M}= 0,  ~\[\Hat
Z_{\hata_1\cdots \hata_5} \]_{\rm M} = +1/2,  ~\[ Q_\a \]_{\rm M} =
+1/2$. For example, the half-integer for ~$\[\Hat Z\]_{\rm
M} = +1/2$~ is required by the vanishing of the $~\Hat Z\-$term in (20a)
for $~R\rightarrow\infty$.  

Upon the compactification for a finite $~R$,
the algebra above becomes what is called 10D de-Sitter algebra     
\holtenproyen,    
where in particular, the components of $~\Hat M_{\hata\hatb}$~ with 
the 10-th coordinate, are identified with the momentum operator $~P_a$~ 
in 10D:
$$ \li{ & \Hat M_{\hata\hatb} 
     = \cases{ \Hat M_{a b} = M_{a b} ~~, \cr 
        \Hat M_{10 b} = R P_b = - \Hat M_{b\, 10} ~~. \cr } 
&(21) \cr } $$  
Thus the algebra (20) becomes the de-Sitter algebra in 10D
\holtenproyen:  
$$ \li{ & \{ Q_\a , Q_\b \} = 
	i a_1 \big(\g\low{11} \g^c \big)_{\a\b} P_c 
	+ a_2 \frac 1 R \big( \g^{a b} \big)_{\a\b} M_{a b} 
	+ i a_5 \frac 1{\sqrt R} \big( \g^{a_1\cdots a_5}\big)_{\a\b}
	Z_{a_1\cdots a_5} ~~, {~~~~~ ~~~~~}
&(22\rma) \cr 
& \[ M_{a b} , M^{c d} \, \] = - 4 \d\du{\[a}{\[c} M\du{b\]}{d\]}
    ~~,~~~~
    \[ M_{a b}, P_c \] = + 2 \eta_{c\[b} P_{a\]} ~~ ,  
&(22\rmb) \cr 
& \[ P_a , P_b \] = i b_1 \frac 1 {R^2} M_{a b} ~,~~~ 
     \[ M_{a b}, Q_\a \]
     = - \frac 12 \big( \g_{a b} \big)\du\a\b Q_\b ~,  ~~~
     \[ P_a , Q_\a
     \] = i b_2 \frac 1 R \big( \g\low{11} \g_a \big) \du\a\b Q_\b ~, 
    {~~~~~ ~~~~~} 
&(22\rmc) \cr 
& \[ \Hat Z_{\hata_1\cdots \hata_5}, Q_\a \] = i b_3 \frac 1 {\sqrt R}
	\big( \Hat\g_{\hata_1\cdots\hata_5} \big) \du\a\b Q_\b ~~, 
     ~~~~ \[ \Hat M_{\hata\hatb} , \Hat Z^{\hatc_1\cdots\hatc_5} \] = 
	   - 10 \d\du{\[ \hata}{\[ \hatc_1} 
	   \Hat Z\du{\hatb \]}{\hatc_2\cdots \hatc_5 \] } ~~, 
&(22{\rm d}) \cr 
& \[ P_b , {\Hat Z}^{\hata_1\cdots\hata_5} \]  
    = -5 \frac 1 R \d\du{10}{\[ \hata_1} 
     \Hat Z\du b{\hata_2\cdots\hata_5 \] } ~~,  ~~~~ 
    \[ \Hat Z_{\[\hat 5\]}, \Hat Z_{\[\hat 5\]'} \] 
    = \frac1{\sqrt R} f\du{\[\hat 5\]\[\hat 5\]'}{\[\hat 5\]''} 
     \Hat Z_{\[\hat 5\]''} ~~, 
&(22{\rm e}) \cr } $$ 
where all other independent commutators are vanishing.  
The index $~{\scst \[\hat 5\]}$~ on $~Z_{\[\hat 5\]}$~ {\it etc.}~is for 
the antisymmetric indices $~{\scst\hata_1\hata_2\cdots\hata_5}$.
We did not decompose the indices in terms of 10D indices for  
(22e), because they are not crucial for our purpose of 
getting the expression for $~g^{-1} d g $~ up to $~{\cal O}(R^{-3/2})$. 
The reason we need an extra $~\g\low{11}$~ in (22a) is explained in 
%%%
\ref\starreport{H.~Nishino and S.J.~Gates, Jr., {\it `The *Report'},
hep-th/9908136.}.
%%%   

In the limit $~R \rightarrow \infty$~ the algebra (22)  recovers the
usual super Poincar\'e algebra in Minkowski 10D, so that we can see the
link with the Green-Schwarz superstring action \superstring.   

We now compute the Maurer-Cartan form $~K=g^{-1} d g$~ up to 
$~{\cal O}(R^{-3/2})$, following the general expansion formula:
$$ \li{ & K=g^{-1} d g = \sum_{n=1}^{\infty}  \fracm 1{n!} 
      \overbrace{\[\, \[ \, \cdots \[\, \[}^n \, d , 
					 \overbrace{ h \,\] , 
     h \, \], \cdots, h \, \] , h \, \] }^n ~~,  
&(23) \cr } $$
where $~g = \exp\big( i X^a P_a + \theta^\a Q_\a\big) \equiv \exp h$.  
We keep at most the $~{\cal O}(R^{-1/2})$~ and $~{\cal
O}(R^{-1})\-$terms, in  order to elucidate the first effect of our 11D
model on the Green-Schwarz superstring action.   As a simple dimensional
consideration, as well as the  commutators that are vanishing, we see
that all the terms at  $~{\cal O}(R^{-1})$~ are exhausted at the fourth
commutator $~(n=4)$~ in (22).   
 
Considering all of these, we can get the expression 
$$ \li{ g^{-1} d g = & \, + i d X^a P_a + d \theta^\a Q_\a \cr 
& + \frac12 \Big[ \, + i a_1 \big(d\Bar\theta\g^a\theta \big)  P_a 
	  + i a_5 \frac1{\sqrt R} \big(d\theta\g^{\[5\]} \theta \big) Z_{\[5\]} 
	  - b_2 \frac 1 R d X^a \big( \Bar\theta \g_a \big)^\a Q_\a \cr 
& ~~~~~ ~~~ + b_2 \frac 1 R \big( d \Bar\theta \g_a)^\a X^a Q_\a 
	  + a_2 \frac 1 R \big(d\Bar\theta\g^{c d} \theta \big) 
	M_{c d} \, \Big] \cr 
& + \frac 1 6 \Big[ \, - b_2 \frac 1 R \big( d\Bar\theta \g^b \theta\big) 
	  \big(\Bar\theta \g_b \big)^\a Q_\a 
	  - i a_1 b_2 \frac 1 R \big( d\Bar\theta \g_b \g^a \theta\big) 
	X^b P_a 
	+ i a_5 b_3 \frac 1 R \big(d\Bar\theta\g^{\[5\]}\theta \big) 
	\big(\Bar\theta\g\low{\[5\]} \big)^\a Q_\a \cr 
& ~~~~~ ~~~ + i a_1 b_2 \frac 1 R d X^a \big( \Bar\theta\theta\big) P_a 
	- 2i a_2 \frac 1 R \big( d\Bar\theta \g^{a b} \theta\big) X_a P_b 
	    - i \frac 1 2 a_2 \frac 1 R \big( d \Bar\theta\g^{b c} \theta \big) 
	    \big( \Bar\theta\g_{b c}\big)^\a Q_\a \, \Big] \cr 
& + \frac1{24} \Big[ \, - i b_2 a_1 \frac 1 R 
     \big( d\Bar\theta \g^a\theta\big)
	   \big( \Bar\theta\theta\big) P_a 
     + \frac12 a_1a_2 \frac1R \big( d\Bar\theta\g^{b c} \theta \big)
     \big( \Bar\theta \g_{b c} \g^d \theta\big) P_d \cr  
& ~~~~~ ~~~ + a_1 a_5 b_3 \frac 1 R \big( d\Bar\theta \g^{\[5\]} 
	\theta \big) \big(\Bar\theta\g\low{11} \g_{\[5\]} \g^a\theta \big) P_a 
	\, \Big] + {\cal O} (R^{-3/2} ) ~~ .  
&(24) \cr } $$   
All of the $~{\cal O}(R^{-1/2})$~ or higher-order terms will disappear in
the limit $~R\rightarrow \infty$, and we are left with the standard factor 
in the Green-Schwarz $~\s\-$model formulation:
$$ \li{ & K \approx i \big( d X^a 
	+ \frac12 a_1 \bar\theta\gamma^a d\theta \big) P_a
	+ d\theta^\alpha Q_\alpha ~~.   
&(25) \cr } $$  

For deriving the Type IIA Green-Schwarz action from ours, the relevant
traces of products of generators in the adjoint representation of
$~OSp(32|1)$~ are $\STr (P P) $, $\STr (P Q)$~ and $\STr (Q Q )$,  while
for the WZNW-term the relevant traces are $\STr (P P P) $, $\STr (P P Q)
$, $\STr (Q Q P) $~ and  $\STr (Q Q Q)$.  Among these, the non-vanishing
ones are normalized as 
$$ \li{ & \STr (P_a P_b) = \eta _{a b}~~,~~~~
	\STr (P_a Q _{\alpha}Q_{\beta} )
	= \frac i 2 a_1 (\g\low{11} \g_a)_{\alpha\beta}~~. 
&(26) \cr } $$

In the process of getting the action for 10D Type IIA Green-Schwarz 
superstring \superstring, the usual fermionic $~\k\-$symmetry
\kappasym, in particular how to understand it in our formulation, plays
a role of an important guiding principle.  We regard the $~\k\-$symmetry
as a restricted gauge transformation given like (7) or (8), under the
condition that the gauge field is vanishing: $A = 0$, which is stronger
than a pure-gauge condition, while  acting only on $~K$, but not on
$~A$, as opposed to eq.~(7).  Since $~K$~ is  a pure-gauge, its form is
to be the same as (24), after the limit $~R\rightarrow \infty$.  We can
identify this $~K$~ with the  `pull-back' used in the Green-Schwarz
$~\s\-$model \superstring: $~K\du i A = \Pi\du i A \equiv
\big(\partial_i Z^M \big) E\du M A$,  where we identify the index
index $~{\scst\a}$~ for $~Q_\a$~ with  that of the fermionic coordinates
in superspace.   In fact, as  a simple algebra reveals,  the pull-back
$~\Pi\du i A$~ satisfies the pure-gauge condition  $~d \Pi + \Pi^2 = 0$~
on a flat 10D background, so that this identification is consistent. 

Thus the $~\k\-$symmetry acts on $~K$~ similarly to (7), 
but leaving $~A$~ intact:  
$$ \li{ & \d_\k K = d \big( \d_\k h \big) + \[ K , \d_\k h \]~~, 
     ~~~~ \d_\k A = 0 ~~, 
&(27) \cr } $$ 
where $~\d_\k h \equiv g^{-1} \d_\k g$.  This also maintains the pure-gauge 
condition of $~K$.  To be more specific, we have 
$$ \li{ &\d_\k h = g^{-1} \d_\k g   
	= \left[\, \big(\d_\k X^m  \big)  E\du m a 
     + \big( \d_\k \theta^\m \big) E\du\m a \, \right] P_a 
     + \big( \d_\k \theta^\m \big) E \du\m\a Q_\a~~. 
&(28) \cr } $$
As usual, the $~P_a\-$term vanishes under the condition 
$~\d_\k X^m = \frac i2 a_1 \big[ \, \big(\d_\k\Bar\theta \big) 
\g\low{11} \g^m \theta \, \big]$, 
while the $~Q_\a\-$term corresponds to the usual $~\k\-$parameter, as a 
shift of the fermionic coordinates $~\theta^\m$.  In this sense, 
$~\d_\k h$~ has only the fermionic component: $~\d_\k h 
= \big(\d_\k h\big)^\a Q_\a$.   

In order to get an explicit link with the Green-Schwarz superstring
action, we take the limit $~R\rightarrow \infty$, 
and restrict the original $~OSp(32|1)$~ group down to the super Poincar\'e 
group.  This corresponds to a truncation of components 
in the $~\STr$-operation.  We use the symbol $~\STr'$~ for such a
truncated trace, in which the generators in 
fermionic contraction $~K_i^\a K_{j\,\a}$~ are excluded.  By
this restriction, the original  $~OSp(32|1)$~ gauge invariance is lost,
but the action $~S_{\rm GS} $~ has the fermionic $~\k\-$symmetry instead
as  a reminiscent of the original $~OSp(32|1)$~ gauge invariance.  
Applying this prescription to our action S in (18) for 
$~d=2$~ (with $\a_d = 0~\hbox{for}~d\neq 2$), we get 
$$ \li{ S_{\rm GS} & \, = \int_{S^2} \left[ \, 
	d^2\zeta \, \frac 1 2 {\sqrt{-\g}} \g^{i j} 
	\STr' \big( K_i K_j \big) + b_2 \, \right] 
	= \int_{S^2} \left[ \, d^2 \zeta \, \frac 1 2 {\sqrt{-\g}} 
	\g^{i j} K_i^a K_{j a} + b_2 \, \right] ~~.  
&(29) \cr } $$ 

\doit0{
We now consider the action for $~d=2$~ 
(with $\a_d = 0~\hbox{for}~d\neq 2$): 
$$ \li{ S_{\rm GS} & \, = \int_{S^2} \left[ \, 
	d^2\zeta \, \frac 1 2 {\sqrt{-\g}} \g^{i j} 
	\STr' \big( K_i K_j \big) + b_2 \, \right] 
	= \int_{S^2} \left[ \, d^2 \zeta \, \frac 1 2 {\sqrt{-\g}} 
	\g^{i j} K_i^a K_{j a} + b_2 \, \right] ~~, 
&(29) \cr } $$ 
which is obtained from $~S$~ in (18) by the limit $~R\rightarrow \infty$, 
and restrict the original $~OSp(32|1)$~ group down to the super Poincar\'e 
group.  This corresponds to some truncations of components 
in the $~\STr'$-operation, 
namely we do not include the fermionic contraction in $\STr'$, {\it i.e.}, 
$~K_i^\a K_{j\,\a} $~ is not included in $\STr'$.  
By this restriction, the original 
$~OSp(32|1)$~ gauge invariance is lost, but the 
action $~S_{\rm GS} $~ has the fermionic $~\k\-$symmetry instead as 
a reminiscent of the original $~OSp(32|1)$~ gauge invariance.  
}

We can replace this sort of rather `artificial' restriction of
$~\STr$~  by considering some rescaling of 
the algebra (23) after the $~R\rightarrow\infty$.  To be specific, 
if we rescale $~P_a~$ and $~Q_\a$~ by 
$$ \li{ & P_a = \xi P{\,}'_a~~, ~~~~Q_\a = {\sqrt\xi} Q{\,}'_\a~~, 
&(30) \cr } $$ 
putting also an additional rescaling factor $~\xi^{-2}$~ in front of our action
(5), and take the limit $~\xi\rightarrow\infty$, then we get exactly the
same action as (29): $\lim_{\xi\rightarrow\infty}
\lim_{R\rightarrow\infty} S = S_{\rm G S}$~ for $~d=2$.  Namely, the
restricted supertrace $~\STr'$~ can be understood as such a limiting
process. 
%vjr change:paragraph added, my point would be :caught the fish, 
%forgotten the trap
Of course we can just {\it define} the supergroup $~G~$ to be
super Poincar\'e and the invariant tensor in the superalgebra to be given
by the trace just considered.  At any rate, our point in obtaining it from
the $~OSp(32|1)$~ model is to show that it appears as the result of a
limiting process, not that the former is equivalent to the later.    

The $~\k\-$invariance of $~S_{\rm G S}$~ can be investigated more explicitly: 
First, we note the gauge transformation of $~b_2$~ is 
$$\li{ & \d_{\rm G} b_2 = - I_2^1(0,K,\l) = + \STr \big(K^ 2 \l \big) ~~.
&(31) \cr } $$ 
Second, since our $~\k\-$transformation is a restricted gauge 
transformation of $~K$, we can read from (31) that 
$$\li{ \d_\k b_2 & \, = d \zeta^i \wedge d \zeta^j 
     \STr\Big( \, K_i^a P_a 
     \[\, K_j^\a Q_\a \, , \, \big(\d_\k h\big)^\b Q_\b \, \] \Big) \cr 
& \, = - i a_1 \e^{i j} d^2 \zeta \, 
	K_i^a K_j^\a \big( \g\low{11} \g_a \big)_{\a\b} \big(\d_\k h\big)^\b 
	~~, 
&(32) \cr } $$
and similarly, 
$$ \li { \d_\k\left[ \, \frac12 {\sqrt{- \g}}
	   \g^{i j} K_i^a K_{j a} 
	   \, \right] &\,  =  {\sqrt{-\g}} \g^{i j} K_i^a 
	   \Big[\, i a_1 K_j^\a \big(\g\low{11}\g_a\big)_{\a\b} 
	   \big( \d_\k h\big)^\b \, \Big] 
     + \frac 1 2 \big[ \, \d_\k \big( {\sqrt{-\g}} 
    \g^{i j} \big) \, \big] \, K_i^a K_{j a}\cr 
& \, = i a_1  {\sqrt{-\g}} \g^{i j} 
	    K_i^a K_j^\a \big( \g\low{11} \g_a \big)_{\a\b}  
     \big(\d_\k h\big)^\b 
     + \frac 1 2 \big[ \, \d_\k 
     \big( {\sqrt{-\g}} \g^{i j} \big) \, \big]  \, K_i^a K_{j a} 
				~~,{~~~~~ ~~~} 
&(33) \cr } $$ 
where we have used (26).  As is easily seen now, the addition of (32) to
(33) yields $~\d_k S_{\rm GS}=0$.  In fact, if we 
set up the $~\k\-$transformation 
%%%
\ref\ikare{S.~Bellucci, S.J.~Gates, Jr., B.~Radak and S.~Vashakidze, 
\mpl{4}{89}{1985}.}
%%%
as 
$$ \li{& \d_\k h = i \P\du - a \big(\g_a \k_+ \big)^\a Q_\a 
	\equiv i \big( \Pisl_- \k_+ \big)^\a Q_\a  ~~, ~~~~\cr 
& \d_\k V\du + i = 4 a_1 \Pi\du + \a \big(\g\low{11}\big)\du \a\b 
	\k_{+\b} 
	\equiv 4 a_1 \big(\Bar\Pi_+\g\low{11} \k_+ \big) ~~,  
	~~~~ \d_\k V\du - i = 0 ~~, ~~~~ 
	\d_\k {\sqrt {-\g}} = \d_\k V = 0 ~~,{~~~~~ ~~~}  
&(34) \cr } $$
where $~{\scst +,~-}$~ are local Lorentz indices in the light-cone frame, 
$~V\du\pm i $~ are the zweibeins: $~\g^{i j} \equiv V\du + j V\du - j 
+ V\du - i V\du + j$, and we identified
$~K_i^A \equiv \Pi\du i A$.   Now we easily see that 
$$ \li{ & \d_\k \left[ \, \frac12 {\sqrt{-\g}}\g^{i j}  \Pi\du i a \Pi_{j a} 
	\, \right]  = - 2a_1 V \big( \Bar\Pi_+ \g\low{11} \k_+ \big) 
	\Pi\du - a\Pi_{-a} ~~, \cr
& \d_\k b_2 = + 2 a_1 V \big( \Bar\Pi_+ \g\low{11} \k_+ \big) 
	\Pi\du - a\Pi_{-a} ~~, 
&(35) \cr} $$ 
{\it i.e.,} ~$ \d_\k S_{\rm G S} = 0$~ for the total action 
in the Type IIA Green-Schwarz superstring \superstring\ikare.  
Notice that the peculiar $\STr'\-$operation excludes the derivative  
term $~d \big(\d_\k h \big)^\a Q_\a$~ as usual in the Green-Schwarz formulation
\superstring. 


%\vfill\eject

\bigskip\bigskip\bigskip



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

In this Letter, we have proposed a new action for extended objects on 
Chern-Simons supergravity \chamseddine\zanone\zantwo\hora, 
and discussed its possible link with 
10D Type IIA Green-Schwarz superstring theory \superstring.    
Our action describe in general a system of interacting branes  with or
without boundaries with a gauge symmetry of supersymmetric type and
generally covariant.  The equations of motion of the system can be
interpreted as if the different branes  carry distributional charges in
their world-volumes. 

As for the quantization of this system, we can quantize  it formally in
the path integral approach where we should in principle  sum over all
possible configurations and topologies of this system.  However such an
ungrateful task might not be required,  as we may expect on the basis of
the (presumed) absence of counterterms\footnotew{Possibly assuming also
additivity under  Stieffel sum and the above mentioned quantization of
the $~\alpha$'s.} with  the required symmetries that our action is in
fact the effective  action already, then all what we need is to consider
a background  field perturbative expansion  around a  solution of the
classical equations of motion (the vacuum).\footnotew{A candidate to a
vacuum that seems interesting is $~AdS^4\times S^7$~ with the spatial
sections of the branes contained into the $~S^7$~ part, except maybe for
the 8-brane which may be absent or have four of its spatial dimensions
in each sector. It may be that the interaction between the branes is
responsable for the smallness of the  $~S^7$~ part.}

We may also get the link with Type IIB Green-Schwarz superstring
\superstring.  To this end, we have to consider a 11D slab with two
10D boundaries for which we identify the gauge parameters with the 10D
coordinates. We chose one half of the 32 components of the spinor to be
zero by imposing  a Weyl condion with respect to the $~\g\low{11}$~ so
the spinor is a 10D Majorana-Weyl spinor.  Both 10D
boundaries are going to be `identified' in the sense that we are going to
assume that the gauge parameters $~X^a$~ correspond to the same 10D
space, however that identification can be done in two ways, related by
parity.  Those ways are equivalent regarding the $~X^a$, but the spinors
in each boundary will have different chiralities.   To be more specific,
suppose we have two parallel boundaries.   The 11D Majorana spinor has 32
components on the first boundary with the components splitting as $~({\bf 32})
= ({\bf 16}_L, {\bf 16}_R)$, where $~{_L}$~ and $~{_R}$~ denote the 
chiralities  in 10D.  Similarly on the second boundary, 
a Majorana spinor has $~({\bf 32}') = ({\bf 16}_L',{\bf 16}_R')$.  
Among these four kinds of Majorana-Weyl spinors on the
two 10D boundaries `identified', we can truncate either the combination 
$~{\bf 16}_R,~{\bf 16}_L'$~ or $~{\bf 16}_R,~{\bf 16}_R'$.  
We have in each case Type IIA and
Type IIB superstring actions, respectively.  


Notice that even though the reduced action looks like an action of a
string embedded into flat superspace, our conceptual frame is quite
different, because the Green-Schwarz coordinates are gauge parameters
in our case.  

%change: I changed the position of the following paragraph
We believe that heterotic strings could be also obtained, if we take the
$~y$'s to be nonzero and combine $~\theta$'s and $~y$'s to get commuting
objects to pass from $~Sp(32)$~ to $~SO(32)$, and alternatively from
$~Sp(16)\times Sp(16)$~  to $~SO(16)\times SO(16)$~  which combined with
a similar `bosonization' of halves of  $~Q_{\alpha}$~ gives $~E_8\times
E_8$. Of course we could get the gauge groups just by adding them to the 
$~OSp(32|1)$~ supergroup, because we have no constraints so far on which
group we use, but that would not be so terse conceptually.  Type I strings
do presumably corresponds to the the degenerate case of a flattened
membrane.  

We could also consider adding a non pure-gauge contribution to the
potential and dimensionally reduce it by assuming the gauge potential
is independent  of the membrane and background coordinates across the
slab. Then proceeding  in the same way as in ref.~\chamseddine\ in
reducing the 3D Chern-Simons action to a 2D topological action, we would
have (for some choices of our symmetric supertrace only) an additional
term in the string action that looks like the dilaton times the
curvature scalar of the string world-sheet, where the dilaton is
essentially the eleventh component of the vielbein and  the 2D curvature
tensor is the one obtained pull-back of the 11D one.  This seems to
provide a heuristic link to the relationship 11D size-dilaton-coupling
strength.  Notice that even though we can order the contributions of
different string configurations to the path integral by the power of
that coupling strength and relate that to the genus of the string
world-sheet, so that it looks like we have a first quantized theory,  we
would actually have a second quantized theory, and our classical strings 
are already first quantized objects.

Chern-Simons supergravity theories \zanone\zantwo\ have so far developed
with no explicit direct link with the conventional supergravity or
superstring theories \superstring, in the sense that the
translation operators in the former $~P_a$~ in supergroups do not
really generate `infinite' dimensional diffeomorphism.  In other
words, all the generators in supegroups generate only `internal'
symmetries, separated from the space-time symmetries. 
It has been only in a certain speculative
limit that the diffeomorphism is supposed to be generated or identified
with the $~P_a$~ operators in Chern-Simons supergravity \hora.  
An important step in that direction has been taken recently by
M.~Ba\~nados 
%%%
\ref\max{M.~Ba\~ nados, {\it `Gravitons and Gauge Fields in 5D
Chern-Simons Supergravity'}, hep-th/9911150.},
%%%
who added a small cosmological constant term to the Chern-Simons
supergravity action in 5D in order to make contact with standard
supergravity in 5D.  It seems plausible that the extended objects of our
model may provide such an additional cosmological constant without breaking any
symmetries.  Considering also the speculation from different grounds in
refs.~\zantwo\hora\max, the 11D part of our model 
might have as a low energy truncation of 11D supergravity.
As a supporting evidence for this, we have provided a more solid link with 
10D superstring \superstring, by taking 
an explicit limit to get the Type IIA Green-Schwarz action in flat 
superspace, with supersymmetry introduced as a part of a supergroup.  

%vjr change: next paragraph could possibly be compacted with previous one
In our formulation, we did not introduce the target supergravity.  
The reason for this is that 
within our conceptual framework, a natural extension to a more general
background is not a string mapped into curved superspace, but the action
given by (5) or (18).  
Even though this looks like a major drawbacks in our formulation at first
sight, we emphasize that this situation is similar to the conventional
M-theory formulation, in which the particular large $~N\-$limit in the 1D  
matrix model is believed to reproduce the curved supergravity background in 11D
%%%
\ref\bfss{T.~Banks, W.~Fischler, S.H.~Shenker and L.~Susskind, 
\pr{55}{97}{5112}; {\it For reviews, see, e.g.,} T.~Banks, 
{\it `TASI Lecture Note on Matrix Theory'}, \hepth{9911068}, 
{\it with references therein}.}. 
%%%
In other words, it is natural to expect that our microscopic model of 
Chern-Simons supergravity may well reproduce the `ordinary' curved 
supergravity background for the target space in
a certain limit for superstring theory.  

It is important to notice that our model is not and does not
in any obvious way reduce to supermembrane theory 
%%%
\ref\supermembrane{E.~Bergshoeff, E.~Sezgin and P.K.~Townsend, 
\ap{185}{88}{330}.},
%%%
which has closed (2+1)-dimensional objects with a quadratic kinetic term
as well as a WZNW-term in the world-volume.  However, quoting from the
first reference  of \townsend\ referring to M-theory
\hulltownsend\witten\townsend, {\it ``... it is not obvious that the
presence of a membrane in D=11 implies the existence of a supermembrane
theory, ... but it is important to appreciate that, however things turn
out, the major premise of M-theory, 
for which the circumstantial evidence is now
overwhelming, is that there exist {\bf some} consistent supersymmetric
quantum theory in D=11 containing membranes and
5-branes,\footnotew{The 5-branes are solitonic objects, while the
membranes are fundamental objects} with D=11 supergravity as its
effective field theory.''} 

Considering all the previous points, 
it may not seem so implausible to conjecture
that a model of the kind presented here for the supergroup $~OSp(32|1)$ and 11D
may actually correspond to M-theory \hulltownsend\witten\townsend, 
understood as the continuum
limit of the underlying pregeometric theory we have mentioned before,
which could be called `covariant matrix theory'.   

%vjr change: another paragraph that could possibly merge with the other 
%ones on standard-CS sugra.  Also be aware of referee's 'hostile' comment
%on the first sentence 
We also stress that our formulation will provide us with a 
link between Chern-Simons supergravity \zanone\zantwo\hora\max\
and the conventional superstring/supergravity or M-theory 
\hulltownsend\witten\townsend, in the sense 
that the Green-Schwarz superstring action for the latter is recovered in
an explicit limit from the former, even though for the present time we
can show that this process works only for flat backgrounds.  

Clearly much more development is expected, especially
concerning  any constraint in the choice of the supergroup $~G$~ and the
dimensionality of the theory and the extended objects coming from
consistency considerations.  It might be useful to try to translate 
the reasoning leading to such constraints in
superstring theory into the formulation 
presented here.  It might also be that these questions have a natural
answer within the context of a future pregeometric theory having a model
of the kind proposed here as its effective theory.

%change: last paragraph gone

%In this Letter, we have introduced a class of models of interacting
%extended objects as a microscopic theory which is also a gauge system for
%a supergroup and generally covariant.  In the particular case that the
%supergroup is $~OSp(32|1)$, the bulk gauge fields are chosen to be pure
%gauge (which would not be the case for a generic solution), and for
%particular choices of the boundary conditions, one recovers different
%string theories.


\bigskip

We are grateful to S.J.~Gates, Jr., for stimulating discussions, and 
reading the manuscript.  We are also indebted to the 
referee who helped to improve an earlier version of the paper.      



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